Among 30 people with different ages, select two groups: the first group with 12 people and the second group with 15 people, such that the oldest person in the first group is younger than the youngest person in the second group. Determine the number of ways to make this selection.

Answer: 4060


From 7:00 to 8:00, the speed of all vehicles in a certain city is halved as it is a peak period. Each morning at 6:50, persons A and B set off from locations A and B, respectively, heading towards each other and meet at a point 24 kilometers from A. If B starts 20 minutes earlier, they will meet at a point 20 kilometers from A. If A starts 20 minutes later, they will meet exactly at the midpoint of A and B. How far apart are locations A and B in kilometers?

Answer: 42


A deck of cards has 54 cards in total, including black spades, red hearts, black clubs, and red diamonds, each with 13 cards, and 2 jokers. What is the minimum number of cards that must be drawn to ensure that there are at least 2 cards of each of the 4 suits?

Answer: 43


The King of Hearts has one clock, and the Queen has another. Both clocks chime every hour. The King's clock chimes faster than the Queen's clock: while the Queen's clock strikes twice, the King's clock strikes three times. One day, both clocks started chiming at the same time. After the King’s clock finished chiming, the Queen's clock chimed two more times.

At what time did this occur?

Answer: 5


The natural numbers 22, 23, and 24 have the following property: the exponents of the prime factors in their factorization are all odd:

$$
22=2^{1} \cdot 11^{1} ; \quad 23=23^{1} ; \quad 24=2^{3} \cdot 3^{1}
$$

What is the largest number of consecutive natural numbers that can have this property? Explain the reasoning for your answer.

Answer: 7


Three problems \( A \), \( B \), and \( C \) were given on a mathematics olympiad. All 25 students solved at least one of these problems. The number of students who solved \( B \) and not \( A \) is twice the number of students who solved \( C \) and not \( A \). The number of students who solved only \( A \) is greater by 1 than the number of students who, along with \( A \), solved at least one other problem. Among the students who solved only one problem, half solved \( A \). How many students solved only \( B \)?

Answer: 6


Given the set
$$
A=\{n|n \in \mathbf{N}, 11| S(n), 11 \mid S(n+1)\} \text {, }
$$
where \(S(m)\) represents the sum of the digits of a natural number \(m\). Find the smallest number in the set \(A\).

Answer: 2899999


A cleaner was mopping the stairs in a skyscraper. To make the work more enjoyable, she counted the cleaned steps. When she had cleaned exactly half of the steps, she took a break. After a while, she started working again and wanted to continue counting the steps. However, when trying to recall the number of steps she had already cleaned, she mistakenly read the correct three-digit number backward, creating a smaller number, and counted from there. After mopping all the stairs, she ended at the number 746. How many steps could she have actually cleaned?

Answer: 1142


An empty \(2020 \times 2020 \times 2020\) cube is given, and a \(2020 \times 2020\) grid of square unit cells is drawn on each of its six faces. A beam is a \(1 \times 1 \times 2020\) rectangular prism. Several beams are placed inside the cube subject to the following conditions:
- The two \(1 \times 1\) faces of each beam coincide with unit cells lying on opposite faces of the cube. (Hence, there are \(3 \cdot 2020^{2}\) possible positions for a beam.)
- No two beams have intersecting interiors.
- The interiors of each of the four \(1 \times 2020\) faces of each beam touch either a face of the cube or the interior of the face of another beam.

What is the smallest positive number of beams that can be placed to satisfy these conditions?

Answer: 3030


In an isosceles triangle \( ABC \) where \( AB = BC \), side \( AC = 10 \). A circle with a diameter of 15 is inscribed in the angle \( ABC \) such that it touches the side \( AC \) at its midpoint. Find the radius of the circle inscribed in the triangle \( ABC \).

Answer: \frac{10}{3}


Let \( N = 30^{2015} \). Find the number of ordered 4-tuples of integers \((A, B, C, D) \in \{1, 2, \ldots, N\}^{4}\) (not necessarily distinct) such that for every integer \( n \), \( A n^3 + B n^2 + 2 C n + D \) is divisible by \( N \).

Answer: 24


3. Two thousand dominoes are arranged in a row at equal (and small) distances from each other. Then, the dominoes are numbered from 1 to 2000. Several more dominoes are lined up in the same way in front of the domino with number 1, and at the other end, $k$ times more $(k>8)$. At the same time and with equal force, the end dominoes are knocked over towards the center. How many dominoes are arranged if, in the end, the domino with number 1984 is left standing?

Answer: 3981


Answer: \dfrac{43}{114}


XLIV OM - I - Problem 11

In six different cells of an $ n \times n $ table, we place a cross; all arrangements of crosses are equally probable. Let $ p_n $ be the probability that in some row or column there will be at least two crosses. Calculate the limit of the sequence $ (np_n) $ as $ n \to \infty $.

Answer: 30


4. If a natural number can be divided by both 20 and 22, and it has exactly 2022 factors, it is called a “2022 Lucky Number”. There are $\qquad$ “2022 Lucky Numbers”.

Answer: 4


Problem 11.1. Twelve different natural numbers are written in a circle, one of which is 1. Any two adjacent numbers differ by either 10 or 7. What is the greatest value that the largest written number can take?

Answer: 58


3. (8 points) The teacher distributed 9 cards, each with a number from $1 \sim 9$, to three students, Jia, Yi, and Bing, giving each of them 3 cards.
Jia said: The numbers on my three cards form an arithmetic sequence;
Yi said: Mine do too;
Bing said: Only mine do not form an arithmetic sequence.
If what they said is all true, then the smallest possible sum of the numbers on the three cards that Bing holds is

Answer: 9


18. Among all tetrahedra with edge lengths 2, 3, 3, 4, 5, 5, what is the maximum volume? Prove your conclusion.
(1983 National Competition Problem)

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


2. The school table tennis championship was held according to the Olympic system. The winner won 6 matches. How many participants in the championship won more matches than they lost? (In the first round of the championship held according to the Olympic system, participants are paired. Those who lost the first game are eliminated from the championship, while those who won in the first round are paired again to play the second round. The losers are eliminated again, the winners are paired for the third round, and so on, until only one champion remains. It is known that in each round of our championship, a pair was found for each participant.)

Answer: 16


29.7. Calculate $\int \sqrt{1+x^{2}} d x$.

Answer: \dfrac{x}{2} \sqrt{1 + x^2} + \dfrac{1}{2} \ln \left( x + \sqrt{1 + x^2} \right) + C


6.1. In February of a non-leap year, Kirill and Vova decided to eat ice cream according to certain rules. If the date was even and the day of the week was Wednesday or Thursday, they each ate seven portions of ice cream. If the day of the week was Monday or Tuesday and the date was odd, they each ate three portions of ice cream. If the day of the week was Friday, the number of portions each ate was equal to the date on the calendar. Ice cream was forbidden on all other days and under other conditions. What is the maximum number of portions of ice cream that Vova could have eaten in February under these conditions?

Answer: 110


8. Through a point $P$ on a sphere with radius 5, three mutually perpendicular chords $P A, P B$, and $P C$ are drawn such that $P A=2 P B$. Then the maximum value of $P A+P B+P C$ is $\qquad$ .

Answer: 2\sqrt{70}


5. Find all functions $f(x): N^{+} \rightarrow N^{+}$, such that for any $x, y \in N^{+}$, we have $f(x+y f(x))=x f(y+1)$.

Answer: f(x) = x


4. In a right-angled triangle, the lengths of all sides are natural numbers, with one of the legs equal to 2012.

a) (7th grade) Find at least one such triangle.

b) (8th-9th grades) Find all such triangles. Answer: b) $(2012,253005,253013); (2012,506016, 506020); (2012,1012035,1012037)$ and $(2012,1509,2515)$.

Answer: (2012, 1509, 2515)


12. Find the smallest natural number greater than 1 that is at least 600 times greater than each of its prime divisors.

Answer: 1944


## Subiectul IV.(20 puncte)

Determinaţi toate funcţiile $f: R \rightarrow(0, \infty)$, primitivabile, ce verifică relaţia: $F(x)+\ln (f(x))=\ln \left(1+\frac{x}{\sqrt{1+x^{2}}}\right)$, $\forall x \in R$, unde $F: R \rightarrow R$ este o primitivă a lui $f$ şi $F(0)=0$.

prof. Cristian Petru Pop,ISJ Cluj

Toate subiectele sunt obligatorii. Se acordă 10 puncte din oficiu.

SUCCES!

Timp efectiv de lucru - 3 ore.

## Barem clasa a XII-a  (OLM 2015-etapa locală)

## Of. $10 p$

Answer: \dfrac{1}{\sqrt{1 + x^2}}


On each OMA lottery ticket there is a $9$-digit number that only uses the digits $1, 2$ and $3$ (not necessarily all three). Each ticket has one of the three colors red, blue or green. It is known that if two banknotes do not match in any of the $9$ figures, then they are of different colors. Bill $122222222$ is red, $222222222$ is green, what color is bill $123123123$?

Answer: red


## Subject 3.

Let $\mathrm{ABCD}$ be a rhombus with $A B=6 \mathrm{~cm}$ and $m(\hat{A})=45^{\circ}$. At point $\mathrm{O}$, the center of the circumcircle of triangle $\mathrm{ABD}$, the perpendicular $\mathrm{OM}$ is raised to the plane of the rhombus, $O M=4 \mathrm{~cm}$. Find:

a) The distance from point O to the plane (MBC).

b) The cosine of the angle between the planes (MBC) and (ABC).

a) Let $d$ be the perpendicular bisector of side (AD), and $\{O\}=d \cap A C$ the center of the circumcircle

of triangle $\mathrm{ABD},\{N\}=d \cap A D,\{P\}=d \cap B C,\{R\}=d \cap A B$.

Geometric figure.

Justification $M P \perp B C, d(O,(M B C))=O Q$

$\triangle A N R$ is a right-angled and isosceles triangle, $A N=N R=3, A R=3 \sqrt{2} ; \triangle B P R$ is a right-angled and isosceles triangle, $R B=A B-A R=3(2-\sqrt{2}), R P=P B=3(\sqrt{2}-1)$

$\triangle A O N \sim \triangle C O P \Rightarrow \frac{N O}{O P}=\frac{A N}{C P} \Leftrightarrow \frac{N O}{O P}=\frac{1}{1+\sqrt{2}} \Leftrightarrow O P=3 \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots$

$\triangle M O P$ is a right-angled triangle, $M P=5, O Q=\frac{O M \cdot O P}{M P}=\frac{12}{5}$

$\mathbf{1 p}$

b) The angle between the planes (MBC) and (ABC) is $\angle O P M$ (justification)

$$
\cos (\angle O P M)=\frac{3}{5}
$$

Answer: \dfrac{12}{5}


A rectangle has a length that is three times its width. It is known that the perimeter of the rectangle is numerically equal to its area. Find the sides of the rectangle.

Answer: \frac{8}{3}


8. (12 points) On the coordinate plane, all points whose coordinates satisfy the condition

$$
|| x|-2|+|y-3| \leq 3
$$

are shaded. Find the area of the resulting figure.

Answer: 34


Given an integer $n>1$, let $a_{1}, a_{2}, \cdots, a_{n}$ be distinct non-negative real numbers, and define the sets
$$
A=\left\{a_{i}+a_{j} \mid 1 \leqslant i \leqslant j \leqslant n\right\}, B=\left\{a_{i} a_{j} \mid 1 \leqslant i \leqslant j \leqslant n\right\} .
$$

Find the minimum value of $\frac{|A|}{|B|}$. Here, $|X|$ denotes the number of elements in the set $X$.

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


In the city of Bukvinsk, people are acquaintances only if their names contain the same letters. Some residents of Bukvinsk were asked how many acquaintances they have in the city. Martin said 20, Klim said 15, Inna said 12, Tamara said 12. What did Kamilla say?

Answer: 15


3. Given $x_{1}, x_{2}, \cdots, x_{100}$ are non-negative real numbers, and for $i=1,2, \cdots, 100$, we have $x_{i}+x_{i+1}+x_{i+2} \leqslant 1$, where $x_{101}=x_{1}, x_{102}=x_{2}$. Find the maximum value of the sum $S=\sum_{i=1}^{100} x_{i} x_{i+2}$.

Answer: \dfrac{25}{2}


11. Given the quadratic function $f(x)=x^{2}+a x+b(a, b \in \mathbf{R})$ has two distinct zeros, if $f\left(x^{2}+2 x-1\right)$ $=0$ has four distinct roots $x_{1}<x_{2}<x_{3}<x_{4}$, and $x_{1}, x_{2}, x_{3}, x_{4}$ form an arithmetic sequence, find the range of $a-b$.

Answer: (-\infty, \dfrac{25}{9}]


6. Let the non-real complex number $z$, satisfy $z^{23}=1$. Then $\sum_{k=0}^{22} \frac{1}{1+z^{k}+z^{2 k}}=$ $\qquad$ .

Answer: \dfrac{46}{3}


In a given circle, we draw two diameters that are perpendicular to each other, and from an arbitrary point $P$ on the circumference of the circle, we drop perpendiculars to these diameters. Determine the position of $P$ such that the sum of the lengths of these perpendiculars is maximized.

Answer: 45^\circ


## Task 12 - V00612

Edgar wrote down a side calculation during a math test so hastily that he can no longer read many of the digits.

Can you figure out the illegible digits? What is the problem?

(The symbol ? is used in place of the illegible digits).

```
? ? 5 ? ? : ? 9 = ? ? ?
1 ? ?
10?
    ? 7
    2?3
    ? ? ?
0
```

Answer: 15573 \div 29 = 537


7. As shown in the figure, in rectangle $A B C D$, $A B=3, A D=4, E$ is a point on $A B$, and $A E=1$. Now, $\triangle B C E$ is folded along $C E$ so that the projection of point $B$ on plane $A E C D$ falls on $A D$. Then, the volume of the pyramid $B-A E C D$ is $\qquad$ .

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


Problem 8.2 Given an isosceles right angle triangle $A B C$ with $\Varangle A C B=90^{\circ}$. Point $P$ lies on $B C, M$ is the midpoint of $A B$ and let $L$ and $N$ be points from the segment $A P$ such that $C N \perp A P$ and $A L=C N$.

a) Find $\Varangle L M N$;

b) If the area of $\triangle A B C$ is 4 times greater than the area of $\triangle L M N$ find $\Varangle C A P$.

Answer: 90^\circ


3. In each cell of a $2017 \times 2017$ grid, there is a lamp, and each lamp has only two states: on or off. A lamp is called "bad" if and only if there is an even number of lamps that are on among its adjacent lamps. Find the minimum possible number of bad lamps in this $2017 \times 2017$ grid.

Note: If two lamps are in cells that share an edge, then these two lamps are considered adjacent.

Answer: 1


Example 9. Derive the equation of a parabola symmetric with respect to the $O y$ axis, given that it passes through the points of intersection of the line $x-y=0$ and the circle $x^{2}+y^{2}-6 y=0$.

Answer: x^2 = 3y


$$
\begin{array}{l}
\text { Three. (25 points) Given the equation in } x \text { : } \\
x^{2}+\sqrt{a-2009} x+\frac{a-2061}{2}=0
\end{array}
$$

has two integer roots. Find all real values of $a$ that satisfy the condition.

Answer: 2013


14. In the workshop, five clocks were checked. It turned out that clock No. 3 is 3 minutes behind clock No. 2 per day. Clock No. 1 is 1 minute behind clock No. 2 per day, clock No. 4 is behind clock No. 3, and clock No. 5 is behind clock No. 1. If the readings of all five clocks are added together and the sum is divided by 5, the exact time is obtained. It is also known that one of these clocks is accurate. Which one exactly?

Answer: 5


247. Two opposite edges of a tetrahedron have lengths $b$ and $c$, while the others are equal to $a$. What is the minimum value of the sum of the distances from an arbitrary point in space to the vertices of this tetrahedron?

Answer: 2\sqrt{a^2 + \dfrac{bc}{2}}


13. A homemade deck of cards consists of 52 cards (including 4 suits: Hearts, Diamonds, Spades, and Clubs. Each suit has one card for each point value from 1 to 13). After shuffling, the cards are placed face down. To ensure that at least 2 cards have the same point value and color, you must draw at least $\qquad$ cards. If you need to ensure that among the drawn cards, there are at least 3 cards with consecutive point values (regardless of color), you must draw at least $\qquad$ cards.

Answer: 27


Petr and Karel played a series of chess games. They agreed that a player would add 3 points for a win, subtract 2 points for a loss, and no points would be awarded for a draw. Their friends wanted to know how many games Petr and Karel had played and who was currently leading, but they only found out that Petr won six times, drew twice, lost several times, and that Karel has exactly 9 points. How many games have the boys played, and who is currently leading?

(M. Volfová)

Answer: 15


1. At $12: 00$ a raft departed from point A to point B, located downstream along the river. At $12: 45$ a motorboat departed from A in pursuit of the raft, caught up with the raft at 13:00, and continued its journey. Upon arriving at point B, the boat turned around and headed back, meeting the raft at 14:00. Determine the time of the raft's arrival at point B.

Answer: 15:00


Lobanov M. S.

Around a round rotating table, on which there are 8 white and 7 black cups, 15 gnomes are sitting. They have put on 8 white and 7 black caps. Each gnome takes a cup whose color matches the color of their cap and places it in front of them, after which the table is rotated randomly. What is the maximum number of matches between the color of the cup and the cap that can be guaranteed after the table is rotated (the gnomes choose how to sit, but they do not know how the table will be rotated)?

Answer: 7


During a chess game, at some point, Hedgehog had twice fewer pieces on the board than Bear, and these pieces were five times fewer than the free squares on the board. How many of Bear's pieces were captured by this moment?

Answer: 0


19. As shown in the figure, points $E$, $F$, $G$, and $H$ are the midpoints of the sides of rectangle $ABCD$, and point $O$ is any point inside the rectangle. It is known that the area of quadrilateral $EBOF$ is 30, the area of quadrilateral $OFCG$ is 40, and the area of quadrilateral $HOGD$ is 35. Then the area of quadrilateral $AEOH$ is $\qquad$.

Answer: 25


Let \(\mathcal{M}\) be the set of \(5 \times 5\) real matrices of rank 3. Given a matrix \(A\) in \(\mathcal{M}\), the set of columns of \(A\) has \(2^{5}-1=31\) nonempty subsets. Let \(k_{A}\) be the number of these subsets that are linearly independent.

Determine the maximum and minimum values of \(k_{A}\), as \(A\) varies over \(\mathcal{M}\).

Answer: 25


GS. 1 In the figure, there are two common tangents. These common tangents meet the circles at points $A, B, C$ and $D$. If $A C=9 \mathrm{~cm}$, $B D=3 \mathrm{~cm}, \angle B A C=60^{\circ}$ and $A B=s \mathrm{~cm}$, find the value of $s$.

Answer: 6


1. In the cube $A B C D-A_{1} B_{1} C_{1} D_{1}$, the size of the dihedral angle $B-A_{1} C-D$ is $\qquad$
(2016, National High School Mathematics League Fujian Province Preliminary)
Hint Establish a spatial rectangular coordinate system as shown in Figure 11.

Answer: 120^\circ


## 16. ESERCIZIO DIMOSTRATIVO

Sia $k \geq 1$ un numero naturale. Determinare in funzione di $k$ il numero di interi positivi $n$ con le seguenti proprietà:

(a) in base dieci si scrivono con $k$ cifre, tutte dispari;

(b) sono divisibili per 5 , e il quoziente $\frac{n}{5}$, scritto in base dieci, ha ancora $k$ cifre, tutte dispari.

Answer: 3^{k-1}


Problem 10.4. Roma thought of a natural number, the sum of the digits of which is divisible by 8. Then he added 2 to the thought number and again got a number, the sum of the digits of which is divisible by 8. Find the smallest number that Roma could have thought of.

Answer: 699


Given a point \(A\) outside a circle, three secants \(ABC\), \(AMN\), and \(AFE\) are drawn through \(A\) intersecting the circle at points \(B\), \(C\), \(M\), \(N\), \(F\), and \(E\) respectively. The chords \(BF\) and \(CE\) intersect the chord \(MN\) at points \(G\) and \(H\). Additionally, \(MN\) intersects \(CF\) at point \(S\) and intersects \(BE\) at point \(T\). Show that \(\frac{1}{AM} + \frac{1}{AN} = \frac{1}{AG} + \frac{1}{AH} = \frac{1}{AS} + \frac{1}{AT}\).

Answer: \frac{1}{AM} + \frac{1}{AN} = \frac{1}{AG} + \frac{1}{AH} = \frac{1}{AS} + \frac{1}{AT}


9.1. Chords $A A^{\prime}, B B^{\prime}$, and $C C^{\prime}$ of a sphere intersect at a common point $S$. Find the sum $S A^{\prime}+S B^{\prime}+S C^{\prime}$, if $A S=6, B S=3, C S=2$, and the volumes of pyramids $S A B C$ and $S A^{\prime} B^{\prime} C^{\prime}$ are in the ratio $2: 9$. If the answer is not an integer, round it to the hundredths.

Answer: 18


**Zaslavsky A.**.

A sphere is circumscribed around a regular tetrahedron $ABCD$. On its faces, as bases, regular pyramids $ABCD', ABD'C, ACD'B, BCD'A$ are constructed outwardly, with their vertices lying on this sphere. Find the angle between the planes $ABC_1$ and $ACD'$.

Answer: 90^\circ


There are $2022$ signs arranged in a straight line. Mark tasks Auto to color each sign with either red or blue with the following condition: for any given sequence of length $1011$ whose each term is either red or blue, Auto can always remove $1011$ signs from the line so that the remaining $1011$ signs match the given color sequence without changing the order. Determine the number of ways Auto can color the signs to satisfy Mark's condition.

Answer: 2^{1011}


146. Find the particular solution of the equation $y^{\prime \prime}-2 y^{\prime}+y=0$, if $y(0)=4, y^{\prime}(0)=2$.

Answer: (4 - 2x)e^x


In base-$2$ notation, digits are $0$ and $1$ only and the places go up in powers of $-2$. For example, $11011$ stands for $(-2)^4+(-2)^3+(-2)^1+(-2)^0$ and equals number $7$ in base $10$. If the decimal number $2019$ is expressed in base $-2$ how many non-zero digits does it contain ?

Answer: 6


Let $D$ be a point inside an acute triangle $\triangle A B C$ such that
$$
\begin{array}{l}
\angle A D B=\angle A C B+90^{\circ}, \\
A C \cdot B D=A D \cdot B C .
\end{array}
$$
(1) Calculate the ratio $\frac{A B \cdot C D}{A C \cdot B D}$. The key related to the perpendicular tangents at point $C$ is to find the relationship between angles and line segments.

Answer: \sqrt{2}


Let $ABC$ be a triangle such that $\angle C < \angle A < 90^\circ$, and $D$ is a point on $AC$ such that $BD = BA$. The points of tangency of the incircle of $ABC$ with the sides $AB$ and $AC$ are denoted by $K$ and $L$ respectively. Let $J$ be the center of the incircle of triangle $BCD$. Show that the line $KL$ intersects the segment $AJ$ at its midpoint.

Answer: \text{The line } KL \text{ intersects the segment } AJ \text{ at its midpoint.}


Example 1 Let $x, y, z > 0$, and $x+y+z=1$. Find
$$
f(x, y, z)=\sum \frac{x(2 y-z)}{1+x+3 y}
$$

the maximum value.

Answer: \dfrac{1}{7}


1. $[\mathbf{3}] 16$ progamers are playing in a single elimination tournament. Each player has a different skill level and when two play against each other the one with the higher skill level will always win. Each round, each progamer plays a match against another and the loser is eliminated. This continues until only one remains. How many different progamers can reach the round that has 2 players remaining?

Answer: 9


фзestiev I.V.

There are three piles of stones. Sisyphus carries one stone from one pile to another. For each transfer, he receives from Zeus an amount of coins equal to the difference in the number of stones in the pile where he places the stone and the number of stones in the pile from which he takes the stone (the stone being moved is not counted). If the specified difference is negative, Sisyphus returns the corresponding amount to Zeus. (If Sisyphus cannot pay, the generous Zeus allows him to make the transfer on credit.) At some point, it turned out that all the stones were in the same piles as they were initially. What is the maximum total earnings of Sisyphus at this moment?

Answer: 0


Let \(a_{0}, a_{1}, a_{2}, \cdots\) be an infinite sequence of real numbers that satisfies the inequality \(\frac{a_{n-1}+a_{n+1}}{2} \geqslant a_{n}\) for all positive integers \(n\). Prove that \(\frac{a_{0}+a_{n+1}}{2} \geqslant \frac{a_{1}+a_{2}+\cdots+a_{n}}{n}\).

Answer: \frac{a_{0} + a_{n+1}}{2} \geqslant \frac{a_{1} + a_{2} + \cdots + a_{n}}{n}


In a certain competition, there are $a$ contestants and $b$ judges, where $b \geq 3$ and $b$ is an odd number. Each judge's decision for each contestant can only be "pass" or "fail". It is known that any two judges have at most $k$ contestants for whom they have the same decision. Prove that $\frac{k}{a} \geq \frac{b-1}{2b}$.

Answer: \dfrac{k}{a} \geq \dfrac{b-1}{2b}


5. a square board consists of $2 n \times 2 n$ squares. You want to mark $n$ of these squares so that no two marked squares are in the same or adjacent rows, and so that no two marked squares are in the same or adjacent columns. In how many ways is this possible?

## Solution:

Answer: (n+1)^2 \cdot n!


3. Given the circle $M: 2 x^{2}+2 y^{2}-8 x-8 y-1=0$, and the line $l: x+y-9=0$, a point $A$ on $l$ is used to construct $\triangle A B C$ such that $\angle B A C=\frac{\pi}{4}$, with side $A B$ passing through the center $M$ of the circle, and $B, C$ on the circle $M$. Then the range of the y-coordinate of point $A$ is $\qquad$

Answer: [3,6]


【Example 5】 $n$ people line up, with $r$ people standing between person A and person B, and also $r$ people standing between person B and person C $(0<2 r<n-3)$. How many different ways can they line up?

Answer: 2(n - 2r - 2) \cdot (n - 3)!


In a class of 60 students, each student's birthday is recorded as a specific month and date. Two questions are asked to each student: How many people in the class have the same birth month as you? How many people in the class have the same birth date as you (for example, January 12 and December 12 have the same date)? It was found that the answers to these questions included all integers from 0 to 14. What is the minimum number of students who share the same birthday in this class?

Answer: 2


5.1. From point $A$ to point $B$, which are 10 km apart, a car departed at 7:00. After traveling $2 / 3$ of the distance, the car passed point $C$, from which a cyclist immediately set off for point $A$. As soon as the car arrived at $B$, a bus immediately departed from $B$ in the opposite direction and arrived at $A$ at 9:00. At what distance from $B$ did the bus catch up with the cyclist, if the cyclist arrived at point $A$ at 10:00 and the speed of each participant in the movement was constant?

Answer: 6


10.6. On the board, three natural numbers were written: two ten-digit numbers $a$ and $b$, as well as their sum $a+b$. What is the maximum number of odd digits that could have been written on the board?

(I. Bogdanov, P. Kozhevnikov)

Answer: 30


Exercise 3. Let $x, y, z$ be non-zero real numbers such that $x+y+z=0$. Suppose that

$$
\frac{x}{y}+\frac{y}{z}+\frac{z}{x}=\frac{x}{z}+\frac{z}{y}+\frac{y}{x}+1
$$

Determine the value of $\frac{x}{y}+\frac{y}{z}+\frac{z}{x}$.

Answer: -1


Problem 9-5. A divisor of a natural number is called proper if it is different from 1 and the number itself. Find all natural numbers for which the difference between the sum of the two largest proper divisors and the sum of the two smallest proper divisors is a prime number.

Answer: 12


1. In each cell of a $75 \times 75$ table, a natural number is written. In each row, there are at least 15 different numbers, and in any three consecutive rows, there are no more than 25 different numbers. What is the maximum number of different numbers that can be in the table?

Answer: 385


We inscribe a sphere in an equilateral cone, to which we lay an tangent plane parallel to the base of the cone. We then inscribe another sphere in the resulting cone, and so on. Determine the sum of the volumes of the spheres if this procedure is continued to infinity. The slant height of the cone is $l=2$.

Answer: \dfrac{2\sqrt{3}}{13} \pi


Example 3 (1999 National High School Competition Question) Given $A(-2,2)$, it is known that $B$ is a moving point on the ellipse $\frac{x^{2}}{25}+\frac{y^{2}}{16}=1$, and $F$ is the left focus. When $|A B|+\frac{5}{3}|B F|$ takes the minimum value, find the coordinates of point $B$.

Answer: \left( -\dfrac{5\sqrt{3}}{2}, 2 \right)


The sequence \(\left\{a_{k}\right\}, k=1, 2, \ldots\) is a sequence of non-negative real numbers, satisfying \(a_{k} - 2 a_{k+1} + a_{k+2} \geq 0\) and \(\sum_{i=1}^{k} a_{i} \leq 1\) for all \(k = 1, 2, \ldots\). Prove that for any natural number \(k\), \(0 \leq a_{k} - a_{k+1} < \frac{2}{k^{2}}\).

Answer: 0 \leq a_{k} - a_{k+1} < \frac{2}{k^{2}}


57. Schools A, B, and C each contributed the same amount of money to buy desks and chairs. As a result, School A took 15 more sets of desks and chairs than School B, and School B took 15 fewer sets of desks and chairs than School C. Therefore, Schools A and C each paid 500 yuan to School B, so the cost of each set of desks and chairs is $\qquad$ yuan.

Answer: 100


Let $ n$ be an even positive integer. Prove that there exists a positive inter $ k$ such that

\[ k \equal{} f(x) \cdot (x\plus{}1)^n \plus{} g(x) \cdot (x^n \plus{} 1)\]

for some polynomials $ f(x), g(x)$ having integer coefficients. If $ k_0$ denotes the least such $ k,$ determine $ k_0$ as a function of $ n,$ i.e. show that $ k_0 \equal{} 2^q$ where $ q$ is the odd integer determined by $ n \equal{} q \cdot 2^r, r \in \mathbb{N}.$

Note: This is variant A6' of the three variants given for this problem.

Answer: 2^q


Tokarev S.I.

Doughnut was snacking at a roadside cafe when a bus drove past him. Three doughnuts after the bus, a motorcycle passed Doughnut, and another three doughnuts later, a car passed. Past Syrup, who was snacking at another cafe on the same road, they drove by in a different order: first the bus, three doughnuts later the car, and another three doughnuts later the motorcycle. It is known that Doughnut and Syrup always eat doughnuts at the same constant speed. Find the speed of the bus if the speed of the car is 60 km/h and the speed of the motorcycle is 30 km/h.

Answer: 40


Example 2 (2006 National High School Mathematics Competition Question) Express 2006 as the sum of 5 positive integers $x_{1}, x_{2}, x_{3}, x_{4}$, $x_{5}$. Let $S=\sum_{1 \leqslant i<j \leqslant 5} x_{i} x_{j}$. Questions:
(1) For what values of $x_{1}, x_{2}, x_{3}, x_{4}, x_{5}$ does $S$ attain its maximum value;
(2) Suppose for any $1 \leqslant i, j \leqslant 5$, $\left|x_{i}-x_{j}\right| \leqslant 2$, for what values of $x_{1}, x_{2}, x_{3}, x_{4}, x_{5}$ does $S$ attain its minimum value.

Answer: 401, 401, 401, 401, 402


[ Inscribed and Circumscribed Circles ] [ Inscribed Angle Subtended by a Diameter ]

A circle with radius 1 is circumscribed around triangle $A P K$. The extension of side $A P$ beyond vertex $P$ intercepts a segment $B K$ from the tangent to the circle at vertex $K$, and $B K$ is equal to 7. Find the area of triangle $A P K$, given that the angle $A B K$ is $\operatorname{arctg} \frac{2}{7}$.

Answer: \dfrac{28}{53}


2B. A four-digit number is a perfect square, such that if the same digit is subtracted from its digits, the resulting number is also a perfect square. Determine all such natural numbers.

Answer: 3136


The numbers \(1, 2, 3, 4,\) and \(5\) are used once each in some order substituting for the letters in the series of powers \(\left.M^{\left(A^{\left(T^{S}\right)}\right)}\right)\). In how many of the arrangements is the units digit of the value of this expression equal to 1?

Answer: 34


As shown in the figure, the small square $EFGH$ is inside the larger square $ABCD$. The total shaded area is 124 square centimeters. Points $E$ and $H$ are on the side $AD$, and $O$ is the midpoint of line segment $CF$. Find the area of the quadrilateral $BOGF$ in square centimeters.

Answer: 31


Let $ABCD$ be a regular tetrahedron. Suppose points $X$, $Y$, and $Z$ lie on rays $AB$, $AC$, and $AD$ respectively such that $XY=YZ=7$ and $XZ=5$. Moreover, the lengths $AX$, $AY$, and $AZ$ are all distinct. Find the volume of tetrahedron $AXYZ$.

[i]Proposed by Connor Gordon[/i]

Answer: \sqrt{122}


5. In the equation below, four small pieces of paper each cover a digit. What is the sum of the four digits that are covered?

The text above is the translation of the provided Chinese text, maintaining the original format and line breaks.

Answer: 23


2. Let $a$ be a real number, and the two parabolas
$$
y=x^{2}+x+a \text { and } x=4 y^{2}+3 y+a
$$

have four intersection points.
(1) Find the range of values for $a$;
(2) Prove that these four intersection points are concyclic, and find the coordinates of the center of the circle.

Answer: (-\infty, -\dfrac{7}{16})


2. King Arthur's Round Table. Three knights are randomly seated on chairs arranged around the table at King Arthur's court. What is the probability that there will be an empty chair on both sides of each knight if:

a) there are a total of 7 chairs? (From 6th grade, 1 point).

b) there are a total of $n$ chairs? (From 7th grade, 2 points).

Answer: \dfrac{1}{5}


11.4. In a glass, the axial section of which is the graph of the function $y=x^{4}$, a "cherry" - a ball of radius $R$ - is placed. For which values of $R$ will the ball touch the bottom of the glass (the point $(0,0)$)?

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


As shown in the figure, Huahua washed 40 handkerchiefs and hung them to dry on 3 ropes. Each handkerchief must be clipped on both ends with clothespins. Huahua used a total of $\qquad$ clothespins.

Answer: 43


I2.2 The surface area of a cube is $b \mathrm{~cm}^{2}$. If the length of each side is increased by $3 \mathrm{~cm}$, its volume is increased by $(2 b-a) \mathrm{cm}^{3}$, find the value of $b$.

Answer: 600


Example 26 Let $n(\geqslant 3)$ be a given positive integer, and for $n$ given real numbers $a_{1}, a_{2}$, $\cdots, a_{n}$, denote the minimum value of $\left|a_{i}-a_{j}\right|(1 \leqslant i<j \leqslant n)$ as $m$. Find the maximum value of the above $m$ under the condition that $a_{1}^{2}+a_{2}^{2}+\cdots+a_{n}^{2}=$ 1.

Answer: \dfrac{2\sqrt{3}}{\sqrt{n(n^{2}-1)}}


5. [5 points] Given the numbers $\log _{\sqrt{x+34}}(2 x+23), \log _{(x+4)^{2}}(x+34), \log _{\sqrt{2 x+23}}(-x-4)$. For which $x$ are two of these numbers equal, and the third one greater than them by 1?

Answer: -9


In a production of Hamlet, several roles are paired, for example, Gertrude and the Queen in the Play can be played by two actors. Before the performance, they decide by drawing lots which of the two will play the Queen in the Play and which will play Gertrude on that day. Similar lotteries decide for the other pairs as well. Sári has already seen one performance, but she wants to see the Gertrude/Queen in the Play, Claudius/Player King, and Ophelia/Fortinbras roles in the other version as well, although not necessarily in the same performance. How many more performances should she buy tickets for so that there is at least a $90 \%$ probability that she will see all three roles in the other casting as well?

Answer: 5


Problem 5.5. The houses of Andrey, Boris, Vova, and Gleb are located in some order on a straight street. The distance between Andrey's and Boris's houses, as well as the distance between Vova's and Gleb's houses, is 600 m. What can the distance in meters between Andrey's and Gleb's houses be, if it is known that it is 3 times greater than the distance between Boris's and Vova's houses? List all possible options.

Answer: 900


[ Touching circles ] [Common tangent to two circles]

Two touching circles are inscribed in an angle of size $2 \alpha$. Find the ratio of the radius of the smaller circle to the radius of a third circle that touches the first two circles and one of the sides of the angle.

Answer: \dfrac{2(1 + \cos \alpha)}{1 + \sin \alpha}


Answer: f_1 < f_2


Find the smallest positive constant $c$ satisfying: For any simple graph $G=G(V,E)$, if $|E|\geq c|V|$, then $G$ contains $2$ cycles with no common vertex, and one of them contains a chord.

Note: The cycle of graph $G(V,E)$ is a set of distinct vertices ${v_1,v_2...,v_n}\subseteq V$, $v_iv_{i+1}\in E$ for all $1\leq i\leq n$ $(n\geq 3, v_{n+1}=v_1)$; a cycle containing a chord is the cycle ${v_1,v_2...,v_n}$, such that there exist $i,j, 1< i-j< n-1$, satisfying $v_iv_j\in E$.

Answer: 4


1. Let the complex number $z=\cos \theta+\mathrm{i} \sin \theta(0<\theta<\pi), \omega=\frac{1-(\bar{z})^{4}}{1+z^{4}}$, given that $|\omega|=\frac{\sqrt{3}}{3}$, $\arg \omega<\frac{\pi}{2}$, find $\theta$.

Answer: \dfrac{\pi}{12}


## Task $4 / 68$

Let $0<a<1, n$ be a natural number. Which numbers $a$ differ from $n$ by their reciprocal?

What value is obtained specifically for $n=1$?

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


2. On 8 balls, numbers are written: $2,3,4,5,6,7,8,9$. In how many ways can the balls be placed into three boxes so that no box contains a number and its divisor?

Answer: 432


Let $p$ be a prime number. Determine the remainder of the Euclidean division of $1^{k}+\cdots+(p-1)^{k}$ for any integer $k \geq 0$ by $p$ (without primitive root).

Answer: 0


Dima went to school in the morning, but after walking exactly halfway, he realized he had forgotten his mobile phone at home. Dima estimated (he had an A in mental math) that if he continued walking at the same speed, he would reach school 3 minutes before the bell for the first lesson. However, if he ran back home for the phone and then ran to school, he would arrive 3 minutes after the bell. Dima decided to go back home, but he got tired while running (he had a C in physical education), and so he walked from home to school at his normal walking speed. As a result, he was 15 minutes late for the first lesson! By how many times is the speed at which he runs greater than the speed at which he walks?

Answer: 2


5. 10 Let $A$ and $B$ be the sums of the odd and even terms, respectively, in the expansion of $(x+a)^{n}$. Find $A^{2}-B^{2}$.

Answer: (x^2 - a^2)^n


13. An ant crawls along the edges of a cube with side length 1 unit. Starting from one of the vertices, in each minute the ant travels from one vertex to an adjacent vertex. After crawling for 7 minutes, the ant is at a distance of $\sqrt{3}$ units from the starting point. Find the number of possible routes the ant has taken.
(2 marks)
An ant crawls along the edges of a cube with side length 1 unit. Starting from one of the vertices, in each minute the ant travels from one vertex to an adjacent vertex. After crawling for 7 minutes, the ant is at a distance of $\sqrt{3}$ units from the starting point. Find the number of possible routes the ant has taken.

Answer: 546


Find the sum of all positive integers $m$ such that $2^m$ can be expressed as a sum of four factorials (of positive integers).

Note: The factorials do not have to be distinct.  For example, $2^4=16$ counts, because it equals $3!+3!+2!+2!$.

Answer: 21


Triangle $ABC$ satisfies $\angle ABC=\angle ACB=78^\circ$.  Points $D$ and $E$ lie on $AB,AC$ and satisfy $\angle BCD=24^\circ$ and $\angle CBE=51^\circ$.  If $\angle BED=x^\circ$, find $x$.

Answer: 12


Suppose the function $\psi$ satisfies $\psi(1)=\sqrt{2+\sqrt{2+\sqrt2}}$ and $\psi(3x)+3\psi(x)=\psi(x)^3$ for all real $x$.  Determine the greatest integer less than $\textstyle\prod_{n=1}^{100}\psi(3^n)$.

Answer: 5792


Answer: 23


You are eating at a fancy restaurant with a person you wish to impress. For some reason, you think that eating at least one spicy course and one meat-filled course will impress the person. The meal is five courses, with four options for each course. Each course has one option that is spicy and meat-filled, one option that is just spicy, one that is just meat-filled, and one that is neither spicy nor meat-filled. How many possible meals can you have?

Answer: 961


Triangle $A_1B_1C_1$ is an equilateral triangle with sidelength $1$.  For each $n>1$, we construct triangle $A_nB_nC_n$ from $A_{n-1}B_{n-1}C_{n-1}$ according to the following rule: $A_n,B_n,C_n$ are points on segments $A_{n-1}B_{n-1},B_{n-1}C_{n-1},C_{n-1}A_{n-1}$ respectively, and satisfy the following: \[\dfrac{A_{n-1}A_n}{A_nB_{n-1}}=\dfrac{B_{n-1}B_n}{B_nC_{n-1}}=\dfrac{C_{n-1}C_n}{C_nA_{n-1}}=\dfrac1{n-1}\]  So for example, $A_2B_2C_2$ is formed by taking the midpoints of the sides of $A_1B_1C_1$.  Now, we can write $\tfrac{|A_5B_5C_5|}{|A_1B_1C_1|}=\tfrac mn$ where $m$ and $n$ are relatively prime integers.  Find $m+n$.  (For a triangle $\triangle ABC$, $|ABC|$ denotes its area.)

Answer: 4891


Charlie noticed his golden ticket was golden in two ways! In addition to being gold, it was a rectangle whose side lengths had ratio the golden ratio $\varphi = \tfrac{1+\sqrt{5}}{2}$. He then folds the ticket so that two opposite corners (vertices connected by a diagonal) coincide and makes a sharp crease (the ticket folds just as any regular piece of paper would). The area of the resulting shape can be expressed as $a + b \varphi$. What is $\tfrac{b}{a}$?

Answer: 2


Find the largest $r$ such that $4$ balls each of radius $r$ can be packed into a regular tetrahedron with side length $1$. In a packing, each ball lies outside every other ball, and every ball lies inside the boundaries of the tetrahedron. If $r$ can be expressed in the form $\frac{\sqrt{a}+b}{c}$ where $a, b, c$ are integers such that $\gcd(b, c) = 1$, what is $a + b + c$?

Answer: 15


We define the function $f(x,y)=x^3+(y-4)x^2+(y^2-4y+4)x+(y^3-4y^2+4y)$. Then choose any distinct $a, b, c \in \mathbb{R}$ such that the following holds: $f(a,b)=f(b,c)=f(c,a)$. Over all such choices of $a, b, c$, what is the maximum value achieved by
\[\min(a^4 - 4a^3 + 4a^2, b^4 - 4b^3 + 4b^2, c^4 - 4c^3 + 4c^2)?\]

Answer: 1


For a positive integer $n$, let $d(n)$ be the number of positive divisors of $n$. What is the smallest positive integer $n$ such that
\[\sum_{t \mid n} d(t)^3\]is divisible by $35$?

Answer: 864


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

Answer: 1291


Andrew and Blair are bored in class and decide to play a game. They pick a pair $(a, b)$ with $1 \le a, b \le 100$. Andrew says the next number in the geometric series that begins with $a,b$ and Blair says the next number in the arithmetic series that begins with $a,b$. For how many pairs $(a, b)$ is Andrew's number minus Blair's number a positive perfect square?

Answer: 281


Given a rational number $r$ that, when expressed in base-$10$, is a repeating, non-terminating decimal, we define $f(r)$ to be the number of digits in the decimal representation of $r$ that are after the decimal point but before the repeating part of $r$. For example, $f(1.2\overline{7}) = 0$ and $f(0.35\overline{2}) = 2$. What is the smallest positive integer $n$ such that $\tfrac{1}{n}, \tfrac{2}{n}$, and $\tfrac{4}{n}$ are non-terminating decimals, where $f\left( \tfrac{1}{n} \right) = 3, f\left( \tfrac{2}{n} \right) = 3$, and $f\left( \tfrac{4}{n} \right) = 2$?.

Answer: 600


A circle with radius $1$ and center $(0, 1)$ lies on the coordinate plane. Ariel stands at the origin and rolls a ball of paint at an angle of $35$ degrees relative to the positive $x$-axis (counting degrees counterclockwise). The ball repeatedly bounces off the circle and leaves behind a trail of paint where it rolled. After the ball of paint returns to the origin, the paint has traced out a star with $n$ points on the circle. What is $n$?

Answer: 36


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

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

Answer: 2


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Arnold`, `Carol`, `Alice`, `Bob`, `Peter`
 - Each mother is accompanied by their child: `Bella`, `Fred`, `Timothy`, `Alice`, `Meredith`, `Samantha`

## Clues:
1. The person's child is named Bella and Peter are next to each other.
2. Bob and the person's child is named Alice are next to each other.
3. The person who is the mother of Timothy is in the fourth house.
4. The person's child is named Alice is somewhere to the right of the person's child is named Fred.
5. Carol is not in the first house.
6. The person's child is named Meredith is in the sixth house.
7. The person's child is named Bella is not in the third house.
8. The person's child is named Samantha is not in the third house.
9. The person's child is named Alice is Arnold.
10. The person's child is named Fred is Peter.
11. The person's child is named Alice is not in the third house.
12. The person's child is named Meredith is Eric.
 What is the value of attribute House for the person whose attribute Name is Arnold? Please reason step by step, and put your final answer within \boxed{}

Answer: 5


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Carol`, `Peter`, `Alice`, `Eric`, `Arnold`
 - They all have a unique favorite flower: `daffodils`, `iris`, `lilies`, `roses`, `tulips`, `carnations`
 - People have unique hair colors: `gray`, `red`, `blonde`, `auburn`, `brown`, `black`
 - Each person has a favorite color: `white`, `blue`, `green`, `purple`, `red`, `yellow`
 - People own unique car models: `tesla model 3`, `ford f150`, `honda civic`, `toyota camry`, `chevrolet silverado`, `bmw 3 series`
 - The mothers' names in different houses are unique: `Janelle`, `Holly`, `Kailyn`, `Sarah`, `Penny`, `Aniya`

## Clues:
1. There is one house between Arnold and the person who loves blue.
2. The person whose mother's name is Janelle is directly left of the person who has auburn hair.
3. The person who owns a Chevrolet Silverado and the person who owns a BMW 3 Series are next to each other.
4. Carol is the person who loves a carnations arrangement.
5. The person who loves the boquet of iris is the person who has gray hair.
6. The person who loves the rose bouquet is The person whose mother's name is Kailyn.
7. Bob is the person who loves the boquet of lilies.
8. The person who loves a bouquet of daffodils is somewhere to the left of the person who owns a Chevrolet Silverado.
9. Bob is in the third house.
10. The person who owns a Honda Civic is in the sixth house.
11. The person who loves the vase of tulips is somewhere to the left of The person whose mother's name is Penny.
12. The person who has brown hair is in the fourth house.
13. The person who has brown hair is somewhere to the left of the person who has blonde hair.
14. The person whose mother's name is Holly is Bob.
15. The person who owns a Ford F-150 is in the third house.
16. Eric is directly left of the person who has auburn hair.
17. The person who owns a Tesla Model 3 is the person who loves yellow.
18. There are two houses between the person who loves purple and The person whose mother's name is Aniya.
19. The person who loves purple is in the first house.
20. The person whose favorite color is green is not in the fourth house.
21. The person who has gray hair is the person who owns a BMW 3 Series.
22. The person who has gray hair is somewhere to the left of Alice.
23. The person whose mother's name is Holly is somewhere to the left of the person who loves the vase of tulips.
24. The person who has red hair is the person who loves white.
 What is the value of attribute HairColor for the person whose attribute Color is white? Please reason step by step, and put your final answer within \boxed{}

Answer: red


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`
 - Each person lives in a unique style of house: `victorian`, `colonial`, `ranch`
 - They all have a unique favorite flower: `daffodils`, `carnations`, `lilies`
 - The people keep unique animals: `cat`, `horse`, `bird`
 - The people are of nationalities: `swede`, `dane`, `brit`

## Clues:
1. The person who loves a carnations arrangement is Eric.
2. The person in a ranch-style home is Arnold.
3. The person in a ranch-style home is not in the first house.
4. The British person is in the first house.
5. The person who loves a bouquet of daffodils is Peter.
6. The Swedish person is Eric.
7. The cat lover is in the second house.
8. The person residing in a Victorian house is directly left of the person who keeps horses.
 What is the value of attribute House for the person whose attribute HouseStyle is ranch? Please reason step by step, and put your final answer within \boxed{}

Answer: 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: `Carol`, `Peter`, `Arnold`, `Bob`, `Eric`, `Alice`
 - People use unique phone models: `iphone 13`, `google pixel 6`, `oneplus 9`, `huawei p50`, `samsung galaxy s21`, `xiaomi mi 11`
 - People have unique hair colors: `gray`, `auburn`, `red`, `brown`, `black`, `blonde`
 - Each person has a unique favorite drink: `coffee`, `water`, `root beer`, `tea`, `milk`, `boba tea`
 - People have unique favorite music genres: `classical`, `jazz`, `rock`, `country`, `hip hop`, `pop`

## Clues:
1. Alice is the person who uses a Huawei P50.
2. The person who uses a Xiaomi Mi 11 is somewhere to the left of the one who only drinks water.
3. The one who only drinks water is somewhere to the left of Bob.
4. Carol is the person who has gray hair.
5. Eric is the tea drinker.
6. The person who has black hair is not in the sixth house.
7. The person who uses a Samsung Galaxy S21 is directly left of the person who loves classical music.
8. The person who uses a OnePlus 9 is not in the third house.
9. Alice is somewhere to the right of Carol.
10. The person who loves jazz music is Alice.
11. The person who loves jazz music is the boba tea drinker.
12. There is one house between the person who uses a OnePlus 9 and Eric.
13. The person who has brown hair is the person who loves hip-hop music.
14. The person who loves pop music is not in the first house.
15. The root beer lover is in the second house.
16. Eric is the person who uses an iPhone 13.
17. The person who has red hair is the person who uses an iPhone 13.
18. The person who has red hair and the coffee drinker are next to each other.
19. Peter is in the first house.
20. Arnold is somewhere to the right of the person who has auburn hair.
21. The person who has gray hair is not in the second house.
22. The person who loves country music is in the fifth house.
 What is the value of attribute PhoneModel for the person whose attribute Drink is root beer? Please reason step by step, and put your final answer within \boxed{}

Answer: oneplus 9


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Arnold`, `Alice`, `Peter`, `Eric`
 - People own unique car models: `tesla model 3`, `toyota camry`, `ford f150`, `honda civic`, `bmw 3 series`
 - The people are of nationalities: `brit`, `norwegian`, `german`, `swede`, `dane`
 - They all have a unique favorite flower: `daffodils`, `carnations`, `lilies`, `tulips`, `roses`

## Clues:
1. The person who owns a Tesla Model 3 is the person who loves a carnations arrangement.
2. The person who owns a Toyota Camry is Alice.
3. The person who owns a BMW 3 Series is somewhere to the right of Arnold.
4. Arnold is not in the third house.
5. The Swedish person is in the first house.
6. The German is the person who loves a bouquet of daffodils.
7. Arnold is not in the second house.
8. The Norwegian is the person who owns a Tesla Model 3.
9. The Dane is somewhere to the left of the person who owns a Ford F-150.
10. The person who owns a BMW 3 Series is somewhere to the right of the person who loves a carnations arrangement.
11. Alice is directly left of the person who loves the rose bouquet.
12. Peter and the person who loves the vase of tulips are next to each other.
13. The British person is somewhere to the right of the Norwegian.
14. The person who loves the vase of tulips is Eric.
15. The person who owns a Ford F-150 is Eric.
 What is the value of attribute Nationality for the person whose attribute Name is Arnold? Please reason step by step, and put your final answer within \boxed{}

Answer: swede


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

## Clues:
1. The person who loves the vase of tulips is the person who has auburn hair.
2. The person who is short is somewhere to the right of Arnold.
3. The person who goes on cultural tours is Arnold.
4. The person who has auburn hair is Bob.
5. The person who is very short is the person who loves a bouquet of daffodils.
6. The person who loves the boquet of iris is the person who is very tall.
7. Peter is somewhere to the right of the root beer lover.
8. The person who loves beach vacations is directly left of the person who prefers city breaks.
9. Bob is directly left of the coffee drinker.
10. The person who is very short is the person who has gray hair.
11. Eric is somewhere to the right of the tea drinker.
12. The person who has red hair is the person who is tall.
13. The person who is tall is somewhere to the right of the person who is short.
14. The person who has black hair is the person who enjoys mountain retreats.
15. The one who only drinks water is the person who prefers city breaks.
16. The person who has brown hair is Carol.
17. The person who has an average height is Arnold.
18. The person who has brown hair is directly left of the person who likes milk.
19. The root beer lover is the person who is very short.
20. The person who is short is somewhere to the left of the coffee drinker.
21. The person who enjoys camping trips is the person who loves a carnations arrangement.
22. The person who is super tall and Arnold are next to each other.
23. There is one house between the person who loves the boquet of lilies and Peter.
24. The person who loves the vase of tulips is directly left of the person who has black hair.
 What is the value of attribute House for the person whose attribute Drink is root beer? Please reason step by step, and put your final answer within \boxed{}

Answer: 1


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Arnold`, `Peter`, `Alice`, `Eric`
 - Each person has a unique favorite drink: `milk`, `root beer`, `coffee`, `water`, `tea`
 - Each person has a unique type of pet: `dog`, `bird`, `hamster`, `cat`, `fish`
 - Each mother is accompanied by their child: `Timothy`, `Meredith`, `Fred`, `Bella`, `Samantha`
 - The mothers' names in different houses are unique: `Kailyn`, `Penny`, `Aniya`, `Janelle`, `Holly`
 - People have unique favorite sports: `tennis`, `soccer`, `swimming`, `baseball`, `basketball`

## Clues:
1. The tea drinker is the person who loves tennis.
2. The person whose mother's name is Aniya is Arnold.
3. The person whose mother's name is Janelle is somewhere to the right of the person who loves swimming.
4. The person whose mother's name is Janelle is the person's child is named Bella.
5. The person's child is named Samantha is the person who likes milk.
6. The coffee drinker is not in the fifth house.
7. Bob is the person's child is named Bella.
8. The person who loves swimming and the person who loves basketball are next to each other.
9. The person who is the mother of Timothy is somewhere to the left of the person who owns a dog.
10. The person whose mother's name is Kailyn is in the fifth house.
11. The root beer lover is Alice.
12. The person whose mother's name is Holly is directly left of the root beer lover.
13. The person who likes milk is not in the second house.
14. There is one house between the person who has a cat and the person who loves baseball.
15. The person whose mother's name is Penny is somewhere to the right of the person who loves tennis.
16. The person's child is named Fred is the person who keeps a pet bird.
17. The person who likes milk is directly left of the person with a pet hamster.
18. The person who keeps a pet bird is Alice.
19. There are two houses between Eric and The person whose mother's name is Aniya.
 What is the value of attribute Children for the person whose attribute Pet is bird? Please reason step by step, and put your final answer within \boxed{}

Answer: Fred


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Peter`, `Bob`, `Eric`, `Arnold`
 - The people keep unique animals: `fish`, `dog`, `horse`, `bird`, `cat`
 - People own unique car models: `honda civic`, `bmw 3 series`, `ford f150`, `toyota camry`, `tesla model 3`
 - Each person has a unique favorite drink: `coffee`, `tea`, `root beer`, `milk`, `water`

## Clues:
1. Arnold is not in the fifth house.
2. Alice and Bob are next to each other.
3. The person who owns a Tesla Model 3 is the person who keeps horses.
4. There are two houses between Arnold and the root beer lover.
5. Eric is in the second house.
6. The person who likes milk is the dog owner.
7. Arnold is the person who owns a Tesla Model 3.
8. The bird keeper is the person who owns a Honda Civic.
9. The one who only drinks water and the person who owns a Toyota Camry are next to each other.
10. The one who only drinks water is directly left of the cat lover.
11. There is one house between the person who owns a Ford F-150 and the fish enthusiast.
12. The dog owner is not in the fifth house.
13. The tea drinker is Bob.
 What is the value of attribute Drink for the person whose attribute CarModel is ford f150? Please reason step by step, and put your final answer within \boxed{}

Answer: milk


There are 2 houses, numbered 1 to 2 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Arnold`
 - Each person has an occupation: `engineer`, `doctor`
 - Each person has a favorite color: `red`, `yellow`
 - Each person prefers a unique type of vacation: `beach`, `mountain`
 - Each person lives in a unique style of house: `victorian`, `colonial`

## Clues:
1. The person living in a colonial-style house is directly left of Arnold.
2. The person whose favorite color is red is Arnold.
3. Arnold is the person who is a doctor.
4. The person who is an engineer is the person who enjoys mountain retreats.
 What is the value of attribute House for the person whose attribute Name is Arnold? Please reason step by step, and put your final answer within \boxed{}

Answer: 2


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Alice`, `Bob`, `Peter`, `Eric`
 - People have unique hair colors: `blonde`, `black`, `gray`, `brown`, `red`
 - They all have a unique favorite flower: `tulips`, `carnations`, `daffodils`, `lilies`, `roses`
 - Each person has a favorite color: `white`, `red`, `green`, `blue`, `yellow`

## Clues:
1. The person who loves the rose bouquet and the person who loves yellow are next to each other.
2. The person who loves a carnations arrangement is Peter.
3. The person who loves the rose bouquet is not in the third house.
4. The person whose favorite color is green is not in the fifth house.
5. The person who loves a bouquet of daffodils is Alice.
6. The person who loves yellow is Eric.
7. The person who has blonde hair is in the fifth house.
8. The person who has black hair is directly left of Arnold.
9. The person who loves yellow is the person who has brown hair.
10. Alice is the person whose favorite color is green.
11. The person who has red hair is the person who loves blue.
12. The person whose favorite color is red is somewhere to the right of the person who loves blue.
13. The person who has black hair is the person who loves the vase of tulips.
 What is the value of attribute House for the person whose attribute Color is yellow? 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`, `Arnold`, `Peter`, `Carol`, `Bob`, `Eric`
 - Each person prefers a unique type of vacation: `mountain`, `beach`, `city`, `cruise`, `camping`, `cultural`
 - Everyone has a unique favorite cigar: `yellow monster`, `pall mall`, `prince`, `dunhill`, `blue master`, `blends`
 - People have unique favorite sports: `swimming`, `soccer`, `tennis`, `basketball`, `volleyball`, `baseball`

## Clues:
1. Arnold is the person who prefers city breaks.
2. There is one house between the person who loves volleyball and the person who loves swimming.
3. Alice is in the fourth house.
4. The person who likes going on cruises is the person who smokes many unique blends.
5. The person who enjoys camping trips is somewhere to the left of Eric.
6. The person who loves volleyball is the person who smokes many unique blends.
7. Arnold is the Prince smoker.
8. The person who loves baseball is Peter.
9. The Prince smoker and the person partial to Pall Mall are next to each other.
10. The person who loves beach vacations is in the sixth house.
11. Carol is the person who loves beach vacations.
12. Carol is the person who smokes Yellow Monster.
13. The person who loves tennis is somewhere to the right of the person who goes on cultural tours.
14. The person who goes on cultural tours is the person who smokes Blue Master.
15. The person who loves basketball is directly left of the person who smokes many unique blends.
16. The person who loves volleyball and Eric are next to each other.
17. The person who loves tennis is Arnold.
 What is the value of attribute FavoriteSport for the person whose attribute House is 4? Please reason step by step, and put your final answer within \boxed{}

Answer: volleyball


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

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

Answer: 3


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Arnold`, `Eric`
 - Each mother is accompanied by their child: `Meredith`, `Bella`, `Fred`
 - The people keep unique animals: `bird`, `cat`, `horse`
 - People own unique car models: `toyota camry`, `tesla model 3`, `ford f150`
 - People have unique favorite book genres: `romance`, `science fiction`, `mystery`
 - Everyone has a favorite smoothie: `desert`, `watermelon`, `cherry`

## Clues:
1. The person who owns a Toyota Camry is in the first house.
2. The person's child is named Bella is in the second house.
3. The Desert smoothie lover is directly left of Peter.
4. The person who likes Cherry smoothies is the person who owns a Ford F-150.
5. The person who loves mystery books is the person's child is named Meredith.
6. The person who loves science fiction books is directly left of Eric.
7. Eric is somewhere to the right of the person who loves mystery books.
8. The cat lover is the person who loves science fiction books.
9. There is one house between the person who keeps horses and the Watermelon smoothie lover.
 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: science fiction


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Peter`, `Arnold`, `Alice`, `Bob`
 - Each person has a unique type of pet: `bird`, `dog`, `hamster`, `fish`, `cat`
 - People have unique favorite book genres: `fantasy`, `romance`, `science fiction`, `biography`, `mystery`
 - People have unique favorite sports: `basketball`, `soccer`, `tennis`, `baseball`, `swimming`
 - Everyone has a unique favorite cigar: `pall mall`, `prince`, `blue master`, `blends`, `dunhill`

## Clues:
1. The Dunhill smoker is in the fifth house.
2. The person with an aquarium of fish is directly left of the person who loves biography books.
3. The person who loves mystery books is somewhere to the left of Arnold.
4. The person who smokes many unique blends is the person who loves soccer.
5. The person who loves romance books is directly left of the person who smokes Blue Master.
6. Arnold is somewhere to the right of the person who loves fantasy books.
7. Alice is not in the third house.
8. The person who loves fantasy books is Peter.
9. The person partial to Pall Mall is somewhere to the left of Eric.
10. The person who keeps a pet bird is in the second house.
11. The person who smokes many unique blends is the person with a pet hamster.
12. Eric is directly left of the person who loves tennis.
13. The person who loves tennis is in the fifth house.
14. The person who loves swimming is directly left of the person who keeps a pet bird.
15. The person partial to Pall Mall is the person who loves baseball.
16. The person who has a cat and Arnold are next to each other.
 What is the value of attribute Cigar for the person whose attribute BookGenre is biography? Please reason step by step, and put your final answer within \boxed{}

Answer: pall mall


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Carol`, `Arnold`, `Eric`, `Alice`, `Bob`, `Peter`
 - Each person has a unique level of education: `bachelor`, `high school`, `master`, `associate`, `trade school`, `doctorate`
 - Each person has a favorite color: `purple`, `yellow`, `white`, `red`, `blue`, `green`
 - The people are of nationalities: `swede`, `dane`, `norwegian`, `chinese`, `brit`, `german`
 - Each person has a unique birthday month: `feb`, `may`, `sept`, `april`, `jan`, `mar`

## Clues:
1. The person who loves yellow is somewhere to the right of the person whose birthday is in January.
2. Bob is the person who loves white.
3. The person who loves blue is the person whose birthday is in April.
4. The person with an associate's degree is Eric.
5. The person whose favorite color is green is the person whose birthday is in May.
6. The person who loves yellow is not in the fifth house.
7. The person with an associate's degree is directly left of the Swedish person.
8. The person with a doctorate is the Dane.
9. The person whose birthday is in January is somewhere to the right of the person whose favorite color is green.
10. There are two houses between the British person and the person whose birthday is in February.
11. The person whose birthday is in February is in the fourth house.
12. Alice is the person with a high school diploma.
13. Carol is in the first house.
14. Bob is the German.
15. The person whose birthday is in May is directly left of the Norwegian.
16. The person whose favorite color is red is not in the fifth house.
17. The person whose birthday is in September is in the second house.
18. There are two houses between the person whose birthday is in May and the person with a master's degree.
19. The Norwegian is Peter.
20. The person who attended trade school is the person who loves purple.
 What is the value of attribute Nationality for the person whose attribute House is 6? Please reason step by step, and put your final answer within \boxed{}

Answer: swede


There are 2 houses, numbered 1 to 2 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Eric`
 - Each person has a unique favorite drink: `water`, `tea`
 - Everyone has a favorite smoothie: `desert`, `cherry`
 - Each person has a favorite color: `yellow`, `red`
 - Everyone has a unique favorite cigar: `pall mall`, `prince`
 - People own unique car models: `tesla model 3`, `ford f150`

## Clues:
1. Eric is not in the second house.
2. The person partial to Pall Mall is not in the first house.
3. The person who owns a Ford F-150 is the one who only drinks water.
4. The Desert smoothie lover is somewhere to the left of the person who owns a Tesla Model 3.
5. The person whose favorite color is red is Eric.
 What is the value of attribute Color for the person whose attribute CarModel is ford f150? Please reason step by step, and put your final answer within \boxed{}

Answer: red


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Peter`, `Eric`
 - Each person has a unique birthday month: `sept`, `april`, `jan`
 - People have unique favorite book genres: `mystery`, `science fiction`, `romance`
 - Everyone has a unique favorite cigar: `prince`, `blue master`, `pall mall`
 - The people keep unique animals: `bird`, `cat`, `horse`
 - People own unique car models: `toyota camry`, `ford f150`, `tesla model 3`

## Clues:
1. Arnold is the Prince smoker.
2. Eric is somewhere to the right of the person who loves romance books.
3. The person who keeps horses is the Prince smoker.
4. The person who smokes Blue Master is the person who loves mystery books.
5. The person who owns a Toyota Camry is the person partial to Pall Mall.
6. The person who owns a Tesla Model 3 is the person whose birthday is in September.
7. Peter is the person partial to Pall Mall.
8. The Prince smoker is somewhere to the right of the person who loves mystery books.
9. The person who keeps horses is the person whose birthday is in January.
10. The person who owns a Toyota Camry is the bird keeper.
 What is the value of attribute BookGenre for the person whose attribute Animal is bird? Please reason step by step, and put your final answer within \boxed{}

Answer: romance


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`, `Eric`, `Bob`, `Peter`, `Arnold`, `Alice`
 - Each person prefers a unique type of vacation: `cultural`, `beach`, `mountain`, `cruise`, `camping`, `city`
 - The people are of nationalities: `chinese`, `brit`, `norwegian`, `swede`, `dane`, `german`
 - Everyone has a favorite smoothie: `lime`, `desert`, `watermelon`, `blueberry`, `dragonfruit`, `cherry`

## Clues:
1. The person who likes Cherry smoothies is the person who enjoys mountain retreats.
2. Eric is the Dane.
3. The German is Peter.
4. The Desert smoothie lover is in the sixth house.
5. The Swedish person is the Dragonfruit smoothie lover.
6. Arnold is the Dragonfruit smoothie lover.
7. The Watermelon smoothie lover is the British person.
8. The Norwegian is the person who enjoys camping trips.
9. The person who likes going on cruises is the British person.
10. There are two houses between Bob and the person who enjoys camping trips.
11. The person who drinks Blueberry smoothies is directly left of the person who goes on cultural tours.
12. Bob is directly left of the Dragonfruit smoothie lover.
13. The person who likes going on cruises is directly left of Peter.
14. The person who prefers city breaks is not in the sixth house.
15. Carol is in the fifth house.
 What is the value of attribute Nationality for the person whose attribute Smoothie is lime? Please reason step by step, and put your final answer within \boxed{}

Answer: norwegian


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

## Clues:
1. The person who prefers city breaks is Bob.
2. The person who has brown hair is the one who only drinks water.
3. Eric is directly left of the person who uses a Samsung Galaxy S21.
4. There is one house between the person who loves beach vacations and the person who enjoys camping trips.
5. The one who only drinks water is somewhere to the left of the person who uses a Xiaomi Mi 11.
6. The person who has brown hair is the Dunhill smoker.
7. The person who has red hair is Arnold.
8. The person who has auburn hair is the person who uses an iPhone 13.
9. The person who has gray hair is not in the sixth house.
10. The person who likes milk is the person who has auburn hair.
11. The person who smokes Blue Master is Alice.
12. The person who uses a OnePlus 9 is the person who likes going on cruises.
13. The coffee drinker is the Prince smoker.
14. The coffee drinker is the person who uses a Huawei P50.
15. Eric is the tea drinker.
16. The Prince smoker is in the third house.
17. The person who smokes Yellow Monster is somewhere to the right of the coffee drinker.
18. The person who enjoys mountain retreats is Peter.
19. The person who uses a Huawei P50 is the person who has blonde hair.
20. There are two houses between the person who loves beach vacations and the person who uses a Google Pixel 6.
21. The one who only drinks water is in the fourth house.
22. Eric is the person partial to Pall Mall.
23. Bob is the boba tea drinker.
 What is the value of attribute PhoneModel for the person whose attribute HairColor is auburn? Please reason step by step, and put your final answer within \boxed{}

Answer: iphone 13


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Arnold`, `Eric`, `Alice`
 - They all have a unique favorite flower: `daffodils`, `carnations`, `roses`, `lilies`
 - People have unique heights: `very short`, `short`, `tall`, `average`
 - The mothers' names in different houses are unique: `Janelle`, `Kailyn`, `Holly`, `Aniya`
 - Each person has an occupation: `engineer`, `doctor`, `teacher`, `artist`
 - People have unique favorite sports: `swimming`, `basketball`, `tennis`, `soccer`

## Clues:
1. The person who loves swimming is the person who loves the rose bouquet.
2. The person who loves the rose bouquet is Eric.
3. Arnold is the person who is tall.
4. The person who loves a bouquet of daffodils is somewhere to the right of the person who is an engineer.
5. The person who loves soccer is the person who is short.
6. The person who is a teacher is in the first house.
7. The person whose mother's name is Janelle is the person who loves a carnations arrangement.
8. The person who loves basketball is the person who has an average height.
9. Arnold is not in the third house.
10. The person whose mother's name is Holly is somewhere to the right of the person who has an average height.
11. Peter is the person who is a doctor.
12. The person whose mother's name is Aniya is Alice.
13. Arnold is the person who loves the boquet of lilies.
 What is the value of attribute Height for the person whose attribute FavoriteSport is swimming? Please reason step by step, and put your final answer within \boxed{}

Answer: very short


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`
 - The people are of nationalities: `dane`, `brit`, `swede`
 - People have unique favorite sports: `basketball`, `tennis`, `soccer`

## Clues:
1. Eric is in the second house.
2. The person who loves soccer is somewhere to the right of the person who loves basketball.
3. The Dane is somewhere to the right of Arnold.
4. The British person is somewhere to the right of the person who loves tennis.
5. The British person is not in the third house.
 What is the value of attribute Nationality for the person whose attribute FavoriteSport is tennis? Please reason step by step, and put your final answer within \boxed{}

Answer: swede


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `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 House for the person whose attribute Name is Arnold? Please reason step by step, and put your final answer within \boxed{}

Answer: 3


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Arnold`, `Eric`, `Alice`
 - They all have a unique favorite flower: `daffodils`, `carnations`, `roses`, `lilies`
 - People have unique heights: `very short`, `short`, `tall`, `average`
 - The mothers' names in different houses are unique: `Janelle`, `Kailyn`, `Holly`, `Aniya`
 - Each person has an occupation: `engineer`, `doctor`, `teacher`, `artist`
 - People have unique favorite sports: `swimming`, `basketball`, `tennis`, `soccer`

## Clues:
1. The person who loves swimming is the person who loves the rose bouquet.
2. The person who loves the rose bouquet is Eric.
3. Arnold is the person who is tall.
4. The person who loves a bouquet of daffodils is somewhere to the right of the person who is an engineer.
5. The person who loves soccer is the person who is short.
6. The person who is a teacher is in the first house.
7. The person whose mother's name is Janelle is the person who loves a carnations arrangement.
8. The person who loves basketball is the person who has an average height.
9. Arnold is not in the third house.
10. The person whose mother's name is Holly is somewhere to the right of the person who has an average height.
11. Peter is the person who is a doctor.
12. The person whose mother's name is Aniya is Alice.
13. Arnold is the person who loves the boquet of lilies.
 What is the value of attribute Height for the person whose attribute Mother is Kailyn? Please reason step by step, and put your final answer within \boxed{}

Answer: tall


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Arnold`, `Eric`
 - Each person lives in a unique style of house: `colonial`, `victorian`, `ranch`
 - People have unique favorite book genres: `science fiction`, `mystery`, `romance`
 - Each person has a unique type of pet: `fish`, `cat`, `dog`
 - Everyone has a unique favorite cigar: `prince`, `blue master`, `pall mall`
 - Each person has a unique hobby: `cooking`, `photography`, `gardening`

## Clues:
1. The person who loves cooking is in the second house.
2. The person who loves romance books is the person who smokes Blue Master.
3. The person who loves mystery books is the person partial to Pall Mall.
4. The person who smokes Blue Master and the person who loves cooking are next to each other.
5. The person who owns a dog is the person in a ranch-style home.
6. The person residing in a Victorian house is the person with an aquarium of fish.
7. The person who has a cat is the Prince smoker.
8. Arnold and the person who has a cat are next to each other.
9. The photography enthusiast is directly left of Arnold.
10. Eric is not in the third house.
11. Eric is the person with an aquarium of fish.
 What is the value of attribute Cigar for the person whose attribute Hobby is cooking? Please reason step by step, and put your final answer within \boxed{}

Answer: pall mall


12.386. The lateral face of a regular triangular pyramid $S A B C$ forms an angle $\alpha$ with the base plane. A plane is drawn through the side $B C$ of the base and a point $D$ on the lateral edge $A S$. Find the angle between this plane and the base plane, if $A D: D S=k$.

Answer: \arctan\left( \dfrac{k \tan \alpha}{k + 3} \right)


8.7. Let's consider the equilateral triangle $A B C$. Points $D, E$, and $F$ are located on the sides $(B C),(C A)$, and, respectively, $(A B)$ such that the lines $D F$ and $A B$ are perpendicular, and the line $D E$ is the perpendicular bisector of the segment $C F$. Find the measure in degrees of the angle $D E F$.

Answer: 45


4. In $\triangle A B C$, $\angle A B C=90^{\circ}, A C=\sqrt[3]{2}, D$ is a point on the extension of $A C$, $C D=A B=1$. Then the degree measure of $\angle C B D$ is $\qquad$

Answer: 30


12. (3 points) Two natural numbers $a$ and $b$, their least common multiple is 60. Then, the difference between these two natural numbers has $\qquad$ possible numerical values.

Answer: 23


$\underline{\text { Rubanov I.S. }}$

Microcalculator MK-97 can perform only three operations on numbers stored in memory:

1) check if two selected numbers are equal,
2) add selected numbers,
3) find the roots of the equation $x^{2}+a x+b=0$ for selected numbers $a$ and $b$, or display a message if there are no roots.

The results of all actions are stored in memory. Initially, the memory contains a single number $x$. How can one use the MK-97 to determine if this number is equal to one?

Answer: 1


Suppose tetrahedron $PABC$ has volume $420$ and satisfies $AB = 13$, $BC = 14$, and $CA = 15$. The minimum possible surface area of $PABC$ can be written as $m+n\sqrt{k}$, where $m,n,k$ are positive integers and $k$ is not divisible by the square of any prime. Compute $m+n+k$.

[i]Ray Li[/i]

Answer: 346


17. Fill in $2013 \times 1, 2013 \times 2, 2013 \times 3, 2013 \times 4, 2013 \times 5, 2013 \times 6, 2013 \times 7, 2013 \times 8, 2013 \times 9, 2013 \times 10, 2013 \times 11$ into the table below, so that the numbers filled in can be divided by the position number of their respective columns. Then there are $\qquad$ different ways to fill in the table.

Answer: 24


3. Let the function odraz map the digits $0,1,2,5,8$ to the digits $0,1,5,2,8$, respectively. A natural number $n=\overline{t_{k} t_{k-1} \cdots t_{1} t_{0}}$ is called reflectable if all its digits are from the set $\{0,1,2,5,8\}$ and $t_{0} \neq 0$, and we define

$$
\operatorname{odraz}(n)=\overline{\operatorname{odraz}\left(t_{0}\right) \operatorname{odraz}\left(t_{1}\right) \cdots \operatorname{odraz}\left(t_{k-1}\right) \operatorname{odraz}\left(t_{k}\right)}
$$

(in other words, the function odraz represents the reflection in a mirror of the number on a calculator screen). Find all natural numbers $n$ with the following properties:

$1^{\circ} n$ is reflectable and $\operatorname{odraz}(n)=n$;

$2^{\circ} n^{2}$ is reflectable and $\operatorname{odraz}\left(n^{2}\right)=n^{2}$.

Answer: 1


5. $S$ is the set of rational numbers $r$ where $0<r<1$, and $r$ has a repeating decimal expansion of the form $0 . a b c a b c a b c \cdots$ $=0 . \dot{a} b \dot{c}, a, b, c$ are not necessarily distinct. Among the elements of $S$ that can be written as a fraction in simplest form, how many different numerators are there?

Answer: 660


Consider a regular $2 n$-gon $P, A_{1} A_{2} \ldots A_{2 n}$ in the plane, where $n$ is a positive integer. We say that a point $S$ on one of the sides of $P$ can be seen from a point $E$ that is external to $P$, if the line segment $S E$ contains no other points that lie on the sides of $P$ except $S$. We color the sides of $P$ in 3 different colors (ignore the vertices of $P$, we consider them colorless), such that every side is colored in exactly one color, and each color is used at least once. Moreover, from every point in the plane external to $P$, points of at most 2 different colors on $P$ can be seen. Find the number of distinct such colorings of $P$ (two colorings are considered distinct if at least one of the sides is colored differently).

Answer: 6n


4. Let $A_{1}, A_{2}, A_{3}, \ldots, A_{11}, A_{12}$ be the consecutive vertices of a regular dodecagon with side $a=2 \text{~cm}$. Let the diagonals $A_{1} A_{10}, A_{2} A_{11}$, and $A_{3} A_{12}$ intersect at points $A, B$, and $C$. Find the area $P_{A B C}$.

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


6. The numbers $112,121,123,153,243,313$, and 322 are among the rows, columns, and diagonals of a $3 \times 3$ square grid of digits (rows and diagonals read left-to-right, and columns read top-to-bottom). What 3-digit number completes the list?

Solution:
524

Answer: 524


1. On the coordinate plane, lines of the form $y=a x+b$ are drawn, where $a$ and $b$ are natural numbers from 1 to 9. Among all the intersection points of these lines, select the point with the greatest sum of coordinates. Write the answer in the form ( $x ; y$ ).

Answer: (8; 73)


6. Given the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ (where $a>b>$ $0$), point $P$ is a point on it, $F_{1} 、 F_{2}$ are the foci of the ellipse, and the external angle bisector of $\angle F_{1} P F_{2}$ is $l$. Construct $F_{1} 、 F_{2}$ perpendicular to $l$ at $R 、 S$. When $P$ traverses the entire ellipse, the trajectory equation of $R 、 S$ is $\qquad$

Answer: x^2 + y^2 = a^2


## Task B-2.4.

The teacher has several candies that she wants to distribute to students who come for additional math lessons so that each student gets an equal number of candies and no candy is left over.

- If two students come, each student cannot get the same number of candies.
- If three students come, each of them will get the same number of candies.
- If four students come, each of them will get the same number of candies.
- If five students come, each of them will get the same number of candies.
- If six students come, each student cannot get the same number of candies.

If it is known that only one of the above statements is false, what is the minimum number of candies the teacher can have?

Answer: 15


## PROBLEM 20. ROLLING DICE

A regular die has the numbers $1,2,3,4,5,6$ on its faces. It is rolled randomly until the sum of the points that come up exceeds the number 12.

What is the most likely total sum of points?

Answer: 13


5. Quadrilateral $ABCD$ is inscribed in a circle, and its diagonals intersect at point $P$. Points $K, L$, and $M$ are the midpoints of sides $AB, BC$, and $CD$ respectively. The radius of the circle circumscribed around triangle $KLP$ is 1. Find the radius of the circle circumscribed around triangle $LMP$.

Problem 1 Answer: 18 passengers.

Answer: 1


10. Given that line segment $A B$ is the diameter of sphere $O$ with radius 2, points $C, D$ are on the surface of sphere $O$, $C D=2, A B \perp C D$, $45^{\circ} \leqslant \angle A O C \leqslant 135^{\circ}$, then the range of the volume of tetrahedron $A B C D$ is $\qquad$ .

Answer: \left[ \dfrac{4}{3}, \dfrac{4\sqrt{3}}{3} \right]


11. Color the 17 squares in the "T" shape on the right using black and white. If each square can only be one color, and no two black squares can be adjacent (share an edge), then there are $\qquad$ coloring schemes.

Paint the 17 squares in the "T" shape on the right using black and white. If each square can only be one color, and no two black squares can be adjacent (share an edge), then there are $\qquad$ coloring schemes.

Answer: 4370


In the fictional country of Mahishmati, there are $50$ cities, including a capital city. Some pairs of cities are connected by two-way flights. Given a city $A$, an ordered list of cities $C_1,\ldots, C_{50}$ is called an [i]antitour[/i] from $A$ if
[list]
[*] every city (including $A$) appears in the list exactly once, and
[*] for each $k\in \{1,2,\ldots, 50\}$, it is impossible to go from $A$ to $C_k$ by a sequence of exactly $k$ (not necessarily distinct) flights.
[/list]
Baahubali notices that there is an antitour from $A$ for any city $A$. Further, he can take a sequence of flights, starting from the capital and passing through each city exactly once. Find the least possible total number of antitours from the capital city.

[i]Proposed by Sutanay Bhattacharya[/i]

Answer: (25!)^2


Let $f$ be a function which takes in $0, 1, 2$ and returns $0, 1, $ or $2$. The values need not be distinct: for instance we could have $f(0) = 1, f(1) = 1, f(2) = 2$. How many such functions are there which satisfy \[f(2) + f(f(0)) + f(f(f(1))) = 5?\]

Answer: 2


Let $P(x)$ be a polynomial with positive integer coefficients and degree 2015. Given that there exists some $\omega \in \mathbb{C}$ satisfying
$$\omega^{73} = 1\quad \text{and}$$
$$P(\omega^{2015}) + P(\omega^{2015^2}) + P(\omega^{2015^3}) + \ldots + P(\omega^{2015^{72}}) = 0,$$
what is the minimum possible value of $P(1)$?

Answer: 2044


1. Cyclists Petya, Vlad, and Timur simultaneously started a warm-up race on a circular cycling track. Their speeds are 27 km/h, 30 km/h, and 32 km/h, respectively. After what shortest time will they all be at the same point on the track again? (The length of the cycling track is 400 meters.)

Answer: 24


For an ${n\times n}$ matrix $A$, let $X_{i}$ be the set of entries in row $i$, and $Y_{j}$ the set of entries in column $j$, ${1\leq i,j\leq n}$. We say that $A$ is [i]golden[/i] if ${X_{1},\dots ,X_{n},Y_{1},\dots ,Y_{n}}$ are distinct sets. Find the least integer $n$ such that there exists a ${2004\times 2004}$ golden matrix with entries in the set ${\{1,2,\dots ,n\}}$.

Answer: 13


There are 5 people standing in a line numbered 1 through 5 in a left to right order.
Each person has the following attributes:  Job, Movie-Genre, Music-Genre, Pet.
The attributes have the following possible values:
- Job: accountant, entrepreneur, mechanic, scientist, teacher
- Movie-Genre: animation, crime, scientific, sports, superhero
- Music-Genre: blues, d&b, electronic, funk, hip-hop
- Pet: dog, frog, guinea-pig, pony, rabbit

Given the following premises about the line of people:
- Music-Genre:funk is not to the left of Job:teacher
- Job:mechanic is not to the right of Movie-Genre:sports
- Pet:frog is not to the right of Music-Genre:blues
- Job:scientist is somewhere between Pet:frog and Pet:guinea-pig
- Job:entrepreneur and Music-Genre:blues have different parity positions
- Music-Genre:d&b is not to the left of Job:scientist
- Job:accountant is somewhere between Job:teacher and Movie-Genre:scientific
- Pet:rabbit is not to the left of Pet:guinea-pig
- Pet:rabbit and Music-Genre:funk have the same parity positions
- Job:accountant == Pet:dog or Music-Genre:electronic == Job:accountant, but not both
- Pet:pony and Job:scientist have different parity positions
- Music-Genre:hip-hop and Movie-Genre:animation have different parity positions
- Movie-Genre:superhero and Music-Genre:d&b have different parity positions
- Movie-Genre:animation and Pet:pony have different parity positions
- Job:entrepreneur is somewhere between Job:scientist and Movie-Genre:superhero
- Movie-Genre:sports is not to the right of Pet:dog

Answer the following question:

Question: Question: What is the Movie-Genre of the person who is mechanic??
 Please provide your answer in the format: \boxed{X}, where X is the answer.

Answer: sports