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

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

Answer: high school


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Alice`, `Peter`, `Arnold`
 - The people are of nationalities: `brit`, `swede`, `norwegian`, `dane`
 - They all have a unique favorite flower: `carnations`, `lilies`, `roses`, `daffodils`
 - Each person has a unique level of education: `high school`, `bachelor`, `master`, `associate`

## Clues:
1. The Norwegian is directly left of the person who loves the rose bouquet.
2. The person with a high school diploma is Eric.
3. Arnold is not in the first house.
4. The person with an associate's degree is the person who loves a carnations arrangement.
5. The Dane is not in the third house.
6. Peter is the person who loves the rose bouquet.
7. The Swedish person and the Norwegian are next to each other.
8. Eric is in the second house.
9. The person who loves the boquet of lilies is the person with a master's degree.
10. The person who loves the boquet of lilies is the Swedish person.
 What is the value of attribute Nationality for the person whose attribute Name is Eric? Please reason step by step, and put your final answer within \boxed{}

Answer: norwegian


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 hobby: `gardening`, `photography`, `cooking`
 - Each person prefers a unique type of vacation: `mountain`, `city`, `beach`
 - Everyone has a unique favorite cigar: `pall mall`, `prince`, `blue master`
 - The mothers' names in different houses are unique: `Janelle`, `Holly`, `Aniya`
 - People have unique heights: `short`, `average`, `very short`

## Clues:
1. The person who loves beach vacations is somewhere to the right of the person who is very short.
2. The person whose mother's name is Aniya is somewhere to the right of the person who loves cooking.
3. Eric is not in the second house.
4. The person who is short is in the third house.
5. The person who prefers city breaks is the photography enthusiast.
6. The person who enjoys gardening is the person who enjoys mountain retreats.
7. The person partial to Pall Mall is the photography enthusiast.
8. The person who enjoys mountain retreats is The person whose mother's name is Holly.
9. The person who enjoys gardening is the person who smokes Blue Master.
10. Peter is The person whose mother's name is Aniya.
 What is the value of attribute Cigar for the person whose attribute Height is short? 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: `Eric`, `Bob`, `Alice`, `Peter`, `Arnold`, `Carol`
 - They all have a unique favorite flower: `daffodils`, `carnations`, `roses`, `tulips`, `iris`, `lilies`
 - People have unique heights: `very tall`, `super tall`, `average`, `short`, `very short`, `tall`
 - Each person has a favorite color: `blue`, `yellow`, `purple`, `red`, `green`, `white`
 - Everyone has a unique favorite cigar: `yellow monster`, `pall mall`, `prince`, `blue master`, `blends`, `dunhill`
 - People use unique phone models: `google pixel 6`, `iphone 13`, `xiaomi mi 11`, `samsung galaxy s21`, `huawei p50`, `oneplus 9`

## Clues:
1. The person who uses an iPhone 13 is the person who loves white.
2. Peter is the person whose favorite color is red.
3. The Prince smoker is directly left of the person partial to Pall Mall.
4. The person who smokes Blue Master and the Dunhill smoker are next to each other.
5. The person who loves blue is the person who uses a Xiaomi Mi 11.
6. The person who loves white is Alice.
7. The person who uses a OnePlus 9 is somewhere to the left of Arnold.
8. The person who is tall is the person who smokes Blue Master.
9. The person who uses a Samsung Galaxy S21 is the Dunhill smoker.
10. Bob is directly left of the person who is very short.
11. The person who loves a bouquet of daffodils is Eric.
12. The person who loves the boquet of iris is the person who uses a OnePlus 9.
13. The person who uses a Huawei P50 is the person who is very short.
14. The person who smokes many unique blends is directly left of Peter.
15. The Prince smoker is not in the first house.
16. The person who is very tall is the person who loves a bouquet of daffodils.
17. The person who uses a Google Pixel 6 is not in the fifth house.
18. The person who is tall is not in the third house.
19. The person who uses a Huawei P50 is somewhere to the left of the person who uses a Samsung Galaxy S21.
20. Arnold is somewhere to the left of the person who is short.
21. The person who loves the vase of tulips is not in the first house.
22. The person who loves the boquet of lilies is directly left of the person who loves yellow.
23. Arnold is the person who loves purple.
24. The person who uses a OnePlus 9 and the person who uses a Samsung Galaxy S21 are next to each other.
25. The person who has an average height is not in the first house.
26. The person who loves the rose bouquet is Alice.
 What is the value of attribute Height for the person whose attribute Flower is lilies? Please reason step by step, and put your final answer within \boxed{}

Answer: very short


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Carol`, `Eric`, `Bob`, `Alice`, `Peter`
 - Each person has a unique birthday month: `feb`, `mar`, `sept`, `jan`, `may`, `april`
 - Everyone has something unique for lunch: `stew`, `soup`, `grilled cheese`, `stir fry`, `spaghetti`, `pizza`
 - People have unique heights: `very short`, `average`, `super tall`, `short`, `very tall`, `tall`
 - People own unique car models: `chevrolet silverado`, `ford f150`, `bmw 3 series`, `tesla model 3`, `toyota camry`, `honda civic`

## Clues:
1. The person who owns a Honda Civic is the person who is short.
2. The person who owns a Ford F-150 is in the fifth house.
3. The person who loves stir fry is somewhere to the left of Eric.
4. The person whose birthday is in May is somewhere to the left of Carol.
5. The person who is very short is somewhere to the left of the person whose birthday is in April.
6. The person who owns a BMW 3 Series is not in the third house.
7. There are two houses between the person who loves stir fry and the person who is a pizza lover.
8. The person who loves the soup is directly left of Eric.
9. The person who loves the spaghetti eater and the person whose birthday is in May are next to each other.
10. Alice is directly left of the person who owns a BMW 3 Series.
11. The person who owns a Tesla Model 3 is somewhere to the left of the person who is tall.
12. The person who is very tall is the person who owns a Toyota Camry.
13. Peter is directly left of the person who is a pizza lover.
14. The person who loves the stew is not in the third house.
15. There is one house between the person whose birthday is in September and the person who is very short.
16. There is one house between the person whose birthday is in March and the person who is super tall.
17. The person who is tall is Bob.
18. The person whose birthday is in May is somewhere to the right of Alice.
19. The person who is very short is in the fourth house.
20. The person whose birthday is in March is the person who is short.
21. Carol is the person who owns a Tesla Model 3.
22. Eric is the person whose birthday is in January.
 What is the value of attribute Food for the person whose attribute Name is Eric? Please reason step by step, and put your final answer within \boxed{}

Answer: stew


A triangle \(ABC\) is drawn on graph paper. Find the measure of angle \(\angle A\).

Answer: 45


In triangle \(ABC\), angle \(A\) is \(40^\circ\). The triangle is randomly thrown onto a table. Find the probability that vertex \(A\) ends up east of the other two vertices.

Answer: \frac{7}{18}


Three not necessarily distinct positive integers between 1 and 99, inclusive, are written in a row on a blackboard. Then, the numbers, without including any leading zeros, are concatenated to form a new integer \( N \). For example, if the integers written, in order, are 25, 6, and 12, then \( N = 25612 \). Determine the number of possible values of \( N \).

Answer: 825957


In trapezoid \(ABCD\) with base \(AD\), the diagonals are the bisectors of angles \(\angle B\) and \(\angle C = 110^{\circ}\). Find the measure of angle \(\angle BAC\).

Answer: 15


Answer: 0


The integers from \(1\) to \(n\) are written in increasing order from left to right on a blackboard. David and Goliath play the following game: starting with David, the two players alternate erasing any two consecutive numbers and replacing them with their sum or product. Play continues until only one number on the board remains. If it is odd, David wins, but if it is even, Goliath wins. Find the 2011th smallest positive integer greater than \(1\) for which David can guarantee victory.

Answer: 4022


Mr. Chan has 8 sons and \(a\) daughters. Each of his sons has 8 sons and \(a\) daughters. Each of his daughters has \(a\) sons and 8 daughters. It is known that the number of his grandsons is one more than the number of his granddaughters and \(a\) is a prime number. Find the value of \(a\).

Let \(\frac{a}{7}=\sqrt[3]{2+\sqrt{b}}+\sqrt[3]{2-\sqrt{b}}\). Find the value of \(b\).

In Figure 1, find the value of \(C\).

Given that \(P_{1}, P_{2}, \ldots, P_{d}\) are \(d\) consecutive prime numbers. If \(P_{1}+P_{2}+\ldots+P_{d-2}=P_{d-1}+P_{d}=C+1\), find the value of \(d\).

Answer: 6


Mr. N, aiming to maximize his profit, holds a patent for a unique product and can sell it only in countries A and B. The demand for his product in these countries is represented by the functions \( q_{A}=40-2P \) and \( q_{B}=26-P \), where \( q_{A} \) and \( q_{B} \) are the quantities of the product in countries A and B, respectively, and \( P \) is its price. Resales of Mr. N's product between the countries involve significant costs for his buyers, so Mr. N can set different prices for his product in each country. Regardless of the production country, the shipping cost to the other country is fixed at 1. Mr. N discovered that the production costs of his product do not depend on the production country and are described by the function \( TC(Q)=8Q+1 \). Mr. N can register his enterprise and produce the product in any of the countries, but they have different taxation systems. In country A, there is a proportional tax system with a fixed rate of 15%. In country B, there is a progressive multi-tier taxation system, with the rates indicated in the table:

| Profit | Up to 30 | From 30 to 100 | From 100 to 150 | Above 150 |
| :--- | :--- | :--- | :--- | :--- |
| Tax, % | Exempt | 10 | 20 | 30 |

What will Mr. N's profit be after taxes? What quantity of the product and at what price will be sold in each country?

Answer: 133.7


Mike and Harry play a game on an \(8 \times 8\) board. For some positive integer \(k\), Mike chooses \(k\) squares and writes an \(M\) in each of them. Harry then chooses \(k+1\) squares and writes an \(H\) in each of them. After Harry is done, Mike wins if there is a sequence of letters forming "HMM" or "MMH," when read either horizontally or vertically, and Harry wins otherwise. Determine the smallest value of \(k\) for which Mike has a winning strategy.

Answer: 16


Three cones with apex $A$ and a slant height of 6 touch each other externally. For two cones, the angle between the slant height and the axis of symmetry is $\frac{\pi}{8}$, and for the third cone, it is $\frac{\pi}{4}$. Find the volume of the pyramid $O_{1} O_{2} O_{3} A$, where $O_{1}, O_{2}, O_{3}$ are the centers of the bases of the cones.

Answer: 9\sqrt{\sqrt{2}+1}


Draw 63 lines on a coordinate plane: \( y = b, y = \sqrt{3} x + 2b, y = -\sqrt{3} x + 2b \), where \( b = -10, -9, -8, \cdots, 8, 9, 10 \). These lines divide the plane into several equilateral triangles. The number of equilateral triangles with side length \( \frac{2}{\sqrt{3}} \) is $\qquad$

Answer: 660


As shown in the figure, in the square $\mathrm{ABCD}$, the area of the isosceles right triangle $\mathrm{AEF}$ is $1 \mathrm{~cm}^2$, and the area of the rectangle $\mathrm{EFGH}$ is $10 \mathrm{~cm}^2$. What is the area of the square $\mathrm{ABCD}$ in $\mathrm{cm}^2$?

Answer: 24.5


Ati has  $ 7$  pots of flower, ordered in  $ P_1,P_2,P_3,P_4,P_5,P_6,P_7$ . She wants to rearrange the position of those pots to  $ B_1,B_2,B_2,B_3,B_4,B_5,B_6,B_7$  such that for every positive integer  $ n<7$ ,  $ B_1,B_2,\dots,B_n$  is not the permutation of  $ P_1,P_2,\dots,P_7$ . In how many ways can Ati do this?

Answer: 3447


Let  $ABCDEF$  regular hexagon of side length  $1$  and  $O$  is its center. In addition to the sides of the hexagon, line segments from  $O$  to the every vertex are drawn, making as total of  $12$  unit segments. Find the number paths of length  $2003$  along these segments that star at  $O$  and terminate at  $O$ .

Answer: \frac{3}{\sqrt{7}}((1+\sqrt{7})^{2002}-(1-\sqrt{7})^{2002})


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}


We call a rectangle of size  $2 \times 3$  (or  $3 \times 2$ ) without one cell in corner a  $P$ -rectangle. We call a rectangle of size  $2 \times 3$  (or  $3 \times 2$ ) without two cells in opposite (under center of rectangle) corners a  $S$ -rectangle. Using some squares of size  $2 \times 2$ , some  $P$ -rectangles and some  $S$ -rectangles, one form one rectangle of size  $1993 \times 2000$  (figures don’t overlap each other). Let  $s$  denote the sum of numbers of squares and  $S$ -rectangles used in such tiling. Find the maximal value of  $s$ .

Answer: 996500


Suppose  $P(x)$  is a monic polynomial of degree  $2023$  such that  $P(k) = k^{2023}P(1-\frac{1}{k})$  for every positive integer  $1 \leq k \leq 2023$ . Then  $P(-1) = \frac{a}{b}$  where  $a$  and  $b$  are relatively prime integers. Compute the unique integer  $0 \leq n < 2027$  such that  $bn-a$  is divisible by the prime  $2027$ .

Answer: 0


For all integers  $n\geq 1$  we define  $x_{n+1}=x_1^2+x_2^2+\cdots +x_n^2$ , where  $x_1$  is a positive integer. Find the least  $x_1$  such that 2006 divides  $x_{2006}$ .

Answer: 531


Several irrational numbers are written on a blackboard. It is known that for every two numbers  $ a$  and  $ b$  on the blackboard, at least one of the numbers  $ a\over b\plus{}1$  and  $ b\over a\plus{}1$  is rational. What maximum number of irrational numbers can be on the blackboard?

*Author: Alexander Golovanov*

Answer: 3


In an office at various times during the day, the boss gives the secretary a letter to type, each time putting the letter on top of the pile in the secretary's in-box.  When there is time, the secretary takes the top letter off the pile and types it.  There are nine letters to be typed during the day, and the boss delivers them in the order 1, 2, 3, 4, 5, 6, 7, 8, 9.

While leaving for lunch, the secretary tells a colleague that letter 8 has already been typed, but says nothing else about the morning's typing.  The colleague wonder which of the nine letters remain to be typed after lunch and in what order they will be typed.  Based upon the above information, how many such after-lunch typing orders are possible?  (That there are no letters left to be typed is one of the possibilities.)

Answer: 256


A  $\pm 1$ -*sequence* is a sequence of  $2022$  numbers  $a_1, \ldots, a_{2022},$  each equal to either  $+1$  or  $-1$ . Determine the largest  $C$  so that, for any  $\pm 1$ -sequence, there exists an integer  $k$  and indices  $1 \le t_1 < \ldots < t_k \le 2022$  so that  $t_{i+1} - t_i \le 2$  for all  $i$ , and   $$ \left| \sum_{i = 1}^{k} a_{t_i} \right| \ge C. $$

Answer: 506


Let  $ \overline{MN}$  be a diameter of a circle with diameter  $ 1$ . Let  $ A$  and  $ B$  be points on one of the semicircular arcs determined by  $ \overline{MN}$  such that  $ A$  is the midpoint of the semicircle and  $ MB\equal{}\frac35$ . Point  $ C$  lies on the other semicircular arc. Let  $ d$  be the length of the line segment whose endpoints are the intersections of diameter  $ \overline{MN}$  with the chords  $ \overline{AC}$  and  $ \overline{BC}$ . The largest possible value of  $ d$  can be written in the form  $ r\minus{}s\sqrt{t}$ , where  $ r$ ,  $ s$ , and  $ t$  are positive integers and  $ t$  is not divisible by the square of any prime. Find  $ r\plus{}s\plus{}t$ .

Answer: 14


Dinesh has several squares and regular pentagons, all with side length  $ 1$ . He wants to arrange the shapes alternately to form a closed loop (see diagram). How many pentagons would Dinesh need to do so?
![Image](https://cdn.artofproblemsolving.com/attachments/8/9/6345d7150298fe26cfcfba554656804ed25a6d.jpg)

Answer: 10


8. For $i=0,1, \ldots, 5$ let $l_{i}$ be the ray on the Cartesian plane starting at the origin, an angle $\theta=i \frac{\pi}{3}$ counterclockwise from the positive $x$-axis. For each $i$, point $P_{i}$ is chosen uniformly at random from the intersection of $l_{i}$ with the unit disk. Consider the convex hull of the points $P_{i}$, which will (with probability 1) be a convex polygon with $n$ vertices for some $n$. What is the expected value of $n$ ?

Answer: 2 + 4 \ln 2


## Task 4 - 110614

Two places $A$ and $B$ are connected by a $999 \mathrm{~km}$ long road.

At intervals of $1 \mathrm{~km}$, milestones are placed along this road, each marked on both sides such that one side indicates the distance from $A$ and the other side the distance from $B$ in km. For example, the stone at the exit of $A$ bears the inscription 0 and 999, and the stone at the entrance of $B$ bears the inscription 999 and 0.

Determine the number of these stones whose inscriptions use at most two different digits (e.g., 722 and 277)!

Answer: 40


On a chess board ($8*8$) there are written the numbers $1$ to $64$: on the first line, from left to right, there are the numbers $1, 2, 3, ... , 8$; on the second line, from left to right, there are the numbers $9, 10, 11, ... , 16$;etc. The $\"+\"$ and $\"-\"$ signs are put to each number such that, in each line and in each column, there are $4$ $\"+\"$ signs and $4$ $\"-\"$ signs. Then, the $64$ numbers are added. Find all the possible values of this sum.

Answer: 0


13. Salvo-control. The artillery system of a missile cruiser fires at a target. If the target is not hit, the system automatically makes a second shot at the same target. A third shot at this target is not made. The cruiser has 10 missiles on board. During exercises, it conducts firings at several targets under identical conditions. The probability of hitting with each single shot is $p$.

a) (from 8th grade, 2 points). If there are only five targets, what is the probability that exactly three unused missiles will remain after firing?

b) (from 9th grade, 3 points). Find the expected number of targets hit if there are nine targets in total.

Answer: 10p^3(1 - p)^2


Initial 281 Given that there exist $k(k \in \mathbf{N}, k \geqslant 2)$ consecutive positive integers, the mean of their squares is a perfect square. Try to find the minimum value of $k$.

Answer: 31


Let \(P\) be a point on the minor arc \( \overparen{BC} \) of the circumcircle of \(\triangle ABC\). The perpendicular projections of \(P\) onto the sides \(BC\), \(CA\), and \(AB\) are denoted as \(L\), \(M\), and \(N\) respectively. Let \(PL = l\), \(PM = m\), \(PN = n\), \(BC = a\), \(CA = b\), and \(AB = c\). Prove that \(mn \cdot a = ln \cdot b + \operatorname{lmc}\).

Answer: mn \cdot a = ln \cdot b + lm \cdot c


5. Magnus and Viswanathan play a game against each other three times:
- In each game, each player's score is a positive integer. The players' scores at the end of the three games are six different integers.
- In each game, the winner's score is at least 25 points. If the winner's score is 25 , their opponent's score must be at most 23 . If the winner's score is greater than 25 , their opponent's score must be exactly 2 less than the winner's score.
- Viswanathan wins either the first game or the second game, but not both.
- Viswanathan wins the third game with a score of 25 .
- Each player's score in the second game is the average of their scores in the first and third games.
What was Magnus's score in the third game?

Answer: 19


10.2. In triangle $A B C$, point $M$ is the midpoint of side $B C$. On the bisector of angle $B A C$, point $N$ is taken such that $B N \perp A N$. Determine the length of side $A C$, if $A B=10 \mathrm{~cm}$ and $M N=2 \mathrm{~cm}$.

Answer: 14


5. To transmit a three-letter word through a channel, the following method is used. Each letter of the word is assigned a pair of digits according to the rule: А -00, Б $-01, \mathrm{~B}-02, \ldots$, Я - 32. After that, the obtained sequence of digits $m_{1}, m_{2}, \ldots, m_{6}$. is transformed by the formula:

$c_{i}=f\left(m_{i}, c_{\{i-1\}}\right), i \in\{1, \ldots, 6\}$.

where $c_{0} \in\{0, \ldots, 9\}$ - a randomly chosen digit and $f(x, y)=r_{10}(x+4 y)-$ the remainder of the division by 10 of the number $x+4 y$. Then the sequence $c_{0}, c_{1}, \ldots, c_{6}$ is transmitted through the channel. Crypto managed to intercept $\left(c_{0}, c_{2}, c_{4}, c_{6}\right)=(1,3,7,1)$, what word could have been transmitted through the channel? Provide the word in your answer (in lowercase letters).

Answer: миф


There are $n$ balls numbered from $1$ to $n$, and $2n-1$ boxes numbered from $1$ to $2n-1$. For each $i$, ball number $i$ can only be put in the boxes with numbers from $1$ to $2i-1$. Let $k$ be an integer from $1$ to $n$. In how many ways we can choose $k$ balls, $k$ boxes and put these balls in the selected boxes so that each box has exactly one ball?

Answer: k! \cdot \dbinom{n}{k}^2


7. A line segment connecting two different points on an ellipse is called a chord of the ellipse. Then, the equation of the locus of the midpoint of a chord of length $\sqrt{5}$ on the ellipse $\frac{x^{2}}{4}+y^{2}=1$ is $\qquad$

Answer: x^4 + 20x^2y^2 + 64y^4 + x^2 - 44y^2 = 0


A triangle's three side lines are tangent to four circles whose radii are four consecutive elements of a geometric sequence. What is the largest angle of the triangle?

Answer: 90^\circ


10、Person A and Person B start from locations $A$ and $B$ respectively at the same time and walk towards each other. When A reaches the halfway point, B doubles his speed. As a result, they meet 1200 meters away from location $B$ and arrive at their destinations simultaneously. The distance between the two locations is $\qquad$ meters.

Answer: 2800


3. (10 points) The expressway from Lishan Town to the provincial capital is 189 kilometers long, passing through the county town. The distance from Lishan Town to the county town is 54 kilometers. At 8:30 AM, a bus departs from Lishan Town to the county town, arriving at 9:15. After a 15-minute stop, it continues to the provincial capital, arriving at 11:00. Another bus departs from the provincial capital straight to Lishan Town at 8:50 AM on the same day, traveling at 60 kilometers per hour. The time when the two buses meet is . $\qquad$

Answer: 10:08


(14) A finite sequence satisfies: the sum of any 3 consecutive terms is negative, and the sum of any 4 consecutive terms is positive. The maximum number of terms in this sequence is $\qquad$ .

Answer: 5


# Problem 5. (3 points)

Given a triangle $ABC$. Points $D$ and $E$ are taken on side $AC$, and segments $BD$ and $BE$ are drawn, dividing triangle $ABC$ into three triangles, one of which is equilateral, and the other two are isosceles. Find the measure of angle $B$. It is known that the sum of the angles in a triangle is 180 degrees.

If there are multiple answers, list all possible answers separated by commas or semicolons.

Answer: 105^\circ


Let  $ABC$  be a triangle with incenter  $I$  and incircle  $\omega$ .  It is given that there exist points  $X$  and  $Y$  on the circumference of  $\omega$  such that  $\angle BXC=\angle BYC=90^\circ$ .  Suppose further that  $X$ ,  $I$ , and  $Y$  are collinear.  If  $AB=80$  and  $AC=97$ , compute the length of  $BC$ .

Answer: 59


Let  $ H$  be the piont of midpoint of the cord  $ PQ$  that is on the circle centered the origin  $ O$  with radius  $ 1.$
Suppose the length of the cord  $ PQ$  is  $ 2\sin \frac {t}{2}$  for the angle  $ t\ (0\leq t\leq \pi)$  that is formed by half-ray  $ OH$  and the positive direction of the  $ x$  axis. Answer the following questions.

(1) Express the coordiante of  $ H$  in terms of  $ t$ .

(2) When  $ t$  moves in the range of  $ 0\leq t\leq \pi$ , find the minimum value of  $ x$  coordinate of  $ H$ .

(3) When  $ t$  moves in the range of  $ 0\leq t\leq \frac {\pi}{2}$ , find the area  $ S$  of the region bounded by the curve drawn by the point  $ H$  and the  $ x$  axis and the  $ y$  axis.

Answer: \frac{\pi+2}{8}


The diagram below shows the regular hexagon  $BCEGHJ$  surrounded by the rectangle  $ADFI$ . Let  $\theta$  be the measure of the acute angle between the side  $\overline{EG}$  of the hexagon and the diagonal of the rectangle  $\overline{AF}$ . There are relatively prime positive integers  $m$  and  $n$  so that  $\sin^2\theta  = \tfrac{m}{n}$ . Find  $m + n$ .

[asy]
import graph; size(3.2cm);
real labelscalefactor = 0.5;
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps);
draw((-1,3)--(-1,2)--(-0.13,1.5)--(0.73,2)--(0.73,3)--(-0.13,3.5)--cycle);
draw((-1,3)--(-1,2));
draw((-1,2)--(-0.13,1.5));
draw((-0.13,1.5)--(0.73,2));
draw((0.73,2)--(0.73,3));
draw((0.73,3)--(-0.13,3.5));
draw((-0.13,3.5)--(-1,3));
draw((-1,3.5)--(0.73,3.5));
draw((0.73,3.5)--(0.73,1.5));
draw((-1,1.5)--(0.73,1.5));
draw((-1,3.5)--(-1,1.5));
label(" $ A $ ",(-1.4,3.9),SE*labelscalefactor);
label(" $ B $ ",(-1.4,3.28),SE*labelscalefactor);
label(" $ C $ ",(-1.4,2.29),SE*labelscalefactor);
label(" $ D $ ",(-1.4,1.45),SE*labelscalefactor);
label(" $ E $ ",(-0.3,1.4),SE*labelscalefactor);
label(" $ F $ ",(0.8,1.45),SE*labelscalefactor);
label(" $ G $ ",(0.8,2.24),SE*labelscalefactor);
label(" $ H $ ",(0.8,3.26),SE*labelscalefactor);
label(" $ I $ ",(0.8,3.9),SE*labelscalefactor);
label(" $ J $ ",(-0.25,3.9),SE*labelscalefactor); [/asy]

Answer: 55


The diameter  $\overline{AB}$  of a circle of radius  $2$  is extended to a point  $D$  outside the circle so that  $BD=3$ . Point  $E$  is chosen so that  $ED=5$  and the line  $ED$  is perpendicular to the line  $AD$ .  Segment  $\overline{AE}$  intersects the circle at point  $C$  between  $A$  and  $E$ . What is the area of  $\triangle ABC$ ? $\textbf{(A) \ } \frac{120}{37}\qquad \textbf{(B) \ } \frac{140}{39}\qquad \textbf{(C) \ } \frac{145}{39}\qquad \textbf{(D) \ } \frac{140}{37}\qquad \textbf{(E) \ } \frac{120}{31}$

Answer: \frac{140}{37}


When drawing all diagonals in a regular pentagon, one gets an smaller pentagon in the middle. What's the ratio of the areas of those pentagons?

Answer: \frac{7-3\sqrt{5}}{2}


Mark and William are playing a game with a stored value. On his turn, a player may either multiply the stored value by  $2$  and add  $1$  or he may multiply the stored value by  $4$  and add  $3$ . The first player to make the stored value exceed  $2^{100}$  wins. The stored value starts at  $1$  and Mark goes first. Assuming both players play optimally, what is the maximum number of times that William can make a move?

(By optimal play, we mean that on any turn the player selects the move which leads to the best possible outcome given that the opponent is also playing optimally. If both moves lead to the same outcome, the player selects one of them arbitrarily.)

Answer: 33


the positive divisors  $d_1,d_2,\cdots,d_k$  of a positive integer  $n$  are ordered
\[1=d_1<d_2<\cdots<d_k=n\]
Suppose  $d_7^2+d_{15}^2=d_{16}^2$ . Find all possible values of  $d_{17}$ .

Answer: 28


Consider  $7$ -gons inscribed in a circle such that all sides of the  $7$ -gon are of different length. Determine the maximal number of  $120^\circ$  angles in this kind of a  $7$ -gon.

Answer: 2


Let  $S$  be a nonempty closed bounded convex set in the plane. Let  $K$  be a line and  $t$  a positive number. Let  $L_1$  and  $L_2$  be support lines for  $S$  parallel to  $K_1$ , and let  $ \overline {L} $  be the line parallel to  $K$  and midway between  $L_1$  and  $L_2$ . Let  $B_S(K,t)$  be the band of points whose distance from  $\overline{L}$  is at most  $ \left( \frac {t}{2} \right) w $ , where  $w$  is the distance between  $L_1$  and  $L_2$ . What is the smallest  $t$  such that \[ S \cap \bigcap_K B_S (K, t) \ne \emptyset \]for all  $S$ ? ( $K$  runs over all lines in the plane.)

Answer: \frac{1}{3}


Four siblings are sitting down to eat some mashed potatoes for lunch: Ethan has 1 ounce of mashed potatoes, Macey has 2 ounces, Liana has 4 ounces, and Samuel has 8 ounces. This is not fair. A blend consists of choosing any two children at random, combining their plates of mashed potatoes, and then giving each of those two children half of the combination. After the children's father performs four blends consecutively, what is the probability that the four children will all have the same amount of mashed potatoes?

Answer: \frac{1}{54}


In the figure below , $4$  of the  $6$  disks are to be colored black and  $2$  are to be colored white. Two colorings that can be obtained from one another by rotation or reflection of the entire figure are considered the same. ![Image](https://i.imgur.com/57nQwBI.jpg)

There are only four such colorings for the given  two colors, as shown.

In how many ways can we color the  $6$  disks such that  $2$  are colored black ,  $2$  are colored white,  $2$  are colored blue with the given identification condition?

Answer: 18


**p1.** Let  $S$  be the sum of the  $99$  terms:   $$ (\sqrt1 + \sqrt2)^{-1},(\sqrt2 + \sqrt3)^{-1}, (\sqrt3 + \sqrt4)^{-1},..., (\sqrt{99} + \sqrt{100})^{-1}. $$   Prove that  $S$  is an integer.**p2.** Determine all pairs of positive integers  $x$  and  $y$  for which  $N=x^4+4y^4$  is a prime. (Your work should indicate why no other solutions are possible.)**p3.** Let  $w,x,y,z$  be arbitrary positive real numbers. Prove each inequality:

(a)  $xy \le \left(\frac{x+y}{2}\right)^2$ (b)  $wxyz \le \left(\frac{w+x+y+z}{4}\right)^4$ (c)  $xyz \le \left(\frac{x+y+z}{3}\right)^3$ **p4.** Twelve points  $P_1$ , $P_2$ ,  $...$ , $P_{12}$  are equally spaaed on a circle, as shown. Prove: that the chords  $\overline{P_1P_9}$ ,  $\overline{P_4P_{12}}$  and  $\overline{P_2P_{11}}$  have a point in common.
![Image](https://cdn.artofproblemsolving.com/attachments/d/4/2eb343fd1f9238ebcc6137f7c84a5f621eb277.png)**p5.** Two very busy men,  $A$  and  $B$ , who wish to confer, agree to appear at a designated place on a certain day, but no earlier than noon and no later than  $12:15$  p.m. If necessary,  $A$  will wait  $6$  minutes for  $B$  to arrive, while  $B$  will wait  $9$  minutes for  $A$  to arrive but neither can stay past  $12:15$  p.m. Express as a percent their chance of meeting.

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

Answer: 74


There's a tape with  $n^2$  cells labeled by  $1,2,\ldots,n^2$ . Suppose that  $x,y$  are two distinct positive integers less than or equal to  $n$ . We want to color the cells of the tape such that any two cells with label difference of  $x$  or  $y$  have different colors. Find the minimum number of colors needed to do so.

Answer: 3


Let  $E$  and  $F$  be the respective midpoints of  $BC,CD$  of a convex quadrilateral  $ABCD$ . Segments  $AE,AF,EF$  cut the quadrilateral into four triangles whose areas are four consecutive integers. Find the maximum possible area of  $\Delta BAD$ .

Answer: 6


Let  $C$  and  $D$  be points on the semicircle with center  $O$  and diameter  $AB$  such that  $ABCD$  is a convex quadrilateral. Let  $Q$  be the intersection of the diagonals  $[AC]$  and  $[BD]$ , and  $P$  be the intersection of the lines tangent to the semicircle at  $C$  and  $D$ . If  $m(\widehat{AQB})=2m(\widehat{COD})$  and  $|AB|=2$ , then what is  $|PO|$ ? $
\textbf{(A)}\ \sqrt 2
\qquad\textbf{(B)}\ \sqrt 3
\qquad\textbf{(C)}\ \frac{1+\sqrt 3} 2
\qquad\textbf{(D)}\ \frac{1+\sqrt 3}{2\sqrt 2}
\qquad\textbf{(E)}\ \frac{2\sqrt 3} 3
$

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


The sum  $\frac{2}{3\cdot 6} +\frac{2\cdot 5}{3\cdot 6\cdot 9} +\ldots +\frac{2\cdot5\cdot \ldots \cdot 2015}{3\cdot 6\cdot 9\cdot \ldots \cdot 2019}$  is written as a decimal number. Find the first digit after the decimal point.

Answer: 6


In a convex quadrilateral  $ABCD$  we are given that  $\angle CAD=10^\circ$ ,  $\angle DBC=20^\circ$ ,  $\angle BAD=40^\circ$ ,  $\angle ABC=50^\circ$ . Compute angle  $BDC$ .

Answer: 40


There are  $\displaystyle{2n}$  students in a school  $\displaystyle{\left( {n \in {\Bbb N},n \geqslant 2} \right)}$ . Each week  $\displaystyle{n}$  students go on a trip. After several trips the following condition was fulfiled: every two students were together on at least one trip. What is the minimum number of trips needed for this to happen?

*Proposed by Oleksandr Rybak, Kiev, Ukraine.*

Answer: 6


3. Given a sequence of positive integers $a_{1}, a_{2}, \cdots, a_{n}, \cdots$, where $a_{1}=2^{2006}$, and for each positive integer $i, a_{i+1}$ is equal to the square of the sum of the digits of $a_{i}$. Find the value of $a_{2006}$.

Answer: 169


9. As shown in Figure 2, in quadrilateral $ABCD$, $AB=BC=CD$, $\angle ABC=78^{\circ}$, $\angle BCD=162^{\circ}$. Let the intersection point of line $AD$ and $BC$ be $E$. Then the size of $\angle AEB$ is

Answer: 21^\circ


Theorem 8: For a function \( y=f(x) \) defined on domain \( D \), if for any \( x \in D \), \( f(a+x) + f(a-x) = 2b \) holds, then the graph of the function \( y=f(x) \) is symmetric about the point \( (a, b) \).

(1) Prove that the graph of the function \( f_{1}(x)=2^{x}-2^{-x} \) is centrally symmetric, while the graph of the function \( f_{2}(x)=x^2 \) is not centrally symmetric.

(2) If the graphs of the functions \( y=f(x) \) and \( y=g(x) \) defined on domain \( \mathbb{R} \) are symmetrical about the point \( (a, b) \), then choose one function among \( y=f(x)+g(x), y=f(x)-g(x), y=f(x)g(x), \) and \( y=\frac{f(x)}{g(x)} \) whose graph is necessarily centrally symmetric, and choose one function whose graph can be not centrally symmetric. Provide reasoning for why the graph is or is not centrally symmetric.

Answer: y = f(x) + g(x)


5. From the odd natural numbers from 1 to 47, 12 fractions less than 1 were formed, using each number exactly once. The resulting fractions were divided into groups of equal values. What is the smallest number of groups that could have been obtained? (I. Rubanov)

Answer: 7


6. The following is a schematic diagram of a computer shooting game. The lengths of line segments $\mathrm{CD}$, $\mathrm{EF}$, and $\mathrm{GH}$ are all 20 centimeters. $0$, $P$, and $Q$ are their midpoints and are located on the same line $A B$. $A 0=45$ centimeters, $\mathrm{OP}=\mathrm{PQ}=20$ centimeters. It is known that the speed of the small ring on $\mathrm{CD}$ is 5 centimeters per second, the speed of the small ring on $\mathrm{EF}$ is 9 centimeters per second, and the speed of the small ring on $\mathrm{GH}$ is 27 centimeters per second. At zero time, a small ring starts from the left endpoint of each of $\mathrm{CD}$, $\mathrm{EF}$, and $\mathrm{GH}$ and moves back and forth uniformly along the line segments. Now, a bullet is fired from point $\mathrm{A}$ toward $\mathrm{B}$ at a uniform speed. To pass through the three rings, what is the maximum speed of the bullet in centimeters per second?

Answer: \dfrac{9}{2}


3. In $\triangle A B C$, $3 B C=A B+C A, T$ is a point on side $A C$ such that $4 A T=A C$, points $K$ and $L$ are on sides $A B$ and $C A$ respectively, satisfying $K L / / B C$, and line $K L$ is tangent to the incircle of $\triangle A B C$. Find the ratio of the two segments into which $B T$ divides $K L$.

Answer: \dfrac{1}{2}


12.197. In triangle $ABC$, the altitude $BM$ is drawn, and a circle is constructed on it as a diameter, intersecting side $AB$ at point $K$ and side $BC$ at point $Z$. Find the ratio of the area of triangle $K Z M$ to the area of triangle $A B C$, if $\angle A=\alpha$ and $\angle C=\beta$.

Answer: \dfrac{1}{4} \sin 2\alpha \sin 2\beta


## Task 23/89

A sequence is given by the formation rule

$$
a_{k}=p \cdot k \cdot(k+1)+1
$$

where $p$ is a prime number and $k>0$. The 7th term is the square of a prime number $P$. Determine all possible pairs $(p ; P)$.

---

Answer: (3, 13)


Problem 3. In a right triangle $ABC$ with a right angle at $C$, the bisector $BD$ and the altitude $CH$ are drawn. A perpendicular $CK$ is dropped from vertex $C$ to the bisector $BD$. Find the angle $HCK$, if $BK: KD=3: 1$.

Answer: 30^\circ


Someone buys 100 cigars for 1 forint. The cigars include half-krajczár cigars, 3-krajczár cigars, and 10-krajczár cigars. How many of each type of cigar can they buy?

Answer: (94, 1, 5)


Question 5 In a $12 \times 9$ rectangular grid, $k$ cells' centers are colored red, such that no 4 red points form the 4 vertices of a rectangle with sides parallel to the grid lines. Find the maximum value of $k$.

In a $12 \times 9$ rectangular grid, $k$ cells' centers are colored red, such that no 4 red points form the 4 vertices of a rectangle with sides parallel to the grid lines. Find the maximum value of $k$.

Answer: 36


Problem 4. On a board, eight numbers equal to 0 are written, one after another. We call an operation the modification of four of the eight numbers as follows: two numbers increase by 3, one number increases by 2, and the fourth number increases by 1.

a) What is the minimum number of operations we need to perform to obtain eight consecutive natural numbers on the board.

b) Is it possible that, after a number of operations, all the numbers written on the board are equal to 2015?

c) Is it possible that, after a sequence of operations, the product of the numbers on the board is 2145?

Answer: 4


[ Arithmetic progression
[ Divisibility of numbers. General properties ]
[ Examples and counterexamples. Constructions ]

Find a finite arithmetic progression with a common difference of 6 and of maximum length, consisting of prime numbers.

Answer: 5


Problem 8.8. Given a triangle $A B C$, in which $\angle A=42^{\circ}$ and $A B<A C$. Point $K$ on side $A C$ is such that $A B=C K$. Points $P$ and $Q$ are the midpoints of segments $A K$ and $B C$ respectively. How many degrees does angle $A C B$ measure, if it is known that $\angle P Q C=110^{\circ} ?$

![](https://cdn.mathpix.com/cropped/2024_05_06_191a5994d6b917032c66g-06.jpg?height=339&width=551&top_left_y=95&top_left_x=451)

Answer: 49


3. A natural number will be called magical if and only if it can be expressed as the sum of two three-digit numbers written with the same digits but in reverse order. For example, the number 1413 is magical because $1413=756+657$; the smallest magical number is 202.

a) Determine the number of all magical numbers.

b) Show that the sum of all magical numbers is equal to 187000.

Answer: 170


Let \( f(x) \) be an odd function defined on \(\mathbf{R}\) and satisfying \( f(x+2) = -f(x) \). Additionally, for \(0 \leqslant x \leqslant 1\), \(f(x) = \frac{1}{2} x\). Determine the set \(\left\{ x \, \left| \, f(x) = -\frac{1}{2} \right. \right\}\).

Answer: \left\{ x \mid x = 4k - 1, \, k \in \mathbb{Z} \right\}


13.140. On one of two machines, a batch of parts is processed 3 days longer than on the other. How many days would it take for each machine to process this batch of parts separately, if it is known that when working together, these machines processed a batch of parts three times larger in 20 days?

Answer: 12


4.1. In an $11 \times 11$ square, the central cell is painted black. Maxim found a rectangular grid of the largest area that is entirely within the square and does not contain the black cell. How many cells does it have?

Answer: 55


9. Consider a triangle $A B C$ and a point $D$ on the side $A C$. If $\overline{A B}=\overline{D C}=1, \angle D B C=30^{\circ}$ and $\angle A B D=90^{\circ}$, calculate the value of $\overline{A D}$.

Answer: \sqrt[3]{2}


Let $f(x)$ be a quadratic function with integer coefficients. If we know that $f(0)$, $f(3)$ and $f(4)$ are all different and elements of the set $\{2, 20, 202, 2022\}$, determine all possible values of $f(1)$.

Answer: -990


1. Given $P A 、 P B 、 P C$ are three non-coplanar rays emanating from point $P$, and the angle between any two rays is $60^{\circ}$. There is a sphere with a radius of 1 that is tangent to all three rays. Find the distance from the center of the sphere $O$ to point $P$.

Answer: \sqrt{3}


Anna, Berta, Vika, Galya, and Dasha (in no particular order) are waiting for the moment when a rare flower blooms in their greenhouse. Each of the girls took a turn observing the flower, with Anna watching for twice as long as Berta, Berta watching twice as long as Vika, and Galya, Dasha, and Vika watching the flower for the same amount of time. The scientific supervisor recorded the girls strictly at the beginning of each hour on the hour, and the changes of the girls did not occur on the hour. Could the scientific supervisor have recorded each girl exactly once?

Answer: \text{No}


As shown in the figure, the vertices of the shaded square are the midpoints of the sides of the larger square $\mathrm{EFGH}$. Semi-circles are drawn outward with half the sides of the larger square as diameters, and then semi-circles are drawn outward with the sides of the shaded square as diameters, forming 8 "crescent shapes". The total area of these 8 "crescent shapes" is 5 square centimeters. What is the area of the larger square EFGH in square centimeters?

Answer: 10


## Task 2 - 201242

In a fish shop, there are three water containers available for keeping live carp.

At the start of sales, there are exactly 20 carp in each of these three containers. At the end of sales, there are a total of 3 carp left. The sold carp were removed one by one.

No exchange of carp between containers took place; no new carp were added during the sales.

Calculate, rounded to 3 decimal places, the probability that at the end of sales, there is exactly one carp in each of the three containers!

Hint:

The probability $p$ to be determined is defined as follows: Let $A$ be the number of all different possibilities for the sequence of removing 57 carp from the three containers. Furthermore, let $G$ be the number of those possibilities among these where at the end of sales, there is exactly one carp in each of the three containers.

Two possible sequences of removal are considered the same if and only if they agree for each $i=1,2, \ldots, 57$ in specifying from which container the $i$-th removal of a carp took place.

With these notations, $p=\frac{G}{A}$.

Answer: 0.108


Ali has three containers: a 4-liter, a 7-liter, and a 10-liter container. The 10-liter container is full of wine, the other two are empty. Someone wants to buy 5 liters of wine from him. Ali wants to measure out the wine, but only his own containers are available. How can he measure out the wine?

And how can he measure out the wine if he has an 8-liter container instead of a 7-liter container?

Answer: 5


10.389 In an isosceles triangle $A B C \quad A B=B C=25$ cm and $A C=14$ cm. Calculate the radius of the circle that touches $B C$ at point $D$, the foot of the altitude $A D$, and passes through the midpoint of $A C$.

Answer: \dfrac{175}{48}


Example 8 One evening, 21 people made $n$ phone calls to each other. It is known that among them, there are $m$ ($m$ is an odd number) people $a_{1}, a_{2}$, $\cdots, a_{m}$ such that $a_{i}$ and $a_{i+1}\left(i=1,2, \cdots, m ; a_{m+1}=a_{1}\right)$ made phone calls. If no three people among these 21 people made phone calls to each other, find the maximum value of $n$.

【Analysis】Construct a graph theory model to characterize the relationships between the elements in the problem. From the conditions given in the problem, we know that there exists an odd cycle in the graph. Since the number of points is finite, by the extreme principle, we know that there must exist the shortest odd cycle $A_{1} A_{2} \cdots A_{2 k+1} A_{1}$. Furthermore, in the shortest odd cycle, any two points $A_{i}$ and $A_{j}(j \neq i \pm 1)$ must not be connected, thus the internal structure of the odd cycle is fully understood. At this point, starting from the shortest odd cycle $C$, the entire graph can be divided into two parts: one part is the odd cycle $A_{1} A_{2} \cdots A_{2 k+1} A_{1}$, and the other part is all the points except $A_{1} A_{2} \cdots A_{2 k+1} A_{1}$. We only need to understand the structure of each part and the connection situation between the two parts to grasp the overall structure.

Answer: 101


In a race from Sparrow Hills to Red Square, three athletes participated. First, Grisha started, then Sasha, and lastly Lena. After the finish, it turned out that during the race Grisha overtook others 10 times, Lena 6 times, and Sasha 4 times, and all three never found themselves at the same point at the same time. In what order did the athletes finish, given that they finished at different times?

Answer: Grisha, Sasha, Lena


1. In the expression $2: 2: 2: 2: 2$ place parentheses so that the result is equal to 2. Find two different ways!

Answer: 2/(2/(2/(2/2)))


Consider the $4\times4$ array of $16$ dots, shown below.

[asy]
size(2cm);
dot((0,0));
dot((1,0));
dot((2,0));
dot((3,0));
dot((0,1));
dot((1,1));
dot((2,1));
dot((3,1));
dot((0,2));
dot((1,2));
dot((2,2));
dot((3,2));
dot((0,3));
dot((1,3));
dot((2,3));
dot((3,3));
[/asy]

Counting the number of squares whose vertices are among the $16$ dots and whose sides are parallel to the sides of the grid, we find that there are nine $1\times1$ squares, four $2\times2$ squares, and one $3\times3$ square, for a total of $14$ squares. We delete a number of these dots. What is the minimum number of dots that must be deleted so that each of the $14$ squares is missing at least one vertex?

Answer: 4


While staying in a 15-story hotel, Polya plays the following game. She enters an elevator on the $6^{\mathrm{th}}$ floor. She flips a fair coin five times to determine her next five stops. Each time she flips heads, she goes up one floor. Each time she flips tails, she goes down one floor. What is the probability that each of her next five stops is on the $7^{\mathrm{th}}$ floor or higher? Express your answer as a common fraction.The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 19


Let $ x$ be a real number such that the five numbers $ \cos(2 \pi x)$, $ \cos(4 \pi x)$, $ \cos(8 \pi x)$, $ \cos(16 \pi x)$, and $ \cos(32 \pi x)$ are all nonpositive.  What is the smallest possible positive value of $ x$?The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 85


Let $z_1,$ $z_2,$ $z_3,$ and $z_4$ be the four distinct complex solutions of the equation
\[
  z^4 - 6z^2 + 8z + 1 = -4(z^3 - z + 2)i.
\]Find the sum of the six pairwise distances between $z_1,$ $z_2,$ $z_3,$ and $z_4$ in the complex plane.The answer is in the form k\sqrt{m}+n,. Please provide the value of k + m + n.

Answer: 15


If $\sin x + \sin y = \frac{96}{65}$ and $\cos x + \cos y = \frac{72}{65}$, then what is the value of $\tan x + \tan y$?The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 619


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Arnold`, `Alice`, `Peter`
 - People have unique favorite sports: `swimming`, `soccer`, `tennis`, `basketball`
 - Each person has a unique type of pet: `bird`, `fish`, `cat`, `dog`
 - Everyone has a favorite smoothie: `dragonfruit`, `desert`, `watermelon`, `cherry`
 - People have unique hair colors: `brown`, `black`, `blonde`, `red`
 - Each person has a unique birthday month: `jan`, `feb`, `sept`, `april`

## Clues:
1. The person whose birthday is in September is the person who likes Cherry smoothies.
2. The person whose birthday is in February is not in the first house.
3. The Desert smoothie lover is the person who loves basketball.
4. Peter is the person who has black hair.
5. The person who loves tennis is somewhere to the right of the person who has a cat.
6. The person with an aquarium of fish is not in the fourth house.
7. Arnold is directly left of the person who has blonde hair.
8. There is one house between the person who has red hair and the person whose birthday is in January.
9. The person whose birthday is in September is directly left of the person who loves soccer.
10. Peter is directly left of the person who loves basketball.
11. The person who has blonde hair is the person whose birthday is in September.
12. Arnold is the person who keeps a pet bird.
13. The person whose birthday is in January is the Watermelon smoothie lover.
14. The person who has blonde hair is Eric.
 What is the value of attribute House for the person whose attribute Smoothie is desert? 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: `Carol`, `Bob`, `Arnold`, `Alice`, `Peter`, `Eric`
 - The people are of nationalities: `swede`, `german`, `dane`, `brit`, `chinese`, `norwegian`
 - Each person has a unique type of pet: `bird`, `cat`, `dog`, `rabbit`, `fish`, `hamster`
 - Each person has a favorite color: `red`, `green`, `purple`, `yellow`, `white`, `blue`

## Clues:
1. The person who has a cat is the British person.
2. The person with a pet hamster is in the fourth house.
3. The person who owns a rabbit is the person who loves purple.
4. Arnold is the person who loves white.
5. The Dane is somewhere to the left of the person who owns a dog.
6. Carol and Peter are next to each other.
7. The Swedish person is somewhere to the right of the person whose favorite color is green.
8. The person who loves blue is the person with a pet hamster.
9. There are two houses between the person whose favorite color is red and Bob.
10. The person who keeps a pet bird is in the second house.
11. Carol is in the first house.
12. The person who loves yellow and the German are next to each other.
13. Alice is the person who loves yellow.
14. The Dane is the person who loves yellow.
15. Bob is the Norwegian.
16. Alice is directly left of the person who loves white.
 What is the value of attribute Nationality for the person whose attribute Name is Bob? Please reason step by step, and put your final answer within \boxed{}

Answer: norwegian


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`, `Alice`, `Eric`
 - People have unique heights: `tall`, `short`, `very short`, `average`
 - Each person has a unique birthday month: `april`, `jan`, `sept`, `feb`
 - People have unique hair colors: `red`, `black`, `brown`, `blonde`
 - People have unique favorite book genres: `science fiction`, `fantasy`, `romance`, `mystery`

## Clues:
1. Eric is the person who has blonde hair.
2. The person who is short is somewhere to the left of the person who has brown hair.
3. Eric is the person who is tall.
4. Arnold is the person who loves fantasy books.
5. The person who loves science fiction books is Eric.
6. The person who is very short is the person who has black hair.
7. The person whose birthday is in April is the person who has brown hair.
8. The person who has blonde hair and the person who has black hair are next to each other.
9. The person who has black hair is the person who loves fantasy books.
10. The person who has red hair is in the first house.
11. The person whose birthday is in September is in the second house.
12. The person whose birthday is in January is the person who has black hair.
13. The person who loves romance books is somewhere to the right of Alice.
 What is the value of attribute HairColor for the person whose attribute Name is Alice? Please reason step by step, and put your final answer within \boxed{}

Answer: red


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Peter`, `Eric`, `Bob`, `Carol`, `Arnold`
 - People have unique favorite sports: `baseball`, `basketball`, `swimming`, `tennis`, `soccer`, `volleyball`
 - Each person has a favorite color: `purple`, `green`, `white`, `red`, `yellow`, `blue`
 - People have unique hair colors: `red`, `auburn`, `blonde`, `black`, `gray`, `brown`
 - Everyone has something unique for lunch: `stew`, `spaghetti`, `stir fry`, `grilled cheese`, `soup`, `pizza`

## Clues:
1. Arnold is somewhere to the left of the person who loves blue.
2. There are two houses between the person whose favorite color is green and Bob.
3. The person who loves white is in the fifth house.
4. The person who loves baseball is directly left of the person who has brown hair.
5. The person who loves tennis is not in the fifth house.
6. The person who has auburn hair is the person who loves the stew.
7. The person whose favorite color is red and the person who loves baseball are next to each other.
8. The person who is a pizza lover is Eric.
9. The person who loves purple is Carol.
10. Peter is in the third house.
11. Carol is the person who loves soccer.
12. The person who loves volleyball is somewhere to the left of the person who has auburn hair.
13. Eric is in the first house.
14. There is one house between the person who has gray hair and the person who loves blue.
15. The person who loves volleyball is directly left of the person who loves baseball.
16. The person who loves the spaghetti eater is in the fifth house.
17. The person who loves eating grilled cheese is the person who has brown hair.
18. The person who loves the soup is in the third house.
19. The person who has blonde hair is somewhere to the left of the person who has gray hair.
20. The person who has red hair is the person who loves white.
21. The person who loves blue is somewhere to the right of the person who loves swimming.
 What is the value of attribute FavoriteSport for the person whose attribute Color is purple? Please reason step by step, and put your final answer within \boxed{}

Answer: soccer


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Eric`, `Peter`, `Bob`, `Arnold`
 - Each person has a favorite color: `red`, `white`, `green`, `blue`, `yellow`
 - Everyone has a favorite smoothie: `watermelon`, `dragonfruit`, `cherry`, `desert`, `lime`
 - Each person has a unique birthday month: `feb`, `sept`, `april`, `mar`, `jan`
 - People have unique favorite book genres: `romance`, `science fiction`, `mystery`, `fantasy`, `biography`
 - People have unique heights: `very short`, `very tall`, `tall`, `short`, `average`

## Clues:
1. The person who drinks Lime smoothies is in the second house.
2. The person who loves fantasy books is directly left of the Dragonfruit smoothie lover.
3. The person whose birthday is in September is not in the third house.
4. There is one house between the person whose birthday is in March and Peter.
5. The person whose birthday is in February is the Desert smoothie lover.
6. The person who is tall is not in the fifth house.
7. The person whose birthday is in January is somewhere to the left of the person whose birthday is in February.
8. The person who is very short is the person who loves blue.
9. The person whose favorite color is red is Arnold.
10. The person who loves science fiction books is directly left of the person who loves biography books.
11. Bob is not in the fifth house.
12. The person whose birthday is in January is the person whose favorite color is green.
13. The person who has an average height is the person who likes Cherry smoothies.
14. The person who loves fantasy books is the person whose birthday is in February.
15. The person whose favorite color is red is directly left of the person who is very short.
16. The person who loves science fiction books is the person whose birthday is in March.
17. The person who is very tall is in the second house.
18. The person who drinks Lime smoothies is somewhere to the left of the person who loves mystery books.
19. Eric is in the third house.
20. The person who loves fantasy books is the person who loves white.
 What is the value of attribute Birthday for the person whose attribute House is 1? Please reason step by step, and put your final answer within \boxed{}

Answer: jan


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`, `Eric`, `Carol`, `Bob`, `Alice`, `Peter`
 - People have unique heights: `short`, `very short`, `very tall`, `tall`, `average`, `super tall`
 - Each person has a unique type of pet: `rabbit`, `bird`, `cat`, `fish`, `hamster`, `dog`
 - People use unique phone models: `huawei p50`, `google pixel 6`, `iphone 13`, `samsung galaxy s21`, `xiaomi mi 11`, `oneplus 9`

## Clues:
1. The person who uses a Google Pixel 6 is somewhere to the right of the person who is very short.
2. The person who has a cat is the person who uses a Google Pixel 6.
3. The person with a pet hamster is Arnold.
4. Bob is the person who has a cat.
5. There is one house between Eric and the person who is tall.
6. The person who owns a rabbit is the person who is super tall.
7. The person who is very tall is Alice.
8. The person who is very tall is somewhere to the right of the person who has an average height.
9. There is one house between the person who is super tall and Carol.
10. The person who owns a dog is not in the sixth house.
11. The person who is short is the person who uses a Samsung Galaxy S21.
12. The person who is tall and the person who uses a Huawei P50 are next to each other.
13. The person who uses a OnePlus 9 is not in the fourth house.
14. The person who uses a Xiaomi Mi 11 is somewhere to the left of the person who uses an iPhone 13.
15. The person who is tall is not in the fourth house.
16. Peter is the person who uses a OnePlus 9.
17. The person who uses a Samsung Galaxy S21 is directly left of Peter.
18. Peter is directly left of the person who is tall.
19. Carol is the person with an aquarium of fish.
 What is the value of attribute PhoneModel for the person whose attribute Name is Arnold? Please reason step by step, and put your final answer within \boxed{}

Answer: xiaomi mi 11


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`, `Alice`, `Eric`, `Peter`, `Arnold`, `Bob`
 - The mothers' names in different houses are unique: `Penny`, `Janelle`, `Aniya`, `Sarah`, `Holly`, `Kailyn`
 - Everyone has a unique favorite cigar: `blends`, `dunhill`, `blue master`, `prince`, `yellow monster`, `pall mall`
 - Each person has a unique type of pet: `cat`, `dog`, `rabbit`, `hamster`, `bird`, `fish`
 - The people keep unique animals: `bird`, `horse`, `cat`, `rabbit`, `dog`, `fish`
 - Everyone has a favorite smoothie: `desert`, `watermelon`, `lime`, `cherry`, `blueberry`, `dragonfruit`

## Clues:
1. The dog owner is not in the third house.
2. The fish enthusiast is not in the fourth house.
3. The person who owns a dog is Eric.
4. The cat lover is not in the second house.
5. The person who smokes Blue Master is somewhere to the right of The person whose mother's name is Sarah.
6. The cat lover is not in the fourth house.
7. There are two houses between Carol and the person who owns a rabbit.
8. The person whose mother's name is Penny is the person with a pet hamster.
9. The person whose mother's name is Kailyn is somewhere to the right of the person with a pet hamster.
10. The dog owner and the person who drinks Lime smoothies are next to each other.
11. The Prince smoker is somewhere to the right of The person whose mother's name is Janelle.
12. There is one house between The person whose mother's name is Penny and Bob.
13. There are two houses between the person partial to Pall Mall and The person whose mother's name is Penny.
14. Peter is somewhere to the left of the person who drinks Lime smoothies.
15. The bird keeper is the Watermelon smoothie lover.
16. The person with a pet hamster is the Prince smoker.
17. The person who smokes Yellow Monster is the person who drinks Lime smoothies.
18. The person whose mother's name is Sarah is the person who smokes Yellow Monster.
19. The Dunhill smoker is somewhere to the left of the Watermelon smoothie lover.
20. The person whose mother's name is Holly is directly left of the person who owns a dog.
21. The bird keeper is The person whose mother's name is Kailyn.
22. The person whose mother's name is Sarah is not in the third house.
23. The person who smokes many unique blends is the rabbit owner.
24. The person who has a cat is somewhere to the right of the person with an aquarium of fish.
25. The person who smokes Yellow Monster is Eric.
26. The person who keeps a pet bird is the Desert smoothie lover.
27. Alice is The person whose mother's name is Kailyn.
28. The Dragonfruit smoothie lover is Carol.
29. The person who keeps horses is the person who likes Cherry smoothies.
 What is the value of attribute Animal for the person whose attribute House is 2? Please reason step by step, and put your final answer within \boxed{}

Answer: fish


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Peter`, `Carol`, `Eric`, `Arnold`, `Bob`
 - People have unique heights: `very tall`, `very short`, `average`, `tall`, `short`, `super tall`

## Clues:
1. Eric is in the second house.
2. Arnold is the person who is short.
3. Alice is somewhere to the left of the person who has an average height.
4. The person who is tall is somewhere to the right of Peter.
5. Arnold is not in the fourth house.
6. The person who is short is somewhere to the right of Eric.
7. There is one house between Peter and Bob.
8. Arnold is somewhere to the left of the person who is very tall.
9. The person who has an average height is Carol.
10. The person who is very short is Eric.
 What is the value of attribute House for the person whose attribute Height is short? Please reason step by step, and put your final answer within \boxed{}

Answer: 3


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Peter`, `Arnold`, `Eric`, `Alice`
 - Everyone has something unique for lunch: `stew`, `stir fry`, `spaghetti`, `grilled cheese`, `pizza`
 - Everyone has a favorite smoothie: `watermelon`, `lime`, `desert`, `dragonfruit`, `cherry`
 - People have unique favorite book genres: `fantasy`, `biography`, `mystery`, `science fiction`, `romance`
 - Each person has a unique hobby: `photography`, `knitting`, `cooking`, `gardening`, `painting`

## Clues:
1. The person who loves stir fry is the Desert smoothie lover.
2. The person who loves the spaghetti eater is the person who enjoys gardening.
3. Bob is somewhere to the left of the person who loves mystery books.
4. The person who paints as a hobby is directly left of Alice.
5. The Desert smoothie lover is the person who loves romance books.
6. The photography enthusiast and the person who loves eating grilled cheese are next to each other.
7. The person who loves cooking is somewhere to the left of the Desert smoothie lover.
8. The person who likes Cherry smoothies is Alice.
9. There is one house between the person who drinks Lime smoothies and the person who enjoys gardening.
10. The Watermelon smoothie lover is Eric.
11. The person who loves eating grilled cheese is directly left of the Dragonfruit smoothie lover.
12. The person who is a pizza lover is Alice.
13. The person who loves romance books is Peter.
14. The person who loves fantasy books is the Watermelon smoothie lover.
15. Alice is the person who loves biography books.
16. Eric is not in the third house.
 What is the value of attribute Hobby for the person whose attribute BookGenre is romance? Please reason step by step, and put your final answer within \boxed{}

Answer: painting


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Eric`, `Alice`, `Arnold`
 - People have unique favorite sports: `swimming`, `tennis`, `basketball`, `soccer`
 - Each person prefers a unique type of vacation: `cruise`, `beach`, `mountain`, `city`
 - Everyone has something unique for lunch: `grilled cheese`, `spaghetti`, `stew`, `pizza`

## Clues:
1. The person who loves the spaghetti eater is somewhere to the left of the person who likes going on cruises.
2. The person who loves the stew is directly left of the person who loves eating grilled cheese.
3. The person who is a pizza lover is in the fourth house.
4. Peter is somewhere to the left of the person who enjoys mountain retreats.
5. The person who likes going on cruises is in the third house.
6. Arnold is the person who loves soccer.
7. The person who loves basketball is not in the first house.
8. There are two houses between the person who prefers city breaks and Eric.
9. The person who loves tennis is the person who likes going on cruises.
10. Alice is in the first house.
 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 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Peter`, `Alice`, `Eric`
 - Each person has a unique type of pet: `bird`, `fish`, `dog`, `cat`
 - Each person lives in a unique style of house: `colonial`, `craftsman`, `ranch`, `victorian`
 - The mothers' names in different houses are unique: `Holly`, `Janelle`, `Kailyn`, `Aniya`
 - Each mother is accompanied by their child: `Fred`, `Samantha`, `Meredith`, `Bella`

## Clues:
1. The person whose mother's name is Kailyn is in the second house.
2. The person living in a colonial-style house is the person's child is named Meredith.
3. There is one house between the person's child is named Samantha and The person whose mother's name is Janelle.
4. The person in a ranch-style home is the person's child is named Fred.
5. Eric is the person's child is named Bella.
6. The person whose mother's name is Janelle is the person with an aquarium of fish.
7. Arnold is the person in a Craftsman-style house.
8. The person who keeps a pet bird and the person with an aquarium of fish are next to each other.
9. The person whose mother's name is Holly is directly left of Peter.
10. The person who has a cat is the person living in a colonial-style house.
11. The person's child is named Fred is somewhere to the left of The person whose mother's name is Janelle.
 What is the value of attribute House for the person whose attribute Name is Alice? Please reason step by step, and put your final answer within \boxed{}

Answer: 4


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Peter`, `Arnold`, `Eric`
 - People have unique heights: `short`, `very short`, `tall`, `average`
 - Each person has a unique type of pet: `fish`, `cat`, `dog`, `bird`
 - Everyone has something unique for lunch: `pizza`, `spaghetti`, `stew`, `grilled cheese`

## Clues:
1. The person who loves the stew is in the second house.
2. The person who is very short is somewhere to the right of Eric.
3. Arnold is the person who is tall.
4. The person with an aquarium of fish is directly left of Arnold.
5. Peter is somewhere to the left of the person who is short.
6. The person who loves eating grilled cheese is directly left of the person who is a pizza lover.
7. The person who has a cat is Arnold.
8. The person with an aquarium of fish is the person who has an average height.
9. Peter is the person who keeps a pet bird.
 What is the value of attribute House for the person whose attribute Pet is fish? 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 Mother for the person whose attribute Drink is water? Please reason step by step, and put your final answer within \boxed{}

Answer: Kailyn


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Peter`, `Arnold`, `Alice`
 - Everyone has a favorite smoothie: `dragonfruit`, `cherry`, `desert`, `watermelon`
 - Everyone has a unique favorite cigar: `blue master`, `pall mall`, `dunhill`, `prince`
 - People have unique heights: `tall`, `average`, `short`, `very short`
 - People use unique phone models: `google pixel 6`, `samsung galaxy s21`, `iphone 13`, `oneplus 9`

## Clues:
1. The Dragonfruit smoothie lover is Eric.
2. The Dunhill smoker is the person who likes Cherry smoothies.
3. The person who uses a Samsung Galaxy S21 is directly left of the person who uses an iPhone 13.
4. The Dunhill smoker is somewhere to the right of the person who is very short.
5. The Watermelon smoothie lover is somewhere to the right of the Desert smoothie lover.
6. The Prince smoker is the person who uses a OnePlus 9.
7. The person who is tall is in the third house.
8. The person who is very short is the person who uses an iPhone 13.
9. The person who smokes Blue Master is not in the first house.
10. The Dunhill smoker is the person who is short.
11. Peter is not in the third house.
12. Arnold is the person who uses a Google Pixel 6.
13. The Dragonfruit smoothie lover is the person partial to Pall Mall.
 What is the value of attribute Height for the person whose attribute House is 3? Please reason step by step, and put your final answer within \boxed{}

Answer: tall


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Alice`, `Eric`, `Arnold`
 - The people keep unique animals: `horse`, `cat`, `fish`, `bird`
 - People use unique phone models: `oneplus 9`, `google pixel 6`, `iphone 13`, `samsung galaxy s21`
 - Each person has a favorite color: `red`, `white`, `yellow`, `green`
 - Each person lives in a unique style of house: `colonial`, `victorian`, `ranch`, `craftsman`

## Clues:
1. The fish enthusiast is the person who uses an iPhone 13.
2. Alice is in the fourth house.
3. Peter is in the first house.
4. The person residing in a Victorian house is the person whose favorite color is red.
5. The person who loves white is the person in a Craftsman-style house.
6. The person whose favorite color is green is in the fourth house.
7. The person living in a colonial-style house is Eric.
8. The bird keeper is Peter.
9. The person who uses a Google Pixel 6 is in the fourth house.
10. The person who loves white is the person who uses a Samsung Galaxy S21.
11. The person living in a colonial-style house is the cat lover.
12. Eric is not in the third house.
 What is the value of attribute Animal for the person whose attribute PhoneModel is google pixel 6? Please reason step by step, and put your final answer within \boxed{}

Answer: horse


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Alice`, `Peter`, `Arnold`
 - They all have a unique favorite flower: `lilies`, `carnations`, `roses`, `daffodils`
 - Each person has a favorite color: `white`, `green`, `yellow`, `red`
 - Each person has a unique level of education: `bachelor`, `associate`, `master`, `high school`
 - Each person has a unique hobby: `painting`, `photography`, `gardening`, `cooking`
 - Everyone has a unique favorite cigar: `dunhill`, `blue master`, `prince`, `pall mall`

## Clues:
1. The person partial to Pall Mall is the person with an associate's degree.
2. The person who loves white is the Prince smoker.
3. The person with a high school diploma is not in the fourth house.
4. The person with a master's degree is not in the second house.
5. Arnold is not in the fourth house.
6. The person whose favorite color is red is not in the third house.
7. The person whose favorite color is red is somewhere to the left of Eric.
8. The person who loves cooking is somewhere to the right of Peter.
9. The person who loves a bouquet of daffodils is somewhere to the left of the person who paints as a hobby.
10. The person who loves white is the person who loves the boquet of lilies.
11. Alice is the photography enthusiast.
12. Arnold is somewhere to the right of the person partial to Pall Mall.
13. The person who loves white is in the second house.
14. The person whose favorite color is green is directly left of the person who enjoys gardening.
15. The Dunhill smoker is the person with a high school diploma.
16. The person who loves the rose bouquet is somewhere to the right of the person whose favorite color is green.
 What is the value of attribute Hobby for the person whose attribute Education is bachelor? Please reason step by step, and put your final answer within \boxed{}

Answer: painting


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 an occupation: `engineer`, `teacher`, `doctor`
 - Each person has a favorite color: `white`, `yellow`, `red`
 - They all have a unique favorite flower: `carnations`, `lilies`, `daffodils`
 - Everyone has something unique for lunch: `pizza`, `spaghetti`, `grilled cheese`
 - People have unique heights: `average`, `short`, `very short`

## Clues:
1. The person who is a pizza lover is the person who is short.
2. The person who loves a carnations arrangement is in the first house.
3. There is one house between Eric and the person who loves eating grilled cheese.
4. The person who loves eating grilled cheese is somewhere to the right of Arnold.
5. The person who loves yellow is the person who is an engineer.
6. The person who loves a bouquet of daffodils is the person who is a teacher.
7. The person who is a pizza lover is the person who is an engineer.
8. The person who loves the spaghetti eater is the person who has an average height.
9. Arnold is somewhere to the right of the person who loves white.
 What is the value of attribute Color for the person whose attribute Name is Peter? Please reason step by step, and put your final answer within \boxed{}

Answer: red


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`, `Eric`, `Bob`, `Peter`
 - Each person has a unique birthday month: `mar`, `feb`, `jan`, `sept`, `april`
 - They all have a unique favorite flower: `tulips`, `lilies`, `carnations`, `daffodils`, `roses`
 - Each person has a unique level of education: `high school`, `bachelor`, `doctorate`, `associate`, `master`
 - Everyone has a favorite smoothie: `desert`, `cherry`, `watermelon`, `dragonfruit`, `lime`

## Clues:
1. There is one house between the Desert smoothie lover and the person whose birthday is in April.
2. The person with a bachelor's degree is Peter.
3. The person who loves a bouquet of daffodils is the person with a high school diploma.
4. Eric is not in the second house.
5. The person who drinks Lime smoothies and the person who loves the rose bouquet are next to each other.
6. Alice is in the third house.
7. The person whose birthday is in February is Peter.
8. The person with a master's degree is the person whose birthday is in March.
9. The person whose birthday is in February is in the fourth house.
10. The person whose birthday is in March is the person who drinks Lime smoothies.
11. Peter is the Dragonfruit smoothie lover.
12. The person with an associate's degree is not in the second house.
13. The person who loves a carnations arrangement is not in the second house.
14. The Watermelon smoothie lover is Arnold.
15. Alice is the person whose birthday is in September.
16. The person who loves the boquet of lilies is not in the second house.
17. The person with an associate's degree is Arnold.
18. The person who loves the vase of tulips is Peter.
19. The person who loves the boquet of lilies is not in the first house.
 What is the value of attribute House for the person whose attribute Education is high school? 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: `Eric`, `Arnold`, `Alice`, `Peter`
 - They all have a unique favorite flower: `lilies`, `roses`, `carnations`, `daffodils`
 - People have unique favorite book genres: `fantasy`, `mystery`, `science fiction`, `romance`
 - Each person prefers a unique type of vacation: `city`, `beach`, `cruise`, `mountain`
 - People have unique hair colors: `black`, `red`, `brown`, `blonde`
 - Everyone has something unique for lunch: `grilled cheese`, `pizza`, `stew`, `spaghetti`

## Clues:
1. The person who likes going on cruises is the person who has brown hair.
2. The person who loves the stew is the person who loves beach vacations.
3. Peter is the person who loves romance books.
4. There is one house between the person who has brown hair and the person who loves the spaghetti eater.
5. The person who is a pizza lover is Arnold.
6. The person who prefers city breaks is somewhere to the right of the person who loves the boquet of lilies.
7. The person who loves the stew is directly left of the person who loves mystery books.
8. The person who has black hair is directly left of the person who loves fantasy books.
9. The person who loves a carnations arrangement is directly left of the person who is a pizza lover.
10. The person who has blonde hair is not in the fourth house.
11. The person who prefers city breaks is directly left of Eric.
12. The person who loves the boquet of lilies is somewhere to the right of Alice.
13. The person who loves a bouquet of daffodils is in the fourth house.
 What is the value of attribute BookGenre for the person whose attribute Food is spaghetti? Please reason step by step, and put your final answer within \boxed{}

Answer: fantasy


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Bob`, `Arnold`, `Eric`, `Peter`
 - People have unique favorite book genres: `romance`, `mystery`, `biography`, `fantasy`, `science fiction`
 - Each person has a unique birthday month: `sept`, `feb`, `jan`, `mar`, `april`
 - Each person prefers a unique type of vacation: `mountain`, `beach`, `city`, `camping`, `cruise`
 - The people keep unique animals: `horse`, `bird`, `fish`, `dog`, `cat`

## Clues:
1. The person whose birthday is in April is the person who loves mystery books.
2. The person whose birthday is in March is somewhere to the left of the person who loves mystery books.
3. The person who loves beach vacations is Arnold.
4. The person who loves beach vacations is the person whose birthday is in January.
5. The bird keeper is directly left of Peter.
6. The person who loves science fiction books is somewhere to the left of the person whose birthday is in February.
7. The fish enthusiast is somewhere to the right of Bob.
8. The fish enthusiast is Eric.
9. The fish enthusiast is the person who loves fantasy books.
10. The person who enjoys mountain retreats is directly left of the person whose birthday is in April.
11. The person whose birthday is in April is the person who enjoys camping trips.
12. The person whose birthday is in September is the person who loves science fiction books.
13. The cat lover is the person who prefers city breaks.
14. The person who loves romance books is somewhere to the right of the person who loves science fiction books.
15. The cat lover is the person whose birthday is in February.
16. The person who keeps horses is Peter.
 What is the value of attribute Birthday for the person whose attribute Animal is bird? Please reason step by step, and put your final answer within \boxed{}

Answer: jan


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Bob`, `Eric`, `Arnold`, `Alice`
 - People have unique hair colors: `brown`, `black`, `gray`, `blonde`, `red`
 - Everyone has a unique favorite cigar: `pall mall`, `dunhill`, `prince`, `blue master`, `blends`
 - Each person has a unique favorite drink: `milk`, `coffee`, `water`, `tea`, `root beer`
 - People have unique favorite music genres: `rock`, `classical`, `pop`, `jazz`, `hip hop`

## Clues:
1. The person who has gray hair and Bob are next to each other.
2. Arnold is not in the first house.
3. The person who has blonde hair is not in the second house.
4. Peter is not in the first house.
5. The Dunhill smoker is the person who likes milk.
6. Arnold is the one who only drinks water.
7. The person who loves classical music is the person partial to Pall Mall.
8. The person who loves pop music is directly left of the person who loves jazz music.
9. The person who has brown hair is Bob.
10. The person who has red hair is the person who loves hip-hop music.
11. The root beer lover is Alice.
12. The person who has black hair is in the third house.
13. There are two houses between the coffee drinker and Arnold.
14. The person who smokes Blue Master is somewhere to the right of the person who loves hip-hop music.
15. The person who has blonde hair is the person who smokes many unique blends.
16. Alice is the person who loves classical music.
17. The Dunhill smoker is the person who loves jazz music.
 What is the value of attribute Drink for the person whose attribute MusicGenre is classical? Please reason step by step, and put your final answer within \boxed{}

Answer: root beer


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Arnold`, `Eric`, `Peter`
 - They all have a unique favorite flower: `daffodils`, `roses`, `lilies`, `carnations`
 - Everyone has a favorite smoothie: `desert`, `watermelon`, `cherry`, `dragonfruit`
 - Each person lives in a unique style of house: `craftsman`, `colonial`, `ranch`, `victorian`
 - Each person has a unique hobby: `cooking`, `photography`, `gardening`, `painting`
 - Each person has a unique favorite drink: `coffee`, `tea`, `milk`, `water`

## Clues:
1. The person in a Craftsman-style house is the Dragonfruit smoothie lover.
2. The tea drinker is the person who loves a bouquet of daffodils.
3. The Desert smoothie lover is the person living in a colonial-style house.
4. There is one house between the Watermelon smoothie lover and the person who loves a bouquet of daffodils.
5. The person who likes Cherry smoothies is directly left of the photography enthusiast.
6. The person who likes milk is the person who loves cooking.
7. The tea drinker is Arnold.
8. The person who loves the boquet of lilies is the person living in a colonial-style house.
9. The person who loves the boquet of lilies is not in the third house.
10. The person living in a colonial-style house is directly left of the person who paints as a hobby.
11. The person in a ranch-style home is the person who likes milk.
12. The person who loves the rose bouquet is the one who only drinks water.
13. Eric is somewhere to the left of the Dragonfruit smoothie lover.
14. The person who loves the rose bouquet is somewhere to the right of the person who loves a carnations arrangement.
15. Peter is the person who loves the boquet of lilies.
 What is the value of attribute Hobby for the person whose attribute Flower is lilies? Please reason step by step, and put your final answer within \boxed{}

Answer: gardening


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Peter`, `Alice`, `Eric`, `Arnold`
 - Each mother is accompanied by their child: `Meredith`, `Bella`, `Samantha`, `Fred`, `Timothy`
 - People have unique favorite music genres: `classical`, `hip hop`, `pop`, `jazz`, `rock`
 - Everyone has a favorite smoothie: `watermelon`, `lime`, `dragonfruit`, `cherry`, `desert`
 - Each person prefers a unique type of vacation: `beach`, `mountain`, `city`, `cruise`, `camping`

## Clues:
1. The person who loves rock music is somewhere to the left of the person who loves classical music.
2. The person who loves beach vacations and the person who loves rock music are next to each other.
3. Eric is the person who is the mother of Timothy.
4. The person who drinks Lime smoothies is not in the second house.
5. The Watermelon smoothie lover is somewhere to the left of the person who loves rock music.
6. Alice is the person who loves beach vacations.
7. The Dragonfruit smoothie lover is somewhere to the right of Peter.
8. The person who loves jazz music is the person's child is named Samantha.
9. Bob is the person's child is named Samantha.
10. Alice is somewhere to the left of Eric.
11. The person who loves pop music is the person who likes going on cruises.
12. The person's child is named Fred is in the fifth house.
13. The person who prefers city breaks is the person who loves jazz music.
14. The Dragonfruit smoothie lover is the person's child is named Samantha.
15. The person's child is named Meredith is the person who likes going on cruises.
16. The Desert smoothie lover is somewhere to the left of the person who likes Cherry smoothies.
17. The person who enjoys mountain retreats is not in the fifth house.
18. The person who drinks Lime smoothies is directly left of the Desert smoothie lover.
 What is the value of attribute Vacation for the person whose attribute Smoothie is desert? Please reason step by step, and put your final answer within \boxed{}

Answer: mountain


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 has a unique level of education: `high school`, `trade school`, `bachelor`, `doctorate`, `master`, `associate`
 - Each person has a favorite color: `purple`, `blue`, `white`, `red`, `green`, `yellow`
 - Each person prefers a unique type of vacation: `cultural`, `cruise`, `beach`, `mountain`, `camping`, `city`
 - People have unique favorite book genres: `science fiction`, `biography`, `historical fiction`, `fantasy`, `romance`, `mystery`

## Clues:
1. Alice is the person with a master's degree.
2. There is one house between the person with an associate's degree and Peter.
3. The person who loves historical fiction books is somewhere to the left of the person who loves biography books.
4. The person who loves historical fiction books is the person with a high school diploma.
5. Bob is the person with a bachelor's degree.
6. Carol is somewhere to the right of Eric.
7. The person with a master's degree is somewhere to the right of the person who enjoys camping trips.
8. The person who loves science fiction books is somewhere to the left of the person who loves yellow.
9. The person who loves historical fiction books is somewhere to the left of the person whose favorite color is green.
10. The person with a high school diploma is not in the second house.
11. The person who loves beach vacations is the person with a doctorate.
12. The person who loves mystery books is somewhere to the left of the person with a bachelor's degree.
13. The person with a doctorate is in the first house.
14. The person with a high school diploma is directly left of the person who enjoys camping trips.
15. The person who loves romance books is not in the third house.
16. There is one house between the person who loves purple and the person with a bachelor's degree.
17. The person who loves biography books is the person whose favorite color is red.
18. There is one house between the person who loves fantasy books and the person who loves blue.
19. The person who goes on cultural tours is the person who loves blue.
20. Carol is not in the fifth house.
21. The person who loves purple is the person with a doctorate.
22. The person who enjoys mountain retreats is somewhere to the right of the person who loves science fiction books.
23. The person who prefers city breaks is the person whose favorite color is green.
 What is the value of attribute BookGenre for the person whose attribute Education is master? Please reason step by step, and put your final answer within \boxed{}

Answer: romance


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Arnold`, `Alice`, `Peter`
 - They all have a unique favorite flower: `lilies`, `roses`, `carnations`, `daffodils`
 - People have unique favorite book genres: `fantasy`, `mystery`, `science fiction`, `romance`
 - Each person prefers a unique type of vacation: `city`, `beach`, `cruise`, `mountain`
 - People have unique hair colors: `black`, `red`, `brown`, `blonde`
 - Everyone has something unique for lunch: `grilled cheese`, `pizza`, `stew`, `spaghetti`

## Clues:
1. The person who likes going on cruises is the person who has brown hair.
2. The person who loves the stew is the person who loves beach vacations.
3. Peter is the person who loves romance books.
4. There is one house between the person who has brown hair and the person who loves the spaghetti eater.
5. The person who is a pizza lover is Arnold.
6. The person who prefers city breaks is somewhere to the right of the person who loves the boquet of lilies.
7. The person who loves the stew is directly left of the person who loves mystery books.
8. The person who has black hair is directly left of the person who loves fantasy books.
9. The person who loves a carnations arrangement is directly left of the person who is a pizza lover.
10. The person who has blonde hair is not in the fourth house.
11. The person who prefers city breaks is directly left of Eric.
12. The person who loves the boquet of lilies is somewhere to the right of Alice.
13. The person who loves a bouquet of daffodils is in the fourth house.
 What is the value of attribute Name for the person whose attribute Vacation is city? Please reason step by step, and put your final answer within \boxed{}

Answer: Peter


Let \( p \) be a prime number such that \( p = a^{2} + b^{2} \). Prove that one of \( \pm a, \pm b \) is congruent to \( \frac{1}{2}\binom{\left(\frac{p-1}{2}\right)}{\frac{p-1}{4}} \bmod{p} \).

Answer: \pm a \text{ or } \pm b \equiv \frac{1}{2} \binom{\left(\frac{p-1}{2}\right)}{\frac{p-1}{4}} \bmod{p}


P.G. Zhenoarove

In a box, there are 111 balls: red, blue, green, and white. It is known that if you pull out 100 balls from the box without looking, there will definitely be four balls of different colors among them. What is the smallest number of balls that need to be pulled out without looking to ensure that there are three balls of different colors among them?

#

Answer: 88


C3 (6-5, Romania) On the plane, given 5 points, the lines connecting these points pairwise are neither parallel, nor perpendicular, nor coincident. From each point, draw perpendiculars to the lines formed by connecting the other four points pairwise. How many intersection points (excluding the given 5 points) can these perpendiculars have at most?

Answer: 310