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

## Clues:
1. The person with a bachelor's degree is the person who enjoys gardening.
2. Arnold is somewhere to the left of the person who loves a carnations arrangement.
3. Eric is not in the second house.
4. The person who loves a bouquet of daffodils is the person who paints as a hobby.
5. The person who loves the rose bouquet is the person who smokes Blue Master.
6. The person partial to Pall Mall is Peter.
7. Arnold is in the third house.
8. Peter is directly left of the person who enjoys gardening.
9. Arnold is the person with a high school diploma.
10. The person who smokes Blue Master is not in the second house.
11. The Prince smoker is the photography enthusiast.
12. The person with a master's degree is somewhere to the left of the person who smokes Blue Master.
 What is the value of attribute House for the person whose attribute Hobby is photography? Please reason step by step, and put your final answer within \boxed{}

Answer: 4


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Peter`, `Arnold`, `Eric`, `Alice`, `Carol`
 - People have unique hair colors: `red`, `gray`, `auburn`, `blonde`, `brown`, `black`
 - People have unique favorite book genres: `historical fiction`, `biography`, `romance`, `fantasy`, `mystery`, `science fiction`
 - People have unique heights: `very short`, `very tall`, `tall`, `short`, `super tall`, `average`
 - Each person has a unique level of education: `associate`, `doctorate`, `high school`, `trade school`, `bachelor`, `master`
 - Each person has a unique favorite drink: `boba tea`, `milk`, `water`, `root beer`, `coffee`, `tea`

## Clues:
1. The person with a doctorate and Bob are next to each other.
2. There is one house between Carol and the person who has black hair.
3. Peter is the person with a doctorate.
4. The person who loves science fiction books is Eric.
5. The one who only drinks water is the person who has an average height.
6. Alice is the person who loves mystery books.
7. The person who loves romance books is Bob.
8. The person who has brown hair is in the second house.
9. The person who has an average height is in the fourth house.
10. The boba tea drinker is the person who is very tall.
11. The person with a high school diploma is Bob.
12. The person who is short is somewhere to the left of the person who is super tall.
13. The person who loves historical fiction books and the person with a bachelor's degree are next to each other.
14. The person who attended trade school is the person who has red hair.
15. Carol is the person who has auburn hair.
16. The person who is very short is somewhere to the right of the person who has gray hair.
17. There are two houses between Eric and the person who loves fantasy books.
18. There are two houses between the tea drinker and the person who is very short.
19. Eric is somewhere to the left of the person with a bachelor's degree.
20. The root beer lover is Alice.
21. The person with an associate's degree is Arnold.
22. The person who is tall is not in the second house.
23. The person who loves science fiction books is the boba tea drinker.
24. The person who has black hair is not in the fourth house.
25. The person who is short is the coffee drinker.
26. The coffee drinker and the person with a high school diploma are next to each other.
27. The coffee drinker is not in the second house.
 What is the value of attribute House for the person whose attribute BookGenre is science fiction? 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`, `Arnold`, `Peter`, `Alice`, `Eric`
 - Each person has a unique birthday month: `sept`, `april`, `feb`, `jan`, `mar`
 - They all have a unique favorite flower: `daffodils`, `lilies`, `tulips`, `carnations`, `roses`

## Clues:
1. The person who loves the vase of tulips is somewhere to the left of the person whose birthday is in January.
2. The person whose birthday is in March and Alice are next to each other.
3. The person whose birthday is in April is Bob.
4. The person whose birthday is in January is not in the fourth house.
5. The person who loves the vase of tulips is not in the first house.
6. The person whose birthday is in February is in the fifth house.
7. The person who loves a carnations arrangement is not in the third house.
8. The person whose birthday is in September is somewhere to the right of Peter.
9. The person who loves the boquet of lilies is the person whose birthday is in February.
10. Eric is not in the third house.
11. The person who loves the rose bouquet is directly left of Alice.
 What is the value of attribute Flower for the person whose attribute Name is Alice? Please reason step by step, and put your final answer within \boxed{}

Answer: tulips


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 Height for the person whose attribute Pet is dog? Please reason step by step, and put your final answer within \boxed{}

Answer: short


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Alice`, `Eric`, `Peter`
 - People have unique favorite music genres: `pop`, `classical`, `rock`, `jazz`
 - Each person has a unique type of pet: `fish`, `dog`, `bird`, `cat`
 - People own unique car models: `tesla model 3`, `toyota camry`, `ford f150`, `honda civic`
 - Each person has a unique favorite drink: `water`, `milk`, `tea`, `coffee`

## Clues:
1. The one who only drinks water is the person who owns a dog.
2. The one who only drinks water is Eric.
3. The tea drinker is somewhere to the right of the person who loves rock music.
4. The person who likes milk is the person who owns a Tesla Model 3.
5. The person who loves rock music and the person who loves classical music are next to each other.
6. There is one house between the person who likes milk and the person who loves classical music.
7. The person who owns a Honda Civic is Arnold.
8. The person who has a cat and the coffee drinker are next to each other.
9. Alice is not in the second house.
10. The person who loves classical music is not in the second house.
11. The person with an aquarium of fish is the tea drinker.
12. The person who owns a Ford F-150 is somewhere to the right of the person with an aquarium of fish.
13. The person who owns a Toyota Camry is not in the third house.
14. The tea drinker and the person who loves jazz music are next to each other.
 What is the value of attribute House for the person whose attribute Drink is water? Please reason step by step, and put your final answer within \boxed{}

Answer: 4


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Arnold`, `Bob`, `Alice`, `Eric`, `Carol`
 - The mothers' names in different houses are unique: `Holly`, `Aniya`, `Kailyn`, `Janelle`, `Penny`, `Sarah`
 - Everyone has something unique for lunch: `stir fry`, `stew`, `spaghetti`, `soup`, `pizza`, `grilled cheese`
 - The people keep unique animals: `dog`, `horse`, `cat`, `fish`, `rabbit`, `bird`
 - Everyone has a favorite smoothie: `watermelon`, `desert`, `lime`, `cherry`, `blueberry`, `dragonfruit`
 - Each person lives in a unique style of house: `mediterranean`, `craftsman`, `victorian`, `colonial`, `modern`, `ranch`

## Clues:
1. The person whose mother's name is Penny is the person residing in a Victorian house.
2. The person in a modern-style house is The person whose mother's name is Kailyn.
3. The person in a modern-style house is somewhere to the right of the person residing in a Victorian house.
4. The person who drinks Blueberry smoothies is not in the sixth house.
5. The person whose mother's name is Sarah is the person who drinks Blueberry smoothies.
6. The person who likes Cherry smoothies is not in the third house.
7. The person who likes Cherry smoothies is Eric.
8. The person whose mother's name is Janelle is not in the third house.
9. The person who loves the spaghetti eater is in the third house.
10. The fish enthusiast is The person whose mother's name is Holly.
11. The person who keeps horses is in the first house.
12. The person in a Mediterranean-style villa is the person who loves the stew.
13. The bird keeper is in the third house.
14. The person living in a colonial-style house is the person who drinks Blueberry smoothies.
15. Carol is the person who keeps horses.
16. There are two houses between the Desert smoothie lover and Peter.
17. The person who loves eating grilled cheese is the Dragonfruit smoothie lover.
18. The person who is a pizza lover is somewhere to the right of the Dragonfruit smoothie lover.
19. The person whose mother's name is Janelle is somewhere to the left of Arnold.
20. Carol is directly left of The person whose mother's name is Aniya.
21. The person in a ranch-style home is The person whose mother's name is Aniya.
22. Arnold is somewhere to the right of the person in a Craftsman-style house.
23. The cat lover is Bob.
24. There is one house between the Watermelon smoothie lover and the person who is a pizza lover.
25. There are two houses between the cat lover and the person who loves stir fry.
26. Arnold is the rabbit owner.
27. Peter is 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: 3


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Arnold`, `Bob`, `Peter`, `Alice`
 - Each person lives in a unique style of house: `modern`, `victorian`, `ranch`, `craftsman`, `colonial`
 - Each person prefers a unique type of vacation: `mountain`, `cruise`, `camping`, `city`, `beach`
 - Everyone has a favorite smoothie: `watermelon`, `desert`, `lime`, `dragonfruit`, `cherry`

## Clues:
1. Arnold is the person who enjoys mountain retreats.
2. Eric is the person residing in a Victorian house.
3. The Desert smoothie lover is the person who enjoys mountain retreats.
4. Bob is the person who likes going on cruises.
5. The person who prefers city breaks is Peter.
6. The person who likes going on cruises is somewhere to the left of the person who enjoys camping trips.
7. The person living in a colonial-style house is somewhere to the right of Eric.
8. There is one house between Bob and the person who drinks Lime smoothies.
9. The person who enjoys camping trips is not in the third house.
10. The person who likes Cherry smoothies is the person in a ranch-style home.
11. Eric is not in the first house.
12. The person who likes Cherry smoothies is not in the fifth house.
13. The person living in a colonial-style house is not in the fourth house.
14. The person who likes going on cruises is not in the fourth house.
15. There is one house between the person living in a colonial-style house and the person who enjoys camping trips.
16. The Dragonfruit smoothie lover is directly left of the person in a Craftsman-style house.
 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: `Arnold`, `Eric`, `Peter`, `Carol`, `Bob`, `Alice`
 - Everyone has a favorite smoothie: `blueberry`, `watermelon`, `cherry`, `lime`, `dragonfruit`, `desert`
 - The people keep unique animals: `bird`, `cat`, `fish`, `horse`, `dog`, `rabbit`

## Clues:
1. The Desert smoothie lover is directly left of Arnold.
2. Carol is the rabbit owner.
3. The Watermelon smoothie lover is the cat lover.
4. The dog owner is somewhere to the right of Peter.
5. The person who keeps horses is directly left of Carol.
6. Eric is not in the fifth house.
7. The person who drinks Lime smoothies is in the sixth house.
8. Carol is in the sixth house.
9. The bird keeper and the person who keeps horses are next to each other.
10. Alice is in the second house.
11. Arnold is in the third house.
12. The Dragonfruit smoothie lover is the dog owner.
13. The person who drinks Blueberry smoothies is not in the fifth house.
 What is the value of attribute Name for the person whose attribute Animal is cat? Please reason step by step, and put your final answer within \boxed{}

Answer: Peter


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`
 - People have unique favorite sports: `basketball`, `tennis`, `soccer`
 - People have unique favorite book genres: `romance`, `science fiction`, `mystery`
 - Each person has a unique birthday month: `sept`, `april`, `jan`
 - They all have a unique favorite flower: `lilies`, `daffodils`, `carnations`

## Clues:
1. Arnold is somewhere to the left of the person who loves romance books.
2. The person who loves soccer is the person who loves a carnations arrangement.
3. The person who loves mystery books is the person whose birthday is in September.
4. The person who loves tennis is in the second house.
5. Peter is the person who loves a bouquet of daffodils.
6. Eric is somewhere to the left of the person whose birthday is in April.
7. The person who loves science fiction books is somewhere to the right of the person who loves tennis.
 What is the value of attribute BookGenre for the person whose attribute Birthday is april? Please reason step by step, and put your final answer within \boxed{}

Answer: science fiction


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Arnold`, `Peter`
 - The people keep unique animals: `horse`, `bird`, `cat`
 - Everyone has a favorite smoothie: `watermelon`, `cherry`, `desert`
 - The mothers' names in different houses are unique: `Holly`, `Janelle`, `Aniya`
 - People have unique favorite music genres: `pop`, `classical`, `rock`
 - Each person has a favorite color: `yellow`, `white`, `red`

## Clues:
1. The cat lover is in the third house.
2. There is one house between the person who loves yellow and the cat lover.
3. The person who loves classical music is directly left of Eric.
4. Arnold is the Desert smoothie lover.
5. The bird keeper is the Watermelon smoothie lover.
6. The person whose mother's name is Janelle is somewhere to the left of the person who loves white.
7. The person who loves yellow is the person who loves rock music.
8. The person whose mother's name is Holly is directly left of Peter.
 What is the value of attribute Mother for the person whose attribute Color is yellow? Please reason step by step, and put your final answer within \boxed{}

Answer: Holly


In a draft of an arithmetic workbook, there was an example in which a given number needed to be multiplied by 3 and 4 subtracted from the result. In the print, an error occurred: instead of the multiplication sign, a division sign was placed, and instead of the minus sign, a plus sign was used. Nevertheless, the final result did not change. What was the original example intended to be placed in the workbook?

Answer: 3 \cdot 3 - 4


We construct a channel from three boards of equal width. The side walls form an angle of $x$ with the base board. What value of the angle $x$ will make the cross-sectional area (and thus the volume) of the channel (which forms the shape of an isosceles trapezoid) the largest possible?

Answer: 60^\circ


\[
\frac{x^{3}}{x^{3}+2 y^{2} z}+\frac{y^{3}}{y^{3}+2 z^{2} x}+\frac{z^{3}}{z^{3}+2 x^{2} y} \geqslant 1
\]

On the table, there is a pile of \( n \) stones. In one move, any existing pile can be split into two smaller piles. At the same time, the sizes of any two piles on the table must be "similar."

Answer: 1


Line \( l \) passes through point \( A(2,0) \) and intersects the parabolic curve \( y = x^{2} + 2 \) at points \( B \) and \( C \). Points \( B_{1} \) and \( C_{1} \) are the feet of the perpendiculars from \( B \) and \( C \) to the \( x \)-axis, respectively. Point \( P \) is on the line segment \( BC \) such that \(\overrightarrow{B P} = \frac{\left|B B_{1}\right|}{\left|C C_{1}\right|} \cdot \overrightarrow{P C}\). Find the equation of the locus of the centroid \( G \) of triangle \( \triangle POA \) (where \( O \) is the origin).

Answer: 12x - 3y - 4 = 0


A 90-degree angle is covered by some angles with the same vertex in two or three layers, and the sum of the angles is 290 degrees. Find the measure of the angle covered in three layers.

Answer: x = 20^\circ


C is a circle and L a line not meeting it. M and N are variable points on L such that the circle diameter MN touches C but does not contain it. Show that there is a fixed point P such that the ∠MPN is constant.

Answer: P


As shown in Figure 1, in a cylindrical container with a height of 10 and a base radius of 1, a ping-pong ball is glued to the upper wall and another to the lower wall. The ping-pong ball on the lower wall is tangent to both the bottom and side surfaces, and the ping-pong ball on the upper wall is tangent to both the top and side surfaces (assume the thickness of both the cylinder and the ping-pong balls is negligible). A plane is tangent to both ping-pong balls. Given that the cross-section of the cylinder's edge by this plane is an ellipse, find the eccentricity of this ellipse.

Answer: \frac{\sqrt{15}}{4}


In a joint-stock company, there are 2017 shareholders, and any group of 1500 of them has a controlling stake (at least 50% of the shares). What is the maximum share of the company that one shareholder can own?

Answer: 32.8\%


Given that \( x \) is a negative real number that satisfies \(\frac{1}{x+\frac{1}{x+2}}=a\). If \( b=x+\frac{7}{2} \), find the value of \( b \).

Answer: 2


An isosceles triangle can be divided into two isosceles triangles (not necessarily equal). Find the possible values for the smallest angle of such a triangle. Provide the smallest of these values in degrees, multiplied by 6006.

Answer: 154440


Max has 2015 cans, numbered from 1 to 2015, and an unlimited supply of coins.

Consider the following initial configurations:

(a) All cans are empty.

(b) Can number 1 contains 1 coin, can number 2 contains 2 coins, and so on, up to can number 2015, which contains 2015 coins.

(c) Can number 1 contains 2015 coins, can number 2 contains 2014 coins, and so on, up to can number 2015, which contains 1 coin.

Max chooses a number \( n \) in each step, with \( 1 \leq n \leq 2015 \), and adds \( n \) coins to every can except can number \( n \).

Determine for each of the three initial configurations (a), (b), and (c) whether Max can achieve a state where all cans contain the same number of coins after a finite number of steps, with at least one step performed.

Answer: Yes


A quadrilateral \(ABCD\) is given, where \(AD \parallel BC\). The angle bisector of \(\angle A\) intersects side \(CD\) at point \(X\) and the extension of side \(BC\) beyond point \(C\) at point \(Y\). It is found that \(\angle AXC = 90^\circ\). Find the length of segment \(AB\), given that \(AD = 19\) and \(CY = 16\).

Answer: 17.5


A circle with a radius of 1 is randomly placed inside a $15 \times 36$ rectangle $ABCD$. Find the probability that this circle does not intersect the diagonal $AC$.

Answer: \frac{375}{442}


Given two circles with radii \( R \) and \( r \). Their common internal tangents are perpendicular to each other. Find the area of the triangle formed by these tangents and the common external tangent to the given circles.

Answer: S = Rr


In the cells of a $5 \times 5$ square, natural numbers from 1 to 5 are placed so that in each column, each row, and each of the two main diagonals, all numbers are different. Can the sum of the numbers in the shaded cells in the drawing be equal to 19?

Answer: \text{No}


In Figure 1, BD, FC, GC, and FE are straight lines. If \( z = a + b + c + d + e + f + g \), find the value of \( z \).

Answer: 540


A and B play a number-changing game on a $5 \times 5$ grid: A starts and both take turns filling empty spaces, with A filling each space with the number 1 and B filling each space with the number 0. After the grid is completely filled, the sum of the numbers in each $3 \times 3$ square is calculated, and the maximum sum among these squares is denoted as $A$. A tries to maximize $A$, while B tries to minimize $A$. What is the maximum value of $A$ that A can achieve?

(The problem is from the 35th IMO selection test)

Answer: 6


a) Consider the $4 \times 4$ board shown in the figure below. The letters and numbers are used to help locate the squares. For example, the square $A1$ is at the top left corner and the $D4$ is at the bottom right corner.

Show a way to choose 12 squares such that exactly 3 squares are selected in each row and each column.

b) Determine the number of different ways to choose 12 squares on the $4 \times 4$ board such that exactly 3 squares are selected in each row and each column.

c) Consider an $8 \times 8$ board with a similar row and column identification as the previous board, but colored alternately in black and white.

Determine the number of ways to choose 56 squares from the board such that all the black squares are chosen and exactly 7 squares are chosen in each row and each column.

Answer: 576


As shown in the figure, the diagonal $BD$ of square $ABCD$ is 20 cm long, and $BDFE$ is a rectangle. What is the area of the pentagon $ABEF D$ in square centimeters?

Answer: 300


In a football match between the teams "Zubilo" and "Shaiba," bets on "Zubilo" winning were accepted at 1 to 2 (i.e., if "Zubilo" wins, the bettor receives twice the amount they bet), and bets on "Shaiba" winning were accepted at 1 to 3. Volka was able to place a bet knowing that he would get back exactly the same amount he bet, regardless of the outcome of the game. What were the odds for a draw?

Answer: 1:6


Let \( D \) be a regular ten-sided polygon with edges of length 1. A triangle \( T \) is defined by choosing three vertices of \( D \) and connecting them with edges. How many different (non-congruent) triangles \( T \) can be formed?

Answer: 8


There are 300 black and white pieces in total. The black crow divides the black and white pieces into 100 piles, each containing 3 pieces. There are 27 piles with exactly $l$ white pieces, 42 piles with 2 or 3 black pieces, and the number of piles with 3 white pieces is equal to the number of piles with 3 black pieces. How many white pieces are there in total?

Answer: 158


As shown in the figure, the line segment $AB$ of length 9 is divided into nine equal parts. What is the total length of all the line segments in the figure?

Answer: 165


A circle having radius \( r_1 \) centered at point \( N \) is tangent to a circle of radius \( r_2 \) centered at \( M \). Let \( l \) and \( j \) be the two common external tangent lines to the two circles. A circle centered at \( P \) with radius \( r_2 \) is externally tangent to circle \( N \) at the point at which \( l \) coincides with circle \( N \), and line \( k \) is externally tangent to \( P \) and \( N \) such that points \( M, N \), and \( P \) all lie on the same side of \( k \). For what ratio \( r_1 / r_2 \) are \( j \) and \( k \) parallel?

Answer: 3


Vladislav Vladimirovich, having less than 100 rubles, went for a walk. When entering a cafe with \( m \) rubles and \( n \) kopecks, he would spend \( n \) rubles and \( m \) kopecks (where \( m \) and \( n \) are natural numbers). What is the maximum number of cafes Vladislav Vladimirovich could visit?

Answer: 6


Given an isosceles triangle $ABC$ with base $AC$. Points $K$ and $N$ are marked on the side $BC$ (with $K$ between $B$ and $N$). It is found that $KN = AN$ and $\angle BAK = \angle NAC$. Find $\angle BAN$.

Answer: 60


A polynomial \( f \in \mathbb{Z}[x] \) is called splitty if and only if for every prime \( p \), there exist polynomials \( g_{p}, h_{p} \in \mathbb{Z}[x] \) with \( \operatorname{deg} g_{p}, \operatorname{deg} h_{p} < \operatorname{deg} f \) and all coefficients of \( f - g_{p} h_{p} \) are divisible by \( p \). Compute the sum of all positive integers \( n \leq 100 \) such that the polynomial \( x^{4} + 16 x^{2} + n \) is splitty.

Answer: 693


As shown in the figure, the square \(ABCD\) is divided into 8 triangles of equal area. If \(DG = 5\), then the area of the square \(ABCD\) is ______.

Answer: 50


We know the following about three natural numbers:

a) all three are different;

b) their sum is 406;

c) their greatest common divisor is a prime number greater than 2 (let's denote it by \( p \));

d) if each number is divided by \( p \), we get three prime numbers again (let these be \( p_{1}, p_{2}, p_{3} \)). What are these numbers?

Note that 1 is not considered a prime number.

Answer: 58,145,203


As shown in the figure, in rectangle $ABCD$, $\triangle AED$ and $\triangle BFC$ are both isosceles right triangles, with $EF = AD = 2$. Find the area of rectangle $ABCD$.

Answer: 8


There are less than \( 4Q \) students in a class. In a mathematics test, \(\frac{1}{3}\) of the students got grade A, \(\frac{1}{7}\) of the students got grade B, half of the students got grade C, and the rest failed. Given that \( R \) students failed in the mathematics test, find the value of \( R \).

Answer: 1


A finite set of integers is called bad if its elements add up to 2010. A finite set of integers is a Benelux-set if none of its subsets is bad. Determine the smallest integer \( n \) such that the set \( \{502, 503, 504, \ldots, 2009\} \) can be partitioned into \( n \) Benelux-sets.

(A partition of a set \( S \) into \( n \) subsets is a collection of \( n \) pairwise disjoint subsets of \( S \), the union of which equals \( S \).)

Answer: 2


A sphere is inscribed in a dihedral angle.

Point \( O \) is located in the section \( A A^{\prime} C^{\prime} C \) of a rectangular parallelepiped \( A B C D A^{\prime} B^{\prime} C^{\prime} D^{\prime} \) with dimensions \( 2 \times 6 \times 9 \) such that \( \angle O A B + \angle O A D + \angle O A A^{\prime} = 180^{\circ} \). The sphere with center at point \( O \) touches the planes \( A^{\prime} B^{\prime} C^{\prime} \) and \( A A^{\prime} B \) and does not intersect the plane \( A A^{\prime} D \). Find the distance from point \( O \) to this plane.

Answer: 3


Given a unit cube $A B C D-A_{1} B_{1} C_{1} D_{1}$, a black and a white ant start crawling from point $A$ along the edges. Each time an ant traverses one edge, it is said to have completed one segment. The white ant's path is $A A_{1} \rightarrow A_{1} D_{1} \rightarrow \cdots$, while the black ant's path is $A B \rightarrow B B_{1} \rightarrow \cdots$. Both ants follow the rule that the $(n+2)$-th segment they crawl must be on a line that is skew to the line of the $n$-th segment. Assume that after each of the black and white ants has crawled 2008 segments, each stops at a vertex of the cube. Determine the distance between the black and white ants at this moment.

Answer: \sqrt{2}


A sequence of 2011 consecutive five-digit numbers is written in a row. It turned out that the sum of the digits of the 21st number is 37, and the sum of the digits of the 54th number is 7. Find the sum of the digits of the 2011th number. (Provide all possible answers and prove that no other answers exist).

Answer: 29


Nina has 7 different chocolate candies, and Kolya has 9 different caramels. In how many ways can they exchange five candies with each other?

Answer: 2646


The ratio of two numbers is $5: a$. If 12 is added to each of them, the ratio becomes $3: 4$. If $b$ is the difference of the original numbers and $b>0$, find the value of $b$.

Answer: 9


Given that \(\alpha\) and \(\beta\) are real numbers, for any real numbers \(x\), \(y\), and \(z\), it holds that
$$
\alpha(x y + y z + z x) \leqslant M \leqslant \beta\left(x^{2} + y^{2} + z^{2}\right),
$$
where \(M = \sum \sqrt{x^{2} + x y + y^{2}} \sqrt{y^{2} + y z + z^{2}}\,\) and \(\sum\) represents cyclic sums. Find the maximum value of \(\alpha\) and the minimum value of \(\beta\).

Answer: 3


On a board, there is an empty $3 \times 57$ table. Masha wants to fill its cells with numbers according to the following rules:

- Each of the numbers $1, 2, 3, \ldots, 171$ must be present in the table;
- The number 1 must be in the bottom-left cell of the table;
- For any natural number $a \leq 170$, the numbers $a$ and $a+1$ must be in adjacent cells.

(a) (1 point) What is the largest number that can be in a cell adjacent to the number 1?

(b) (3 points) A cell is called good if the number 171 can be placed in it. How many good cells are there in total?

Answer: 85


In triangle \( \triangle ABC \), the base \( AB \) lies on the x-axis. The altitudes \( AM, BN \), and \( CK \) are drawn. Find the length of the base \( AB \), given the coordinates of points \( M(2,2) \) and \( N(4,4) \).

Answer: 4\sqrt{5}


For \( t \in [0, \pi] \), let \( S = \sin^{2} x + \sin^{2} y + \left(8 \sin^{2} t + 2\right) \sin x \sin y \). The maximum value of \( S \) is denoted as \( M(t) \) and the minimum value is denoted as \( N(t) \), where \( x \) and \( y \) are non-negative real numbers and \( x + y = t \). Let \( g(t) = M(t) - N(t) \). Find the maximum value of \( g(t) \) and the corresponding value of \( t \).

Answer: \frac{27}{4}


Two siblings sold their flock of sheep. Each sheep was sold for as many florins as the number of sheep originally in the flock. They divided the revenue by giving out 10 florins at a time. First, the elder brother took 10 florins, then the younger brother, then the elder again, and so on. In the end, the younger brother received less than 10 florins, so the elder brother gave him his knife, making their earnings equal. How much is the knife worth in florins?

Answer: 2


Two unit squares \( S_{1} \) and \( S_{2} \) have horizontal and vertical sides. Let \( x \) be the minimum distance between a point in \( S_{1} \) and a point in \( S_{2} \), and let \( y \) be the maximum distance between a point in \( S_{1} \) and a point in \( S_{2} \). Given that \( x = 5 \), the difference between the maximum and minimum possible values for \( y \) can be written as \( a + b \sqrt{c} \), where \( a, b \), and \( c \) are integers and \( c \) is positive and square-free. Find \( 100a + 10b + c \).

Answer: 472


1. A rhombus with a side length of 1 and an acute angle of $60^{\circ}$ is covered by three equal circles. Find the smallest possible radius.

The rhombus with a side length of 1 and an acute angle of $60^{\circ}$ is covered by three equal circles. Find the smallest possible radius.

Answer: \dfrac{1}{2}


I Noticed that 2023 China Western Mathematical Olympiad has not added to collection yet so I made it [here](https://artofproblemsolving.com/community/c3513189).
I am not sure if other problems were posted. If it was, please add the link under here, thanks!

Answer: 2023


The sides of rectangle  $ABCD$  have lengths 10 and 11.  An equilateral triangle is drawn so that no point of the triangle lies outside  $ABCD.$   The maximum possible area of such a triangle can be written in the form  $p\sqrt{q}-r,$  where  $p, q,$  and  $r$  are positive integers, and  $q$  is not divisible by the square of any prime number.  Find  $p+q+r.$

Answer: 554


Let  $ABCD$  be a cyclic quadrilateral where  $AB = 4$ ,  $BC = 11$ ,  $CD = 8$ , and  $DA = 5$ . If  $BC$  and  $DA$  intersect at  $X$ , find the area of  $\vartriangle XAB$ .

Answer: 6\sqrt{5}


In each of the  $9$  small circles of the following figure we write positive integers less than  $10$ , without repetitions. In addition, it is true that the sum of the  $5$  numbers located around each one of the  $3$  circles is always equal to  $S$ . Find the largest possible value of  $S$ .
![Image](https://cdn.artofproblemsolving.com/attachments/6/6/2db2c1ac7f45022606fb0099f24e6287977d10.png)

Answer: 28


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

Answer: 13


The cards in a stack of  $2n$  cards are numbered consecutively from  $1$  through  $2n$  from top to bottom. The top  $n$  cards are removed, kept in order, and form pile  $A$ . The remaining cards form pile  $B$ . The cards are then restacked by taking cards alternately from the tops of pile  $B$  and  $A$ , respectively. In this process, card number  $(n+1)$  becomes the bottom card of the new stack, card number  $1$  is on top of this card, and so on, until piles  $A$  and  $B$  are exhausted. If, after the restacking process, at least one card from each pile occupies the same position that it occupied in the original stack, the stack is named *magical*. Find the number of cards in the magical stack in which card number  $131$  retains its original position.

Answer: 130


A right rectangular prism  $P$  (i.e., a rectangular parallelpiped) has sides of integral length  $a, b, c,$  with  $a\le b\le c.$   A plane parallel to one of the faces of  $P$  cuts  $P$  into two prisms, one of which is similar to  $P,$  and both of which have nonzero volume.  Given that  $b=1995,$  for how many ordered triples  $(a, b, c)$  does such a plane exist?

Answer: 40


In an  $m \times n$  grid, each square is either filled or not filled. For each square, its *value* is defined as  $0$  if it is filled and is defined as the number of neighbouring filled cells if it is not filled. Here, two squares are neighbouring if they share a common vertex or side. Let  $f(m,n)$  be the largest total value of squares in the grid. Determine the minimal real constant  $C$  such that   $$ \frac{f(m,n)}{mn} \le C $$  holds for any positive integers  $m,n$ *CSJL*

Answer: 2


The road that goes from the town to the mountain cottage is  $76$  km long. A group of hikers finished it in  $10$  days, never travelling more than  $16$  km in two consecutive days, but travelling at least  $23$  km in three consecutive days. Find the maximum ammount of kilometers that the hikers may have traveled in one day.

Answer: 9


Alice and Bob are each secretly given a real number between 0 and 1 uniformly at random. Alice states, “My number is probably greater than yours.” Bob repudiates, saying, “No, my number is probably greater than yours!” Alice concedes, muttering, “Fine, your number is probably greater than mine.” If Bob and Alice are perfectly reasonable and logical, what is the probability that Bob’s number is actually greater than Alice’s?

Answer: \frac{11}{12}


Among all pairs of real numbers  $(x, y)$  such that  $\sin \sin x = \sin \sin y$  with  $-10 \pi \le x, y \le 10 \pi$ , Oleg randomly selected a pair  $(X, Y)$ .  Compute the probability that  $X = Y$ .

Answer: \frac{1}{20}


Let  $n>2$  be an integer. In a country there are  $n$  cities and every two of them are connected by a direct road. Each road is assigned an integer from the set  $\{1, 2,\ldots ,m\}$  (different roads may be assigned the same number). The *priority* of a city is the sum of the numbers assigned to roads which lead to it. Find the smallest  $m$  for which it is possible that all cities have a different priority.

Answer: 3


There exist positive integers  $N, M$  such that  $N$ 's remainders modulo the four integers  $6, 36,$   $216,$  and  $M$  form an increasing nonzero geometric sequence in that order. Find the smallest possible value of  $M$ .

Answer: 2001


On the evening, more than  $\frac 13$  of the students of a school are going to the cinema. On the same evening, More than  $\frac {3}{10}$  are going to the theatre, and more than  $\frac {4}{11}$  are going to the concert. At least how many students are there in this school?

Answer: 173


A polygon on a coordinate grid is built of  $ 2005$  dominoes  $ 1 \times 2$ . What is the smallest number of sides of an even length such a polygon can have?

Answer: 2


A quadrilateral  $ABCD$  is inscribed in a circle. Suppose that  $|DA| =|BC|= 2$  and $ |AB|  = 4$ . Let  $E $  be the intersection  point of lines  $BC$  and  $DA$ . Suppose that  $\angle AEB = 60^o$  and that  $|CD| <|AB|$ . Calculate the radius of the circle.

Answer: 2


Let  $ABCD$  be a convex quadrilateral such that  $AB=4$ ,  $BC=5$ ,  $CA=6$ , and  $\triangle{ABC}$  is similar to  $\triangle{ACD}$ . Let  $P$  be a point on the extension of  $DA$  past  $A$  such that  $\angle{BDC}=\angle{ACP}$ . Compute  $DP^2$ .

*2020 CCA Math Bonanza Lightning Round #4.3*

Answer: 169


Let  $ ABC$  be a triangle with  $ AB \equal{} AC$  . The angle bisectors of  $ \angle C AB$  and  $ \angle AB C$  meet the sides  $ B C$  and  $ C A$  at  $ D$  and  $ E$  , respectively. Let  $ K$  be the incentre of triangle  $ ADC$ . Suppose that  $ \angle B E K \equal{} 45^\circ$  . Find all possible values of  $ \angle C AB$  .

*Jan Vonk, Belgium, Peter Vandendriessche, Belgium and Hojoo Lee, Korea*

Answer: 90


Let  $\triangle ABC$  be a triangle with  $AB=3$  and  $AC=5$ . Select points  $D, E,$  and  $F$  on  $\overline{BC}$  in that order such that  $\overline{AD}\perp \overline{BC}$ ,  $\angle BAE=\angle CAE$ , and  $\overline{BF}=\overline{CF}$ . If  $E$  is the midpoint of segment  $\overline{DF}$ , what is  $BC^2$ ?

Let  $T = TNYWR$ , and let  $T = 10X + Y$  for an integer  $X$  and a digit  $Y$ . Suppose that  $a$  and  $b$  are real numbers satisfying  $a+\frac1b=Y$  and  $\frac{b}a=X$ . Compute  $(ab)^4+\frac1{(ab)^4}$ .

Answer: 9602


8. If 4 lines in a plane intersect each other pairwise and no three lines are concurrent, then there are $\qquad$ pairs of consecutive interior angles.

Answer: 24


3. In triangle $A B C$, side $B C$ is 19 cm. The perpendicular $D F$, drawn from the midpoint of side $A B$ - point $D$, intersects side $B C$ at point $F$. Find the perimeter of triangle $A F C$, if side $A C$ is $10 \, \text{cm}$.

Answer: 29


3. A circle is inscribed in trapezoid $A B C D$, touching the lateral side $A D$ at point $K$. Find the area of the trapezoid if $A K=16, D K=4$ and $C D=6$.

Answer: 432


Problem 6. In an acute-angled triangle $A B C$ with side $A C=1$, the altitude $B H$ is drawn, in triangle $B H C$ - the bisector $C L$, in triangle $B L C$ - the median $B M$. The line $A L$ intersects side $B C$ at point $K$, and $\angle B H K=\angle M H C=15^{\circ}$. Find the area of quadrilateral $K L H M$.

Answer: \dfrac{3(2 - \sqrt{3})}{8}


6. Let $\mathbb{R}^{+}$ denote the set of all positive real numbers. Find all functions $f$ : $\mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ satisfying for all $x, y \in \mathbb{R}^{+}$ the equation

$$
f(x f(y))=f(x y)+x .
$$

(P. Kaňovský)

Answer: f(x) = x + 1


3. Given two geometric progressions $a_{n}$ and $b_{n}$. It is known that $a_{0}=4$ and the sum of the denominators of these two progressions is 5. Find what $b_{0}$ can be, if it is known that $a_{0}, b_{0}, a_{1}, b_{1}$ are consecutive terms of some arithmetic progression (in that order)

Answer: -2


8. Let $a_{1}, a_{2}, \cdots, a_{105}$ be a permutation of $1,2, \cdots, 105$, satisfying: for any $m \in\{3,5,7\}$, for all $n$ such that $1 \leqslant n<n+m \leqslant 105$, we have $m \mid\left(a_{n+m}-a_{n}\right)$. Then the number of different permutations that meet the requirement is $\qquad$ (answer with a specific number).

Answer: 3628800


2 [ Inequalities problems. Case analysis ]

Sasha invited Petya to visit, saying that he lives in the 10th entrance in apartment No. 333, but forgot to mention the floor. Approaching the building, Petya noticed that the building is nine-story. Which floor should he go to? (The number of apartments on each floor is the same, apartment numbers in the building start from one.)

#

Answer: 3


3. On Eeyore's Birthday, Winnie-the-Pooh, Piglet, and Owl came to visit. When Owl left, the average age in this company decreased by 2 years, and when Piglet left, the average age decreased by another 1 year. How many years older is Owl than Piglet?

Answer: Owl is 6 years older than Piglet.

Answer: 6


For example. (Beauty Giant: 26th Y Students Mathematics Competition Questions) In $\triangle A B C$, $\angle A<\angle C<90^{\circ}<\angle B$, construct the external angle bisectors of $\angle A$ and $\angle B$. If the lengths of these two external angle bisectors (from the vertex to the intersection point with the extended opposite side) are both equal to $A B$, find $\angle A$.

Answer: 12^\circ


Example 1.4. Calculate $\int e^{x} \cos x d x$.

Answer: \frac{e^{x}(\sin x + \cos x)}{2}


6. In the Cartesian coordinate system $x O y$, the point set $K=\{(x, y) \mid x, y=-1,0,1\}$. If three points are randomly selected from $K$, then the probability that the distance between each pair of these three points is no more than 2 is $\qquad$ .

Answer: \dfrac{5}{14}


Let $k\ge 1$ be a positive integer.

We consider $4k$ chips, $2k$ of which are red and $2k$ of which are blue. A sequence of those $4k$ chips can be transformed into another sequence by a so-called move, consisting of interchanging a number (possibly one) of consecutive red chips with an
equal number of consecutive blue chips.  For example, we can move from $r\underline{bb}br\underline{rr}b$ to $r\underline{rr}br\underline{bb}b$ where $r$ denotes a red chip and $b$ denotes a blue chip.

Determine the smallest number $n$ (as a function of $k$) such that starting from any initial sequence of the $4k$ chips, we need at most $n$ moves to reach the state in which the first $2k$ chips are red.

Answer: k


Vasily is going to finish his institute in a year. Only 270 out of 300 third-year students successfully pass the exams and complete their bachelor's degree. If Vasily ends up among the 30 expelled students, he will have to go to work with a salary of 25,000 rubles per month. It is also known that annually every fifth graduate gets a job with a salary of 60,000 rubles per month, every tenth receives 80,000 rubles per month, every 20th graduate cannot get a job in his profession and their average salary is 25,000 rubles per month, and the salary of all others is 40,000 rubles. When Vasily finished school, he could have chosen not to go to university and instead work as a real estate assistant, like his friend Fedor. Fedor's salary increases by 3,000 rubles every year. What is Vasily's expected salary? Whose salary will be higher in a year and by how much — Vasily’s expected salary or Fedor’s actual salary, if Fedor started working with a salary of 25,000 rubles at the same time Vasily entered the university? Note: Bachelor’s education lasts 4 years.

Answer: 45025


Example 7. Reduce the equation

$$
x^{2} \frac{d^{2} y}{d x^{2}}-3 x \frac{d y}{d x}+\left(x^{4}-12\right) y=0
$$

to a Bessel equation and find its general solution.

Answer: y = x^{2} \left[ C_1 J_{2}\left( \frac{x^{2}}{2} \right) + C_2 Y_{2}\left( \frac{x^{2}}{2} \right) \right]


The colonizers of a spherical planet have decided to build $N$ towns, each having area $1/1000$ of the total area of the planet. They also decided that any two points belonging to different towns will have different latitude and different longitude. What is the maximal value of $N$?

Answer: 31


181 Given a regular quadrilateral pyramid with edge lengths and base side lengths both equal to 1. The radius $r=$ $\qquad$ of the escribed sphere that is tangent to the extensions of the four side faces and the base.

Answer: \dfrac{\sqrt{6} + \sqrt{2}}{4}


## 11. How Many Are We?

To help you answer this question, let's say that the probability that at least two of us have the same birthday is less than $1 / 2$, but this would be false if one more person joined us.

Answer: 22


4. There are four positive numbers, one of which is $\frac{1}{2}$. If any two numbers are taken from these four, among the remaining two numbers, there must be one number such that the sum of these three numbers is 1. Then these four numbers are

Answer: \frac{1}{2}


A ship in the fog is attempting to dock at the shore. The crew does not know the direction of the shore but can see a lighthouse located on a small island 1 km from the shore and understands that the distance from the ship to the lighthouse does not exceed 1 km (the exact distance to the lighthouse is unknown). The lighthouse is surrounded by reefs, so approaching it is not possible. Can the ship reach the shore by traveling no more than 1 km? (The shoreline is straight, the trajectory is plotted on a computer display before the ship starts moving, and then the autopilot navigates the ship along this trajectory.)

Answer: Yes


19. Find the smallest number, written only using ones and zeros, that is divisible by 225.

Answer: 11111111100


[u]level C[/u]

p10. Construct the following figure, where $ABCD$ is a rectangle, $PQC$ is equilateral and $2 PD = PA$.
[img]https://cdn.artofproblemsolving.com/attachments/3/d/6fd25309c373e91b8d837df7378725b6b6fd7d.gif[/img]

p11. Let $ABC$ be an isosceles triangle, with base $AB = 10$ cm. Let $M$ and $N$ be the midpoints of the sides $AC$ and $BC$ respectively. Let $G$ be the point of intersection of $BM$ and $NA$. If the angle $AGB$ is right, find the area of $ABC$.

p12. Let $A, B, C$, and $D$ be four collinear points such that $AB = BC = CD$.
Let $P$ be a point on the plane such that $2 \angle APB = \angle BPC = 2 \angle PD$.
Find the measure of angle $\angle APB$.

p13. Let S and $R$ be two circles with centers $O_1$ and $O_2$ respectively. Let $P$ be a point on $S$. A parallel to $O_1O_2$ is drawn by $P$, which intersects $R$ at $A$ and $B$. Lines $PO_1$ and $AO_2$ intersect at point $E$. Find the locus of $E$ as  $P$ moves on $S$.

Answer: 75


The sum of the distances from a point inside an equilateral triangle of perimeter length \( p \) to the sides of the triangle is \( s \). Show that \( s \sqrt{12} = p \).

Answer: s \sqrt{12} = p


9.2. On the board, a certain natural number $N$ was written nine times (one under the other). Petya appended a non-zero digit to the left or right of each of the 9 numbers; all the appended digits are different. What is the maximum number of prime numbers that could result among the 9 obtained numbers?

(I. Efremov)

Answer: 6


3. Felix chooses a positive integer as the starting number and writes it on the board. He then repeatedly performs the following step: he replaces the number $n$ on the board with $\frac{1}{2} n$ if $n$ is even and with $n^{2}+3$ if $n$ is odd.

(a) Prove that if Felix starts with an odd number, the next two numbers he writes will be even.

(b) For how many choices of starting numbers less than 2023 will Felix never write a number with more than four digits on the board?

Answer: 21


3. Solve in the set of integers the equation $x^{2}+16 x+55=3^{y^{2}-2 y}$.

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

Note: The note itself is not part of the translation but is provided to clarify the instruction. Here is the translation as requested:

3. Solve in the set of integers the equation $x^{2}+16 x+55=3^{y^{2}-2 y}$.

Answer: (-2, -1)


7. In the Cartesian coordinate system, the ellipse $\Omega: \frac{x^{2}}{4}+y^{2}=1$, $P$ is a moving point on $\Omega$, $A, B$ are two fixed points, where the coordinates of $B$ are $(0,3)$. If the minimum area of $\triangle P A B$ is 1 and the maximum area is 5, then the length of line segment $A B$ is $\qquad$.

Answer: \sqrt{7}


9.3. The number 2019 is represented as the sum of different odd natural numbers. What is the maximum possible number of addends?

Answer: 43


Example 9 On the square $A B C D$: there are 10 points, 8 of which are inside $\triangle A B C$, and 2 points are on the side $\mathrm{I}$ of the square (not at the vertices). [L These 10 points, together with points $A, B, C, D$, are not collinear. Find how many small triangles these 10 points, along with the 4 vertices of the square, can divide the square into.

Answer: 20


We define a positive integer as "special" if it is divisible by all of its non-zero digits. Determine the maximum number of consecutive special numbers that can exist.

Answer: 12


Let triangle $ABC$ be a right triangle in the $xy$ -plane with a right angle at $C_{}$ . Given that the length of the hypotenuse $AB$ is $60$ , and that the medians through $A$ and $B$ lie along the lines $y=x+3$ and $y=2x+4$ respectively, find the area of triangle $ABC$ .
Please give the answer directly without any intermediate steps.

Answer: 400


6. In a convex quadrilateral $ABCD$, $\angle BAD + \angle ADC = 240^{\circ}$, $E$ and $F$ are the midpoints of sides $AD$ and $BC$, respectively, and $EF = \sqrt{7}$. If two squares $A_1$ and $A_2$ are drawn with sides $AB$ and $CD$, respectively, and a rectangle $A_3$ is drawn with length $AB$ and width $CD$, find the sum of the areas of the three figures $A_1$, $A_2$, and $A_3$.

Answer: 28


Pythagorean Theorem (direct and inverse). Trapezoids (miscellaneous).

In a convex quadrilateral $M N L Q$, the angles at vertices $N$ and $L$ are right angles, and $\operatorname{tg} \angle Q M N = 2 / 3$. Find the diagonal $N Q$, if it is known that side $L Q$ is half the length of side $M N$ and 2 units longer than side $L N$.

Answer: 2\sqrt{13}


Problem 3. Let $\mathcal{C}$ be the set of functions $f:[0,1] \rightarrow \mathbb{R}$, twice differentiable on $[0,1]$, which have at least two zeros, not necessarily distinct, in $[0,1]$, and $\left|f^{\prime \prime}(x)\right| \leq 1$ for all $x$ in $[0,1]$. Determine the maximum value that the integral

$$
\int_{0}^{1}|f(x)| \mathrm{d} x
$$

can take as $f$ ranges over the set $\mathcal{C}$, and the functions that achieve this maximum.

(A function $f$ has two zeros at the same point $a$ if $f(a)=f^{\prime}(a)=0$.)

Answer: \dfrac{1}{6}


Answer: 1


4. Given $O$ as the origin, $M$ as a point on the moving chord $AB$ of the parabola $y^{2}=2 p x$ $(p>0)$. If $O A \perp O B, O M \perp A B$, denote the area of $\triangle O A B$ as $S$, and $O M=h$. Then the range of $\frac{S}{h}$ is . $\qquad$

Answer: [2p, +\infty)


Let $n>0$ be an integer. Each of the $n$ girls in a group is the only one who knows a certain piece of gossip. To share their information, they call each other two at a time, but each time, only one speaks and the other just listens to all the information her friend tells her.

Determine the minimum number of phone calls sufficient for each girl to know all the gossip.

Answer: 2n - 2


3. Given a parallelogram $D A B C$. On the side $D C$ is a point $L$ such that 3. $\overline{D L}=\overline{D C}$, and on the side $D A$ is a point $K$ such that $4 \cdot \overline{D K}=\overline{D A}$. Through the points $L$ and $K$ is drawn a line $p$ which intersects the diagonal $D B$ at point $M$. What fraction of the diagonal $D B$ is the segment $D M$?

Answer: \dfrac{1}{7}


Rosa won a perfume bottle in the shape of a cylinder with a base radius of $7 \mathrm{~cm}$ and a height of $10 \mathrm{~cm}$. After two weeks of using the perfume, $0.45$ liters remained in the bottle. What fraction represents the volume that Rosa has already used?

Answer: \dfrac{49\pi - 45}{49\pi}


A regular three-sided pyramid (its base is an equilateral triangle) is inscribed in a sphere, the diameter of which is nine times the height of the pyramid. What angle do two side faces enclose?

Answer: 120^\circ


12. In a regular tetrahedron $ABCD$, $MN$ is the line segment connecting the midpoint of $AD$ and the center of $\triangle BCD$, and $PQ$ is the line segment connecting the midpoint of $CD$ and the center of $\triangle ABC$. Find the angle formed by the lines $MN$ and $PQ$.

Answer: \arccos\left(\dfrac{1}{18}\right)


Let $M$ be a finite sum of numbers, such that among any three of its elements there are two whose sum belongs to $M$.  Find the greatest possible number of elements of $M$.

Answer: 7


1. Given a convex quadrilateral $A B C D$ in which $\angle B A C=20^{\circ}, \angle C A D=60^{\circ}$, $\angle A D B=50^{\circ}$, and $\angle B D C=10^{\circ}$. Find $\angle A C B$.

Answer: 80


Let \( g(\theta) = \lambda_1 \cos \theta + \lambda_2 \cos 2\theta + \cdots + \lambda_n \cos n\theta \), where \( \lambda_1, \lambda_2, \cdots, \lambda_n \) and \( \theta \) are all real numbers. If \( g(\theta) \geq -1 \) for all real numbers \( \theta \), prove that \( \lambda_1 + \lambda_2 + \cdots + \lambda_n \leq n \).

Answer: \lambda_1 + \lambda_2 + \cdots + \lambda_n \leq n


$$
\begin{aligned}
& \text { [Isosceles, inscribed, and circumscribed trapezoids] } \\
& \text { [ Right triangle with an angle of \$30^\circ\$ ] } \\
& \text { [ Pythagorean theorem (direct and inverse). ] } \\
& \text { [ Equilateral (equiangular) triangle ] }
\end{aligned}
$$

Quadrilateral $A B C D$ is inscribed in a circle with center $O, \angle B O A=\angle C O D=60^{\circ}$. The perpendicular $B K$, dropped to side $A D$, is equal to $6 ; A D=3 B C$.

Find the area of triangle $C O D$.

Answer: \dfrac{63\sqrt{3}}{4}


9. Sleepy Flies (7th grade. 1 point). One day in October, Nikita glanced at the clock hanging on the wall and saw that four flies had fallen asleep on the clock face. The first one was sleeping precisely at the 12 mark, while the others were also precisely positioned at the 3, 6, and 9 marks.

- How beautiful! - Nikita marveled and even felt sorry for the flies, noticing that the hour hand posed no threat to them, but the minute hand would definitely sweep them all away one by one.

Find the probability that exactly 20 minutes after Nikita looked at the clock, two flies had already been swept away.

Answer: \dfrac{1}{3}


A cyclic $n$-gon is divided by non-intersecting (inside the $n$-gon) diagonals to $n-2$ triangles. Each of these triangles is similar to at least one of the remaining ones. For what $n$ this is possible?

Answer: 4


Given an acute triangle \( \triangle ABC \) with its circumcircle denoted as \( \odot O \). The tangent to \( \odot O \) at point \( A \) intersects \( BC \) at point \( D \). Let \( P \) be any point on the line \( OD \). Perpendiculars from \( P \) to \( AB \) and \( AC \) intersect these sides at points \( M \) and \( N \), respectively. Prove that the orthocenter \( H \) of triangle \( PMN \) lies on \( BC \).

Answer: H \text{ lies on } BC


A partition of a positive integer is even if all its elements are even numbers. Similarly, a partition
is odd if all its elements are odd. Determine all positive integers  $n$  such that the number of even partitions of $n$  is equal to the number of odd partitions of  $n$ .
Remark: A partition of a positive integer  $n$  is a non-decreasing sequence of positive integers whose sum of
elements equals  $n$ . For example,  $(2; 3; 4), (1; 2; 2; 2; 2)$  and  $(9) $  are partitions of  $9.$

Answer: 2


6. Let the side length of rhombus $A_{1} A_{2} A_{3} A_{4}$ be $1, \angle A_{1} A_{2} A_{3}=$ $\frac{\pi}{6}, P$ be a point in the plane of rhombus $A_{1} A_{2} A_{3} A_{4}$. Then the minimum value of $\sum_{1 \leqslant i<j \leqslant 4} \overrightarrow{P A_{i}} \cdot \overrightarrow{P A_{j}}$ is $\qquad$ .

Answer: -1


# Task No. 7.1

## Condition:

On the Misty Planet, santiks, kubriks, and tugriks are in circulation. One santik can be exchanged for 1 kubrik or 1 tugrik, 1 kubrik can be exchanged for 3 santiks, and 1 tugrik can be exchanged for 4 santiks. No other exchanges are allowed. Jolly U, initially having 1 santik, made 20 exchanges, and now he has 25 santiks (and no other money). How many times did he exchange kubriks for santiks?

Answer: 6


10. The Seventh Story

- Here's a frying pan, - said the King. Will you finally bake me some pretzels or not?
- Without a recipe? - asked the Queen.
- Use your usual recipe, shouted the King impatiently. - Last time your pretzels turned out wonderfully!

![](https://cdn.mathpix.com/cropped/2024_05_21_5afa4e43618f5800e92dg-025.jpg?height=745&width=948&top_left_y=264&top_left_x=514)

- I can't, - answered the Queen. - Last time I baked pretzels using a recipe from my cookbook, and it was stolen!

The greatest suspicion fell on the Duchess's cook. The cookbook was indeed found in the Duchess's kitchen. It could have been stolen only by the cook, the Duchess, and the Cheshire Cat.

- The cookbook was stolen by the Cheshire Cat! - declared the Duchess at the trial.
- Oh yes, I did steal the cookbook! - agreed the Cheshire Cat with a smile.
- I didn't steal the cookbook! - asserted the cook.

It turned out that the one who stole the cookbook was lying, and at least one of the other accused told the truth.

Who stole the cookbook?

Answer: The\ Duchess


Is it possible to use the digits \(1,2,3,4,5\) to form one two-digit number and one three-digit number such that the second number is divisible by the first? Each digit must be used exactly once.

Answer: \text{Yes}


127. A truncated cone is described around a sphere. The total surface area of this cone is $S$. A second sphere touches the lateral surface of the cone along the circumference of the cone's base. Find the volume of the truncated cone, given that the part of the surface of the second sphere that is inside the first has an area of $Q$.

Answer: \dfrac{S \sqrt{Q}}{3 \sqrt{\pi}}


Given four positive numbers: $a, b, c, d$. Among the products $a b, a c, a d, b c, b d, c d$, we know the values of five of them, which are 2, 3, 4, 5, and 6. What is the value of the sixth product?

Answer: \dfrac{12}{5}


Let $P$ be a polynomial of degree greater than or equal to 2 and $\left(q_{n}\right)_{n \geq 0}$ a sequence of rational numbers satisfying $q_{n}=P\left(q_{n+1}\right)$ for all $n \geq 0$. Prove that the sequence $\left(q_{n}\right)$ is periodic from a certain rank.

Answer: \text{The sequence }(q_n)\text{ is periodic from a certain rank.}


Example 2 (17th All-Soviet Union Mathematical Olympiad, 9th Grade) One side of a rectangle is $1 \mathrm{~cm}$. It is known that it is divided into four smaller rectangles by two perpendicular lines, with the area of three of them being no less than $1 \mathrm{~cm}^{2}$, and the area of the fourth one being no less than $2 \mathrm{~cm}^{2}$. How long must the other side of the original rectangle be at least?

Answer: 3 + 2\sqrt{2}


XXV - I - Task 1

During World War I, a battle took place near a certain castle. One of the shells destroyed a statue of a knight with a spear standing at the entrance to the castle. This happened on the last day of the month. The product of the day of the month, the month number, the length of the spear expressed in feet, half the age of the battery commander firing at the castle expressed in years, and half the time the statue stood expressed in years equals 451,066. In which year was the statue erected?

Answer: 1714


Problem 3. Determine the natural numbers $p$ with the property that the sum of the first $p$ non-zero natural numbers is a four-digit natural number with prime factorization $2^{m} \cdot 3^{n} \cdot (m+n)$, where $m, n \in \mathbb{N}^{*}$.

Answer: 63


8. (10 points) On the beach, there is a pile of chestnuts, which belong to four monkeys. They want to divide them equally. The first monkey came, waited and waited but the other monkeys did not come, so it divided the chestnuts into four equal piles, with one left over. It threw the leftover one into the sea and took one of the four piles. The second monkey came, and without waiting for the others, it divided the remaining chestnuts into four equal piles, with one left over. It threw away the leftover one and took one pile. The third monkey did the same, dividing the remaining chestnuts into four equal piles, throwing away the leftover one, and taking one pile. The last monkey also divided the remaining chestnuts into four equal piles, threw away the leftover one, and took one pile. So, the original pile of chestnuts had at least $\qquad$ chestnuts.

Answer: 253


\section*{Problem 2 - 261032}

In a domino game with the numbers \(0,1, \ldots, 6\), each tile is divided into two halves, each half bearing one of the numbers.

In a domino game, all combinations of two of the numbers \(0,1, \ldots, 6\) appear exactly once (including those where the same number appears on both halves of a tile). According to this, a domino game consists of exactly 28 tiles.

A chain is formed when multiple tiles are placed next to each other in a sequence such that adjacent halves of neighboring tiles always bear the same number (domino game rule).

A chain is called closed if the two outer halves at the free ends of the chain also bear the same number.

a) Determine the smallest number \(k>1\) of different tiles from a domino game that can form a closed chain!

b) From a domino game, closed chains consisting of \(k\) different tiles each are to be formed (\(k\) being the number mentioned in a)). Each tile of the domino game should be used in at most one of these chains.

Determine the maximum number \(g\) of chains that can be formed in this way!

c) Determine the number of all different closed chains consisting of \(k\) different tiles that can be formed! (Thus, each tile of the domino game can appear in any number of these chains.)

Two closed chains are considered the same if they show the same sequence of numbers when a suitable starting tile and direction are chosen.

For example, the chains \((2,4),(4,5),(5,5),(5,1),(1,2)\) and \((5,4),(4,2),(2,1),(1,5),(5,5)\) are considered the same.

Answer: 3


Let $\mathbb{N}=\{1,2,3, \ldots\}$ be the set of positive integers. Find all functions $f$, defined on $\mathbb{N}$ and taking values in $\mathbb{N}$, such that $(n-1)^{2}<f(n) f(f(n))<n^{2}+n$ for every positive integer $n$.

Answer: f(n) = n


4. Let the sequence of positive numbers be $\left\{a_{n}\right\}$, with the sum of the first $n$ terms being $S_{n}$, and there exists a positive number $t$, such that for all natural numbers $n$, $\sqrt{t S_{n}}=\frac{t+a_{n}}{2}$ holds. If $\lim _{n \rightarrow \infty} \frac{\sqrt{S_{n}}}{a_{n}}<t$, then the range of values for $t$ is

Answer: (\sqrt[3]{\frac{1}{4}}, \infty)


11. A. For the function $y=x^{2}+(2 k-1) x+k^{2}$, are the two intersection points with the $x$-axis both to the left of the line $x=1$? If so, please explain the reason; if not, find the range of values for $k$ when both intersection points are to the right of the line $x=1$.

Answer: (-\infty, -2)


Example 1-14 There are 7 male athletes and 3 female athletes lining up to enter the venue. If it is required that the first and last athletes must be male, and female athletes must not be adjacent, how many arrangements are possible? If the female athletes are together and placed at either the beginning or the end of the line, how many arrangements are possible? How many arrangements are possible if the female athletes are together, regardless of their position?

Answer: 604800


11. Andrew, Bob, and Chris are working together to finish a group project. If Andrew doesn't help, it would take 2 hours. If Bob doesn't help, it would take 3 hours. If Chris doesn't help, it would take 4 hours. Now they just learned that Dave can also help them out. If Dave works on the project alone, it would take him half a day. What is the least amount of time it would take for them to finish the group project?

Answer: \dfrac{8}{5}


1. Let $a_{1}, a_{2}, \ldots, a_{n}$ be integers $(n>1)$ satisfying $a_{1}+a_{2}+\cdots+a_{n}=a_{1} a_{2} \cdots a_{n}=2005$. Find the smallest possible value of $n$.
(1 mark)
Let $a_{1}, a_{2}, \ldots, a_{n}$ be integers $(n>1)$, such that $a_{1}+a_{2}+\cdots+a_{n}=a_{1} a_{2} \cdots a_{n}=2005$. Find the smallest possible value of $n$.

Answer: 5


From all the numbers with a digit sum of 25, find the one whose product of digits is maximum. If there are several such numbers, provide the smallest one among them.

Answer: 33333334


Example 28 (2005 British Mathematical Olympiad) Determine the smallest possible value of the largest term in an arithmetic sequence consisting of 7 distinct prime numbers.

Determine the smallest possible value of the largest term in an arithmetic sequence consisting of 7 distinct prime numbers.

Answer: 907


26. For each integer $n \geqslant 2$, determine the minimum value of
$$
a_{0}+a_{1}+\cdots+a_{n}
$$
satisfying the conditions
$$
a_{0}=1, a_{i} \leqslant a_{i+1}+a_{i+2}, i=0,1, \cdots, n-2
$$
where $a_{0}, a_{1}, \cdots, a_{n}$ are non-negative numbers.

Answer: \dfrac{F_{n+2} - 1}{F_n}


In the convex quadrilateral $ABCD$, the intersection points of the opposite sides are $K$ and $L$, and the intersection point of the diagonals is $M$. The diagonal $AC$ intersects the line $KL$ at point $N$. Determine the length of $CN$ if it is known that $AM=3 \text{~cm}$ and $MC=2 \text{~cm}$.

Answer: 10


1. People numbered $1,2, \cdots, 2015$ are arranged in a line, and a position-swapping game is played among them, with the rule that each swap can only occur between adjacent individuals. Now, the person numbered 100 and the person numbered 1000 are to swap positions, with the minimum number of swaps required being $\qquad$ times.

Answer: 1799


Let $n$ be a positive integer. $2n+1$ tokens are in a row, each being black or white. A token is said to be [i]balanced[/i] if the number of white tokens on its left plus the number of black tokens on its right is $n$. Determine whether the number of [i]balanced[/i] tokens is even or odd.

Answer: odd


1. On the island of knights and liars (liars always lie, knights always tell the truth), each resident supports exactly one football team. In a survey, all residents of the island participated. To the question "Do you support 'Rostov'?", 40% of the residents answered "Yes". To a similar question about 'Zенit', 30% answered affirmatively, about 'Locomotive' - 50%, and about CSKA - 0%. What percentage of the island's residents actually support 'Rostov'?

Answer: 30


3. The sequence $c_{n}$ is the sum of a geometric progression $b_{n}$ with the first term 3 and some arithmetic progression. It is known that $c_{1}+c_{3}=141, c_{2}=33$. Find the common ratio of the progression $b_{n}$. If there are multiple possible answers, write them in any order separated by a semicolon.

Answer: -4


Petya wants to write out all possible sequences of 100 natural numbers, each of which contains the number three at least once, and any two neighboring members differ by no more than 1. How many sequences will he have to write out?

Answer: 3^{100} - 2^{100}


4. From the midpoint $O$ of the height $SE$ of a right regular quadrilateral pyramid with apex $S$ and foot $E$ of the height, perpendiculars $OM$ are dropped to the lateral edge and $OK$ to the lateral face of the pyramid. If the lengths of these perpendiculars are $\overline{OM}=p$ and $\overline{OK}=q$, calculate the volume of the pyramid.

Answer: \dfrac{16p^3q^3}{3(p^2 - q^2)\sqrt{2q^2 - p^2}}


Exercise 1. The integers $1,2, \ldots, 2018$ are written on the board. Then 2017 operations are performed as follows: choose two numbers $a$ and $b$, erase them, and write $a+b+2ab$ in their place. At the end, only one integer remains on the board.

What are the possible values that the units digit of this integer can take?

Answer: 2


Let \( n \) be a natural number. For any real numbers \( x, y, z \), if the inequality \(\left(x^{2}+y^{2}+z^{2}\right) \leqslant n\left(x^{4}+y^{4}+z^{4}\right)\) always holds, then the smallest value of \( n \) is _____.

Answer: 3


6. As shown in the figure, in right $\triangle A C B$, $C D$ is the altitude on the hypotenuse $A B$, and $D E$ is the altitude on the hypotenuse $B C$ of $\mathrm{Rt} \triangle C D B$. If $B E=$ $6, C E=4$, then the length of $A D$ is $\qquad$

Answer: \dfrac{4\sqrt{15}}{3}


2. The sequence $\left\{x_{n}\right\}$ satisfies
$$
\begin{array}{l}
x_{0}=1, x_{1}=6, \\
x_{2}=x_{1}+\sin x_{1}=5.72 \cdots, \\
x_{3}=x_{2}+\cos x_{2}=6.56 \cdots .
\end{array}
$$

In general,
if $x_{n} \geqslant x_{n-1}$, then $x_{n+1}=x_{n}+\sin x_{n}$;
if $x_{n}<x_{n-1}$, then $x_{n+1}=x_{n}+\cos x_{n}$. For any $n \in \mathbf{Z}_{+}$, find the smallest constant $c$ such that $x_{n}<c$.

Answer: 3\pi


## Task B-3.7.

The lengths of the sides of a cyclic quadrilateral $A B C D$ are $|A B|=3 \mathrm{~cm},|B C|=4 \mathrm{~cm},|C D|=5$ $\mathrm{cm}$ and $|D A|=6 \mathrm{~cm}$. Determine the cosines of the acute angles and the area of the quadrilateral $A B C D$.

Answer: \dfrac{3}{7}


Authors: Berlov S.L., Petrov F., Akopyan A.V.

On the side $B C$ of the rhombus $A B C D$, a point $M$ is chosen. Lines drawn through $M$ perpendicular to the diagonals $B D$ and $A C$ intersect the line $A D$ at points $P$ and $Q$ respectively. It turns out that the lines $P B, Q C$ and $A M$ intersect at one point. What can the ratio $B M: M C$ be?

Answer: \dfrac{1}{2}


9. In the tetrahedron $O A B C$, $\angle A O B=45^{\circ}, \angle A O C=\angle B O C=30^{\circ}$, then the cosine value of the dihedral angle $A-O C-B$'s plane angle $\alpha$ is $\qquad$.

Answer: 2\sqrt{2} - 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: `Arnold`, `Eric`, `Alice`, `Peter`
 - Each person has a unique birthday month: `feb`, `jan`, `april`, `sept`
 - People have unique hair colors: `black`, `blonde`, `brown`, `red`
 - Each person lives in a unique style of house: `colonial`, `ranch`, `victorian`, `craftsman`
 - Each person has an occupation: `engineer`, `artist`, `doctor`, `teacher`
 - Everyone has a favorite smoothie: `cherry`, `dragonfruit`, `desert`, `watermelon`

## Clues:
1. The person who is an engineer is the person whose birthday is in February.
2. The person who likes Cherry smoothies is somewhere to the left of the person who has red hair.
3. The person who is a doctor is the person who has blonde hair.
4. The person whose birthday is in January is the person living in a colonial-style house.
5. Alice is the person residing in a Victorian house.
6. Peter is the Desert smoothie lover.
7. Peter is the person whose birthday is in September.
8. The Dragonfruit smoothie lover is the person who is a teacher.
9. The Watermelon smoothie lover is the person residing in a Victorian house.
10. The person living in a colonial-style house is directly left of the person who has black hair.
11. The person whose birthday is in February is in the fourth house.
12. The person who has black hair is not in the third house.
13. Eric is the person in a ranch-style home.
14. The person who has blonde hair is directly left of the person who likes Cherry smoothies.
 What is the value of attribute Birthday for the person whose attribute Smoothie is desert? Please reason step by step, and put your final answer within \boxed{}

Answer: sept


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`, `Bob`, `Arnold`, `Carol`, `Eric`, `Peter`
 - People use unique phone models: `xiaomi mi 11`, `google pixel 6`, `samsung galaxy s21`, `oneplus 9`, `huawei p50`, `iphone 13`
 - People have unique favorite book genres: `romance`, `historical fiction`, `fantasy`, `mystery`, `science fiction`, `biography`
 - They all have a unique favorite flower: `tulips`, `roses`, `carnations`, `daffodils`, `iris`, `lilies`
 - The people keep unique animals: `dog`, `cat`, `bird`, `fish`, `rabbit`, `horse`
 - Each person has a unique level of education: `trade school`, `high school`, `master`, `associate`, `bachelor`, `doctorate`

## Clues:
1. The bird keeper is the person who uses a Google Pixel 6.
2. The person who loves a bouquet of daffodils is the person with an associate's degree.
3. The person who uses a OnePlus 9 is not in the fourth house.
4. There are two houses between the person with a master's degree and the rabbit owner.
5. The person who loves a carnations arrangement is Arnold.
6. The person who loves the boquet of lilies is not in the sixth house.
7. The person who loves the vase of tulips is Carol.
8. The person with a high school diploma is Bob.
9. The person who attended trade school is somewhere to the left of the person who keeps horses.
10. The person who loves mystery books is Eric.
11. The fish enthusiast is directly left of Carol.
12. Bob is the fish enthusiast.
13. Eric is the dog owner.
14. The person who loves a carnations arrangement is the person who uses a Google Pixel 6.
15. The person who loves fantasy books is in the second house.
16. The person who uses a Samsung Galaxy S21 is the person who loves the boquet of iris.
17. Alice is the person with an associate's degree.
18. The person who uses a Google Pixel 6 is not in the second house.
19. The dog owner is the person with a doctorate.
20. The fish enthusiast is the person who uses a Samsung Galaxy S21.
21. The person who uses an iPhone 13 is somewhere to the right of the person who loves fantasy books.
22. The person who loves mystery books is somewhere to the right of the person who loves the boquet of iris.
23. Alice is the person who uses an iPhone 13.
24. The person who attended trade school is not in the fourth house.
25. The person who loves biography books is directly left of Eric.
26. There is one house between the rabbit owner and the person who loves historical fiction books.
27. The person who loves the boquet of iris is somewhere to the right of the person who uses a Xiaomi Mi 11.
28. The person who loves science fiction books is the person who uses a Samsung Galaxy S21.
29. The person who loves the boquet of lilies is not in the second house.
 What is the value of attribute Animal for the person whose attribute Name is Bob? 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: `Peter`, `Eric`, `Alice`, `Bob`, `Arnold`, `Carol`
 - People have unique favorite book genres: `mystery`, `fantasy`, `romance`, `historical fiction`, `science fiction`, `biography`
 - Everyone has a favorite smoothie: `cherry`, `desert`, `lime`, `watermelon`, `blueberry`, `dragonfruit`
 - The people keep unique animals: `fish`, `rabbit`, `bird`, `cat`, `horse`, `dog`
 - People have unique favorite music genres: `classical`, `hip hop`, `country`, `jazz`, `rock`, `pop`
 - Everyone has a unique favorite cigar: `prince`, `dunhill`, `blends`, `pall mall`, `blue master`, `yellow monster`

## Clues:
1. The person who smokes many unique blends is the person who loves fantasy books.
2. The person who loves biography books is the Desert smoothie lover.
3. There are two houses between the person who loves pop music and the person who loves country music.
4. The rabbit owner is in the third house.
5. Arnold is the bird keeper.
6. The person who loves country music is in the fifth house.
7. The person who loves mystery books is not in the first house.
8. The person who drinks Lime smoothies is not in the third house.
9. The fish enthusiast is somewhere to the right of the person who loves rock music.
10. The Prince smoker is somewhere to the left of the person who likes Cherry smoothies.
11. Bob is the cat lover.
12. Eric is the person who likes Cherry smoothies.
13. The person who smokes Yellow Monster is in the sixth house.
14. The Prince smoker is the Dragonfruit smoothie lover.
15. Carol is directly left of Arnold.
16. The person who smokes Blue Master is directly left of the person who loves classical music.
17. The person who loves classical music is the person who loves historical fiction books.
18. The Dunhill smoker and the person who loves science fiction books are next to each other.
19. The person who keeps horses is directly left of Peter.
20. The Watermelon smoothie lover is the cat lover.
21. The person who smokes many unique blends is directly left of the person who loves jazz music.
22. The cat lover is in the first house.
23. The Desert smoothie lover is the person who loves jazz music.
24. The person who drinks Lime smoothies is directly left of the fish enthusiast.
25. The person who smokes Blue Master is not in the fifth house.
 What is the value of attribute House for the person whose attribute Name is Peter? Please reason step by step, and put your final answer within \boxed{}

Answer: 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`, `Peter`, `Arnold`, `Alice`
 - Each person lives in a unique style of house: `victorian`, `ranch`, `colonial`, `craftsman`
 - Each person has an occupation: `teacher`, `engineer`, `doctor`, `artist`
 - Everyone has a unique favorite cigar: `prince`, `dunhill`, `pall mall`, `blue master`
 - People have unique favorite music genres: `pop`, `rock`, `classical`, `jazz`
 - People have unique heights: `very short`, `tall`, `average`, `short`

## Clues:
1. The person who loves classical music is the person living in a colonial-style house.
2. The person who smokes Blue Master is the person in a ranch-style home.
3. The person who loves rock music is the person who smokes Blue Master.
4. The Dunhill smoker is directly left of Peter.
5. Eric is in the first house.
6. The person residing in a Victorian house is in the third house.
7. The person who loves rock music and the person who loves classical music are next to each other.
8. The person who is a teacher is the person who is short.
9. The person who loves classical music is the Prince smoker.
10. The person who is a doctor is somewhere to the left of the person who is very short.
11. Arnold is the person who is an artist.
12. The person who smokes Blue Master is in the first house.
13. The person who is tall is directly left of the person who loves pop music.
14. The person residing in a Victorian house is the person who is a teacher.
 What is the value of attribute Occupation for the person whose attribute HouseStyle is victorian? Please reason step by step, and put your final answer within \boxed{}

Answer: teacher


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `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 Name for the person whose attribute Smoothie is desert? Please reason step by step, and put your final answer within \boxed{}

Answer: Peter


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

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

Answer: red


There are 2 houses, numbered 1 to 2 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Eric`
 - The people keep unique animals: `horse`, `cat`
 - Each person has a favorite color: `yellow`, `red`
 - Everyone has a favorite smoothie: `cherry`, `desert`
 - People have unique favorite book genres: `science fiction`, `mystery`
 - Everyone has something unique for lunch: `pizza`, `grilled cheese`

## Clues:
1. Eric is the person whose favorite color is red.
2. The Desert smoothie lover is in the second house.
3. The cat lover is the person who loves eating grilled cheese.
4. Arnold is in the second house.
5. The person who keeps horses is somewhere to the left of the person who loves mystery books.
 What is the value of attribute House for the person whose attribute Animal is horse? Please reason step by step, and put your final answer within \boxed{}

Answer: 1


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Bob`, `Arnold`, `Carol`, `Eric`, `Peter`
 - People use unique phone models: `xiaomi mi 11`, `google pixel 6`, `samsung galaxy s21`, `oneplus 9`, `huawei p50`, `iphone 13`
 - People have unique favorite book genres: `romance`, `historical fiction`, `fantasy`, `mystery`, `science fiction`, `biography`
 - They all have a unique favorite flower: `tulips`, `roses`, `carnations`, `daffodils`, `iris`, `lilies`
 - The people keep unique animals: `dog`, `cat`, `bird`, `fish`, `rabbit`, `horse`
 - Each person has a unique level of education: `trade school`, `high school`, `master`, `associate`, `bachelor`, `doctorate`

## Clues:
1. The bird keeper is the person who uses a Google Pixel 6.
2. The person who loves a bouquet of daffodils is the person with an associate's degree.
3. The person who uses a OnePlus 9 is not in the fourth house.
4. There are two houses between the person with a master's degree and the rabbit owner.
5. The person who loves a carnations arrangement is Arnold.
6. The person who loves the boquet of lilies is not in the sixth house.
7. The person who loves the vase of tulips is Carol.
8. The person with a high school diploma is Bob.
9. The person who attended trade school is somewhere to the left of the person who keeps horses.
10. The person who loves mystery books is Eric.
11. The fish enthusiast is directly left of Carol.
12. Bob is the fish enthusiast.
13. Eric is the dog owner.
14. The person who loves a carnations arrangement is the person who uses a Google Pixel 6.
15. The person who loves fantasy books is in the second house.
16. The person who uses a Samsung Galaxy S21 is the person who loves the boquet of iris.
17. Alice is the person with an associate's degree.
18. The person who uses a Google Pixel 6 is not in the second house.
19. The dog owner is the person with a doctorate.
20. The fish enthusiast is the person who uses a Samsung Galaxy S21.
21. The person who uses an iPhone 13 is somewhere to the right of the person who loves fantasy books.
22. The person who loves mystery books is somewhere to the right of the person who loves the boquet of iris.
23. Alice is the person who uses an iPhone 13.
24. The person who attended trade school is not in the fourth house.
25. The person who loves biography books is directly left of Eric.
26. There is one house between the rabbit owner and the person who loves historical fiction books.
27. The person who loves the boquet of iris is somewhere to the right of the person who uses a Xiaomi Mi 11.
28. The person who loves science fiction books is the person who uses a Samsung Galaxy S21.
29. The person who loves the boquet of lilies is not in the second house.
 What is the value of attribute Flower for the person whose attribute Education is trade school? Please reason step by step, and put your final answer within \boxed{}

Answer: roses


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Eric`, `Arnold`
 - Everyone has a unique favorite cigar: `blue master`, `prince`, `pall mall`
 - Each mother is accompanied by their child: `Fred`, `Meredith`, `Bella`
 - The mothers' names in different houses are unique: `Holly`, `Aniya`, `Janelle`
 - The people keep unique animals: `cat`, `horse`, `bird`

## Clues:
1. The Prince smoker is The person whose mother's name is Janelle.
2. The person's child is named Bella is somewhere to the left of the person's child is named Meredith.
3. Peter is the person partial to Pall Mall.
4. There is one house between the cat lover and Eric.
5. The person whose mother's name is Aniya is the person's child is named Bella.
6. The person's child is named Meredith is Peter.
7. The bird keeper is somewhere to the right of Peter.
 What is the value of attribute Animal for the person whose attribute Children is Meredith? Please reason step by step, and put your final answer within \boxed{}

Answer: horse


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Arnold`, `Eric`, `Peter`, `Bob`, `Carol`
 - Everyone has a favorite smoothie: `cherry`, `blueberry`, `desert`, `lime`, `dragonfruit`, `watermelon`
 - People have unique favorite music genres: `rock`, `classical`, `jazz`, `pop`, `country`, `hip hop`
 - People have unique favorite book genres: `science fiction`, `romance`, `historical fiction`, `fantasy`, `mystery`, `biography`
 - Each person has a favorite color: `yellow`, `white`, `green`, `blue`, `red`, `purple`

## Clues:
1. The person who loves purple and the Dragonfruit smoothie lover are next to each other.
2. The person who loves classical music is in the fifth house.
3. Eric is somewhere to the left of the person who loves classical music.
4. The person who loves hip-hop music is directly left of the Desert smoothie lover.
5. The person who loves yellow is the person who loves science fiction books.
6. Arnold is the person who loves rock music.
7. Carol is the person who likes Cherry smoothies.
8. There are two houses between the person who loves jazz music and the Watermelon smoothie lover.
9. The person whose favorite color is green is somewhere to the right of the person who loves jazz music.
10. The person who drinks Blueberry smoothies is directly left of the Watermelon smoothie lover.
11. Arnold is somewhere to the left of the Watermelon smoothie lover.
12. Bob is somewhere to the left of the person who loves yellow.
13. Bob is the person who loves historical fiction books.
14. The person who loves biography books is the person who loves pop music.
15. The person who loves romance books is in the fifth house.
16. There are two houses between Bob and the person who loves mystery books.
17. The person who loves white is the person who loves historical fiction books.
18. Peter is somewhere to the left of the person who loves blue.
19. There is one house between the person who loves biography books and the person who loves yellow.
 What is the value of attribute Smoothie for the person whose attribute Color is blue? Please reason step by step, and put your final answer within \boxed{}

Answer: watermelon


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`
 - People have unique favorite sports: `soccer`, `baseball`, `swimming`, `tennis`, `basketball`
 - Each person lives in a unique style of house: `modern`, `victorian`, `colonial`, `craftsman`, `ranch`
 - Each person prefers a unique type of vacation: `beach`, `city`, `cruise`, `mountain`, `camping`
 - Each person has an occupation: `artist`, `engineer`, `lawyer`, `doctor`, `teacher`
 - The mothers' names in different houses are unique: `Penny`, `Aniya`, `Janelle`, `Holly`, `Kailyn`

## Clues:
1. There is one house between The person whose mother's name is Janelle and Eric.
2. Arnold is not in the fifth house.
3. The person whose mother's name is Janelle is Peter.
4. The person who enjoys mountain retreats is The person whose mother's name is Holly.
5. The person whose mother's name is Aniya is the person in a modern-style house.
6. The person who likes going on cruises is The person whose mother's name is Kailyn.
7. The person who prefers city breaks is the person living in a colonial-style house.
8. The person who loves soccer is The person whose mother's name is Holly.
9. The person who is an artist is Eric.
10. The person who is a doctor is in the second house.
11. The person who is a teacher is somewhere to the left of the person in a modern-style house.
12. The person whose mother's name is Penny is not in the second house.
13. The person in a ranch-style home is in the third house.
14. The person who loves swimming is The person whose mother's name is Kailyn.
15. Arnold is not in the third house.
16. The person who loves swimming is in the fourth house.
17. The person who loves baseball and the person who prefers city breaks are next to each other.
18. The person who loves basketball is somewhere to the left of the person who loves soccer.
19. Bob is somewhere to the left of the person in a modern-style house.
20. The person who enjoys camping trips is in the fifth house.
21. The person in a Craftsman-style house is the person who is an engineer.
22. There are two houses between the person living in a colonial-style house and the person in a modern-style house.
 What is the value of attribute Mother for the person whose attribute Name is Arnold? Please reason step by step, and put your final answer within \boxed{}

Answer: Penny


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Eric`, `Alice`, `Peter`
 - Each mother is accompanied by their child: `Samantha`, `Meredith`, `Bella`, `Fred`
 - Everyone has a favorite smoothie: `dragonfruit`, `cherry`, `watermelon`, `desert`
 - Everyone has something unique for lunch: `stew`, `grilled cheese`, `spaghetti`, `pizza`
 - The mothers' names in different houses are unique: `Kailyn`, `Holly`, `Janelle`, `Aniya`
 - They all have a unique favorite flower: `daffodils`, `carnations`, `roses`, `lilies`

## Clues:
1. The Watermelon smoothie lover is the person's child is named Bella.
2. The person whose mother's name is Aniya and the person's child is named Fred are next to each other.
3. Alice is the person's child is named Meredith.
4. Eric is the person who likes Cherry smoothies.
5. The person who loves a bouquet of daffodils is directly left of The person whose mother's name is Janelle.
6. The person who loves the spaghetti eater is in the first house.
7. The person who likes Cherry smoothies is the person who loves the boquet of lilies.
8. The person's child is named Meredith is The person whose mother's name is Kailyn.
9. There is one house between The person whose mother's name is Kailyn and The person whose mother's name is Janelle.
10. Arnold is in the second house.
11. The person who loves eating grilled cheese is somewhere to the right of The person whose mother's name is Aniya.
12. The person whose mother's name is Janelle is somewhere to the right of The person whose mother's name is Kailyn.
13. The person who loves the stew is the person who loves a carnations arrangement.
14. Eric is somewhere to the left of the Dragonfruit smoothie lover.
 What is the value of attribute Name for the person whose attribute Flower is carnations? Please reason step by step, and put your final answer within \boxed{}

Answer: Peter