Let $n\ge3$ be a positive integer. Find the real numbers $x_1\ge0,\ldots,x_n\ge 0$, with $x_1+x_2+\ldots +x_n=n$, for which the expression \[(n-1)(x_1^2+x_2^2+\ldots+x_n^2)+nx_1x_2\ldots x_n\] takes a minimal value.

Answer: n^2


## Task A-4.5.

Let $n$ be a natural number. In how many ways can we fill an $n \times n$ table with the numbers $1, 2, -1, -2$ such that the product of the numbers in each row is equal to -2 and the product of the numbers in each column is also equal to -2?

Answer: n! \cdot 2^{(n-1)^2}


Among the points corresponding to number $1,2,...,2n$ on the real line, $n$ are colored in blue and $n$ in red. Let $a_1,a_2,...,a_n$ be the blue points and $b_1,b_2,...,b_n$ be the red points. Prove that the sum $\mid a_1-b_1\mid+...+\mid a_n-b_n\mid$ does not depend on coloring , and compute its value. :roll:

Answer: n^2


(solved by Juliette Fournier). Let $\lambda$ be the positive root of the equation $t^{2}-1998 t-1=0$. Let the sequence $\left(x_{n}\right)$ be defined by $x_{0}=1$ and, for all $n \geqslant 0$, by:

$$
x_{n+1}=\left[\lambda x_{n}\right]
$$

where $[x]$ is the integer part of $x$. Calculate the remainder of the Euclidean division of $x_{1998}$ by 1998.

Answer: 1000


2. For a given value $t$, we consider number sequences $a_{1}, a_{2}, a_{3}, \ldots$ such that $a_{n+1}=\frac{a_{n}+t}{a_{n}+1}$ for all $n \geqslant 1$.

(a) Suppose that $t=2$. Determine all starting values $a_{1}>0$ such that $\frac{4}{3} \leqslant a_{n} \leqslant \frac{3}{2}$ holds for all $n \geqslant 2$.

(b) Suppose that $t=-3$. Investigate whether $a_{2020}=a_{1}$ for all starting values $a_{1}$ different from -1 and 1 .

Answer: [1, 2]


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

Answer: 13


6. Let's introduce the following notations: $a$ - the side of the base, $b$ - the lateral edge, $N$ - the midpoint of side $AB$, $M$ - the midpoint of side $AD$, $K$ - the midpoint of edge $SC$ of the regular quadrilateral pyramid $SABCD$.

![](https://cdn.mathpix.com/cropped/2024_05_06_f36d8ea35c99389d52e3g-05.jpg?height=1057&width=680&top_left_y=1156&top_left_x=745)

We construct the section of the pyramid $SABCD$ by a plane passing through points $N$ and $M$ and parallel to the median $DK$ of the lateral face $SDC$: through points $N$ and $M$ we draw the line $NM$ until it intersects with lines $BC$ and $CD$ at points $N_1$ and $M_1$ respectively; through point $M_1$ we draw a line parallel to the median $DK$, the points of intersection of this line with edges $SD$ and $SC$ are denoted as $P$ and $R$ respectively; we connect points $R$ and $N_1$; the point of intersection of $RN_1$ and $SB$ is denoted as $Q$; we connect $Q$ and $N$, $P$ and $M$. The section $NMPRQ$ is constructed. Let's find its area.

By the condition, $AM = MD = AN = NB = \frac{a}{2}$. Therefore, triangle $AMN$ is a right isosceles triangle and $MN = \frac{a}{\sqrt{2}}$. Note that triangles $MDM_1$ and $NBN_1$ are right triangles. They are equal to triangle $AMN$ by the leg and angle. Therefore, $M_1D = N_1B = \frac{a}{2}$. Consider the plane of the lateral face $SDC$. Since $M_1R \parallel DK$, then $\frac{RK}{CK} = \frac{M_1D}{CD} = \frac{1}{2}$. Then $RK = \frac{CK}{2} = \frac{b}{4}$, $SR = SK - RK = \frac{b}{4}$. Moreover, $PR \parallel DK$, $\frac{SP}{PD} = \frac{SR}{RK} = 1$. Therefore, $SP = PD = \frac{b}{2}$. Since $P$ is the midpoint of edge $SD$ and $M$ is the midpoint of base $AD$, then $PM$ is the midline of triangle $ADS$ and $PM = \frac{AS}{2} = \frac{b}{2}$, $PM \parallel AS$. Similarly, we can obtain that $QN = \frac{AS}{2} = \frac{b}{2}$, $QN \parallel AS$. Therefore, quadrilateral $PMQN$ is a parallelogram. We will prove that it is a rectangle. Indeed, the projection of $AC$ of edge $AS$ is the diagonal of the square $ABCD$. Therefore, $AC$ is perpendicular to the diagonal $BD$, and since $BD \parallel MN$, then $AC$ is perpendicular to $MN$. Hence, $AS$ is perpendicular to $MN$. Since $PM \parallel AS$, it follows that $PM$ is perpendicular to $MN$. Thus, the section $NMPRQ$ consists of a rectangle $PMQN$ with sides $MN = \frac{a}{\sqrt{2}}$ and $PM = \frac{b}{2}$ and an isosceles triangle $PQR$ with base $PQ = \frac{b}{2}$ and height $RF$. Let's find the area of the rectangle: $MN \cdot PM = \frac{ab}{2\sqrt{2}}$. Since $\frac{CR}{RS} = \frac{CO}{OA} = 3$, then $\frac{OR}{AS} = \frac{CO}{OA} = \frac{3}{4}$. From this, we get $OR = \frac{3b}{4}$, and $FR = \frac{3b}{4} - \frac{b}{2} = \frac{b}{4}$. Then the area of triangle $PQR$ is $\frac{1}{2} \cdot \frac{a}{\sqrt{2}} \cdot \frac{b}{4} = \frac{ab}{8\sqrt{2}}$. The area of the section $NMPRQ$ is: $\frac{ab}{2\sqrt{2}} + \frac{ab}{8\sqrt{2}} = \frac{5ab}{8\sqrt{2}} = \frac{5ab\sqrt{2}}{16}$. Substituting $a = 1$, $b = 2$, we find $S = \frac{5\sqrt{2}}{8}$.

Answer: \dfrac{5\sqrt{2}}{8}


$4 \cdot 212$ The river water is flowing, and at point $Q$ it flows into a still lake. A swimmer travels downstream from $P$ to $Q$, and then across the lake to $R$, taking a total of 3 hours. If he travels from $R$ to $Q$ and then to $P$, it takes a total of 6 hours. If the lake water also flows, with a speed equal to that of the river, then traveling from $P$ to $Q$ and then to $R$ would take $\frac{5}{2}$ hours. Under these conditions, how long would it take to travel from $R$ to $Q$ and then to $P$?

Answer: \dfrac{15}{2}


Five lighthouses are located, in order, at points $A, B, C, D$, and $E$ along the shore of a circular lake with a diameter of $10$ miles. Segments $AD$ and $BE$ are diameters of the circle. At night, when sitting at $A$, the lights from $B, C, D$, and $E$ appear to be equally spaced along the horizon. The perimeter in miles of pentagon $ABCDE$ can be written $m +\sqrt{n}$, where $m$ and $n$ are positive integers. Find $m + n$.

Answer: 95


## 125. Math Puzzle 10/75

"What time is it?" Marie-Luise was asked. She jokingly replies: "Until the end of the day, twice two fifths of what has already elapsed since its beginning remain." What time was it at that moment?

Answer: 13:20


$N$  straight lines are drawn on a plane. The  $N$  lines can be partitioned into set of lines such that if a line  $l$  belongs to a partition set then all lines parallel to  $l$  make up the rest of that set. For each  $n>=1$ ,let  $a_n$  denote the number of partition sets of size  $n$ . Now that  $N$  lines intersect at certain points on the plane. For each  $n>=2$  let  $b_n$  denote the number of points that are intersection of exactly  $n$  lines. Show that $\sum_{n>= 2}(a_n+b_n)$  $\binom{n}{2}$   $=$   $\binom{N}{2}$

Answer: \sum_{n \geq 2} (a_n + b_n) \binom{n}{2} = \binom{N}{2}


$4 \cdot 44$ Find the smallest natural number $n$, such that in a simple graph with 10 vertices, $n$ edges, and two-colored, there always exists a monochromatic triangle or a monochromatic quadrilateral.

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

Answer: 31


73. On a given circle, two fixed points $A$ and $B$ are taken, $\overrightarrow{A B}=\alpha$. A random circle passes through points $A$ and $B$. A line $l$ is also drawn through $A$, intersecting the circles again at points $C$ and $D$ ($C$ is on the given circle). The tangents to the circles at points $C$ and $D$ (where $C$ and $D$ are the points of tangency) intersect at point $M$; $N$ is a point on $l$ such that $|C N|=|A D|,|D N|=|C A|$. What values can $\angle CMN$ take?

Answer: \dfrac{\alpha}{2}


4. In space, there are 4 points $A, B, C, D$, satisfying $A B=B C=C D$. If $\angle A B C=\angle B C D=\angle C D A=36^{\circ}$, then the angle formed by line $A C$ and line $B D$ is $\qquad$

Answer: 36^\circ


Four, (50 points) 11 interest classes, several students participate (can participate repeatedly), and each interest class has the same number of students (full, unknown number). It is known that any nine interest classes include all students, while any eight interest classes do not include all students. Find the minimum total number of students.

Answer: 165


2. Find the largest integer $n$ such that
$$
\frac{n^{2007}+n^{2006}+\cdots+n^{2}+n+1}{n+2007}
$$
is an integer.

Answer: \dfrac{2007^{2008} - 1}{2008} - 2007


In an Olympic camp, there are \( a \) trainees and \( b \) examiners, where \( b \) is odd and at least 3. Each examiner decides for each trainee whether they have passed or failed the camp. Let \( k \) be an integer such that no two examiners can have the same opinion on more than \( k \) trainees. Show the inequality

$$
\frac{k}{a} \geqslant \frac{b-1}{2b}
$$

Answer: \frac{k}{a} \geqslant \frac{b-1}{2b}


There are $20$ geese numbered $1-20$ standing in a line. The even numbered geese are standing at the front in the order $2,4,\dots,20,$ where $2$ is at the front of the line. Then the odd numbered geese are standing behind them in the order, $1,3,5,\dots ,19,$ where $19$ is at the end of the line. The geese want to rearrange themselves in order, so that they are ordered $1,2,\dots,20$ (1 is at the front), and they do this by successively swapping two adjacent geese. What is the minimum number of swaps required to achieve this formation?

[i]Author: Ray Li[/i]

Answer: 55


3. As shown in the figure, let the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$ have left and right foci $F_{1}, F_{2}$, respectively. A line through $F_{2}$ intersects the ellipse at points $A\left(x_{1}, y_{1}\right)$ and $B\left(x_{2}, y_{2}\right)$. If the area of the incircle of $\triangle A F_{1} B$ is $\pi$, and $\left|y_{1}-y_{2}\right|=4$, then the eccentricity of the ellipse is $\qquad$

Answer: \dfrac{1}{2}


17. There are 15 students, numbered from 1 to 15. Student 1 wrote a five-digit natural number. Student 2 said: “This number is divisible by 2.” Student 3 said: “This number is divisible by 3.”... and so on, each student said the number is divisible by their number. In fact, only two students with adjacent numbers were incorrect. What is the five-digit number written by Student 1? $\qquad$

Answer: 60060


47 (1289). Two brothers walk together from school to home at the same speed. One day, 15 minutes after leaving school, the first brother ran back to school and, upon reaching it, immediately set off to catch up with the second. Left alone, the second continued walking home at half his usual speed. When the first brother caught up with the second, they resumed their initial speed and arrived home 6 minutes later than usual. How many times faster is the first brother's running speed compared to their usual walking speed?

Answer: 3


Two regular hexagons of side length $2$ are laid on top of each other such that they share the same center point and one hexagon is rotated $30^\circ$ about the center from the other. Compute the area of the union of the two hexagons.

Answer: 48\sqrt{3} - 72


For each positive integer $n$, let $S(n)$ be the sum of the digits of $n$. Determines the smallest positive integer $a$ such that there are infinite positive integers $n$ for which you have $S (n) -S (n + a) = 2018$.

Answer: 7


5. Consider all pairs $(x, y)$ of real numbers such that $0 \leq x \leq$ $y \leq 1$. Let $M(x, y)$ be the maximum value of the set

$$
A=\{x y, x y-x-y+1, x+y-2 x y\}
$$

Find the minimum value that $M(x, y)$ can take for all these pairs $(x, y)$.

Answer: \dfrac{4}{9}


4. In $\triangle A B C$, $B C=2, C A=3, A B=4, P$ is a point inside $\triangle A B C$, $D, E, F$ are on $A B, B C, C A$ respectively, and $P D \parallel B C, P E \parallel A C, P F \parallel A B$. If $P D=P E=P F=l$, then $l=$

Answer: \dfrac{12}{13}


The three scales shown below are balanced. The mass of $\lambda$ is $1 \mathrm{~kg}$. Which of the other objects (circle, square, and triangle) has a mass of $1 \mathrm{~kg}$?

![Balanced Scales](https://cdn.mathpix.com/cropped/2024_04_17_d0cbae19e125c7a52478g-1.jpg?height=282&width=1659&top_left_y=2057&top_left_x=295)

Answer: \text{circle}


7th Putnam 1947 Problem B3 Let O be the origin (0, 0) and C the line segment { (x, y) : x ∈ [1, 3], y = 1 }. Let K be the curve { P : for some Q ∈ C, P lies on OQ and PQ = 0.01 }. Let k be the length of the curve K. Is k greater or less than 2? Solution

Answer: k < 2


5. On the table, there are 2018 playing cards (2018 stacks, each with one card). Petya wants to combine them into one deck of 2018 cards in 2017 operations. Each operation consists of merging two stacks. When Petya merges stacks of $a$ and $b$ cards, Vasily Ivanovich pays Petya $a \cdot b$ rubles. What is the maximum amount of money Petya can earn by performing all 2017 operations?

Answer: \dfrac{2018 \times 2017}{2}


## Task A-4.5.

Let $ABC$ be a triangle and $O$ the center of its circumscribed circle. The line $p$ is perpendicular to the bisector of angle $\varangle BAC$, passes through the midpoint of side $\overline{BC}$, and the midpoint of segment $\overline{AO}$. Determine the measure of angle $\varangle BAC$.

Answer: 120^\circ


3. In a circle with radius $R$, determine the central angle corresponding to a circular segment with the following property: the chord of the segment is equal to the perimeter of the circle with the largest radius that is inscribed in it.

Answer: 4 \arctan\left(\frac{2}{\pi}\right)


B1. After breakfast, the sisters Anna and Birgit go to school, each to a different school. Their house is located on a bike path that runs between both schools. Anna cycles at a constant speed of $12 \mathrm{~km}$ per hour and Birgit walks in the opposite direction at a constant speed of $4 \mathrm{~km}$ per hour. They leave at the same time. Shortly after, their mother notices that the girls have forgotten their lunch and decides to catch up with them. Exactly 10 minutes after Anna and Birgit have left, mother sets off on her electric bike. She catches up with Anna first (and hands Anna her lunch) and then immediately turns around towards Birgit. When she catches up with Birgit, she gives Birgit her lunch and immediately turns around again to head back home. Mother always rides at a constant speed of $24 \mathrm{~km}$ per hour.

How many minutes after Anna and Birgit's departure is mother back home?

Answer: 42


4. In the theater, there are $k$ rows of seats. 770 spectators came to the theater and sat down (possibly not occupying all the seats). After the intermission, all the spectators forgot which seats they had and sat down differently. For what largest $k$ will there definitely be 4 spectators who sat in the same row both before and after the intermission?

Answer: 16


447. Find the greatest common divisor of the numbers 111111 and 111111111.

Answer: 111


12.233. A cube is inscribed in a regular quadrilateral pyramid; the vertices of its upper base lie on the lateral edges, and the vertices of its lower base are in the plane of the pyramid's base. Find the ratio of the volume of the cube to the volume of the pyramid, if the lateral edge of the pyramid makes an angle $\alpha$ with the base plane.

Answer: \dfrac{3\sqrt{2} \tan^2 \alpha}{(\tan \alpha + \sqrt{2})^3}


Task 3. (15 points) Point $A$ lies on side $L M$ of triangle $K L M$ with an angle of $60^{\circ}$ at vertex $K$. Circles are inscribed in triangles $A K L$ and $A K M$ with centers $F$ and $O$ respectively. Find the radius of the circumcircle of triangle $F K O$, if $A O=7, A F=4$.

Answer: \sqrt{65}


Three, (25 points) A chemical plant, starting from January this year, if it does not improve its production environment and continues to produce as it is, will earn 700,000 yuan per month. At the same time, it will receive penalties from the environmental protection department, with the first month's penalty being 30,000 yuan, and increasing by 20,000 yuan each subsequent month. If the plant invests 5,000,000 yuan to add recycling and purification equipment (assuming no time for equipment renovation), it can not only improve the environment but also significantly reduce raw material costs. According to estimates, the cumulative net income for the first 5 months after production is a quadratic function of production time \( n \) (in months). The cumulative income for the first 1, first 2, and first 3 months can reach 1,010,000 yuan, 2,040,000 yuan, and 3,090,000 yuan, respectively, and will stabilize at the level of the fifth month thereafter. At the same time, the plant will not only avoid penalties but also receive a one-time reward of 1,000,000 yuan from the environmental protection department. How many months will it take for the investment to start paying off, i.e., for the net income after the investment to exceed the net income without the investment?

Answer: 9


68. An electric vehicle, after being fully charged, travels from location A to location B and then returns to A, following a specific route. If the car uses the air conditioner on the way to B and turns it off on the way back, the car will run out of power exactly when it returns to A. If the car does not use the air conditioner for the entire journey, it will still have $40 \%$ of its battery left after completing the round trip. If the car uses the air conditioner for the entire journey, it will run out of power 36 kilometers away from A while returning from B. The distance between A and B is $\qquad$ kilometers.

Answer: 63


On the sides $AB$ and $BC$ of an equilateral triangle $ABC$, points $M$ and $N$ are taken such that $MN \parallel AC$. Let $E$ be the midpoint of segment $AN$, and $D$ be the centroid of triangle $BMN$. Find the angles of triangle $CDE$.

Answer: 30^\circ


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

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


3. Hua Hua writes letters to Yuan Yuan with a ballpoint pen. When the 3rd pen refill is used up, she is writing the 4th letter; when she finishes the 5th letter, the 4th pen refill is not yet used up; if Hua Hua uses the same amount of pen refill for each letter, then to finish writing 16 letters, Hua Hua needs to prepare at least how many pen refills.

Answer: 13


5. The numerator and denominator are positive integers and their sum is 99. Among such proper fractions, those that can be converted into repeating decimals are
$\qquad$ in number.

Answer: 38


A six-digit number \( A \) is divisible by 19. The number obtained by removing its last digit is divisible by 17, and the number obtained by removing the last two digits in \( A \) is divisible by 13. Find the largest \( A \) that satisfies these conditions.

Answer: 998412


The weight \( w(p) \) of a polynomial \( p \), \( p(x)=\sum_{i=0}^{m} a_{i} x^{i} \), with integer coefficients \( a_{i} \) is defined as the number of its odd coefficients. For \( i=0,1,2, \ldots \), let \( q_{i}(x)=(1+x)^{4} \). Prove that for any finite sequence \( 0 \leq i_{1}<i_{2}<\cdots<i_{n} \), the inequality
\[ w\left(q_{i_{1}}+\cdots+q_{i_{n}}\right) \geq w\left(q_{i_{1}}\right) \]
holds.

Answer: w\left(q_{i_1} + \cdots + q_{i_n}\right) \geq w\left(q_{i_1}\right)


## Task 23/90

We are looking for the smallest natural number $n>1000$ with the following properties:

1. The digit sequence of $n$ is symmetric.
2. The prime factorization of $n$ contains exactly two prime factors $p_{1}$ and $p_{2}$ in 1st power.
3. When divided by $p_{1}$, $p_{2}$ leaves a remainder of 5.

Answer: 1991


5. In trapezoid $A B C D$, $\overrightarrow{A B}=2 \overrightarrow{D C},|\overrightarrow{B C}|=6, P$ is a point on the plane of trapezoid $A B C D$, and satisfies $\overrightarrow{A P}+\overrightarrow{B P}+4 \overrightarrow{D P}=0, \overrightarrow{D A} \cdot \overrightarrow{C B}=|\overrightarrow{D A}| \cdot|\overrightarrow{D P}|, Q$ is a moving point on side $A D$, then the minimum value of $|\overrightarrow{P Q}|$ is $\qquad$.

Answer: \dfrac{4\sqrt{2}}{3}


A right, quadrilateral pyramid is constructed around a sphere with radius $r$. What is the volume of this pyramid if the center of the sphere divides the height of the pyramid in the ratio $p: q$?

Answer: \dfrac{4 (p + q)^2 r^3}{3 p (q - p)}


1. Solve the inequality $\lg \left(5^{x}+x-20\right)>x-x \lg 2$.

Answer: (20, \infty)


1. Calculate the sum

$$
S=\frac{1-i}{\sqrt{2}}+\left(\frac{1-i}{\sqrt{2}}\right)^{2}+\left(\frac{1-i}{\sqrt{2}}\right)^{3}+\ldots+\left(\frac{1-i}{\sqrt{2}}\right)^{1987}
$$

where \( i \) is the imaginary unit.

Answer: - (1 + \sqrt{2})i


14.A. Choose $n$ numbers from $1,2, \cdots, 9$. Among them, there must be some numbers (at least one, or possibly all) whose sum is divisible by 10. Find the minimum value of $n$.

Answer: 5


[ Ratio of areas of similar triangles ] Law of Sines

In an acute-angled triangle $A B C$, heights $A P$ and $C Q$ are dropped from vertices $A$ and $C$ to sides $B C$ and $A B$. It is known that the area of triangle $A B C$ is 18, the area of triangle $B P Q$ is 2, and $P Q=2 \sqrt{2}$. Find the radius of the circumcircle of triangle $A B C$.

Answer: \dfrac{9}{2}


5. The center of a circle with radius 2 lies on the circumference of a circle with radius 3. Find the area of the intersection of the circles bounded by these circumferences.

Answer: 4 \arccos\left(\frac{1}{3}\right) + 9 \arccos\left(\frac{7}{9}\right) - 4\sqrt{2}


12. (18 points) Let $n (n \geqslant 11)$ be a positive integer. The set $A$ consists of the sums of 10 consecutive positive integers not greater than $n$, and the set $B$ consists of the sums of 11 consecutive positive integers not greater than $n$. If the number of elements in $A \cap B$ is 181, find the maximum and minimum values of $n$.

Answer: 2001


For a given triangle, we construct an equilateral triangle outward and inward on each of its sides. In how many ways can 4 points be selected from the 9 points found in the figure so that they are the vertices of a parallelogram?

Would our findings remain valid if, instead of equilateral triangles, we constructed isosceles right triangles where the hypotenuse is always the corresponding side of the original triangle?

[^0]:    ${ }^{1}$ We remind the contestants that the findings must be justified, as always. First-year students can answer the second question based on precise observations made from a detailed drawing this time.

Answer: 9


4. Let $A B C D$ be a cyclic quadrilateral with $A B=3, B C=2, C D=2, D A=4$. Let lines perpendicular to $\overline{B C}$ from $B$ and $C$ meet $\overline{A D}$ at $B^{\prime}$ and $C^{\prime}$, respectively. Let lines perpendicular to $\overline{A D}$ from $A$ and $D$ meet $\overline{B C}$ at $A^{\prime}$ and $D^{\prime}$, respectively. Compute the ratio $\frac{\left.\mid B C C^{\prime} B^{\prime}\right]}{\left[D A A^{\prime} D^{\prime}\right]}$, where $[\varpi]$ denotes the area of figure $\varpi$.

Answer: \dfrac{37}{76}


Task 10.5. Vika has 60 cards with numbers from 1 to 60. She wants to divide all the cards into pairs so that the modulus of the difference of the numbers in all pairs is the same. How many ways are there to do this?

Answer: 8


[ Extremal Properties (other) ] [ Examples and Counterexamples. Constructions ]

The number of edges of a convex polyhedron is 99. What is the maximum number of edges that a plane, not passing through its vertices, can intersect?

Answer: 66


## Task 2 - 130622

Four opaque cubes with edge lengths $a_{1}=24 \mathrm{~cm}, a_{2}=12 \mathrm{~cm}, a_{3}=6 \mathrm{~cm}$ and $a_{4}=3 \mathrm{~cm}$ are to be stacked on top of each other on an opaque tabletop, with the largest at the bottom, the next largest on top of it, and so on, until the smallest cube is on top, where each cube rests completely on the top face of the one below it (or on the tabletop), without protruding beyond this surface.

Determine the total area of the visible (i.e., not covered) surface parts of these cubes!

Answer: 3636


B1. A teacher makes a large fraction from the numbers 1 to 12. She writes part of the numbers in the numerator and puts $x$-signs between those numbers. The remaining numbers she puts in the denominator, again with $x$-signs between those numbers. Both in the numerator and in the denominator, there is at least one number. She does this in such a way that the fraction is equal to the smallest possible integer. What is this number?

Answer: 231


Auto: Witness $M$.

In quadrilateral $A B C D$, sides $A B, B C$ and $C D$ are equal, $M$ is the midpoint of side $A D$. It is known that $\angle B M C=90^{\circ}$. Find the angle between the diagonals of quadrilateral $A B C D$.

Answer: 30^\circ


Ruda received an alarm clock for his birthday. He was happy about it and set it according to the exact time. Since then, every morning when he got up (Saturdays, Sundays, and holidays included), he pressed the button that lights up the display for exactly 4 seconds. He noticed that during the time the button was pressed, the time on the alarm clock was stopped. Otherwise, the alarm clock neither slowed down nor sped up. On the afternoon of December 11th, Ruda looked at his alarm clock and found that it was exactly 3 minutes behind the correct time.

When did Ruda receive this alarm clock?

(M. Petrová)

Hint. How many times did Ruda press the button?

Answer: October 27


Answer: \dfrac{a^3 + c^2}{b}


4. In how many ways can two knights, two bishops, two rooks, a queen, and a king be arranged on the first row of a chessboard so that the following conditions are met:
1) The bishops stand on squares of the same color;
2) The queen and the king stand on adjacent squares. (20 points).

Answer: 504


A semicircle has diameter $\overline{AD}$ with $AD = 30$. Points $B$ and $C$ lie on $\overline{AD}$, and points $E$ and $F$ lie on the arc of the semicircle. The two right triangles $\vartriangle BCF$ and $\vartriangle CDE$ are congruent. The area of $\vartriangle  BCF$ is $m\sqrt{n}$, where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any prime. Find $m + n$.
[img]https://cdn.artofproblemsolving.com/attachments/b/c/c10258e2e15cab74abafbac5ff50b1d0fd42e6.png[/img]

Answer: 52


Given positive integers$ m,n$ such that $ m < n$. Integers $ 1,2,...,n^2$ are arranged in $ n \times n$ board. In each row, $ m$ largest number colored red. In each column $ m$ largest number colored blue. Find the minimum number of cells such that colored both red and blue.

Answer: m^2


5. Garden workers need to dig holes at equal distances along the edge of a circular flower bed with a circumference of 300 meters. They started digging holes every 3 meters along the edge of the flower bed. When they had dug 30 holes, they received instructions from their superiors to change to planting trees every 5 meters. How many more holes do they need to dig to complete the task?

Answer: 54


9. (16 points) Given the function
$$
f(x)=a \cos x+b \cos 2 x+c \cos 3 x,
$$

and $f(x) \geqslant-1$ always holds. Find the maximum value of $a-b+c$.

Answer: 1


Let $ABC$ be a triangle with $\angle A = 60^\circ$, $AB = 12$, $AC = 14$. Point $D$ is on $BC$ such that $\angle BAD = \angle CAD$. Extend $AD$ to meet the circumcircle at $M$. The circumcircle of $BDM$ intersects $AB$ at $K \neq B$, and line $KM$ intersects the circumcircle of $CDM$ at $L \neq M$. Find $\frac{KM}{LM}$.

Answer: \dfrac{13}{8}


A continuous function $f : R  \to R$ satisfies the conditions $f(1000) = 999$ and $f(x)f(f(x)) = 1$ for all real $x$. Determine $f(500)$.

Answer: \dfrac{1}{500}


\( k \) is a real number greater than 1. A uniform wire consists of the curve \( y = e^x \) between \( x = 0 \) and \( x = k \), and the horizontal line \( y = e^k \) between \( x = k - 1 \) and \( x = k \). The wire is suspended from \( (k - 1, e^k) \) and a horizontal force is applied at the other end, \( (0, 1) \), to keep it in equilibrium. Show that the force is directed towards increasing \( x \).

Answer: \text{The force is directed towards increasing } x


At the base of the pyramid $S A B C$ lies an equilateral triangle $A B C$, and all lateral faces have equal areas. The edge $S A$ is equal to 2, and the edge $S B$ is equal to $\sqrt{2}$. A section of the pyramid is made through vertex $B$ perpendicular to the edge $S C$. Find the area of this section.

Answer: \dfrac{3\sqrt{6}}{40}


Let $A=\{1,2,\cdots,6\}$, function $f:A \rightarrow A$. Mark $p(f)=f(1)\cdots f(6)$. Then the number of functions that make $p(f)|36$ is \_\_\_\_\_.

Answer: 580


Let  $F_m$  be the  $m$ 'th Fibonacci number, defined by  $F_1=F_2=1$  and  $F_m = F_{m-1}+F_{m-2}$  for all  $m \geq 3$ . Let  $p(x)$  be the polynomial of degree 1008 such that  $p(2n+1)=F_{2n+1}$  for  $n=0,1,2,\ldots,1008$ . Find integers  $j$  and  $k$  such that  $p(2019) = F_j - F_k$ .

Answer: (2019,1010)


Mary told John her score on the American High School Mathematics Examination (AHSME), which was over 80.  From this, John was able to determine the number of problems Mary solved correctly.  If Mary's score had been any lower, but still over 80, John could not have determined this.  What was Mary's score?  (Recall that the AHSME consists of 30 multiple-choice problems and that one's score,  $s$ , is computed by the formula  $s = 30 + 4c - w$ , where  $c$  is the number of correct and  $w$  is the number of wrong answers; students are not penalized for problems left unanswered.)

Answer: 119


Nine tiles are numbered  $1, 2, 3, \ldots, 9,$  respectively.  Each of three players randomly selects and keeps three of the tile, and sums those three values.  The probability that all three players obtain an odd sum is  $m/n,$  where  $m$  and  $n$  are relatively prime positive integers.  Find  $m+n.$

Answer: 17


Determine the smallest positive real number  $ k$  with the following property. Let  $ ABCD$  be a convex quadrilateral, and let points  $ A_1$ ,  $ B_1$ ,  $ C_1$ , and  $ D_1$  lie on sides  $ AB$ ,  $ BC$ ,  $ CD$ , and  $ DA$ , respectively. Consider the areas of triangles  $ AA_1D_1$ ,  $ BB_1A_1$ ,  $ CC_1B_1$  and  $ DD_1C_1$ ; let  $ S$  be the sum of the two smallest ones, and let  $ S_1$  be the area of quadrilateral  $ A_1B_1C_1D_1$ . Then we always have  $ kS_1\ge S$ .

*Author: Zuming Feng and Oleg Golberg, USA*

Answer: 1


Bob’s Rice ID number has six digits, each a number from  $1$  to  $9$ , and any digit can be used any number of times. The ID number satifies the following property: the first two digits is a number divisible by  $2$ , the first three digits is a number divisible by  $3$ , etc. so that the ID number itself is divisible by  $6$ . One ID number that satisfies this condition is  $123252$ . How many different possibilities are there for Bob’s ID number?

Answer: 324


Triangle  $ABC$  has  $\overline{AB} = \overline{AC} = 20$  and  $\overline{BC} = 15$ . Let  $D$  be the point in  $\triangle ABC$  such that  $\triangle ADB \sim \triangle BDC$ . Let  $l$  be a line through  $A$  and let  $BD$  and  $CD$  intersect  $l$  at  $P$  and  $Q$ , respectively. Let the circumcircles of  $\triangle BDQ$  and  $\triangle CDP$  intersect at  $X$ . The area of the locus of  $X$  as  $l$  varies can be expressed in the form  $\tfrac{p}{q}\pi$  for positive coprime integers  $p$  and  $q$ . What is  $p + q$ ?

Answer: 1291


Consider the number obtained by writing the numbers  $1,2,\ldots,1990$  one after another. In this number every digit on an even position is omitted; in the so obtained number, every digit on an odd position is omitted; then in the new number every digit on an even position is omitted, and so on. What will be the last remaining digit?

Answer: 9


In the picture there are six coins, each with radius 1cm. Each coin is tangent to exactly two other coins next to it (as in the picture). Between the coins, there is an empty area whose boundary is a star-like shape. What is the perimeter of this shape?
![Image](https://i.imgur.com/aguQRVd.png)

Answer: 4\pi


In  $\triangle ABC$ , the sides have integers lengths and  $AB=AC$ . Circle  $\omega$  has its center at the incenter of  $\triangle ABC$ . An *excircle* of  $\triangle ABC$  is a circle in the exterior of  $\triangle ABC$  that is tangent to one side of the triangle and tangent to the extensions of the other two sides. Suppose that the excircle tangent to  $\overline{BC}$  is internally tangent to  $\omega$ , and the other two excircles are both externally tangent to  $\omega$ . Find the minimum possible value of the perimeter of  $\triangle ABC$ .

Answer: 20


**p1.** A square is tiled by smaller squares as shown in the figure. Find the area of the black square in the middle if the perimeter of the square  $ABCD$  is  $14$  cm.
![Image](https://cdn.artofproblemsolving.com/attachments/1/1/0f80fc5f0505fa9752b5c9e1c646c49091b4ca.png)**p2.** If  $a, b$ , and  $c$  are numbers so that  $a + b + c = 0$  and  $a^2 + b^2 + c^2 = 1$ . Compute  $a^4 + b^4 + c^4$ .**p3.** A given fraction  $\frac{a}{b}$  ( $a, b$  are positive integers,  $a \ne b$ ) is transformed by the following rule: first,  $1$  is added to both the numerator and the denominator, and then the numerator and the denominator of the new fraction are each divided by their greatest common divisor (in other words, the new fraction is put in simplest form). Then the same transformation is applied again and again. Show that after some number of steps the denominator and the numerator differ exactly by  $1$ .**p4.** A goat uses horns to make the holes in a new  $30\times 60$  cm large towel. Each time it makes two new holes. Show that after the goat repeats this  $61$  times the towel will have at least two holes whose distance apart is less than  $6$  cm.**p5.** You are given  $555$  weights weighing  $1$  g,  $2$  g,  $3$  g,  $...$  ,  $555$  g. Divide these weights into three groups whose total weights are equal.**p6.** Draw on the regular  $8\times 8$  chessboard a circle of the maximal possible radius that intersects only black squares (and does not cross white squares). Explain why no larger circle can satisfy the condition.

PS. You should use hide for answers. Collected [here](https://artofproblemsolving.com/community/c5h2760506p24143309).

Answer: \frac{1}{2}


Suppose you have  $27$  identical unit cubes colored such that  $3$  faces adjacent to a vertex are red and the other  $3$  are colored blue. Suppose further that you assemble these  $27$  cubes randomly into a larger cube with  $3$  cubes to an edge (in particular, the orientation of each cube is random). The probability that the entire cube is one solid color can be written as  $\frac{1}{2^n}$  for some positive integer  $n$ . Find  $n$ .

Answer: 53


You are given three lists A, B, and C.  List A contains the numbers of the form  $10^{k}$  in base 10, with  $k$  any integer greater than or equal to 1.  Lists B and C contain the same numbers translated into base 2 and 5 respectively:  \[\begin{array}{lll}A & B & C  10 & 1010 & 20  100 & 1100100 & 400  1000 & 1111101000 & 13000  \vdots & \vdots & \vdots \end{array}.\] Prove that for every integer  $n > 1$ , there is exactly one number in exactly one of the lists B or C that has exactly  $n$  digits.

Answer: 1


The number  $2017$  is prime. Let  $S=\sum_{k=0}^{62}\binom{2014}{k}$ . What is the remainder when  $S$  is divided by  $2017$ ? $\textbf{(A) }32\qquad
\textbf{(B) }684\qquad
\textbf{(C) }1024\qquad
\textbf{(D) }1576\qquad
\textbf{(E) }2016\qquad$

Answer: 1024


A bee starts flying from point  $P_0$ . She flies 1 inch due east to point  $P_1$ . For  $j \ge 1$ , once the bee reaches point  $P_j$ , she turns  $30^{\circ}$  counterclockwise and then flies  $j+1$  inches straight to point  $P_{j+1}$ . When the bee reaches  $P_{2015}$  she is exactly  $a\sqrt{b} + c\sqrt{d}$  inches away from  $P_0$ , where  $a$ ,  $b$ ,  $c$  and  $d$  are positive integers and  $b$  and  $d$  are not divisible by the square of any prime. What is  $a+b+c+d$ ? $ \textbf{(A)}\ 2016 \qquad\textbf{(B)}\ 2024 \qquad\textbf{(C)}\ 2032 \qquad\textbf{(D)}\ 2040 \qquad\textbf{(E)}\ 2048$

Answer: 2024


There is a stone on each square of  $n\times n$  chessboard. We gather  $n^2$  stones and distribute them to the squares (again each square contains one stone)  such that any two adjacent stones are again adjacent. Find all distributions such that at least one stone at the corners remains at its initial square. (Two squares are adjacent if they share a common edge.)

Answer: 3


Find the number of positive integers less than or equal to  $2017$  that have at least one pair of adjacent digits that are both even. For example, count the numbers  $24$ ,  $1862$ , and  $2012$ , but not  $4$ ,  $58$ , or  $1276$ .

Answer: 738


Any five points are taken inside or on a square with side length \(1\). Let \(a\) be the smallest possible number with the property that it is always possible to select one pair of points from these five such that the distance between them is equal to or less than \(a\). The original answer is in the format \(\frac{\sqrt{n}}{m}\). Please calculate the value of \(k + m + n\).

Answer: 5


In the figure, equilateral hexagon $ABCDEF$ has three nonadjacent acute interior angles that each measure $30^\circ$. The enclosed area of the hexagon is $6\sqrt{3}$. The original answer for the perimeter of the hexagon is in the form $k\sqrt{3}$. Please find the value of $k$.

Answer: 12


Answer: 1


A circle with center $O$ is tangent to the coordinate axes and to the hypotenuse of the $30^\circ$-$60^\circ$-$90^\circ$ triangle $ABC$ as shown, where $AB = 1$. If the radius of the circle is $r$, find the value of $100r$ to the nearest integer.

Answer: 237


A circular disc with diameter $D$ is placed on an $8 \times 8$ checkerboard with width $D$ so that the centers coincide. Find the number of checkerboard squares which are completely covered by the disc.

Answer: 32


One, (40 points) Given a set $A=\left\{a_{1}, a_{2}, \cdots, a_{n}\right\}$ consisting of $n$ positive integers that satisfies: for any two different subsets of set $A$, the sums of their respective elements are not equal. Find the minimum value of $\sum_{i=1}^{n} \sqrt{a_{i}}$.

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


In a circle of radius $R$, a chord $AB$ and a diameter $AC$ are drawn. Chord $PQ$, which is perpendicular to the diameter $AC$, intersects chord $AB$ at point $M$. Given that $AB = a$ and the ratio $PM : MQ = 3$, find $AM$.

Answer: \dfrac{4aR^2}{16R^2 - 3a^2}


The ratio of Mary's age to Alice's age is $3:5$ . Alice is $30$ years old. How old is Mary?
$\textbf{(A)}\ 15\qquad\textbf{(B)}\ 18\qquad\textbf{(C)}\ 20\qquad\textbf{(D)}\ 24\qquad\textbf{(E)}\  50$

Answer: 18


Example 8 Given 4 spheres in space, their radii are $2, 2, 3, 3$, and each sphere is externally tangent to the other 3 spheres. There is another smaller sphere that is externally tangent to these 4 spheres. Find the radius of the smaller sphere.

Answer: \dfrac{6}{11}


3. In a tournament among six teams, every team plays against each other team exactly once. When a team wins, it receives 3 points and the losing team receives 0 points. If the game is a draw, the two teams receive 1 point each.

Can the final scores of the six teams be six consecutive numbers $a, a+1, \ldots, a+5$ ? If so, determine all values of $a$ for which this is possible.

Answer: 4


In an acute triangle $\triangle ABC$, with $AB \neq AC$, let $H$ and $G$ be the orthocenter and centroid of the triangle, respectively. The necessary and sufficient condition for $\angle AGH = 90^{\circ}$ is $\frac{1}{S_{\triangle HAB}} + \frac{1}{S_{\triangle HAC}} = \frac{2}{S_{\triangle HBC}}$.

Answer: \frac{1}{S_{\triangle HAB}} + \frac{1}{S_{\triangle HAC}} = \frac{2}{S_{\triangle HBC}}


[ Higher degree equations (other).]

Write down the equation of which the root will be the number $\alpha=\frac{1}{2}(\sqrt[3]{5 \sqrt{2}+7}-\sqrt[3]{5 \sqrt{2}-7})$. Write the number $\alpha$ without using radicals.

Answer: 1


Example 2 Let $A$ be a subset of the set of positive integers $\mathbf{N}^{*}$. For any $x, y \in A, x \neq y$, we have $\mid x- y \left\lvert\, \geqslant \frac{x y}{25}\right.$. Find the maximum value of $|A|$. (26th IMO Shortlist)

Answer: 9


Leonid has a white checkered rectangle. First, he painted every other column gray, starting from the leftmost one, and then he painted every other row gray, starting from the topmost one. All the cells adjacent to the border of the rectangle ended up being painted.

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

An example of a painted $5 \times 7$ rectangle is shown below.

Answer: 301


3. In triangle $ABC$, the median $BM$ is twice as short as side $AB$ and forms an angle of 40 degrees with it. Find angle $ABC$.

Answer: 110^\circ


On the bisector of a $60^{\circ}$ angle, a line passing through a point located $1/2$ units away from the vertex of the angle cuts off a triangle with an area of $\sqrt{3} / 4$. How far are the points of intersection of this line with the arms of the angle from the vertex of the angle?

Answer: \sqrt{3} + \sqrt{2}


2. (mathematics) Thirteen circles of the same radius $r=2 \sqrt{2-\sqrt{3}}$ are cut out of colored paper and laid out on a plane so that their centers lie on one straight line. The distance between the centers of adjacent circles is the same and equals 2. Find the perimeter of the figure on the plane formed by these circles.

Answer: 22\pi(\sqrt{6} - \sqrt{2})


50th Putnam 1989 Problem B5 A quadrilateral is inscribed in a circle radius 1. Two opposite sides are parallel. The difference between their lengths is d > 0. The distance from the intersection of the diagonals to the center of the circle is h. Find sup d/h and describe the cases in which it is attained.

Answer: 2


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Peter`, `Eric`, `Arnold`, `Carol`, `Bob`
 - People have unique heights: `very short`, `very tall`, `short`, `super tall`, `tall`, `average`
 - Everyone has something unique for lunch: `grilled cheese`, `stir fry`, `pizza`, `spaghetti`, `soup`, `stew`
 - The mothers' names in different houses are unique: `Holly`, `Janelle`, `Penny`, `Aniya`, `Kailyn`, `Sarah`
 - Each person has a unique favorite drink: `tea`, `water`, `milk`, `coffee`, `boba tea`, `root beer`
 - Each person has a unique type of pet: `rabbit`, `fish`, `bird`, `hamster`, `cat`, `dog`

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

Answer: fish


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`, `Carol`, `Alice`, `Bob`, `Eric`
 - Each mother is accompanied by their child: `Alice`, `Timothy`, `Bella`, `Meredith`, `Fred`, `Samantha`
 - Everyone has a favorite smoothie: `desert`, `cherry`, `watermelon`, `blueberry`, `lime`, `dragonfruit`

## Clues:
1. The person's child is named Fred and the Desert smoothie lover are next to each other.
2. The person who drinks Blueberry smoothies is somewhere to the left of the person's child is named Fred.
3. Alice is not in the fifth house.
4. The person's child is named Samantha is not in the second house.
5. The Watermelon smoothie lover is somewhere to the right of the person who likes Cherry smoothies.
6. Alice is the person's child is named Alice.
7. Alice is the Watermelon smoothie lover.
8. Peter is somewhere to the right of the person's child is named Samantha.
9. Arnold is not in the second house.
10. Bob is the person who is the mother of Timothy.
11. Arnold is directly left of Carol.
12. The person who likes Cherry smoothies is directly left of the person's child is named Samantha.
13. The person's child is named Meredith is in the sixth house.
14. The Dragonfruit smoothie lover is the person's child is named Meredith.
 What is the value of attribute House for the person whose attribute Children is Bella? Please reason step by step, and put your final answer within \boxed{}

Answer: 1


There are 2 houses, numbered 1 to 2 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Arnold`
 - Each person has a unique birthday month: `sept`, `april`
 - Each mother is accompanied by their child: `Bella`, `Fred`
 - Each person has a unique level of education: `associate`, `high school`

## Clues:
1. The person whose birthday is in April is the person's child is named Bella.
2. The person with an associate's degree is directly left of the person with a high school diploma.
3. The person with an associate's degree and the person's child is named Fred are next to each other.
4. Eric is the person with a high school diploma.
 What is the value of attribute Birthday for the person whose attribute Children is Fred? Please reason step by step, and put your final answer within \boxed{}

Answer: sept


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`, `Arnold`, `Peter`, `Bob`, `Eric`
 - Each mother is accompanied by their child: `Samantha`, `Meredith`, `Bella`, `Timothy`, `Fred`
 - Everyone has a favorite smoothie: `lime`, `dragonfruit`, `watermelon`, `desert`, `cherry`
 - Each person has a unique type of pet: `hamster`, `bird`, `fish`, `cat`, `dog`
 - Each person has a unique favorite drink: `tea`, `coffee`, `milk`, `root beer`, `water`
 - The mothers' names in different houses are unique: `Penny`, `Aniya`, `Holly`, `Janelle`, `Kailyn`

## Clues:
1. Alice is the person's child is named Samantha.
2. The person whose mother's name is Penny is somewhere to the right of the person's child is named Meredith.
3. Peter is The person whose mother's name is Penny.
4. The person who is the mother of Timothy is directly left of The person whose mother's name is Kailyn.
5. The person who likes Cherry smoothies is Eric.
6. There is one house between the person who drinks Lime smoothies and The person whose mother's name is Holly.
7. There is one house between the person who keeps a pet bird and the coffee drinker.
8. The person with a pet hamster is Arnold.
9. The tea drinker is in the fifth house.
10. There are two houses between the one who only drinks water and the tea drinker.
11. The person who likes Cherry smoothies is the person who keeps a pet bird.
12. The person whose mother's name is Holly is the person who owns a dog.
13. There is one house between the person who keeps a pet bird and the person's child is named Bella.
14. The person whose mother's name is Kailyn and The person whose mother's name is Penny are next to each other.
15. Peter is not in the fourth house.
16. The person with a pet hamster is not in the second house.
17. The person who likes Cherry smoothies is directly left of the person's child is named Meredith.
18. The Dragonfruit smoothie lover is the person who has a cat.
19. The root beer lover is in the first house.
20. The person's child is named Fred is not in the fourth house.
21. The Desert smoothie lover and The person whose mother's name is Aniya are next to each other.
 What is the value of attribute Smoothie for the person whose attribute House is 5? Please reason step by step, and put your final answer within \boxed{}

Answer: watermelon


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`, `Alice`, `Peter`
 - People own unique car models: `honda civic`, `tesla model 3`, `toyota camry`, `ford f150`
 - They all have a unique favorite flower: `roses`, `daffodils`, `lilies`, `carnations`
 - Each person has a unique favorite drink: `milk`, `water`, `tea`, `coffee`
 - People have unique heights: `very short`, `average`, `tall`, `short`
 - Each person has a favorite color: `red`, `white`, `yellow`, `green`

## Clues:
1. The person who owns a Toyota Camry is somewhere to the left of the person who owns a Ford F-150.
2. The coffee drinker is the person who owns a Ford F-150.
3. The person who has an average height is in the third house.
4. The person whose favorite color is green and the person who owns a Tesla Model 3 are next to each other.
5. There are two houses between the person who loves the rose bouquet and the person who owns a Tesla Model 3.
6. The person who is very short is the tea drinker.
7. Alice is the person who loves white.
8. The person who loves the boquet of lilies is the person who owns a Tesla Model 3.
9. Peter is the person whose favorite color is red.
10. The person who owns a Toyota Camry is Eric.
11. The person who is short is somewhere to the left of the person who loves the boquet of lilies.
12. The person who loves a bouquet of daffodils is the person who loves white.
13. The coffee drinker is in the second house.
14. The person who owns a Tesla Model 3 is the person who likes milk.
 What is the value of attribute House for the person whose attribute Color is red? Please reason step by step, and put your final answer within \boxed{}

Answer: 4


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Arnold`, `Peter`
 - People own unique car models: `ford f150`, `tesla model 3`, `toyota camry`
 - Each person prefers a unique type of vacation: `beach`, `mountain`, `city`
 - The people are of nationalities: `dane`, `swede`, `brit`
 - People have unique hair colors: `blonde`, `black`, `brown`
 - Each person lives in a unique style of house: `colonial`, `ranch`, `victorian`

## Clues:
1. The Swedish person is the person who owns a Ford F-150.
2. The person who has blonde hair is not in the second house.
3. The person who owns a Toyota Camry and Arnold are next to each other.
4. The person who has black hair is somewhere to the right of the person in a ranch-style home.
5. The person who has black hair is the Dane.
6. The person residing in a Victorian house is not in the second house.
7. Peter is directly left of the person who loves beach vacations.
8. The person living in a colonial-style house is the person who has blonde hair.
9. The person who has black hair is Arnold.
10. The Dane is the person who loves beach vacations.
11. Peter is the person who prefers city breaks.
 What is the value of attribute HouseStyle for the person whose attribute Vacation is beach? Please reason step by step, and put your final answer within \boxed{}

Answer: victorian


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Arnold`, `Eric`
 - Each mother is accompanied by their child: `Bella`, `Meredith`, `Fred`
 - People use unique phone models: `samsung galaxy s21`, `iphone 13`, `google pixel 6`
 - People have unique favorite book genres: `romance`, `science fiction`, `mystery`
 - Everyone has a unique favorite cigar: `blue master`, `prince`, `pall mall`
 - People have unique heights: `short`, `very short`, `average`

## Clues:
1. Eric is directly left of the Prince smoker.
2. The person who has an average height is somewhere to the right of Peter.
3. The person who uses a Samsung Galaxy S21 is in the second house.
4. The person who smokes Blue Master is in the third house.
5. The person who uses a Google Pixel 6 is Eric.
6. Arnold is somewhere to the right of the person who loves science fiction books.
7. The person who uses an iPhone 13 is the person's child is named Bella.
8. The person who is short is the Prince smoker.
9. The person who loves mystery books is the person who uses a Google Pixel 6.
10. The person's child is named Meredith and the person who is short are next to each other.
 What is the value of attribute Height for the person whose attribute PhoneModel is samsung galaxy s21? Please reason step by step, and put your final answer within \boxed{}

Answer: short


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

## Clues:
1. The person who loves the spaghetti eater is the Dragonfruit smoothie lover.
2. The person who drinks Lime smoothies is in the second house.
3. Carol is somewhere to the left of the Desert smoothie lover.
4. The person who is an engineer is not in the second house.
5. The person who loves the soup is in the second house.
6. The person who owns a dog is not in the second house.
7. The person who is a pizza lover is Alice.
8. Arnold is somewhere to the left of Eric.
9. The person who loves classical music is the Dragonfruit smoothie lover.
10. The person who owns a rabbit is somewhere to the left of the Desert smoothie lover.
11. The person who is a nurse is not in the sixth house.
12. There are two houses between the person who is a nurse and the person who loves the spaghetti eater.
13. The person who likes Cherry smoothies is the person with an aquarium of fish.
14. The person who has a cat is Alice.
15. The person who likes Cherry smoothies is somewhere to the right of the Desert smoothie lover.
16. There is one house between the person who drinks Blueberry smoothies and the person who loves eating grilled cheese.
17. The person who loves classical music is somewhere to the left of the person who loves the stew.
18. The person who is a lawyer is the person who loves classical music.
19. The person who loves jazz music is Bob.
20. Bob is the person who drinks Lime smoothies.
21. The person who loves pop music is the person who is a teacher.
22. The person who is an engineer is directly left of the person who loves rock music.
23. The person who loves hip-hop music is not in the first house.
24. The person with an aquarium of fish is Peter.
25. The person who drinks Blueberry smoothies is the person who keeps a pet bird.
26. The person who has a cat is in the first house.
27. The person who is a doctor is not in the sixth house.
28. The person who owns a rabbit is not in the second house.
29. Carol is not in the fourth house.
 What is the value of attribute MusicGenre for the person whose attribute Smoothie is blueberry? 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: `Peter`, `Alice`, `Arnold`, `Eric`
 - People own unique car models: `toyota camry`, `tesla model 3`, `honda civic`, `ford f150`
 - People have unique heights: `average`, `very short`, `short`, `tall`
 - The mothers' names in different houses are unique: `Aniya`, `Kailyn`, `Holly`, `Janelle`
 - Each person has a unique level of education: `bachelor`, `high school`, `master`, `associate`
 - The people keep unique animals: `cat`, `fish`, `horse`, `bird`

## Clues:
1. The bird keeper is The person whose mother's name is Kailyn.
2. The person who owns a Ford F-150 is the person who is tall.
3. The person whose mother's name is Kailyn is not in the third house.
4. The person with a master's degree is the fish enthusiast.
5. The person who is very short is somewhere to the left of the cat lover.
6. The fish enthusiast is the person who is short.
7. The person whose mother's name is Janelle is directly left of the person who owns a Toyota Camry.
8. The person whose mother's name is Kailyn is not in the first house.
9. The person who is short is somewhere to the right of Alice.
10. The person with a high school diploma is the person who has an average height.
11. There is one house between the person who is very short and The person whose mother's name is Holly.
12. The person with a master's degree is somewhere to the right of the person with a bachelor's degree.
13. The person with a master's degree and Peter are next to each other.
14. The person who owns a Tesla Model 3 is Eric.
15. The person who has an average height is somewhere to the right of Arnold.
16. The person who keeps horses is directly left of the person with a master's degree.
 What is the value of attribute Education for the person whose attribute Height is short? Please reason step by step, and put your final answer within \boxed{}

Answer: master


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Arnold`, `Carol`, `Alice`, `Peter`, `Eric`
 - The mothers' names in different houses are unique: `Sarah`, `Janelle`, `Aniya`, `Kailyn`, `Holly`, `Penny`
 - Each mother is accompanied by their child: `Fred`, `Samantha`, `Bella`, `Meredith`, `Alice`, `Timothy`
 - Each person prefers a unique type of vacation: `city`, `mountain`, `camping`, `beach`, `cruise`, `cultural`
 - People have unique favorite book genres: `romance`, `mystery`, `historical fiction`, `science fiction`, `biography`, `fantasy`

## Clues:
1. The person who loves beach vacations is not in the second house.
2. The person who loves fantasy books is somewhere to the left of Peter.
3. The person whose mother's name is Sarah is the person who prefers city breaks.
4. The person who enjoys camping trips is somewhere to the right of Peter.
5. The person who likes going on cruises is the person's child is named Meredith.
6. There is one house between the person who is the mother of Timothy and Eric.
7. The person whose mother's name is Janelle is not in the second house.
8. The person's child is named Fred is somewhere to the left of Eric.
9. The person who goes on cultural tours is in the fourth house.
10. The person whose mother's name is Janelle is not in the first house.
11. The person whose mother's name is Holly is somewhere to the right of the person who loves historical fiction books.
12. The person's child is named Bella is somewhere to the left of Alice.
13. Arnold is somewhere to the right of the person who loves fantasy books.
14. The person who loves mystery books is in the fourth house.
15. The person's child is named Alice is the person who enjoys camping trips.
16. The person whose mother's name is Kailyn is the person who likes going on cruises.
17. There are two houses between the person who loves fantasy books and The person whose mother's name is Aniya.
18. The person who loves fantasy books is Carol.
19. The person who likes going on cruises is the person who loves biography books.
20. The person who loves fantasy books is in the third house.
21. The person whose mother's name is Aniya is the person who loves romance books.
22. The person whose mother's name is Janelle is not in the fourth house.
23. The person's child is named Fred is not in the fourth house.
24. The person who loves biography books is not in the second house.
25. There are two houses between The person whose mother's name is Holly and Eric.
 What is the value of attribute House for the person whose attribute BookGenre is fantasy? Please reason step by step, and put your final answer within \boxed{}

Answer: 3


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Arnold`, `Carol`, `Bob`, `Peter`, `Alice`
 - Each person lives in a unique style of house: `ranch`, `victorian`, `modern`, `colonial`, `craftsman`, `mediterranean`
 - Each person has a unique favorite drink: `coffee`, `water`, `root beer`, `tea`, `milk`, `boba tea`
 - Each person has an occupation: `engineer`, `nurse`, `doctor`, `artist`, `lawyer`, `teacher`
 - The people keep unique animals: `horse`, `fish`, `dog`, `cat`, `rabbit`, `bird`
 - Each person has a favorite color: `red`, `purple`, `white`, `green`, `yellow`, `blue`

## Clues:
1. The fish enthusiast is in the first house.
2. Carol is somewhere to the left of the person in a Mediterranean-style villa.
3. The bird keeper is the person who loves purple.
4. The person in a ranch-style home is the one who only drinks water.
5. The coffee drinker is not in the fourth house.
6. The person living in a colonial-style house and the boba tea drinker are next to each other.
7. The cat lover is the person who is a teacher.
8. The person in a modern-style house is in the third house.
9. The person who is an engineer and Peter are next to each other.
10. Arnold is the person who is an artist.
11. The person whose favorite color is red is Arnold.
12. The coffee drinker is somewhere to the right of the person who is a teacher.
13. The bird keeper is the person in a ranch-style home.
14. The rabbit owner is in the sixth house.
15. The boba tea drinker is Peter.
16. Eric is the cat lover.
17. The person who is a nurse is directly left of the person in a modern-style house.
18. The bird keeper is Carol.
19. There are two houses between the cat lover and the boba tea drinker.
20. Bob is the tea drinker.
21. The person residing in a Victorian house is the dog owner.
22. The person whose favorite color is green is in the fourth house.
23. The person who loves blue is the root beer lover.
24. The person who loves white is not in the sixth house.
25. Peter is the person who is a doctor.
 What is the value of attribute Color for the person whose attribute House is 4? Please reason step by step, and put your final answer within \boxed{}

Answer: green


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Arnold`, `Peter`, `Eric`
 - People have unique favorite sports: `tennis`, `swimming`, `basketball`, `soccer`
 - Each person lives in a unique style of house: `victorian`, `colonial`, `craftsman`, `ranch`
 - Everyone has a favorite smoothie: `watermelon`, `cherry`, `desert`, `dragonfruit`
 - Each person has an occupation: `teacher`, `artist`, `engineer`, `doctor`

## Clues:
1. Alice and the person in a Craftsman-style house are next to each other.
2. The Desert smoothie lover is in the first house.
3. The Dragonfruit smoothie lover is Eric.
4. The person in a ranch-style home is somewhere to the right of the person living in a colonial-style house.
5. The person who is a doctor is not in the second house.
6. Peter is the person who likes Cherry smoothies.
7. The person who is an artist is the Dragonfruit smoothie lover.
8. Alice is in the third house.
9. The person who is an engineer is somewhere to the right of the person who loves swimming.
10. The person who loves basketball is the person who is an artist.
11. The Dragonfruit smoothie lover is not in the second house.
12. The person residing in a Victorian house is directly left of the person who loves soccer.
13. The person who loves tennis is somewhere to the left of the person who loves swimming.
 What is the value of attribute House for the person whose attribute Occupation is artist? Please reason step by step, and put your final answer within \boxed{}

Answer: 4


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Alice`, `Peter`, `Eric`, `Bob`
 - Each person has an occupation: `lawyer`, `artist`, `engineer`, `doctor`, `teacher`
 - The people are of nationalities: `norwegian`, `dane`, `swede`, `german`, `brit`
 - They all have a unique favorite flower: `daffodils`, `lilies`, `roses`, `carnations`, `tulips`

## Clues:
1. There is one house between the person who loves the rose bouquet and the person who is an artist.
2. The person who is an engineer is in the second house.
3. The Swedish person is the person who is an artist.
4. The Dane is somewhere to the right of Bob.
5. The person who loves a bouquet of daffodils is Alice.
6. The Dane is the person who is a doctor.
7. The person who loves the boquet of lilies is not in the fourth house.
8. The person who loves the rose bouquet is Peter.
9. The person who is an engineer is somewhere to the left of the person who is a teacher.
10. The Norwegian is in the third house.
11. Arnold is the British person.
12. The person who loves a carnations arrangement is the Dane.
13. Alice is the Norwegian.
 What is the value of attribute Flower for the person whose attribute Name is Arnold? Please reason step by step, and put your final answer within \boxed{}

Answer: lilies


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

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

Answer: pall mall


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Arnold`, `Eric`, `Alice`
 - Each person prefers a unique type of vacation: `mountain`, `cruise`, `beach`, `city`
 - Each person lives in a unique style of house: `craftsman`, `ranch`, `victorian`, `colonial`
 - The mothers' names in different houses are unique: `Kailyn`, `Holly`, `Aniya`, `Janelle`
 - They all have a unique favorite flower: `daffodils`, `roses`, `lilies`, `carnations`
 - Each person has a favorite color: `green`, `white`, `yellow`, `red`

## Clues:
1. Alice is directly left of the person whose favorite color is red.
2. The person who loves a bouquet of daffodils is The person whose mother's name is Janelle.
3. Eric is not in the first house.
4. The person who loves the boquet of lilies is directly left of the person living in a colonial-style house.
5. The person residing in a Victorian house is in the second house.
6. The person who loves the rose bouquet is the person who loves yellow.
7. The person whose mother's name is Aniya is the person who loves yellow.
8. The person who enjoys mountain retreats is in the first house.
9. The person who likes going on cruises is the person whose favorite color is red.
10. The person who loves yellow is in the second house.
11. The person who loves white is the person who prefers city breaks.
12. The person whose mother's name is Holly is not in the fourth house.
13. The person whose favorite color is red is directly left of Peter.
14. The person who loves a bouquet of daffodils is the person in a ranch-style home.
 What is the value of attribute Mother for the person whose attribute Color is red? Please reason step by step, and put your final answer within \boxed{}

Answer: Holly


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Peter`, `Arnold`
 - Everyone has a favorite smoothie: `desert`, `cherry`, `watermelon`
 - Each person prefers a unique type of vacation: `city`, `mountain`, `beach`
 - The people are of nationalities: `dane`, `brit`, `swede`
 - People have unique favorite sports: `tennis`, `soccer`, `basketball`

## Clues:
1. The Dane is directly left of the person who prefers city breaks.
2. The person who loves soccer is Peter.
3. Eric is directly left of the Desert smoothie lover.
4. The person who loves basketball is directly left of the person who prefers city breaks.
5. The Desert smoothie lover is not in the third house.
6. Arnold is somewhere to the left of Peter.
7. The British person is in the third house.
8. The person who likes Cherry smoothies is the Dane.
9. The person who likes Cherry smoothies is the person who enjoys mountain retreats.
 What is the value of attribute House for the person whose attribute Nationality is swede? Please reason step by step, and put your final answer within \boxed{}

Answer: 2


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`, `Arnold`, `Alice`, `Peter`
 - Everyone has a unique favorite cigar: `blue master`, `dunhill`, `pall mall`, `prince`
 - Everyone has something unique for lunch: `stew`, `spaghetti`, `pizza`, `grilled cheese`
 - People have unique favorite book genres: `fantasy`, `mystery`, `science fiction`, `romance`

## Clues:
1. Arnold is the person who loves eating grilled cheese.
2. The person who smokes Blue Master is Alice.
3. Arnold is in the second house.
4. The person who is a pizza lover and the person who loves the stew are next to each other.
5. Arnold is the person who loves science fiction books.
6. The person partial to Pall Mall is the person who loves eating grilled cheese.
7. Eric is somewhere to the left of Alice.
8. The person who is a pizza lover is somewhere to the left of the Dunhill smoker.
9. The person who loves fantasy books is the person who loves the stew.
10. The person who loves mystery books is not in the first house.
 What is the value of attribute House for the person whose attribute Cigar is blue master? Please reason step by step, and put your final answer within \boxed{}

Answer: 3


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Arnold`, `Eric`
 - Each person has a unique level of education: `associate`, `bachelor`, `high school`
 - People have unique heights: `short`, `very short`, `average`
 - Each person has a unique hobby: `cooking`, `gardening`, `photography`
 - People have unique favorite book genres: `romance`, `mystery`, `science fiction`
 - Everyone has a favorite smoothie: `cherry`, `desert`, `watermelon`

## Clues:
1. The Watermelon smoothie lover is the person who is very short.
2. The person who likes Cherry smoothies is the person who loves romance books.
3. The Desert smoothie lover is the person who enjoys gardening.
4. The Watermelon smoothie lover is somewhere to the right of the person with a high school diploma.
5. The person with an associate's degree is directly left of the person with a high school diploma.
6. The photography enthusiast is the person with an associate's degree.
7. The person who likes Cherry smoothies is Arnold.
8. The person who loves science fiction books is somewhere to the right of the person who enjoys gardening.
9. The person who is short is not in the first house.
10. The Watermelon smoothie lover is somewhere to the right of Eric.
 What is the value of attribute Height for the person whose attribute Name is Eric? Please reason step by step, and put your final answer within \boxed{}

Answer: short


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

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

Answer: lilies


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Carol`, `Eric`, `Alice`, `Peter`, `Arnold`
 - Each person prefers a unique type of vacation: `camping`, `cruise`, `mountain`, `cultural`, `city`, `beach`
 - Each person has a unique type of pet: `fish`, `hamster`, `bird`, `rabbit`, `dog`, `cat`
 - The people are of nationalities: `brit`, `swede`, `german`, `norwegian`, `dane`, `chinese`
 - People have unique favorite book genres: `biography`, `romance`, `mystery`, `fantasy`, `science fiction`, `historical fiction`
 - People own unique car models: `tesla model 3`, `bmw 3 series`, `honda civic`, `chevrolet silverado`, `toyota camry`, `ford f150`

## Clues:
1. The person who loves fantasy books is Peter.
2. The Chinese is somewhere to the right of the Dane.
3. The person who owns a Tesla Model 3 is the person with a pet hamster.
4. Carol is the person who loves beach vacations.
5. The person who has a cat is in the second house.
6. There is one house between the person who loves mystery books and the person who prefers city breaks.
7. The person who owns a Honda Civic is in the first house.
8. The person who loves science fiction books is directly left of the person who prefers city breaks.
9. The British person is the person who owns a Honda Civic.
10. There is one house between the person who loves fantasy books and the person who owns a Toyota Camry.
11. The person who keeps a pet bird is the person who owns a Toyota Camry.
12. The Dane is not in the third house.
13. Eric is not in the sixth house.
14. The person who loves romance books is the person who enjoys camping trips.
15. Arnold is the person who likes going on cruises.
16. The person who owns a dog is in the fifth house.
17. The person who enjoys camping trips is directly left of the person who owns a rabbit.
18. The person who owns a Ford F-150 is the person who owns a rabbit.
19. The person who loves fantasy books is directly left of Eric.
20. The German is the person who owns a BMW 3 Series.
21. The person who goes on cultural tours is not in the first house.
22. The person who loves mystery books is not in the sixth house.
23. The person who loves historical fiction books is somewhere to the left of the person who enjoys camping trips.
24. The Dane is Bob.
25. There is one house between the Dane and the Chinese.
26. The Swedish person is somewhere to the right of the person who likes going on cruises.
27. Eric is not in the third house.
 What is the value of attribute CarModel for the person whose attribute BookGenre is fantasy? Please reason step by step, and put your final answer within \boxed{}

Answer: honda civic


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Alice`, `Eric`, `Arnold`, `Bob`
 - Everyone has a favorite smoothie: `lime`, `cherry`, `desert`, `dragonfruit`, `watermelon`
 - Everyone has something unique for lunch: `grilled cheese`, `spaghetti`, `pizza`, `stew`, `stir fry`
 - People have unique hair colors: `gray`, `black`, `red`, `brown`, `blonde`

## Clues:
1. The person who has blonde hair is the person who loves eating grilled cheese.
2. The Desert smoothie lover is somewhere to the left of the person who has red hair.
3. The person who loves the spaghetti eater is the person who has red hair.
4. Bob and the person who has blonde hair are next to each other.
5. The person who likes Cherry smoothies is in the fourth house.
6. Arnold is the person who has black hair.
7. The Desert smoothie lover is in the third house.
8. The Desert smoothie lover is the person who loves eating grilled cheese.
9. Eric is the person who has blonde hair.
10. The person who has gray hair is somewhere to the left of the person who has black hair.
11. The person who has black hair is the Dragonfruit smoothie lover.
12. Bob is the Watermelon smoothie lover.
13. The person who loves stir fry is the person who has black hair.
14. The person who is a pizza lover is the person who has brown hair.
15. Peter is the person who has gray hair.
 What is the value of attribute House for the person whose attribute Food is grilled cheese? 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: `Peter`, `Alice`, `Eric`, `Bob`, `Arnold`
 - Each person has a favorite color: `white`, `red`, `green`, `blue`, `yellow`
 - Each person lives in a unique style of house: `craftsman`, `modern`, `colonial`, `victorian`, `ranch`
 - People have unique favorite book genres: `fantasy`, `science fiction`, `biography`, `romance`, `mystery`

## Clues:
1. Alice is not in the fourth house.
2. Arnold is not in the fourth house.
3. The person who loves blue is Bob.
4. The person who loves white is in the fifth house.
5. The person who loves romance books is not in the fifth house.
6. The person who loves science fiction books is not in the fifth house.
7. The person whose favorite color is green is not in the fourth house.
8. The person who loves romance books is somewhere to the right of the person in a ranch-style home.
9. The person residing in a Victorian house is in the first house.
10. The person whose favorite color is red is in the second house.
11. The person in a modern-style house is somewhere to the right of Arnold.
12. Eric is the person living in a colonial-style house.
13. The person who loves science fiction books is somewhere to the right of Bob.
14. The person who loves fantasy books is the person residing in a Victorian house.
15. The person in a Craftsman-style house is not in the fifth house.
16. The person who loves biography books is the person whose favorite color is red.
17. Eric is in the third house.
18. The person who loves romance books is somewhere to the right of Eric.
 What is the value of attribute HouseStyle for the person whose attribute BookGenre is fantasy? Please reason step by step, and put your final answer within \boxed{}

Answer: victorian


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Arnold`, `Eric`, `Peter`
 - Each person has a unique favorite drink: `water`, `tea`, `coffee`, `milk`
 - Each person has a unique level of education: `master`, `associate`, `high school`, `bachelor`
 - The mothers' names in different houses are unique: `Kailyn`, `Holly`, `Aniya`, `Janelle`
 - Each person has a favorite color: `yellow`, `white`, `green`, `red`

## Clues:
1. Eric is the person with an associate's degree.
2. The person whose mother's name is Kailyn is the person who loves yellow.
3. The person who likes milk is in the third house.
4. The person who likes milk is somewhere to the right of the person who loves white.
5. The person with a bachelor's degree is in the first house.
6. The person whose mother's name is Aniya is not in the second house.
7. The one who only drinks water is Arnold.
8. The person whose favorite color is green is Peter.
9. The person whose favorite color is red is somewhere to the left of the person who loves white.
10. The person with a high school diploma is directly left of The person whose mother's name is Janelle.
11. The person with a high school diploma is the tea drinker.
 What is the value of attribute Drink for the person whose attribute House is 1? Please reason step by step, and put your final answer within \boxed{}

Answer: water


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Peter`, `Carol`, `Arnold`, `Bob`, `Eric`
 - Each mother is accompanied by their child: `Alice`, `Meredith`, `Samantha`, `Bella`, `Fred`, `Timothy`
 - Each person prefers a unique type of vacation: `cruise`, `camping`, `mountain`, `cultural`, `beach`, `city`
 - The people are of nationalities: `german`, `dane`, `chinese`, `norwegian`, `swede`, `brit`
 - Each person has a unique type of pet: `rabbit`, `fish`, `cat`, `bird`, `dog`, `hamster`

## Clues:
1. The person who loves beach vacations is directly left of Arnold.
2. The Swedish person is the person who keeps a pet bird.
3. Peter is not in the sixth house.
4. There is one house between the Chinese and the person with a pet hamster.
5. The German is directly left of the person's child is named Samantha.
6. The Swedish person is the person who enjoys camping trips.
7. The Dane is the person who prefers city breaks.
8. The person's child is named Alice is somewhere to the right of the person who enjoys mountain retreats.
9. The Dane is Carol.
10. The person who owns a dog is not in the sixth house.
11. The Swedish person is not in the sixth house.
12. The Swedish person and Peter are next to each other.
13. The person's child is named Alice is Peter.
14. Arnold is the Norwegian.
15. Bob is somewhere to the right of the Chinese.
16. Carol is the person who owns a dog.
17. The person with an aquarium of fish is not in the third house.
18. The person who owns a rabbit is in the second house.
19. The person's child is named Bella is somewhere to the left of the British person.
20. The person's child is named Fred is in the third house.
21. There is one house between Carol and the person who goes on cultural tours.
22. Eric is the person who is the mother of Timothy.
 What is the value of attribute House for the person whose attribute Pet is hamster? 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: `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 Nationality for the person whose attribute Vacation is mountain? Please reason step by step, and put your final answer within \boxed{}

Answer: swede


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Peter`, `Eric`, `Bob`, `Arnold`, `Carol`
 - Everyone has a unique favorite cigar: `pall mall`, `yellow monster`, `dunhill`, `blue master`, `prince`, `blends`
 - People have unique favorite music genres: `hip hop`, `jazz`, `country`, `pop`, `classical`, `rock`
 - Each person has a unique favorite drink: `water`, `milk`, `boba tea`, `tea`, `root beer`, `coffee`
 - The mothers' names in different houses are unique: `Kailyn`, `Penny`, `Janelle`, `Holly`, `Sarah`, `Aniya`
 - Everyone has something unique for lunch: `soup`, `pizza`, `spaghetti`, `stir fry`, `stew`, `grilled cheese`

## Clues:
1. Carol is directly left of the person who loves eating grilled cheese.
2. Eric is not in the second house.
3. The person whose mother's name is Holly is somewhere to the right of Carol.
4. The person who loves eating grilled cheese is somewhere to the right of the person who loves rock music.
5. Eric is directly left of Carol.
6. The person who loves pop music is not in the third house.
7. Eric is the person who loves country music.
8. The person who loves classical music is in the sixth house.
9. The coffee drinker is Bob.
10. The person who smokes many unique blends is Peter.
11. The person who loves the stew is not in the fifth house.
12. The root beer lover is directly left of The person whose mother's name is Janelle.
13. There are two houses between The person whose mother's name is Sarah and the person who smokes Yellow Monster.
14. Eric is the tea drinker.
15. The person partial to Pall Mall is somewhere to the right of the person who loves stir fry.
16. The person who loves the soup is Bob.
17. The person who loves hip-hop music is directly left of The person whose mother's name is Kailyn.
18. Arnold is somewhere to the right of The person whose mother's name is Kailyn.
19. The one who only drinks water is directly left of the person who smokes Blue Master.
20. The person who loves the spaghetti eater is somewhere to the left of the person who smokes many unique blends.
21. The person whose mother's name is Sarah is directly left of the person who loves jazz music.
22. The person who loves hip-hop music is directly left of the root beer lover.
23. The one who only drinks water is the person who loves the stew.
24. The Dunhill smoker is not in the second house.
25. The person who likes milk is The person whose mother's name is Janelle.
26. Eric is The person whose mother's name is Aniya.
 What is the value of attribute Name for the person whose attribute Drink is root beer? Please reason step by step, and put your final answer within \boxed{}

Answer: Peter


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Arnold`, `Carol`, `Bob`, `Eric`, `Peter`
 - People use unique phone models: `iphone 13`, `samsung galaxy s21`, `xiaomi mi 11`, `oneplus 9`, `google pixel 6`, `huawei p50`
 - Each person has a unique birthday month: `april`, `mar`, `feb`, `may`, `jan`, `sept`
 - Each person has an occupation: `nurse`, `doctor`, `engineer`, `teacher`, `artist`, `lawyer`

## Clues:
1. The person who uses a Google Pixel 6 is directly left of the person whose birthday is in March.
2. The person who is a nurse is Eric.
3. Arnold is directly left of the person whose birthday is in April.
4. Alice and the person whose birthday is in January are next to each other.
5. The person who is an artist is the person whose birthday is in May.
6. The person whose birthday is in September is not in the sixth house.
7. There is one house between the person whose birthday is in March and the person who is a nurse.
8. The person whose birthday is in March is the person who is a lawyer.
9. The person who uses a Xiaomi Mi 11 is directly left of the person who is a nurse.
10. The person who is a teacher is not in the sixth house.
11. Eric is the person who uses an iPhone 13.
12. The person who is an engineer and Peter are next to each other.
13. The person whose birthday is in January is in the third house.
14. The person who uses a Samsung Galaxy S21 is Bob.
15. The person who is a lawyer and the person whose birthday is in April are next to each other.
16. The person who uses a OnePlus 9 is somewhere to the right of the person who is a doctor.
 What is the value of attribute Name for the person whose attribute Occupation is doctor? Please reason step by step, and put your final answer within \boxed{}

Answer: Peter


Two spheres, each with a volume of $V$, are positioned relative to each other such that the center of one sphere is on the surface of the other. What is the volume of the common part of the two spheres?

Answer: \dfrac{5}{16}V


Bob has invented the Very Normal Coin (VNC). When the VNC is flipped, it shows heads $\textstyle\frac{1}{2}$ of the time and tails $\textstyle\frac{1}{2}$ of the time - unless it has yielded the same result five times in a row, in which case it is guaranteed to yield the opposite result. For example, if Bob flips five heads in a row, then the next flip is guaranteed to be tails.

Bob flips the VNC an infinite number of times. On the $n$th flip, Bob bets $2^{-n}$ dollars that the VNC will show heads (so if the second flip shows heads, Bob wins $\$0.25$, and if the third flip shows tails, Bob loses $\$0.125$).

Assume that dollars are infinitely divisible. Given that the first flip is heads, the expected number of dollars Bob is expected to win can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a, b$. Compute $100a + b$.

[i]Proposed by Lewis Chen[/i]

Answer: 34783


24. S2 (POL) $)^{1 \mathrm{MO}}$ The positive real numbers $x_{0}, x_{1}, \ldots, x_{1996}$ satisfy $x_{0}=$ $x_{1995}$ and
$$
x_{i-1}+\frac{2}{x_{i-1}}=2 x_{i}+\frac{1}{x_{i}}
$$
for $i=1,2, \ldots, 1995$. Find the maximum value that $x_{0}$ can have.

Answer: 2^{997}


Let \( n \in \mathbf{N}^{*} \). Consider the set \( S = \{1, 2, \ldots, 2n\} \) and its \( k \) subsets \( A_{1}, A_{2}, \ldots, A_{k} \) that satisfy the following conditions:

1. For any \( i \neq j \) (with \( i, j \in \{1, 2, \ldots, k\} \)), the intersection \( A_{i} \cap A_{j} \) has exactly an odd number of elements.
2. For any \( i \) (where \( i = 1, 2, \ldots, k \)), \( i \notin A_{i} \).
3. If \( i \in A_{j} \), then \( j \in A_{i} \).

Determine the maximum value of \( k \).

Answer: 2n - 1


5. (2000 World Inter-City League) A bus, starting a 100-kilometer journey at 12:20 PM, has a computer that at 1:00 PM, 2:00 PM, 3:00 PM, 4:00 PM, 5:00 PM, and 6:00 PM, says: “If the average speed from now on is the same as the average speed so far, then it will take one more hour to reach the destination.” Is this possible? If so, how far has the bus traveled by 6:00 PM?

Answer: 85


Example 4 Let $x_{i}>0(i=1,2, \cdots, n), \mu>0$, $\lambda-\mu>0$, and $\sum_{i=1}^{n} x_{i}=1$. For a fixed $n\left(n \in \mathbf{N}_{+}\right)$, find the minimum value of $f\left(x_{1}, x_{2}, \cdots, x_{n}\right)=\sum_{i=1}^{n} \frac{x_{i}}{\lambda-\mu x_{i}}$.

Answer: \dfrac{1}{\lambda - \dfrac{\mu}{n}}


10. In the tetrahedron $P-ABC$, $PA=BC=\sqrt{6}$, $PB=AC=\sqrt{8}$, $PC=AB=\sqrt{10}$. Then the radius of the circumscribed sphere of the tetrahedron is $\qquad$.

Answer: \sqrt{3}