André, Bianca, Carlos, and Dalva want to raffle a book among themselves. To do this, they place 3 white balls and 1 black ball in a box and agree that, in alphabetical order of their names, each of them will draw a ball, without returning it to the box. Whoever draws the black ball will win the book.

a) What is the probability that André wins the book?

b) What is the probability that Dalva wins the book?

To raffle another book among them, André suggested using 2 black balls and 6 white balls. As before, the first person to draw a black ball will win the book; if the first four balls drawn are white, they will continue to draw balls in the same order. In this new raffle:

c) What is the probability that André wins the book?

d) What is the probability that Dalva wins the book?

Answer: \frac{1}{7}


The store sells 20 items, each priced uniquely in whole numbers from 1 to 20 rubles. The store has a promotion: when purchasing any 5 items, one of them is given for free, and the customer can choose which one. Vlad wants to buy all 20 items in this store, paying as little as possible. How many rubles does he need? (Each of the 20 items is sold in a single unit.)

Answer: 136


The integer sequence \(\left\{a_{n}\right\}\) is defined by \(a_{1}=1\), \(a_{2}=2\), and \(a_{n+2}=5a_{n+1}+a_{n}\). Determine the value of the expression \(\left[\frac{a_{2}}{a_{1}}\right]\left\{\left[\frac{a_{3}}{a_{2}}\right\}\left\{\frac{a_{4}}{a_{3}}\right\} \cdots \left\{\left\{\frac{a_{20225}}{a_{2024}}\right\}\left[\frac{a_{20224}}{a_{2}}\right]\right\}\right.\).

Answer: 1


Rectangle \(ABCD\) is congruent to rectangle \(ABEF\), and \(D-AB-E\) forms a dihedral angle. \(M\) is the midpoint of \(AB\). \(FM\) makes an angle \(\theta\) with \(BD\), where \(\sin \theta = \frac{\sqrt{78}}{9}\). Find the value of \(\frac{AB}{BC}\).

Answer: \frac{\sqrt{2}}{2}


According to legend, the poet Li Bai from the Tang Dynasty went to buy wine, singing as he walked down the street. Each time he passed a store, the amount of his wine doubled, and each time he saw a flower, he drank 2 cups. After encountering four stores and flowers in total, he still had 2 cups of wine left at the end. How many cups of wine did he originally have?

Answer: 2


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

Answer: 13


There are three types of plants in the garden: sunflowers, lilies, and peonies.

1. There is only one day in a week when all three types of flowers bloom simultaneously.
2. No type of flower can bloom for three consecutive days.
3. In one week, the number of days when any two types of flowers do not bloom together will not exceed one day.
4. Sunflowers do not bloom on Tuesday, Thursday, and Sunday.
5. Lilies do not bloom on Thursday and Saturday.
6. Peonies do not bloom on Sunday.

On what day of the week do all three types of flowers bloom simultaneously? (Represent the days from Monday to Sunday as numbers 1 to 7.)

Answer: 5


a) Consider a prime $p$ that divides $10^{n} + 1$ for some positive integer $n$. For example, $p = 7$ divides $10^{3} + 1$. By analyzing the main period of the decimal representation of $\frac{1}{p}$, verify that the number of times the digit $i$ appears is equal to the number of times the digit $9 - i$ appears for each $i \in \{0, 1, 2, \ldots, 9\}$.

b) Consider a prime number $p$ that does not divide 10 and suppose that the period of the decimal representation of $\frac{1}{p}$ is $2k$. Is it always possible to break the period into two numbers that sum up to $10^k - 1$? For example, the period of $\frac{1}{7}$ has a length of $6 = 2k$, as it is equal to 142857. See that $142 + 857 = 999 = 10^3 - 1 = 10^k - 1$.

c) Given

$$
x = \frac{1}{1998} + \frac{1}{19998} + \frac{1}{199998} + \ldots
$$

when we write $2x$ as a decimal number, what is the $59^{\text{th}}$ digit after the decimal point?

Answer: 1


Answer: 933


Denote the circles around $A$ and $B$ in problem 1448 by $k_{1}$ and $k_{2}$, respectively, with radii $r_{1}$ and $r_{2}$. Let $k$ be a circle that is tangent to $k_{1}$ and $k_{2}$ from the inside, with radius $r_{3}$. What does the ratio $r_{3} / r_{1}$ approach when, fixing $A$ and $B$, the point $C$ approaches $A$, i.e., as $r_{1}$ approaches 0?

Answer: \frac{1}{2}


If a natural number \( n \) can be uniquely represented as the sum \( a_1 + a_2 + \cdots + a_k \) and the product \( a_1 a_2 \cdots a_k \) of \( k \) natural numbers (\( k \geq 2 \)), then this natural number is called a "good" number. For example, 10 is a "good" number because \( 10 = 5 + 2 + 1 + 1 + 1 = 5 \times 2 \times 1 \times 1 \times 1 \), and this representation is unique. Use the notion of prime numbers to determine which natural numbers are "good" numbers.

Answer: n


As shown in the figure, two ants, A and B, move on the following circumferences. \( AC \) is the diameter of the larger circle. Point \( B \) is on \( AC \), with \( AB \) and \( BC \) being the diameters of two smaller circles respectively. Ant A moves clockwise on the larger circle, while Ant B moves in a "8" pattern along the two smaller circles in the direction indicated by the arrow (A → B → C → B → A). Both ants start from point \( A \) at the same time and keep moving continuously. The speed ratio of Ant A to Ant B is \( v_{\text{A}} : v_{\text{B}} = 3 : 2 \). After \( T_1 \) minutes, the two ants meet. Then, Ant A increases its speed by \(\frac{1}{3}\) while Ant B's speed remains unchanged, and they continue to move along their original paths. After \( T_2 \) minutes, the two ants meet again. Given that \( T_1 + T_2 = 100^3 - 99^3 + 98^3 - 97^3 + \dots + 2^3 - 1^3 \), determine how many minutes it takes for Ant A to complete one round of the larger circle at its original speed (express the answer as an improper fraction).

Answer: \frac{1015000}{9}


5. In an acute triangle $A B C$, let $D$ be the foot of the altitude from vertex $C$ and $P, Q$ be the feet of the perpendiculars from point $D$ to sides $A C$ and $B C$. The areas of triangles $A D P, D C P, D B Q, C D Q$ are denoted by $S_{1}, S_{2}, S_{3}, S_{4}$, respectively. Calculate $S_{1}: S_{3}$, given that $S_{1}: S_{2}=2: 3, S_{3}: S_{4}=3: 8$.

Answer: \dfrac{88}{45}


3. Perica stands on one of the four floors of a building. In one move, he moves to an adjacent floor (the floor above or the floor below, if it exists). In how many ways can Perica make $n$ moves, where $n$ is a given non-negative integer, if he can start on any floor but must end on the last one?

Answer: F_{n+1}


Given the positive integer $m \geq 2$, $n \geq 3$. Define the following set
$$S = \left\{(a, b) | a \in \{1, 2, \cdots, m\}, b \in \{1, 2, \cdots, n\} \right\}.$$Let $A$ be a subset of $S$. If there does not exist positive integers $x_1, x_2, y_1, y_2, y_3$ such that $x_1 < x_2, y_1 < y_2 < y_3$ and
$$(x_1, y_1), (x_1, y_2), (x_1, y_3), (x_2, y_2) \in A.$$Determine the largest possible number of elements in $A$.

Answer: 2m + n - 2


Let $S$ be a finite set of points on the plane, with no three points collinear. Each point in $S$ is colored either blue or red. It is assumed that any triangle whose three vertices are blue contains at least one red point inside, and any triangle whose three vertices are red contains at least one blue point inside.

Determine the largest possible value of $|S|$.

Answer: 8


10. For positive integer $n$ and real number $x (0 \leqslant x < n)$, define
$$
f(n, x)=(1-\{x\}) \cdot C_{n}^{[x]}+\{x\} \cdot C_{n}^{[x]+1},
$$

where $[x]$ denotes the greatest integer not exceeding the real number $x$, and $\{x\}=x-[x]$. If integers $m, n \geqslant 2$ satisfy
$$
f\left(m, \frac{1}{n}\right)+f\left(m, \frac{2}{n}\right)+\cdots+f\left(m, \frac{m n-1}{n}\right)=123,
$$

find the value of $f\left(n, \frac{1}{m}\right)+f\left(n, \frac{2}{m}\right)+\cdots+f\left(n, \frac{m n-1}{m}\right)$.

Answer: 74


13. (15 points) In the sequence $\left\{a_{n}\right\}$,
$$
a_{n}=2^{n} a+b n-80\left(a 、 b \in \mathbf{Z}_{+}\right) \text {. }
$$

It is known that the minimum value of the sum of the first $n$ terms $S_{n}$ is obtained only when $n=6$, and $7 \mid a_{36}$. Find the value of $\sum_{i=1}^{12}\left|a_{i}\right|$.

Answer: 8010


4. [6] Joe B. is frustrated with chess. He breaks the board, leaving a $4 \times 4$ board, and throws 3 black knights and 3 white kings at the board. Miraculously, they all land in distinct squares! What is the expected number of checks in the resulting position? (Note that a knight can administer multiple checks and a king can be checked by multiple knights.)

Answer: \dfrac{9}{5}


6. For an $n$-element set $H_{n}=\left\{x_{1}, x_{2}, \cdots, x_{n}\right\}$, the number of subsets is $a_{n}=2^{n}$. For all these subsets, we design a method to conveniently arrange them in a sequence, such as the empty set $\varnothing$ being placed at position 0, $\left\{x_{1}\right\}$ at position 1, $\left\{x_{2}\right\}$ at position 2, $\left\{x_{1}, x_{2}\right\}$ at position 3, $\left\{x_{3}\right\}$ at position 4, and so on. Then, the subset at position 100 is . $\qquad$

Answer: \{x_3, x_6, x_7\}


10. Given a regular 2017-gon inscribed in a unit circle, let $S$ be the set of distances between any two distinct vertices of the 2017-gon, and $Q=\prod_{s \in S} s$. Then $Q^{2}=$ $\qquad$

Answer: 2017


Answer: 2


Example 2 A class participated in a math competition, with a total of $a$, $b$, and $c$ three questions. Each question either scores full marks or 0 points, where question $a$ is worth 20 points, and questions $b$ and $c$ are worth 25 points each. After the competition, every student answered at least one question correctly, and those who answered all questions correctly are $\mathrm{i}$ people, and 15 people answered two questions correctly. The sum of the number of people who answered question $a$ correctly and the number of people who answered question $b$ correctly is 29; the sum of the number of people who answered question $a$ correctly and the number of people who answered question $c$ correctly is 25; the sum of the number of people who answered question $b$ correctly and the number of people who answered question $c$ correctly is 20. What is the average score of the class?
(1909, National Junior High School Mathematics Competition)

Answer: 42


13. $A B C D E F G H$ is a cube in which $A B C D$ is the top face, with vertices $H, G, F$ and $E$ directly below the vertices $A, B$, $C$ and $D$ respectively. A real number is assigned to each vertex. At each vertex, the average of the numbers in the three adjacent vertices is then computed. The averages obtained at $A, B, C, D, E, F, G, H$ are 1, 2, 3, 4, 5, 6, 7, 8 respectively. Find the number assigned to vertex $F$.
(2 marks)
13. $A B C D E F G H$ is a cube, where $A B C D$ is the top face, and $H, G, F, E$ are directly below $A, B, C, D$ respectively. A real number is assigned to each vertex, and then at each vertex, the average of the numbers in the three adjacent vertices is computed. The averages obtained at $A, B, C, D, E, F, G, H$ are 1, 2, 3, 4, 5, 6, 7, 8 respectively. Find the number assigned to vertex $F$.

Answer: 13


6. Let the ellipse $C: \frac{x^{2}}{4}+\frac{y^{2}}{3}=1$, and the line $l: y=4x+m$. Then, the range of values for $m$ such that there are two points on the ellipse $C$ that are symmetric with respect to the line $l$ is $\qquad$.

Answer: \left( -\dfrac{2\sqrt{13}}{13}, \dfrac{2\sqrt{13}}{13} \right)


7. Given the parabola $y^{2}=4 x$, with its focus at $F$, a line passing through the focus $F$ and with an inclination angle of $\theta\left(0<\theta<\frac{\pi}{2}\right)$ intersects the parabola at points $A$ and $B$, $A O$ (where $O$ is the origin) intersects the directrix at point $B^{\prime}$, and $B O$ intersects the directrix at point $A^{\prime}$. Then the area of quadrilateral $A B B^{\prime} A^{\prime}$ is $\qquad$

Answer: \dfrac{8}{\sin^3 \theta}


4. In $\triangle A B C$, $A B=A C$, the internal angle bisectors of $\angle C A B$ and $\angle A B C$ intersect the sides $B C$ and $C A$ at points $D$ and $E$, respectively. Let $K$ be the incenter of $\triangle A D C$. If $\angle B E K=45^{\circ}$, find all possible values of $\angle C A B$. (Belgium provided)

Answer: 60^\circ


4. In a triangular pyramid, two of the three faces are isosceles right triangles, and the other is an equilateral triangle with a side length of 1. Then, the number of such triangular pyramids is $\qquad$.

Answer: 3


In the convex quadrilateral $ABCD$, point $X$ is selected on side $AD$, and the diagonals intersect at point $E$. It is known that $AC = BD$, $\angle ABX = \angle AX B = 50^o$, $\angle CAD = 51^o$, $\angle AED = 80^o$. Find the value of angle $\angle AXC$.

Answer: 80^\circ


A team of six people is laying a 40-meter long pipe in the ground. For preparing the trench, they decided to take turns, with each person digging for the amount of time it would take the remaining five to collectively dig a quarter of the trench. The trench was completed when the last member of the team finished their allotted time. How many meters of the trench could they have dug in that same amount of time if they all worked together?

Answer: 100


9. Given real numbers $x, y$ satisfy $2^{x}+3^{y}=4^{x}+9^{y}$, try to find the range of $U=8^{x}+27^{y}$.

Answer: (1,2]


## Task B-3.7.

In the figure, a parallelogram $A B C D$ is shown, with the shorter side $\overline{A B}$ being $2 \mathrm{~cm}$ long, and the height to this side is $12 \mathrm{~cm}$. A line $p$ passes through the vertex $D$ of the parallelogram, which is perpendicular to the line $C D$, and forms an angle $\alpha$ with the side $\overline{A D}$ such that $\sin \alpha=\frac{3}{5}$. Determine the volume of the solid formed by rotating the parallelogram $A B C D$

![](https://cdn.mathpix.com/cropped/2024_05_30_803c5e1b47d2c202a3a9g-22.jpg?height=408&width=369&top_left_y=1235&top_left_x=1369)
around the line $p$.

Answer: 264\pi


6. In triangle $A B C$ with angle $A$ equal to $60^{\circ}$, the angle bisector $A D$ is drawn. The radius of the circumcircle of triangle $A D C$ with center at point $O$ is $\sqrt{3}$. Find the length of the segment $O M$, where $M$ is the intersection point of segments $A D$ and $B O$, if $A B=1.5$.

Answer: \dfrac{\sqrt{21}}{3}


Four. (20 points) Given the function $f_{n}(x)=n^{2} x^{2}(1-$ $x)^{n}, x \in[0,1], n \in \mathbf{N}_{+}$. If the maximum value of $f_{n}(x)$ is denoted as $a_{n}$, try to find the minimum term of the sequence $\left\{a_{n}\right\}$.

Answer: \dfrac{4}{27}


1. An $(l, a)$-design of a set is a collection of subsets of that set such that each subset contains exactly $l$ elements and that no two of the subsets share more than $a$ elements. How many $(2,1)$-designs are there of a set containing 8 elements?

Answer: 2^{28}


90. Two circles with radii $R$ and $r$ intersect, $A$ is one of the points of intersection. $B C$ is a common tangent ( $B$ and $C$ are the points of tangency). Find the radius of the circle circumscribed about triangle $A B C$.

Answer: \sqrt{Rr}


44th Putnam 1983 Problem B5 Define ||x|| as the distance from x to the nearest integer. Find lim n→∞ 1/n ∫ 1 n ||n/x|| dx. You may assume that ∏ 1 ∞ 2n/(2n-1) 2n/(2n+1) = π/2. Solution

Answer: \ln \frac{4}{\pi}


The base of the regular, unit volume pyramid $A B C D E$ is the rectangle $A B C D$. The pyramid is cut by a plane passing through the edge $B C$, and the intersection domain intersects the edges $E A, E D$ at points $A_{1}$ and $D_{1}$, respectively. The intersection of the lines $B A_{1}$ and $C D_{1}$ is point $M$. What is the volume of the tetrahedron $E M A_{1} D_{1}$, if $E M=2 A B$?

Answer: \dfrac{4}{9}


3. Let $D$ be the foot of the altitude from $A$ to side $BC$ of triangle $ABC$. The angle bisector of the internal angle at $C$ intersects the opposite side at point $E$. What is the measure of angle $\Varangle EDB$, if $\Varangle CEA=\frac{\pi}{4}$?

Answer: \dfrac{\pi}{4}


According to the Gregorian calendar, a total of 97 leap days must be inserted over 400 consecutive years. After how many years will there be a 1-day discrepancy between the Gregorian calendar and the "exact" calendar if a year has a length of 365 days, 5 hours, 48 minutes, and 46 seconds?

Answer: 3323


In $\triangle ABC$ points $M,O$ are midpoint of $AB$ and circumcenter. It is true, that $OM=R-r$. Bisector of external $\angle A$ intersect $BC$ at $D$ and bisector of external $\angle C$ intersect $AB$ at $E$.
Find possible values of $\angle CED$

[i]D. Shiryaev [/i]

Answer: 90^\circ


12.140. In what ratio does the point $O$, from which all three sides of the isosceles triangle $ABC$ are seen at the same angle $(\angle AOB = \angle BOC = \angle COA)$, divide the height of the isosceles triangle $ABC$ if the angle at the base of the triangle is $\alpha\left(\alpha>\frac{\pi}{6}\right)$?

Answer: \sqrt{3} \tan \alpha - 1


7. As shown in the figure, in triangle $A B C$, $B E=E D=D C$, connect $A D$ and $A E$. Take a point $F$ on $A D$, and connect $C F$ and $B F$. If the areas of triangle $A C F$ and triangle $E F G$ are 28 and 12, respectively, then the area of triangle $A B C$ is $\qquad$

Answer: 210


An ant starts at the origin of the Cartesian coordinate plane. Each minute, it moves randomly one unit in one of the directions: up, down, left, or right, with all four directions being equally likely. Its direction each minute is independent of its direction in any previous minutes. It stops when it reaches a point $(x,y)$ such that $|x|+|y|=3$. The expected number of moves it makes before stopping can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $100m+n$.

Answer: 3907


$A_1, A_2, \cdots, A_8$ are fixed points on a circle. Determine the smallest positive integer $n$ such that among any $n$ triangles with these eight points as vertices, two of them will have a common side.

Answer: 9


A positive integer $n > 1$ is called beautiful if $n$ can be written in one and only one way as $n = a_1 + a_2 + \cdots + a_k = a_1 \cdot a_2 \cdots a_k$ for some positive integers $a_1, a_2, \ldots, a_k$, where $k > 1$ and $a_1 \geq a_2 \geq \cdots \geq a_k$. For example, 6 is beautiful since $6 = 3 \cdot 2 \cdot 1 = 3 + 2 + 1$, and this is unique. But 8 is not beautiful since $8 = 4 + 2 + 1 + 1 = 4 \cdot 2 \cdot 1 \cdot 1$ as well as $8 = 2 + 2 + 2 + 1 + 1 = 2 \cdot 2 \cdot 2 \cdot 1 \cdot 1$, so uniqueness is lost. Find the largest beautiful number less than 100.

Answer: 95


In a country, there are $28$ cities, and between some cities, there are two-way flights. Each city has exactly one airport, which can be either small, medium, or big. For every route that:
1. Contains more than two cities,
2. Does not repeat any city,
3. Ends where it begins,

It must include all types of airports. What is the maximum number of flights in this country?

Answer: 53


Let $ABCD$ be an isosceles trapezoid with $AD \parallel BC$ and $BC > AD$ such that the distance between the incenters of $\triangle ABC$ and $\triangle DBC$ is $16$. If the perimeters of $ABCD$ and $ABC$ are $120$ and $114$ respectively, then the area of $ABCD$ can be written as $m\sqrt{n}$, where $m$ and $n$ are positive integers with $n$ not divisible by the square of any prime. Find $100m+n$.

Answer: 7114


Ted loves creating haikus (Japanese three-line poems with $5$, $7$, $5$ syllables each), but his vocabulary is rather limited. For integers $1 \le n \le 7$, he knows $2n$ words with $n$ syllables. Words cannot cross between lines but may be repeated. Compute the number of distinct haikus Ted can make, and find the remainder when this number is divided by $100$.

Answer: 28


Amandine and Brennon play a turn-based game, with Amandine starting. On their turn, a player must select a positive integer which cannot be represented as a sum of multiples of any of the previously selected numbers. For example, if $3, 5$ have been selected so far, only $1, 2, 4, 7$ are available to be picked; if only $3$ has been selected so far, all numbers not divisible by three are eligible. A player loses immediately if they select the integer $1$.

Call a number $n$ \textit{feminist} if $\gcd(n, 6) = 1$ and if Amandine wins if she starts with $n$. Compute the sum of the \textit{feminist} numbers less than $40$.

Answer: 192


There is a sequence of numbers $+1$ and $-1$ of length $n$. It is known that the sum of every $10$ neighboring numbers in the sequence is $0$, and that the sum of every $12$ neighboring numbers in the sequence is not zero. What is the maximal value of $n$?

Answer: 15


A positive integer is written on each corner of a square such that numbers on opposite vertices are relatively prime, while numbers on adjacent vertices are not relatively prime. What is the smallest possible value of the sum of these $4$ numbers?

Answer: 60


Define a sequence $a_i$ as follows: \( a_1 = 181 \) and for \( i \ge 2 \), \( a_i = a_{i-1}^2 - 1 \) if \( a_{i-1} \) is odd and \( a_i = \frac{a_{i-1}}{2} \) if \( a_{i-1} \) is even. Find the least \( i \) such that \( a_i = 0 \).

Answer: 105


Let $\sigma_1 : \mathbb{N} \to \mathbb{N}$ be a function that takes a natural number $n$, and returns the sum of the positive integer divisors of $n$. For example, $\sigma_1(6) = 1 + 2 + 3 + 6 = 12$. What is the largest number $n$ such that $\sigma_1(n) = 1854$?

Answer: 1234


Let $\triangle ABC$ be an equilateral triangle with side length $s$ and $P$ a point in the interior of this triangle. Suppose that $PA$, $PB$, and $PC$ are the roots of the polynomial $t^3-18t^2+91t-89$. Then $s^2$ can be written in the form $m+\sqrt{n}$ where $m$ and $n$ are positive integers. Find $100m+n$.

Answer: 7208


All three-digit natural numbers, whose first digits are odd and greater than 1, are written on the board. What is the maximum number of quadratic equations of the form \( a x^{2} + b x + c = 0 \) that can be created by using these numbers as \( a \), \( b \), and \( c \), each no more than once, such that all these equations have roots?

Answer: 100


As shown in Figure 1, a line segment of length 1 is divided into two parts, $x$ and $y$. Then, the segment of length $x$ is bent into a semicircular arc $ACB$, and the segment of length $y$ is folded into a rectangle $ABDE$ along three sides $(BD, DE, EA)$, forming a closed "curved shape" $ACBDEA$. What is the maximum area of this curved shape?

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


A group of students forms a hollow square formation. The outer layer consists of 52 people, and the inner layer consists of 28 people. How many students are in this group?

Answer: 160


King Arthur has two equally wise advisors – Merlin and Percival. Each of them finds the correct answer to any question with a probability \( p \), or an incorrect answer with a probability \( q = 1-p \).

If both advisors give the same answer, the king listens to them. If they give opposite answers, the king makes a decision by flipping a coin.

One day, Arthur wondered whether he needed two advisors or if one would suffice. He called the advisors and said:

- I think the probability of making the right decisions won't decrease if I only keep one advisor and follow him. If this is correct, I should fire one of you. If not, I will leave everything as it is. Answer me, should I fire one of you?

- Whom exactly are you going to fire, King Arthur? – asked the advisors.
- If I decide to fire one of you, I will choose by drawing lots, flipping a coin.

The advisors left to think about the answer. The advisors, it is to be repeated, are equally wise, but not equally honest.

Percival is very honest and will try to give the correct answer, even if he risks being fired. But Merlin, honest in all other matters, in this situation decides to give an answer that maximizes the probability of not being fired. What is the probability that Merlin will be fired?

Answer: \frac{1}{4}


A school's emergency telephone tree operates as follows: The principal calls two students. In the first level, there is only the principal. In the second level, two students are contacted. For each subsequent level, each student contacts two new students who have not yet been contacted. This process continues with each student from the previous level contacting two new students. After the 8th level, how many students in total have been contacted?

Answer: 254


In the movie "Monkey King: Hero Is Back," there is a scene where Sun Wukong battles mountain demons. Some of the demons are knocked down, such that the number of knocked-down demons is one-third more than the standing demons. After a while, 2 more demons are knocked down, but 10 demons stand back up. At this point, the number of standing demons is one-fourth more than the knocked-down demons. How many demons are standing now?

Answer: 35


The cube $A B C D A^{\prime} B^{\prime} C^{\prime} D^{\prime}$ shown in Figure 2 is cut into two parts by a plane passing through point $A$, the trisection point $H$ on the edge $C C^{\prime}$ closer to $C$, and the midpoint $F$ on the edge $D D^{\prime}$. Determine the volume ratio of the two parts.

Answer: \frac{19}{89}


Four squares with integer side lengths are arranged as shown in the diagram, with three vertices of square \( A \) being the centers of squares \( B \), \( C \), and \( D \). If the total area of the red parts is equal to the area of the green part, what is the minimum side length of square \( A \)?

Answer: 3


Two circles with radii 4 and 3 touch each other externally. Common external tangents \(PQ\) and \(RS\) are drawn to these circles such that points \(P\) and \(S\) belong to the circle with the larger radius, and points \(Q\) and \(R\) belong to the circle with the smaller radius. Find the radius of the circle that is tangent to the segments \(RS\), \(SP\), and \(PQ\).

Answer: \frac{24}{7}


Find \((11a + 2b, 18a + 5b)\) if \((a, b) = 1\).

Note: The notation \((x, y)\) represents the greatest common divisor (gcd) of \(x\) and \(y\).

Answer: 1


How old is Willy?

- "Did your teacher give you this problem?" I asked Willy. "It looks rather tedious."
- "No," Willy replied, "I came up with it myself. See this polynomial? My age is one of its roots. What I mean is, if you substitute the number of years I was when I last celebrated my birthday for $x$, the polynomial will equal zero."
- "Wait a minute," I remarked, "this might not be too hard to do. Our equation has integer coefficients, so we need to find an integer root... Let me try $x=7 \ldots$ No, we get 77.
- "Do I look that young?" Willy asked.
- "Alright, let's try a larger integer... No, now we get 85, which is not zero."
- "You always joke," Willy said. "You know very well I'm not such a child."

How old is Willy?

Answer: 14


In an isosceles trapezoid \(ABCD\) with bases \(AD\) and \(BC\) (\(AD > BC\)), the lateral side is 40 cm, and the angle \(\angle BAC\) is \(45^\circ\). Let \(O\) be the center of the circle circumscribed around \(ABCD\). It turns out that point \(O\) lies on the line connecting point \(D\) to the midpoint of side \(AB\). Find the length of the base \(AD\) of the trapezoid.

Answer: 20(\sqrt{6}+\sqrt{2})


Peter, Paul, and David joined a table tennis tournament. On the first day, two of them were randomly chosen to play a game against each other. On each subsequent day, the loser of the game on the previous day would take a rest, and the other two persons would play a game against each other. After a certain number of days, it was found that Peter had won 22 games, Paul had won 20 games, and David had won 32 games. What was the total number of games that Peter had played?

Answer: 48


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`, `Peter`, `Alice`
 - Everyone has something unique for lunch: `grilled cheese`, `spaghetti`, `pizza`, `stew`
 - People use unique phone models: `iphone 13`, `google pixel 6`, `samsung galaxy s21`, `oneplus 9`
 - Each person has a unique hobby: `painting`, `cooking`, `photography`, `gardening`

## Clues:
1. Peter is the person who is a pizza lover.
2. The person who is a pizza lover and the person who loves the spaghetti eater are next to each other.
3. The person who loves eating grilled cheese is directly left of Eric.
4. Eric is the person who loves cooking.
5. The person who uses a Google Pixel 6 is in the fourth house.
6. The photography enthusiast is in the second house.
7. The person who is a pizza lover is the person who uses a OnePlus 9.
8. The person who uses a Samsung Galaxy S21 is in the third house.
9. Arnold is directly left of the person who enjoys gardening.
 What is the value of attribute Food for the person whose attribute Hobby is cooking? Please reason step by step, and put your final answer within \boxed{}

Answer: stew


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Alice`, `Eric`, `Peter`, `Arnold`
 - People use unique phone models: `iphone 13`, `samsung galaxy s21`, `huawei p50`, `google pixel 6`, `oneplus 9`
 - People have unique favorite sports: `soccer`, `swimming`, `baseball`, `basketball`, `tennis`
 - Everyone has a favorite smoothie: `cherry`, `dragonfruit`, `lime`, `desert`, `watermelon`
 - Each person prefers a unique type of vacation: `city`, `cruise`, `beach`, `mountain`, `camping`
 - The people keep unique animals: `horse`, `fish`, `cat`, `dog`, `bird`

## Clues:
1. Peter is somewhere to the right of the person who uses an iPhone 13.
2. The person who loves tennis is directly left of the person who uses a Google Pixel 6.
3. There is one house between the Watermelon smoothie lover and the person who loves beach vacations.
4. The person who loves baseball is in the fourth house.
5. The dog owner is Peter.
6. There is one house between the person who prefers city breaks and the person who keeps horses.
7. The fish enthusiast is directly left of the cat lover.
8. Arnold is the Desert smoothie lover.
9. The person who likes Cherry smoothies is the person who enjoys mountain retreats.
10. The person who loves swimming is not in the first house.
11. Bob is the person who prefers city breaks.
12. The person who uses a Huawei P50 is the fish enthusiast.
13. The person who loves tennis is the person who enjoys mountain retreats.
14. The person who loves baseball is somewhere to the left of the person who loves soccer.
15. The person who uses a OnePlus 9 is Peter.
16. There is one house between the person who prefers city breaks and Alice.
17. The person who enjoys camping trips is the Dragonfruit smoothie lover.
18. The person who loves baseball is the person who enjoys camping trips.
19. The person who loves beach vacations is directly left of the person who enjoys camping trips.
20. Bob is the fish enthusiast.
 What is the value of attribute House for the person whose attribute PhoneModel is samsung galaxy s21? Please reason step by step, and put your final answer within \boxed{}

Answer: 5


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Arnold`, `Eric`
 - Everyone has a unique favorite cigar: `prince`, `blue master`, `pall mall`

## Clues:
1. Arnold is not in the third house.
2. Arnold is somewhere to the right of the person partial to Pall Mall.
3. There is one house between Eric and the person who smokes Blue Master.
 What is the value of attribute House for the person whose attribute Name is Eric? Please reason step by step, and put your final answer within \boxed{}

Answer: 1


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

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

Answer: mystery


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`, `Carol`, `Eric`, `Alice`, `Arnold`
 - Each person has a unique type of pet: `bird`, `dog`, `cat`, `rabbit`, `fish`, `hamster`
 - Each person lives in a unique style of house: `victorian`, `ranch`, `modern`, `mediterranean`, `colonial`, `craftsman`
 - Each person has a unique birthday month: `mar`, `sept`, `may`, `feb`, `jan`, `april`

## Clues:
1. The person with a pet hamster is somewhere to the right of the person whose birthday is in March.
2. The person whose birthday is in January is somewhere to the left of the person whose birthday is in September.
3. The person whose birthday is in May is in the second house.
4. The person living in a colonial-style house is in the second house.
5. Carol is in the third house.
6. The person in a Mediterranean-style villa is not in the sixth house.
7. The person with an aquarium of fish is somewhere to the right of Bob.
8. Eric is in the sixth house.
9. There is one house between the person who has a cat and the person residing in a Victorian house.
10. There are two houses between the person residing in a Victorian house and the person with a pet hamster.
11. The person in a Craftsman-style house is Arnold.
12. The person living in a colonial-style house is somewhere to the left of the person in a modern-style house.
13. The person with an aquarium of fish is not in the second house.
14. Peter is the person living in a colonial-style house.
15. The person whose birthday is in January is directly left of the person whose birthday is in April.
16. There is one house between the person who keeps a pet bird and the person in a modern-style house.
17. Carol is the person whose birthday is in March.
18. The person in a Craftsman-style house is in the fourth house.
19. The person who owns a dog is in the fourth house.
 What is the value of attribute HouseStyle for the person whose attribute Birthday is april? Please reason step by step, and put your final answer within \boxed{}

Answer: modern


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`, `Peter`, `Alice`
 - They all have a unique favorite flower: `lilies`, `roses`, `daffodils`, `carnations`
 - People have unique favorite sports: `swimming`, `tennis`, `soccer`, `basketball`
 - People have unique hair colors: `black`, `brown`, `blonde`, `red`
 - Each person has a unique birthday month: `feb`, `jan`, `april`, `sept`
 - Each person prefers a unique type of vacation: `cruise`, `city`, `mountain`, `beach`

## Clues:
1. The person who enjoys mountain retreats is in the fourth house.
2. The person whose birthday is in April and the person who loves a bouquet of daffodils are next to each other.
3. The person who likes going on cruises is the person who loves a carnations arrangement.
4. The person whose birthday is in September is the person who loves soccer.
5. Arnold is the person who loves a bouquet of daffodils.
6. Alice is in the third house.
7. The person who loves basketball is the person whose birthday is in April.
8. The person whose birthday is in February is the person who has brown hair.
9. The person who has black hair is somewhere to the left of the person who likes going on cruises.
10. The person who prefers city breaks is Eric.
11. The person who has brown hair is Arnold.
12. The person who has blonde hair is the person who loves tennis.
13. The person who loves tennis is the person who likes going on cruises.
14. The person who loves beach vacations is not in the third house.
15. The person who loves the boquet of lilies is in the first house.
 What is the value of attribute Name for the person whose attribute House is 4? 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: `Bob`, `Eric`, `Carol`, `Arnold`, `Alice`, `Peter`
 - People use unique phone models: `google pixel 6`, `samsung galaxy s21`, `iphone 13`, `huawei p50`, `oneplus 9`, `xiaomi mi 11`
 - Each person has a favorite color: `white`, `yellow`, `blue`, `green`, `red`, `purple`

## Clues:
1. The person whose favorite color is green is the person who uses a Google Pixel 6.
2. The person who uses a Xiaomi Mi 11 is somewhere to the right of the person who uses a Huawei P50.
3. There are two houses between Peter and the person whose favorite color is red.
4. Eric is somewhere to the right of the person whose favorite color is green.
5. Carol is the person who uses an iPhone 13.
6. The person who loves blue is not in the second house.
7. The person who uses a Samsung Galaxy S21 is somewhere to the left of the person who loves yellow.
8. The person who loves blue is somewhere to the left of the person who loves white.
9. Carol and the person whose favorite color is red are next to each other.
10. The person who uses a Google Pixel 6 is Bob.
11. Alice is the person who uses a Xiaomi Mi 11.
12. The person whose favorite color is green is not in the second house.
13. The person who loves purple is directly left of Carol.
14. Alice is the person who loves purple.
 What is the value of attribute PhoneModel for the person whose attribute Name is Carol? Please reason step by step, and put your final answer within \boxed{}

Answer: iphone 13


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Arnold`, `Alice`, `Bob`, `Peter`
 - Each person lives in a unique style of house: `colonial`, `modern`, `ranch`, `victorian`, `craftsman`
 - People have unique favorite sports: `tennis`, `basketball`, `soccer`, `baseball`, `swimming`
 - People have unique favorite music genres: `pop`, `classical`, `jazz`, `rock`, `hip hop`

## Clues:
1. The person in a Craftsman-style house and the person who loves basketball are next to each other.
2. Alice is the person living in a colonial-style house.
3. Peter is somewhere to the right of the person who loves swimming.
4. The person living in a colonial-style house is the person who loves soccer.
5. The person in a ranch-style home is the person who loves baseball.
6. The person who loves pop music is somewhere to the left of the person who loves swimming.
7. The person who loves baseball is not in the second house.
8. The person in a modern-style house is in the fourth house.
9. The person who loves basketball and Peter are next to each other.
10. The person in a Craftsman-style house is somewhere to the left of Arnold.
11. The person who loves jazz music is not in the fourth house.
12. There are two houses between the person who loves tennis and the person in a ranch-style home.
13. Eric is not in the third house.
14. The person who loves hip-hop music is the person in a ranch-style home.
15. The person who loves tennis is the person who loves classical music.
 What is the value of attribute MusicGenre for the person whose attribute House is 1? Please reason step by step, and put your final answer within \boxed{}

Answer: pop


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Eric`, `Arnold`
 - Everyone has a favorite smoothie: `desert`, `watermelon`, `cherry`
 - Each person has a unique hobby: `photography`, `cooking`, `gardening`
 - The people are of nationalities: `dane`, `swede`, `brit`

## Clues:
1. The photography enthusiast is the British person.
2. Arnold is directly left of the British person.
3. The person who enjoys gardening is somewhere to the right of the Dane.
4. The Swedish person is the Desert smoothie lover.
5. The photography enthusiast is somewhere to the left of Eric.
6. Arnold is directly left of the Watermelon smoothie lover.
 What is the value of attribute Nationality for the person whose attribute Name is Eric? Please reason step by step, and put your final answer within \boxed{}

Answer: swede


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

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

Answer: tall


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`, `Alice`, `Arnold`, `Peter`, `Bob`
 - People use unique phone models: `samsung galaxy s21`, `huawei p50`, `oneplus 9`, `iphone 13`, `google pixel 6`
 - Everyone has a favorite smoothie: `desert`, `dragonfruit`, `watermelon`, `cherry`, `lime`
 - The mothers' names in different houses are unique: `Janelle`, `Holly`, `Penny`, `Aniya`, `Kailyn`
 - Each person has an occupation: `doctor`, `lawyer`, `engineer`, `teacher`, `artist`

## Clues:
1. The person whose mother's name is Kailyn is Peter.
2. The Dragonfruit smoothie lover is not in the fourth house.
3. The person who uses an iPhone 13 is the person who is an artist.
4. The person whose mother's name is Aniya is Arnold.
5. The person whose mother's name is Aniya is the Desert smoothie lover.
6. The person whose mother's name is Aniya is in the second house.
7. The person who drinks Lime smoothies is in the first house.
8. The person who uses a Huawei P50 is the person who is a doctor.
9. Eric is the person who drinks Lime smoothies.
10. There is one house between the person who is a teacher and Bob.
11. The person whose mother's name is Aniya is the person who is a lawyer.
12. The person who uses an iPhone 13 is the person who likes Cherry smoothies.
13. Arnold is the person who uses a OnePlus 9.
14. The person who is a doctor is The person whose mother's name is Penny.
15. Peter is the person who uses a Samsung Galaxy S21.
16. Bob is not in the third house.
17. The person whose mother's name is Holly is directly left of Peter.
 What is the value of attribute Smoothie for the person whose attribute House is 1? Please reason step by step, and put your final answer within \boxed{}

Answer: lime


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Peter`, `Eric`, `Arnold`
 - Each person has a unique favorite drink: `coffee`, `tea`, `milk`, `water`
 - People have unique favorite book genres: `fantasy`, `science fiction`, `mystery`, `romance`
 - People have unique hair colors: `red`, `black`, `blonde`, `brown`

## Clues:
1. The coffee drinker is the person who has blonde hair.
2. The person who has brown hair is the person who likes milk.
3. Arnold is the person who loves fantasy books.
4. The person who has red hair is in the fourth house.
5. The person who has red hair is the person who loves science fiction books.
6. The person who loves romance books is the tea drinker.
7. The person who has blonde hair is not in the third house.
8. The tea drinker is directly left of the person who loves mystery books.
9. The person who has black hair is not in the second house.
10. The person who loves romance books is somewhere to the left of Eric.
11. The person who loves fantasy books is somewhere to the left of Peter.
 What is the value of attribute Drink for the person whose attribute House is 4? Please reason step by step, and put your final answer within \boxed{}

Answer: water


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 level of education: `master`, `bachelor`, `associate`, `high school`
 - People have unique favorite sports: `tennis`, `soccer`, `swimming`, `basketball`
 - Each person has an occupation: `teacher`, `engineer`, `doctor`, `artist`
 - They all have a unique favorite flower: `carnations`, `daffodils`, `lilies`, `roses`
 - People have unique heights: `very short`, `average`, `tall`, `short`

## Clues:
1. Eric is the person who loves a carnations arrangement.
2. The person who loves swimming is somewhere to the left of the person who is an artist.
3. Arnold is directly left of the person who is tall.
4. The person who loves swimming and the person who is a teacher are next to each other.
5. The person who loves basketball is the person with a bachelor's degree.
6. The person who is a doctor is the person who loves the rose bouquet.
7. The person who loves the boquet of lilies is the person who loves tennis.
8. The person who loves a carnations arrangement is the person with a master's degree.
9. The person who is a teacher is directly left of Alice.
10. The person who is an artist is the person with an associate's degree.
11. The person who loves a bouquet of daffodils is the person who is an artist.
12. The person who loves soccer is not in the fourth house.
13. The person who loves a bouquet of daffodils is somewhere to the right of the person who is short.
14. The person with a master's degree is the person who is very short.
 What is the value of attribute Occupation for the person whose attribute Education is high school? Please reason step by step, and put your final answer within \boxed{}

Answer: teacher


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`, `Bob`, `Eric`, `Peter`
 - Each person has a favorite color: `green`, `red`, `white`, `yellow`, `blue`
 - Everyone has something unique for lunch: `grilled cheese`, `stew`, `pizza`, `spaghetti`, `stir fry`
 - People use unique phone models: `google pixel 6`, `huawei p50`, `oneplus 9`, `samsung galaxy s21`, `iphone 13`
 - Each person has a unique level of education: `master`, `associate`, `doctorate`, `bachelor`, `high school`

## Clues:
1. The person who uses a Samsung Galaxy S21 is somewhere to the left of the person whose favorite color is green.
2. The person whose favorite color is red is Alice.
3. Bob is directly left of the person whose favorite color is red.
4. Alice is the person with an associate's degree.
5. The person with a high school diploma is the person who loves eating grilled cheese.
6. The person who uses a Google Pixel 6 is the person who loves yellow.
7. The person who loves stir fry is the person with a bachelor's degree.
8. The person who is a pizza lover is not in the first house.
9. The person who loves white is the person with a master's degree.
10. The person who loves blue is Eric.
11. The person who uses a Huawei P50 is the person who loves the stew.
12. The person who uses an iPhone 13 is the person with a high school diploma.
13. The person whose favorite color is green is the person who loves stir fry.
14. The person with a doctorate and the person who loves white are next to each other.
15. The person who loves the stew is not in the second house.
16. Arnold is somewhere to the right of the person who loves eating grilled cheese.
17. The person who loves yellow is in the first house.
 What is the value of attribute Education for the person whose attribute Name is Bob? Please reason step by step, and put your final answer within \boxed{}

Answer: master


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Peter`, `Alice`, `Eric`
 - Each person has a unique type of pet: `bird`, `fish`, `dog`, `cat`
 - Each person lives in a unique style of house: `colonial`, `craftsman`, `ranch`, `victorian`
 - The mothers' names in different houses are unique: `Holly`, `Janelle`, `Kailyn`, `Aniya`
 - Each mother is accompanied by their child: `Fred`, `Samantha`, `Meredith`, `Bella`

## Clues:
1. The person whose mother's name is Kailyn is in the second house.
2. The person living in a colonial-style house is the person's child is named Meredith.
3. There is one house between the person's child is named Samantha and The person whose mother's name is Janelle.
4. The person in a ranch-style home is the person's child is named Fred.
5. Eric is the person's child is named Bella.
6. The person whose mother's name is Janelle is the person with an aquarium of fish.
7. Arnold is the person in a Craftsman-style house.
8. The person who keeps a pet bird and the person with an aquarium of fish are next to each other.
9. The person whose mother's name is Holly is directly left of Peter.
10. The person who has a cat is the person living in a colonial-style house.
11. The person's child is named Fred is somewhere to the left of The person whose mother's name is Janelle.
 What is the value of attribute Name for the person whose attribute House is 2? 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: `Bob`, `Arnold`, `Peter`, `Eric`, `Alice`, `Carol`
 - People have unique favorite music genres: `jazz`, `rock`, `pop`, `hip hop`, `classical`, `country`
 - Each person has a unique birthday month: `april`, `feb`, `jan`, `mar`, `sept`, `may`
 - Each person has a unique favorite drink: `root beer`, `coffee`, `tea`, `boba tea`, `water`, `milk`
 - Each person has an occupation: `nurse`, `doctor`, `teacher`, `lawyer`, `artist`, `engineer`

## Clues:
1. The person who loves pop music is not in the fourth house.
2. The boba tea drinker is the person whose birthday is in April.
3. The person whose birthday is in January is somewhere to the right of the person whose birthday is in May.
4. Arnold is not in the third house.
5. Eric is in the first house.
6. Carol is the person who loves jazz music.
7. The person who loves classical music is directly left of the root beer lover.
8. The person who loves country music is somewhere to the left of the person who is a teacher.
9. The person who is a teacher is somewhere to the left of the person who is a nurse.
10. The person who is a doctor is somewhere to the left of the person who is a teacher.
11. The root beer lover is the person who loves hip-hop music.
12. Arnold is the person who is a lawyer.
13. Carol is not in the fourth house.
14. The person who is an artist is somewhere to the left of the person who loves pop music.
15. The person whose birthday is in April is Bob.
16. The person who likes milk is not in the second house.
17. The one who only drinks water is somewhere to the right of the person whose birthday is in March.
18. The tea drinker is the person whose birthday is in September.
19. The person whose birthday is in January is the person who loves pop music.
20. The person who loves jazz music is directly left of the person who loves rock music.
21. Alice is the person who is a nurse.
22. The tea drinker is not in the sixth house.
23. The person who is a doctor is the root beer lover.
 What is the value of attribute Birthday for the person whose attribute Name is Alice? Please reason step by step, and put your final answer within \boxed{}

Answer: feb


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Arnold`, `Peter`, `Eric`, `Alice`
 - They all have a unique favorite flower: `lilies`, `daffodils`, `carnations`, `roses`, `tulips`
 - Each person has an occupation: `artist`, `doctor`, `lawyer`, `teacher`, `engineer`
 - Everyone has a favorite smoothie: `lime`, `watermelon`, `cherry`, `dragonfruit`, `desert`
 - People own unique car models: `honda civic`, `ford f150`, `bmw 3 series`, `toyota camry`, `tesla model 3`

## Clues:
1. Arnold is the person who is an artist.
2. The person who is a doctor is the person who loves the rose bouquet.
3. Bob is the person who loves a carnations arrangement.
4. The person who owns a BMW 3 Series is Arnold.
5. The person who loves the boquet of lilies is somewhere to the right of Alice.
6. The Watermelon smoothie lover is somewhere to the left of the person who drinks Lime smoothies.
7. The person who is a lawyer is the person who owns a Toyota Camry.
8. The person who loves the boquet of lilies is Arnold.
9. There are two houses between Bob and the person who owns a Honda Civic.
10. The person who owns a Tesla Model 3 is in the third house.
11. Peter is directly left of the person who loves the boquet of lilies.
12. The Desert smoothie lover is the person who is a teacher.
13. The person who is an artist and the Dragonfruit smoothie lover are next to each other.
14. The person who is a doctor is the Watermelon smoothie lover.
15. The person who loves the vase of tulips is directly left of the person who owns a Toyota Camry.
 What is the value of attribute House for the person whose attribute CarModel is tesla model 3? Please reason step by step, and put your final answer within \boxed{}

Answer: 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: `Alice`, `Peter`, `Eric`, `Bob`, `Carol`, `Arnold`
 - Everyone has a unique favorite cigar: `dunhill`, `blends`, `blue master`, `yellow monster`, `pall mall`, `prince`
 - People have unique favorite book genres: `fantasy`, `romance`, `science fiction`, `mystery`, `historical fiction`, `biography`

## Clues:
1. The person partial to Pall Mall is Alice.
2. The Prince smoker is not in the sixth house.
3. Carol is directly left of the person who loves fantasy books.
4. The Prince smoker is not in the fifth house.
5. Peter is in the first house.
6. The person partial to Pall Mall is directly left of the Dunhill smoker.
7. The person partial to Pall Mall is in the second house.
8. Carol is the person who loves mystery books.
9. Bob is not in the fourth house.
10. The person who smokes Yellow Monster is not in the sixth house.
11. The person who loves biography books is not in the third house.
12. The person who loves historical fiction books is in the first house.
13. The person who smokes many unique blends is not in the fifth house.
14. The person who loves romance books is somewhere to the right of Eric.
15. There is one house between Arnold and the person who smokes Yellow Monster.
16. There is one house between the person who smokes many unique blends and Bob.
 What is the value of attribute Name for the person whose attribute House is 1? 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: `Carol`, `Bob`, `Alice`, `Arnold`, `Eric`, `Peter`
 - Each person prefers a unique type of vacation: `camping`, `cultural`, `beach`, `city`, `mountain`, `cruise`
 - People have unique favorite sports: `swimming`, `volleyball`, `baseball`, `basketball`, `tennis`, `soccer`

## Clues:
1. The person who loves tennis is somewhere to the left of Peter.
2. Arnold is not in the fifth house.
3. The person who prefers city breaks and Peter are next to each other.
4. The person who enjoys camping trips is the person who loves basketball.
5. The person who likes going on cruises is Carol.
6. The person who enjoys mountain retreats is the person who loves swimming.
7. The person who loves baseball is the person who loves beach vacations.
8. Bob is the person who loves swimming.
9. The person who goes on cultural tours is the person who loves soccer.
10. The person who loves tennis is directly left of the person who enjoys mountain retreats.
11. The person who enjoys camping trips is Alice.
12. There are two houses between the person who loves tennis and Alice.
13. The person who goes on cultural tours is not in the fifth house.
14. The person who loves beach vacations is in the third house.
 What is the value of attribute Vacation for the person whose attribute Name is Bob? Please reason step by step, and put your final answer within \boxed{}

Answer: mountain


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Arnold`, `Alice`, `Peter`, `Eric`
 - People use unique phone models: `oneplus 9`, `google pixel 6`, `iphone 13`, `huawei p50`, `samsung galaxy s21`
 - Each person has a favorite color: `blue`, `red`, `yellow`, `green`, `white`
 - The mothers' names in different houses are unique: `Janelle`, `Penny`, `Holly`, `Aniya`, `Kailyn`
 - The people are of nationalities: `german`, `norwegian`, `swede`, `dane`, `brit`
 - Everyone has something unique for lunch: `spaghetti`, `grilled cheese`, `stew`, `pizza`, `stir fry`

## Clues:
1. The person who uses a OnePlus 9 and Arnold are next to each other.
2. There are two houses between the person who uses a OnePlus 9 and The person whose mother's name is Kailyn.
3. The person whose mother's name is Penny is somewhere to the left of the Swedish person.
4. The person who loves yellow is the person who uses a Google Pixel 6.
5. The person who loves eating grilled cheese is The person whose mother's name is Holly.
6. Arnold is the person who loves the spaghetti eater.
7. Bob is the person who uses a Huawei P50.
8. The German is the person who is a pizza lover.
9. Eric is directly left of the person who loves blue.
10. Alice is The person whose mother's name is Holly.
11. The person who loves stir fry is the British person.
12. The person whose mother's name is Penny is not in the third house.
13. Arnold is directly left of the person who uses a Huawei P50.
14. The British person is somewhere to the left of the Dane.
15. The person who uses a Google Pixel 6 is Arnold.
16. The person whose mother's name is Aniya is the person who uses a Samsung Galaxy S21.
17. The person who loves blue is the Norwegian.
18. The person who loves white is the person who uses a Huawei P50.
19. The person whose favorite color is red is directly left of the person who loves blue.
 What is the value of attribute PhoneModel for the person whose attribute Mother is Holly? Please reason step by step, and put your final answer within \boxed{}

Answer: oneplus 9


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Alice`, `Peter`, `Eric`
 - They all have a unique favorite flower: `roses`, `carnations`, `daffodils`, `lilies`
 - Each person lives in a unique style of house: `victorian`, `craftsman`, `colonial`, `ranch`
 - Everyone has a unique favorite cigar: `prince`, `dunhill`, `blue master`, `pall mall`

## Clues:
1. The person in a ranch-style home is in the third house.
2. The person who smokes Blue Master is somewhere to the left of the person residing in a Victorian house.
3. There are two houses between Eric and the person who loves the rose bouquet.
4. The Prince smoker is Peter.
5. The person in a Craftsman-style house is the Dunhill smoker.
6. The Dunhill smoker is the person who loves a carnations arrangement.
7. Arnold is in the fourth house.
8. The Prince smoker is in the second house.
9. The person who loves the boquet of lilies is the person who smokes Blue Master.
 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 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Bob`, `Alice`, `Eric`, `Peter`, `Carol`
 - People have unique favorite sports: `volleyball`, `soccer`, `basketball`, `baseball`, `swimming`, `tennis`
 - Each person has a unique hobby: `gardening`, `cooking`, `painting`, `woodworking`, `knitting`, `photography`
 - Each mother is accompanied by their child: `Bella`, `Meredith`, `Alice`, `Samantha`, `Fred`, `Timothy`
 - Each person has a unique birthday month: `feb`, `jan`, `mar`, `may`, `sept`, `april`
 - Everyone has something unique for lunch: `stir fry`, `grilled cheese`, `pizza`, `spaghetti`, `stew`, `soup`

## Clues:
1. The person who loves soccer is somewhere to the right of Eric.
2. The person who paints as a hobby is somewhere to the left of the woodworking hobbyist.
3. The person whose birthday is in February is not in the third house.
4. The person who is the mother of Timothy is directly left of Arnold.
5. The person who loves cooking is somewhere to the left of the person's child is named Fred.
6. Alice is the person who paints as a hobby.
7. The person's child is named Alice is the person who loves cooking.
8. The person whose birthday is in March is the person's child is named Samantha.
9. The person who loves the soup is the person whose birthday is in May.
10. Carol is somewhere to the right of the person whose birthday is in May.
11. The person whose birthday is in March and the person who is a pizza lover are next to each other.
12. The person whose birthday is in February is not in the fourth house.
13. There are two houses between the person who loves baseball and the person who loves the stew.
14. The person whose birthday is in May is somewhere to the left of the person who loves volleyball.
15. The person whose birthday is in January is somewhere to the right of Arnold.
16. The person who loves cooking is not in the third house.
17. The person who enjoys knitting is the person whose birthday is in September.
18. The person who enjoys gardening is the person's child is named Meredith.
19. The person who is a pizza lover is the person who loves swimming.
20. There is one house between the person who loves basketball and the person whose birthday is in May.
21. The person who loves the soup is directly left of the person who loves the spaghetti eater.
22. The person who loves the stew is the person's child is named Bella.
23. The person who enjoys knitting is somewhere to the left of the person who loves the soup.
24. Bob is somewhere to the left of the person who loves the stew.
25. Arnold is the person who loves stir fry.
26. The woodworking hobbyist is directly left of the person who loves the soup.
 What is the value of attribute Food for the person whose attribute Birthday is sept? Please reason step by step, and put your final answer within \boxed{}

Answer: pizza


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Bob`, `Arnold`, `Eric`, `Alice`
 - Each person has a favorite color: `red`, `white`, `green`, `blue`, `yellow`
 - Each person has a unique hobby: `gardening`, `cooking`, `painting`, `photography`, `knitting`
 - People have unique heights: `very tall`, `tall`, `average`, `short`, `very short`
 - People have unique hair colors: `black`, `blonde`, `brown`, `gray`, `red`

## Clues:
1. The person whose favorite color is green is somewhere to the right of the person who has black hair.
2. The person who has an average height is somewhere to the right of the person who loves blue.
3. The person who has red hair is Alice.
4. The person who loves cooking is not in the first house.
5. The person who loves yellow is the person who has gray hair.
6. The person who is short is somewhere to the left of the person who loves yellow.
7. The person who enjoys gardening is the person who loves white.
8. Arnold is somewhere to the left of the person who has black hair.
9. The person who is short is somewhere to the right of the person who has blonde hair.
10. Eric and the person who loves cooking are next to each other.
11. The person who has black hair is directly left of the person who loves blue.
12. The person who is very short is the person who has black hair.
13. Bob is the person who loves blue.
14. The person who is tall is directly left of the photography enthusiast.
15. The person who loves white is not in the first house.
16. The person who loves yellow is the person who enjoys knitting.
17. The person who loves blue is the person who has blonde hair.
 What is the value of attribute House for the person whose attribute HairColor is brown? Please reason step by step, and put your final answer within \boxed{}

Answer: 1


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Peter`, `Eric`, `Carol`, `Alice`, `Bob`
 - The people are of nationalities: `swede`, `brit`, `chinese`, `dane`, `german`, `norwegian`
 - People have unique favorite book genres: `science fiction`, `mystery`, `fantasy`, `biography`, `historical fiction`, `romance`
 - Everyone has a favorite smoothie: `blueberry`, `dragonfruit`, `desert`, `watermelon`, `cherry`, `lime`
 - People use unique phone models: `oneplus 9`, `samsung galaxy s21`, `huawei p50`, `xiaomi mi 11`, `iphone 13`, `google pixel 6`

## Clues:
1. Alice is the British person.
2. There is one house between Eric and the Dragonfruit smoothie lover.
3. The Chinese is somewhere to the left of Alice.
4. The person who loves historical fiction books is directly left of the person who loves biography books.
5. The person who loves biography books is somewhere to the right of Carol.
6. The person who uses an iPhone 13 is somewhere to the right of the Desert smoothie lover.
7. The person who loves mystery books is the British person.
8. Bob is somewhere to the right of the person who uses a OnePlus 9.
9. Carol is the person who drinks Blueberry smoothies.
10. There are two houses between the person who loves romance books and the Watermelon smoothie lover.
11. The person who likes Cherry smoothies is the Swedish person.
12. The person who loves science fiction books is the person who uses a Google Pixel 6.
13. The person who drinks Lime smoothies is Eric.
14. Bob is in the fifth house.
15. Carol is in the second house.
16. The Norwegian and Alice are next to each other.
17. The Swedish person is not in the fifth house.
18. The German and the Norwegian are next to each other.
19. The person who loves fantasy books is the Dragonfruit smoothie lover.
20. The British person is somewhere to the left of Peter.
21. The person who loves romance books is the person who uses a Samsung Galaxy S21.
22. Alice is the person who uses a Xiaomi Mi 11.
23. The person who loves romance books is Eric.
 What is the value of attribute PhoneModel for the person whose attribute House is 1? Please reason step by step, and put your final answer within \boxed{}

Answer: samsung galaxy s21


Anya, Vanya, Danya, Sanya, and Tanya were collecting apples. It turned out that each of them collected a whole percentage of the total number of apples, and all these numbers were different and greater than zero. Then Tanya, who collected the most apples, ate her apples. After that, it turned out that each of the remaining children still had a whole percentage, but now from the number of apples left. What is the minimum number of apples that could have been collected?

Answer: 20


When adding two natural numbers, Alyosha accidentally added an extra digit, so instead of the number 23456, he got the number 55555. Determine which digit Alyosha accidentally added.

A flask is filled with a 96% hydrochloric acid solution. 12 liters of acid were poured out and the flask was refilled with water. Then, 18 more liters were poured out and the flask was refilled with water again, after which the concentration of the acid in the flask was 32%. Find the volume of the flask.

Answer: 5


A pattern of Figures is shown below. Figure 1 is a regular pentagon with side length 1. Figure 2 is a regular pentagon of side length 2 drawn around Figure 1 so that the two shapes share the top vertex, \( T \), and the sides on either side of \( T \) overlap. The pattern continues so that for each \( n > 1 \), Figure \( n \) is a regular pentagon of side length \( n \) drawn around the previous Figure so that the two shapes share the top vertex, \( T \), and the sides on either side of \( T \) overlap. The ink length of each Figure is the sum of the lengths of all of the line segments in the Figure.
(a) Determine the ink length of Figure 4.
(b) Determine the difference between the ink length of Figure 9 and the ink length of Figure 8.
(c) Determine the ink length of Figure 100.

Answer: 500


When the boy Clive approached his grandfather's wall cuckoo clock, it was 12:05.

Clive started spinning the minute hand with his finger until the hour hand returned to its original position. How many "cuckoos" did the grandfather count in the neighboring room during this time?

Answer: 78


A team of five people is laying a 30-meter pipe into the ground. To prepare the trench, they decided to dig in turns with the condition that each person digs for the amount of time it takes the other four together to dig one-third of the trench. The 30-meter trench was completed when the last team member had worked their allotted time. How many meters of the trench could they have dug in the same time if they all worked together?

Answer: 80


Three swimmers have to swim a 50-meter lane in the pool, immediately turn back and return to the starting point. The first swimmer starts first, the second swimmer starts 5 seconds later, and the third swimmer starts another 5 seconds after the second. At some point in time, before reaching the end of the lane, all three swimmers were equidistant from the start. The third swimmer, having reached the end of the lane and turning back, met the second swimmer 4 meters from the end of the lane, and the first swimmer 7 meters from the end of the lane. Find the speed of the third swimmer.

Answer: \frac{22}{15}\,


In a warehouse, there are 8 cabinets, each containing 4 boxes, and each box contains 10 mobile phones. The warehouse, each cabinet, and each box are locked with a key. The manager is tasked with retrieving 52 mobile phones. What is the minimum number of keys the manager must take with him?

Answer: 9


At a round table, 30 people are seated - knights and liars (knights always tell the truth, and liars always lie). It is known that each of them at this table has exactly one friend, and for knights, this friend is a liar, and for liars, this friend is a knight (friendship is always mutual). When asked, "Is your friend sitting next to you?", those sitting every other seat answered "Yes". How many of the rest could also have answered "Yes"?

Answer: 0


Given a cube with a non-negative real number placed at each vertex such that the sum of all the real numbers is 1, two players, A and B, play a game. Player A first selects any face of the cube, then Player B selects another face, and finally Player A selects a third face, with the condition that faces parallel to those already selected cannot be chosen.

Prove that Player A can always ensure that the sum of the numbers at the common vertex of the three chosen faces is no more than $\frac{1}{6}$.

Answer: \frac{1}{6}


Wesyu is a farmer, and she's building a cow pasture. She starts with a triangle \( A_{0} A_{1} A_{2} \) where angle \( A_{0} \) is \( 90^\circ \), angle \( A_{1} \) is \( 60^\circ \), and \( A_{0} A_{1} \) is 1. She then extends the pasture. First, she extends \( A_{2} A_{0} \) to \( A_{3} \) such that \( A_{3} A_{0} = \frac{1}{2} A_{2} A_{0} \) and the new pasture is triangle \( A_{1} A_{2} A_{3} \). Next, she extends \( A_{3} A_{1} \) to \( A_{4} \) such that \( A_{4} A_{1} = \frac{1}{6} A_{3} A_{1} \). She continues, each time extending \( A_{n} A_{n-2} \) to \( A_{n+1} \) such that \( A_{n+1} A_{n-2} = \frac{1}{2^{n} - 2} A_{n} A_{n-2} \). What is the smallest \( K \) such that her pasture never exceeds an area of \( K \)?

Answer: \sqrt{3}


1500 products are packed into 15 boxes for shipment. After sealing the boxes, it is discovered that one product was not packed into any box. Due to strict sealing requirements, boxes cannot be reopened for inspection. Someone suggests using a scale to weigh the boxes to find which box is missing a product (because the box missing a product will be lighter). Normally, it would take 14 weighings to find the box missing the product. Is there a way to reduce the number of weighings and still determine which box is missing a product?

Answer: 4


Naomi has a broken calculator. All it can do is either add one to the previous answer, or square the previous answer. (It performs the operations correctly.) Naomi starts with 2 on the screen. In how many ways can she obtain an answer of 1000?

Answer: 128


There are 111 lamps, each with its own switch. It is allowed to flip 13 of them at the same time. Initially, some lamps are on and some are off.

a) Is it possible to turn off all the lamps?

b) How many flips are required to do this if all the lamps were initially on?

Answer: 9


A decorator will distribute flowers at eight points around a square flower arrangement, as indicated in the figure below. He wants to do this in such a way that on each side of the arrangement, people always see the same number of flowers. In the example below, there are a total of 11 flowers, and on each of the 4 sides of the square, exactly 4 flowers are seen.

a) What is the maximum number of flowers that can be used, considering that exactly 9 flowers should be seen on each side of the square?

b) What is the minimum number of flowers that can be used, considering that exactly 12 flowers should be seen on each side of the square?

Answer: 24


A circle passes through the vertices $A$ and $C$ of triangle $ABC$, intersecting side $AB$ at point $E$ and side $BC$ at point $F$. The angle $AEC$ is 5 times the angle $BAF$, and the angle $ABC$ is $72^{\circ}$. Find the radius of the circle if $AC=6$.

Answer: 3


In a football tournament, 15 teams participated, and each team played against every other team exactly once. A win earns 3 points, a draw earns 1 point, and a loss earns 0 points.

After the tournament ended, it turned out that some 6 teams each scored at least \( N \) points. What is the highest integer value that \( N \) can take?

Answer: 34


A circle is tangent to the extensions of two sides \(A B\) and \(A D\) of square \(A B C D\) with side length \(6+2 \sqrt{5}\) cm. From point \(C\), two tangents are drawn to this circle. Find the radius of the circle if the angle between the tangents is \(36^\circ\) and it is known that \(\sin 18^\circ = \frac{\sqrt{5}-1}{4}\).

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


Assume a set of points \( S \) in the plane has the properties: (i) any three points are not collinear; (ii) the distances between any two points are all distinct. For two points \( A \) and \( B \) in \( S \), if there exists \( C \in S \) such that \( |AC| < |AB| < |BC| \), then \( AB \) is called a median edge of \( S \). For three points \( A \), \( B \), and \( C \) in \( S \), if \( AB \), \( AC \), and \( BC \) are all median edges of \( S \), then \( \triangle ABC \) is called a median triangle of \( S \). Find the smallest \( n \) such that any \( n \)-point set \( S \) satisfying properties (i) and (ii) must contain a median triangle.

Answer: 6


Let \(f(x) = \frac{x+1}{x-1}\). Then for \(x^2 \neq 1\), find the expression for \(f(-x)\). The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\).

Answer: 2


A spider has one sock and one shoe for each of its eight legs. In how many different orders can the spider put on its socks and shoes, assuming that, on each leg, the sock must be put on before the shoe? Provide your answer in the form of \(\frac{m}{n}\) and find the value of m + n.

Answer: 20922789888256


If \(\frac{x}{x-1} = \frac{y^2+2y-1}{y^2-2y-2}\), find the value of \(x\). The original answer can be expressed in the form \(ay^2 + by + c\). Please find the value of \(a + b + c\).

Answer: 2


Find the first step in finding the product \((3x + 2)(x - 5)\) by use of the distributive property in the form \(a(b + c) = ab + ac\). The original answer is in the form \((a+b)x + (a+b)(c)\). Please give the value of a + b + c.

Answer: 0


Equilateral $\triangle ABC$ is inscribed in a circle. A second circle is tangent internally to the circumcircle at $T$ and tangent to sides $AB$ and $AC$ at points $P$ and $Q$. If side $BC$ has length $12$, find the length of segment $PQ$.

Answer: 8


A rectangular box measures $a \times b \times c$, where $a,$ $b,$ and $c$ are integers and $1 \leq a \leq b \leq c$. The volume and surface area of the box are numerically equal. Find the number of ordered triples $(a,b,c)$ that are possible.

Answer: 10


After simple interest for two months at $5\%$ per annum was credited, a Boy Scout Troop had a total of $\textdollar 255.31$ in the Council Treasury. The interest credited was a number of dollars plus the following number of cents. If the answer is \( x \) cents, what is the value of \( x \)?

Answer: 11


Let $r$ be the number that results when both the base and the exponent of $a^b$ are tripled, where $a, b>0$. If $r$ equals the product of $a^b$ and $x^b$ where $x>0$, find the value of $x$. The original answer is in the form of $k \cdot m^n$, where $k$, $m$, and $n$ are constants. Please find the value of $k + m + n$.

Answer: 30


Let \(ABCD\) be a regular tetrahedron and let \(E\) be a point inside the face \(ABC\). Denote by \(s\) the sum of the distances from \(E\) to the faces \(DAB\), \(DBC\), \(DCA\), and by \(S\) the sum of the distances from \(E\) to the edges \(AB\), \(BC\), \(CA\). The original answer is in \(\frac{k\sqrt{m}}{n}\) format. Please give the value of \(k + m + n\).

Answer: 7


Every cell of an $8\times8$ chessboard contains either $1$ or $-1$. It is known that there are at least four rows such that the sum of numbers inside the cells of those rows is positive. Find the maximum number of columns such that the sum of numbers inside the cells of those columns is less than $-3$.

Answer: 6


A 16-step path is to go from $( -4, -4)$ to $(4,4)$ with each step increasing either the $x$-coordinate or the $y$-coordinate by 1. Find the number of such paths that stay outside or on the boundary of the square $-2 \le x \le 2$, $-2 \le y \le 2$ at each step.

Answer: 1698


A rectangle is inscribed in a triangle such that its base coincides with the base $b$ of the triangle. If the altitude of the triangle is $h$, and the altitude $x$ of the rectangle is half the base of the rectangle, find the expression for $x$ in terms of $b$ and $h$. The original answer is in the form \frac{kb}{mh+b}, please give the value of k + m.

Answer: 3


7.1. Write the number 1.11 ten times and the number 1.01 eleven times. Cross out one or several numbers so that the sum of the remaining numbers is equal to 20.19.

Answer: 20.19


Given the circle \(\Gamma: (x-2)^{2} + y^{2} = 3\), let the line \(l_{1}: x - m y - 1 = 0\) intersect the circle \(\Gamma\) at points \(A\) and \(C\), and the line \(l_{2}: m x + y - m = 0\) intersect the circle \(\Gamma\) at points \(B\) and \(D\). The segments \(AB\) and \(CD\) are located above and below the x-axis, respectively. When the slope of \(CD\) is \(-1\), find the length of the segment \(AB\).

Answer: 2\sqrt{2}


X OM - I - Task 8

The sides of a triangle are the medians of another triangle. Calculate the ratio of the areas of both triangles. Can a triangle be constructed from the medians of any triangle?

Answer: \dfrac{3}{4}


Let $a_1$, $a_2, \dots, a_{2015}$ be a sequence of positive integers in $[1,100]$.
Call a nonempty contiguous subsequence of this sequence [i]good[/i] if the product of the integers in it leaves a remainder of $1$ when divided by $101$.
In other words, it is a pair of integers $(x, y)$ such that $1 \le x \le y \le 2015$ and \[a_xa_{x+1}\dots a_{y-1}a_y \equiv 1 \pmod{101}. \]Find the minimum possible number of good subsequences across all possible $(a_i)$.

[i]Proposed by Yang Liu[/i]

Answer: 19320


The eleventh question: A certain meeting has 30 participants, each of whom has at most 5 acquaintances among the others, and in any group of 5 people, there exist two who are not acquaintances. Find the maximum positive integer $\mathrm{k}$, such that in the 30 people satisfying the above conditions, there always exist $\mathrm{k}$ people, none of whom are acquaintances with each other.

Answer: 6


On an \( n \times m \) grid, Alice and Bob take turns placing \( 1 \times 2 \) dominos in any direction. The player who cannot place a domino loses. Who has a winning strategy:

- If both \( n \) and \( m \) are even?
- If \( n \) is even and \( m \) is odd?

Answer: \text{Bob}


34. Taiyi Zhenren and Shen Gongbao competed for the position of Golden Immortal, making their disciples Nezha and Ao Bing take the same exam. After one incense stick, both of them finished all the questions. Nezha got $\frac{1}{3}$ of the questions wrong, and Ao Bing got 6 questions wrong. Taiyi Zhenren said: “The questions you both got wrong account for $\frac{1}{5}$ of all the questions.”
Shen Gongbao said: “Fortunately, the questions you both got right exceeded half of all the questions.”
Smart students, calculate how many questions Nezha and Ao Bing both got right. $\qquad$ questions.

Answer: 20


5. On the website of the football club "Rostov," a poll is being conducted to determine which of the $m$ football players the website visitors consider the best at the end of the season. Each visitor votes once for one player. The website displays the rating of each player, which is the percentage of votes cast for them, rounded to the nearest whole number. After several visitors have voted, the total rating of the nominees was $95 \%$. What is the smallest $m$ for which this is possible?

Answer: 11


In a right-angled triangle, one of the angles is $30^{\circ}$. Prove that the segment of the perpendicular drawn to the hypotenuse through its midpoint, intersecting the longer leg, is three times shorter than the longer leg.

Answer: \frac{1}{3}


Let \( A \) be a nonempty set of positive integers. Suppose that there are positive integers \( b_1, \ldots, b_n \) and \( c_1, \ldots, c_n \) such that:
(i) For each \( i \), the set \( b_i A + c_i = \{ b_i a + c_i \mid a \in A \} \) is a subset of \( A \), and
(ii) The sets \( b_i A + c_i \) and \( b_j A + c_j \) are disjoint whenever \( i \neq j \).

Prove that
\[ \frac{1}{b_1} + \cdots + \frac{1}{b_n} \leq 1. \]

Answer: \frac{1}{b_1} + \cdots + \frac{1}{b_n} \leq 1


Example 4 Draw the following lines on the coordinate plane $y=k, y=\sqrt{3} x+2 k, y=-\sqrt{3} x+2 k$, where $k=$ $0, \pm 1, \pm 2, \cdots, \pm 10$. These 63 lines can divide the plane into several equilateral triangles. Find the number of equilateral triangles with side length $\frac{2}{\sqrt{3}}$.
(1994 12th American Invitational Mathematics Examination Problem)

Answer: 660


Two precocious students from Level 3 participated in a university chess tournament. Each participant plays against all the others exactly once. A win is worth 1 point, a draw is worth 0.5 points, and a loss is worth 0 points. The sum of the scores of the two Level 3 students is 6.5. All university students scored the same. How many university students participated in the competition?

Answer: 11


450. For what values of $x$ is the equality $|x+6|=$ $=-(x+6)$ true?

Answer: (-\infty, -6]


A natural number $n$ has the following property:
For arbitrary real numbers $a_{1}, a_{2}, \ldots, a_{d}$, which both satisfy $a_{1}+a_{2}+\ldots+a_{d}=2013$ and $0 \leq a_{i} \leq 1$ for $i=1,2, \ldots, d$, there exists a partition of the set of these real numbers into $n$ pairwise disjoint subsets (some of which may be empty), such that the sum of the numbers in each subset is at most 1.
Determine the smallest number $n$ with this property.

Answer: 2013


19.7.17 ** Find the smallest integer greater than 2 that can be expressed both as the sum of the squares of two positive integers and as the sum of the cubes of two positive integers. Also prove that there are infinitely many positive integers that can be expressed both as the sum of the squares of two coprime positive integers and as the sum of the cubes of two coprime positive integers.

Answer: 65


11. A paper rectangle $A B C D(A B=3, B C=9)$ was folded so that vertices $A$ and $C$ coincided. What is the area of the resulting pentagon?

Answer: \dfrac{39}{2}


14. (3 points) There is a clock that now shows 10 o'clock. After $\qquad$ minutes, the minute hand and the hour hand overlap for the first time; after another $\qquad$ minutes, the minute hand and the hour hand overlap for the second time.

Answer: 54\frac{6}{11}


1. The production of ceramic items consists of 3 sequential stages: forming a ceramic item on a potter's wheel for 15 minutes, drying for 10 minutes, and firing for 30 minutes. It is required to produce 75 items. How should 13 masters be distributed between molders and firers to work on stages 1 and 3 respectively for the entire duration of the stage (drying does not require workers), to complete the work in the shortest time possible? In the answer, write the shortest time (in minutes) required to complete the entire job. Provide the answer as a number without indicating the unit. (5 points)

Answer: 325


2. In triangle $A B C$, the following holds: $|B C|=4,|C A|=5, \angle B C A=2 \angle C A B$. Calculate the radii of the circumscribed and inscribed circles of this triangle.

Answer: \dfrac{8\sqrt{7}}{7}


16. (6 points) In the rabbit figure formed by the tangram, the area of the rabbit's ears (shaded part) is 10 square centimeters. Then the area of the rabbit figure is $\qquad$ square centimeters.

Answer: 80


：Given $M$ as the midpoint of the height $S O$ of a regular tetrahedron $S-A B C$ with edge length 1. A plane is drawn through $M$ parallel to the lateral edge $S A$ and the base triangle $B C$. The area of the section formed by this plane cutting through the tetrahedron is $\qquad$
$\qquad$ .

Answer: \dfrac{2}{9}


Three, a football invitational tournament involves sixteen cities, each sending two teams, Team A and Team B. According to the competition rules, each pair of teams plays at most one match. Additionally, the two teams from the same city do not play against each other. After a period of matches, it was found that, except for Team A from City A, all other teams have played a different number of matches. How many matches has Team B from City A played? Please prove your conclusion.

Answer: 15


4A. Given a convex quadrilateral $A B C D$ for which: $\angle A B D=50^{\circ}$, $\measuredangle A D B=80^{\circ}, \measuredangle A C B=40^{\circ}$ and $\measuredangle D B C=\measuredangle B D C+30^{\circ}$. Find $\measuredangle D B C$.

Answer: 70^\circ


Shapovalovo $A . B$. The numbers from 1 to 37 were written in a row such that the sum of any initial segment of numbers is divisible by the next number in the sequence.

What number is in the third position if the first position is occupied by the number 37, and the second by 1? #

Answer: 2


Given a sequence of positive real numbers $x_{0}, x_{1}, x_{2}, \cdots, x_{1995}$ that satisfies the following conditions:
1. $x_{0}=x_{1995}$;
2. For $i=1, 2, \cdots, 1995$, the equation $x_{i-1}+\frac{2}{x_{i-1}}=2 x_{i}+\frac{1}{x_{i}}$ holds.

Find the maximum value of $x_{0}$.

Answer: 2^{997}


We consider sports tournaments with $n \ge 4$ participating teams and where every pair of teams plays against one another at most one time. We call such a tournament [i]balanced [/i] if any four participating teams play exactly three matches between themselves. So, not all teams play against one another.
Determine the largest value of $n$ for which a balanced tournament with $n$ teams exists.

Answer: 5


Vasya has chosen a natural number \( n \leq 2020 \). Petya tries to guess it by naming a natural number \( x \) and asking if the chosen number is greater than \( x \) (whether \( x < n \)). Vasya answers either "yes" or "no". Petya wins if he determines the number. However, Petya loses if he receives a "no" response for the second time and cannot name the chosen number. What is the minimum number of questions Petya needs to ask to guarantee a win?

Answer: 64


8. [6] Let $O$ be the point $(0,0)$. Let $A, B, C$ be three points in the plane such that $A O=15, B O=15$, and $C O=7$, and such that the area of triangle $A B C$ is maximal. What is the length of the shortest side of $A B C$ ?

Answer: 20


9. For what values of the parameter $\boldsymbol{a}$ does the equation $|f(x)-4|=p(x)$, where $f(x)=\left|\frac{x^{2}+3 x}{x+3}-\frac{x^{2}-4 x+4}{2-x}\right|$, $p(x)=a$ have three solutions? If there is more than one value of the parameter, indicate their product in the answer.

Answer: 2


[ Equilateral (Regular) Triangle ]

In an equilateral triangle $D E F$, a circle of radius $r$ is inscribed. This circle is externally tangent to three other circles of the same radius at the points of tangency with the sides of the triangle. The centers of the external circles are $O_{1}, O_{2}, O_{3}$, respectively. Find the area of the hexagon formed by the intersection of triangles $D E F$ and $O_{1}, O_{2}, O_{3}$.

Answer: 2\sqrt{3} r^2


## 265. Math Puzzle $6 / 87$

For a motorcycle, the optimal rotational speed of the engine shaft is $6000 \mathrm{U} / \mathrm{min}$.

What is the gear ratio of the rear axle to the engine shaft produced by the transmission if the optimal rotational speed results in a speed of $60 \mathrm{~km} / \mathrm{h}$?

The diameter of the rear wheel is $53 \mathrm{~cm}$ (circumference 1.67 m).

Answer: \dfrac{1}{10}


2. $V, W, X, Y, Z$ are 5 digits in base 5. The three three-digit numbers $(V Y Z)_{5},(V Y X)_{5},(V V W)_{5}$ in base 5 increase sequentially with a common difference of 1. What is the three-digit number $(X Y Z)_{5}$ in base 10?

Answer: 108


The director of the school decided to take a photo of the 2008 graduates. He arranged the students in parallel rows, all with the same number of students, but this arrangement was too wide for the camera's field of view. To solve this problem, the director noticed that it was enough to remove one student from each row and place them in a new row. This arrangement did not please the director because the new row had 4 students fewer than the others. He then decided to remove one more student from each row and place them in the new row he created, and found that this way all the rows had the same number of students, and finally took the photo. How many students appeared in the photo?

Answer: 24


Problem 1. Let $n$ be a natural number. Find the least natural number $k$ for which there exist $k$ sequences of 0 's and 1 's of length $2 n+2$ with the following property: any sequence of 0 's and 1 's of length $2 n+2$ coincides in at least $n+2$ positions with some of these $k$ sequences.

Answer: 4


## 122. Math Puzzle $7 / 75$

Peter rides his bike from school to the sports field. When he had covered 3/4 of the distance, he met his friend Hans, who was also riding at the same speed.

How fast are both of them riding if the teacher, traveling by moped ( $v=40 \mathrm{~km} / \mathrm{h}$ ), meets Peter and Hans exactly at their departure from the school and the sports field, respectively?

(Since the route passes through a sparsely trafficked and open area, the speeds can be considered constant.)

Answer: 20


75. Let positive real numbers $a, b, c$ satisfy the condition $(a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)=13$. Find the maximum and minimum values of the following expression.
$$P=\frac{a^{3}+b^{3}+c^{3}}{a b c} \text { (Pham Kim Hung) }$$

Answer: 11 + 2\sqrt{3}


Through the vertices of triangle \(ABC\) and a point \(P\) inside it, lines are drawn meeting the opposite sides at points \(A_1, B_1,\) and \(C_1\), respectively. A line drawn through point \(C_1\) parallel to \(AA_1\) intersects sides \(AC\) and \(BC\) at points \(M\) and \(N\), respectively, and intersects segment \(BB_1\) at point \(D\). Prove that \(NC_1^2 = MN \cdot ND\).

Answer: NC_1^2 = MN \cdot ND


C4. We have a group of $n$ kids. For each pair of kids, at least one has sent a message to the other one. For each kid $A$, among the kids to whom $A$ has sent a message, exactly $25 \%$ have sent a message to $A$. How many possible two-digit values of $n$ are there?

Answer: 26


6. Given a positive integer $k$ that satisfies for any positive integer $n$, the smallest prime factor of $n^{2}+n-k$ is no less than 11. Then, $k_{\text {min }}$ $=$ . $\qquad$

Answer: 43


5. Given a grid board $3 \times 2021$ (3 cells vertically and 2021 cells horizontally) and an unlimited supply of cardboard strips $1 \times 3$. Petya and Vasya play the following game. They take turns placing the strips on the board (cell by cell) without overlapping, one strip per turn. Petya places his strips horizontally, while Vasya places his strips vertically. The player who cannot make a move loses, and Petya starts. Who of the players can ensure their victory regardless of the opponent's actions?

Answer: Petya


17. At 6:20:25, a certain electronic watch displays 6:20:25. How many times within the hour from 5:00 to 6:00 does this watch display 5 different digits? $\qquad$ kinds.

Answer: 840


10. In quadrilateral $A B C D$, $A B=B C=C D=$ $26, A D=30 \sqrt{3}, A C$ and $B D$ intersect at point $O, \angle A O B=$ $60^{\circ}$. Then $S_{\text {quadrilateral } A B C D}=$ $\qquad$

Answer: 506\sqrt{3}


11. (20 points) Let there be $m$ distinct positive even numbers and $n$ distinct positive odd numbers whose sum is 2015. Find the maximum value of $20 m+15 n$.

Answer: 1105


The sum of the areas of all triangles whose vertices are also vertices of a 1 by 1 by 1 cube is $m + \sqrt{n} + \sqrt{p},$ where $m, n,$ and $p$ are integers. Find $m + n + p.$
Please give the answer directly without any intermediate steps.

Answer: 348


1373. Integrate the equation

$$
x d y=(x+y) d x
$$

and find the particular solution that satisfies the initial conditions $y=2$ when $x=-1$.

Answer: y = x \ln|x| - 2x