3. $A B C A_{1} B_{1} C_{1}$ - a right triangular prism with a circumscribed sphere. The perimeter of the base $A B C$ is 32 units, and the product of the sides is 896 cubic units. The surface area of the prism is 192 square units. Find the square of the radius of its circumscribed sphere.

Answer: 53


Question 228, Set $S=\{1,2, \ldots, 10\}$ has several five-element subsets satisfying: any two elements in $S$ appear together in at most two five-element subsets. Ask: What is the maximum number of five-element subsets?

Answer: 8


## Zadatak B-1.6.

U nizu brojeva

$$
20,202,2020,20202,202020, \ldots
$$

svaki se sljedeći broj dobije dopisivanjem znamenke 2 ili 0 prethodnom broju, naizmjence. Izračunajte zbroj znamenaka prvih sto brojeva toga niza koji su djeljivi s 202.

Answer: 10200


7. In how many different ways can we fill the 9 fields of a $3 \times 3$ table with the first 9 natural numbers (one in each field) so that the sums of the numbers written in each row of the table are equal?

The use of a pocket calculator or any reference materials is not allowed.

Answer: 2592


8.2. Find the largest natural number with all distinct digits such that the sum of any two of its digits is a prime number.

Answer: 520


12.070. The lateral edges of a regular triangular pyramid are pairwise perpendicular. Find the angle between a lateral face and the plane of the base.

Answer: \arctan{\sqrt{2}}


The National Bank of Hungary withdrew the 1 and 2 forint coins from circulation on March 1, 2008. In cash purchases in stores, the prices of individual items are not rounded separately, but the final amount is rounded to the nearest five. The following table can be seen in the issued information about the rounding rules:

amounts ending in 1 or 2 are rounded down to 0;

amounts ending in 3 or 4 are rounded up to 5;

amounts ending in 6 or 7 are rounded down to 5;

amounts ending in 8 or 9 are rounded up to 0.

Marci buys two croissants every morning at the local bakery. After a few days, due to rounding, he saves exactly the price of one croissant. We know that the price of a croissant is more than 10 Ft. What could be the unit price of the croissant if this saving occurs as quickly as possible?

Answer: 16


11.58 (1) In a square consisting of $7 \times 7$ cells, mark the centers of $k$ cells such that no four of the marked points form the vertices of a rectangle with sides parallel to the grid lines. Find the maximum value of $k$.
(2) Solve the same problem for a square consisting of $13 \times 13$ cells.

Answer: 21


4. The following infinite sequence composed of 1s and 2s has an interesting property: starting from the first term, group the same consecutive numbers together, and then write down the number of terms in each group in order. The infinite sequence formed by these numbers is exactly the sequence itself. This sequence is called the Kolakoski sequence. According to this property, continue to write the next 8 terms of this sequence (from the 13th term to the 20th term).

Answer: 1


Josefina the Quail dances by the marsh, using steps of double length — short steps measure $45 \mathrm{~cm}$, long steps $60 \mathrm{~cm}$. Over time, she has worn an oval path, which she dances around repeatedly during long nights. If she repeats three long steps forward and one short step back, then the ninetieth step will bring her exactly back to where she started. If she repeats three short steps forward and one long step back, she also ends up exactly where she started.

Which step will bring Josefina back to the original spot in the second case?

(M. Petrová)

Answer: 162


11.181. The total surface area of the cone is $\pi S$ sq. units. The lateral surface of the cone, when unfolded onto a plane, forms a sector with an angle of $60^{\circ}$. Determine the volume of the cone.

Answer: \dfrac{\pi S \sqrt{5S}}{21}


Let $ABCD$ be a convex quadrilateral with $\angle A=60^o$. Let $E$ and $Z$ be the symmetric points of $A$ wrt $BC$ and $CD$ respectively. If the points $B,D,E$ and $Z$ are collinear, then calculate the angle $\angle BCD$.

Answer: 60^\circ


From point A, 100 planes (the flagship and 99 additional planes) take off simultaneously. With a full tank of fuel, a plane can fly 1000 km. In flight, planes can transfer fuel to each other. A plane that has transferred fuel to others makes a gliding landing. How should the flight be organized so that the flagship flies as far as possible?

Answer: 5187


Let \( O \) be the point of intersection of the medians of triangle \( ABC \). Find the length of the median drawn from vertex \( A \) if \( \angle BAC = 35^\circ \), \( \angle BOC = 145^\circ \), and \( BC = a \).

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


6. Find all functions $f: \mathbf{Z}_{+} \rightarrow \mathbf{Z}_{+}$, such that for all positive integers $m, n$, $f(m)+f(n)-m n$ is non-zero and divides $m f(m)+n f(n)$.

Answer: f(n) = n^2


33. Today is January 18, 2020, and it is displayed on the electronic calendar as 20200118. If there is a multi-digit number $202001188462 \cdots \cdots$, starting from the 9th digit, each digit is the unit digit of the product of the three preceding digits. According to this rule, what is the 2020th digit of this multi-digit number? $\qquad$ .

Answer: 2


If $|x|$ is less than 0.7, then if we use fifth Maclaurin polynomial approximate $sin(x)$ the error is less than 0.0001. Is this correct? Answer True or False.
Please give the answer directly without any intermediate steps.

Answer: True


8. (10 points) Each letter in the table represents a number, and different letters represent different numbers. The leading digit of each number cannot be zero. Each row from left to right forms an arithmetic sequence, and each column from top to bottom also forms an arithmetic sequence. Therefore, the five-digit number $\overline{\mathrm{CDEFG}}=$ $\qquad$
\begin{tabular}{|l|l|l|}
\hline$A$ & $B A$ & $A A$ \\
\hline
\end{tabular}
\begin{tabular}{|c|c|c|}
\hline$A B$ & $C A$ & $E F$ \\
\hline$C D$ & $G A$ & $B D C$ \\
\hline
\end{tabular}

Answer: 40637


Let \( r \) be some real constant, and \( P(x) \) a polynomial which has remainder 2 when divided by \( x-r \), and remainder \(-2 x^{2}-3 x+4\) when divided by \(\left(2 x^{2}+7 x-4\right)(x-r)\). Find all values of \( r \).

Answer: \frac{1}{2}


A coin is flipped until either $F F F$ or $F I F$ appears in three consecutive flips (where $F$ represents heads and $I$ represents tails). What is the probability that $F F F$ appears before $F I F$?

Answer: \dfrac{2}{5}


2. In the expression $1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9=100$, replace the asterisks with arithmetic operation signs to obtain a correct equation.

Answer: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 \times 9 = 100


124. Intersection of Great Circles. It is known that on a sphere, $n$ great circles intersect in $n(n-1)$ points in general. How should the numbers $1,2, \ldots, n(n-1)$ be placed in these points so that the sums of the numbers located on each circle are equal to each other. (Recall that the great circles lie in planes passing through the center of the sphere.)

Answer: (n-1)(n(n-1) + 1)


G3.3 In Figure $1, \angle A=60^{\circ}, \angle B=\angle D=90^{\circ} . B C=2, C D=3$ and $A B=x$, find the value of $x$.

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


7. Let $\triangle A B C$ be an acute-angled triangle with the lengths of the sides opposite to its internal angles being $a$, $b$, and $c$ respectively. If $2 a^{2}=2 b^{2}+c^{2}$, then the minimum value of $\tan A+\tan B+\tan C$ is $\qquad$.

Answer: 6


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

## Clues:
1. The person whose favorite color is green is somewhere to the left of the person with a high school diploma.
2. Eric is the person whose favorite color is red.
3. There are two houses between Alice and the person who loves white.
4. The person who smokes Blue Master and the person partial to Pall Mall are next to each other.
5. The person who loves white is the person with a master's degree.
6. Arnold is in the fourth house.
7. The person who loves yellow is the person who smokes Blue Master.
8. The Dunhill smoker is Alice.
9. The person partial to Pall Mall is the person with a master's degree.
10. The person who loves yellow is the person with a bachelor's degree.
 What is the value of attribute Name for the person whose attribute Cigar is blue master? 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 Drink for the person whose attribute Vacation is city? Please reason step by step, and put your final answer within \boxed{}

Answer: water


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Eric`, `Carol`, `Arnold`, `Bob`, `Alice`
 - Each person has a favorite color: `blue`, `red`, `green`, `yellow`, `purple`, `white`
 - Each person prefers a unique type of vacation: `city`, `beach`, `cruise`, `cultural`, `mountain`, `camping`
 - People have unique hair colors: `blonde`, `black`, `gray`, `brown`, `red`, `auburn`
 - Each person has an occupation: `artist`, `lawyer`, `engineer`, `teacher`, `doctor`, `nurse`
 - They all have a unique favorite flower: `roses`, `iris`, `daffodils`, `carnations`, `tulips`, `lilies`

## Clues:
1. The person who loves a bouquet of daffodils is the person who is an artist.
2. The person who enjoys camping trips is the person whose favorite color is green.
3. The person who has brown hair is in the first house.
4. The person who loves blue is somewhere to the right of the person who enjoys mountain retreats.
5. The person who loves the vase of tulips is the person who enjoys mountain retreats.
6. Eric is in the second house.
7. The person who likes going on cruises and Arnold are next to each other.
8. The person who loves yellow and the person who enjoys camping trips are next to each other.
9. The person who loves yellow is somewhere to the left of Peter.
10. The person who is a lawyer and Arnold are next to each other.
11. The person who has red hair is Arnold.
12. Alice is the person who is a doctor.
13. The person who loves purple is directly left of the person who loves the vase of tulips.
14. The person who has brown hair is somewhere to the left of the person who is an engineer.
15. The person who has gray hair is somewhere to the left of the person whose favorite color is red.
16. The person who is an artist is somewhere to the right of Arnold.
17. Alice is the person who loves the boquet of lilies.
18. The person whose favorite color is red is directly left of the person who prefers city breaks.
19. The person who prefers city breaks is somewhere to the right of the person who has blonde hair.
20. There is one house between Alice and Bob.
21. The person who loves the rose bouquet is directly left of the person who is a nurse.
22. The person whose favorite color is green is in the fourth house.
23. The person who loves beach vacations is somewhere to the left of the person who has red hair.
24. The person who has black hair is Alice.
25. The person who loves blue is the person who loves a carnations arrangement.
 What is the value of attribute Flower for the person whose attribute HairColor is brown? Please reason step by step, and put your final answer within \boxed{}

Answer: iris


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Carol`, `Alice`, `Bob`, `Peter`, `Arnold`
 - Each person prefers a unique type of vacation: `cultural`, `city`, `cruise`, `mountain`, `camping`, `beach`
 - Everyone has a favorite smoothie: `blueberry`, `watermelon`, `lime`, `desert`, `dragonfruit`, `cherry`
 - People have unique favorite book genres: `fantasy`, `mystery`, `science fiction`, `biography`, `historical fiction`, `romance`

## Clues:
1. The person who loves science fiction books is the person who enjoys camping trips.
2. Arnold is not in the fifth house.
3. The person who loves fantasy books is Peter.
4. The Dragonfruit smoothie lover is the person who loves beach vacations.
5. Alice is somewhere to the left of Eric.
6. The person who drinks Blueberry smoothies is somewhere to the left of the person who enjoys mountain retreats.
7. The person who likes Cherry smoothies is in the fifth house.
8. The person who loves fantasy books is the person who prefers city breaks.
9. There is one house between Peter and the person who loves mystery books.
10. There is one house between the Dragonfruit smoothie lover and the person who drinks Lime smoothies.
11. The person who likes Cherry smoothies is directly left of the person who loves romance books.
12. The person who loves science fiction books is directly left of Eric.
13. The person who loves mystery books is somewhere to the left of Carol.
14. Carol is somewhere to the right of the person who enjoys mountain retreats.
15. The person who likes going on cruises is the person who likes Cherry smoothies.
16. Eric is the person who loves historical fiction books.
17. The person who prefers city breaks is the Desert smoothie lover.
18. The Desert smoothie lover is somewhere to the right of the Dragonfruit smoothie lover.
 What is the value of attribute House for the person whose attribute Name is Arnold? Please reason step by step, and put your final answer within \boxed{}

Answer: 4


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Peter`, `Eric`, `Alice`, `Bob`
 - Each person prefers a unique type of vacation: `beach`, `mountain`, `cruise`, `city`, `camping`
 - Each person lives in a unique style of house: `ranch`, `modern`, `craftsman`, `colonial`, `victorian`
 - Each person has a unique level of education: `bachelor`, `doctorate`, `master`, `associate`, `high school`
 - The mothers' names in different houses are unique: `Aniya`, `Kailyn`, `Janelle`, `Penny`, `Holly`

## Clues:
1. The person whose mother's name is Aniya is the person in a ranch-style home.
2. The person in a Craftsman-style house is Bob.
3. The person with a high school diploma is not in the fourth house.
4. Arnold and the person in a Craftsman-style house are next to each other.
5. Arnold is The person whose mother's name is Penny.
6. The person with an associate's degree is somewhere to the left of the person with a master's degree.
7. Bob and Peter are next to each other.
8. The person living in a colonial-style house is The person whose mother's name is Penny.
9. The person whose mother's name is Holly is the person who enjoys camping trips.
10. The person with a doctorate is the person who prefers city breaks.
11. The person with a bachelor's degree is The person whose mother's name is Aniya.
12. The person with an associate's degree is somewhere to the right of Eric.
13. Arnold is directly left of the person who likes going on cruises.
14. The person who enjoys camping trips is somewhere to the right of the person residing in a Victorian house.
15. The person who enjoys mountain retreats is in the first house.
16. The person whose mother's name is Janelle is the person with an associate's degree.
17. The person with an associate's degree is directly left of Arnold.
 What is the value of attribute House for the person whose attribute Name is Eric? Please reason step by step, and put your final answer within \boxed{}

Answer: 1


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

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

Answer: 3


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Peter`, `Arnold`, `Bob`, `Eric`, `Carol`
 - Each person has a unique hobby: `cooking`, `gardening`, `painting`, `knitting`, `photography`, `woodworking`
 - Each person has a unique birthday month: `april`, `sept`, `mar`, `jan`, `feb`, `may`
 - Everyone has a unique favorite cigar: `blends`, `prince`, `pall mall`, `dunhill`, `yellow monster`, `blue master`

## Clues:
1. The person whose birthday is in April is Eric.
2. Arnold is in the first house.
3. The person whose birthday is in January is the person who smokes many unique blends.
4. The Dunhill smoker is Alice.
5. Bob is the person whose birthday is in May.
6. The person whose birthday is in September and the photography enthusiast are next to each other.
7. Bob is the person who smokes Blue Master.
8. The person who loves cooking is the person whose birthday is in May.
9. The woodworking hobbyist and the person whose birthday is in March are next to each other.
10. The person who enjoys knitting is directly left of the person who smokes Yellow Monster.
11. Peter and the person who paints as a hobby are next to each other.
12. The person whose birthday is in March is somewhere to the left of the Prince smoker.
13. The person whose birthday is in February is somewhere to the left of the person who smokes many unique blends.
14. The person whose birthday is in March is somewhere to the right of the person whose birthday is in September.
15. The person who smokes many unique blends is directly left of the person whose birthday is in May.
16. The photography enthusiast and Peter are next to each other.
 What is the value of attribute Name for the person whose attribute Cigar is prince? Please reason step by step, and put your final answer within \boxed{}

Answer: Peter


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

## Clues:
1. The person whose birthday is in September is the person who likes Cherry smoothies.
2. The person whose birthday is in February is not in the first house.
3. The Desert smoothie lover is the person who loves basketball.
4. Peter is the person who has black hair.
5. The person who loves tennis is somewhere to the right of the person who has a cat.
6. The person with an aquarium of fish is not in the fourth house.
7. Arnold is directly left of the person who has blonde hair.
8. There is one house between the person who has red hair and the person whose birthday is in January.
9. The person whose birthday is in September is directly left of the person who loves soccer.
10. Peter is directly left of the person who loves basketball.
11. The person who has blonde hair is the person whose birthday is in September.
12. Arnold is the person who keeps a pet bird.
13. The person whose birthday is in January is the Watermelon smoothie lover.
14. The person who has blonde hair is Eric.
 What is the value of attribute House for the person whose attribute FavoriteSport is soccer? Please reason step by step, and put your final answer within \boxed{}

Answer: 4


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Alice`, `Arnold`, `Eric`
 - The people keep unique animals: `horse`, `bird`, `fish`, `cat`
 - Each person has a unique level of education: `high school`, `bachelor`, `associate`, `master`
 - Each person has a unique favorite drink: `tea`, `water`, `coffee`, `milk`
 - People have unique favorite book genres: `fantasy`, `science fiction`, `romance`, `mystery`
 - Each person has a unique birthday month: `april`, `feb`, `sept`, `jan`

## Clues:
1. Peter is the person whose birthday is in February.
2. The person with a master's degree is the bird keeper.
3. The person who keeps horses is not in the second house.
4. The person whose birthday is in January is the person who loves science fiction books.
5. The person who loves mystery books is Alice.
6. There is one house between the person who likes milk and Alice.
7. The person whose birthday is in April is the person who loves fantasy books.
8. The person with a high school diploma is not in the second house.
9. Eric is directly left of the person who likes milk.
10. The fish enthusiast is the person whose birthday is in February.
11. The tea drinker is somewhere to the left of Peter.
12. The one who only drinks water is the person with a master's degree.
13. The person with a bachelor's degree is somewhere to the right of the person whose birthday is in January.
14. Peter is in the third house.
15. The person who loves science fiction books is Arnold.
 What is the value of attribute Drink for the person whose attribute Birthday is jan? Please reason step by step, and put your final answer within \boxed{}

Answer: milk


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

## Clues:
1. The person who loves classical music is Peter.
2. The person in a ranch-style home is not in the fourth house.
3. The person who loves a bouquet of daffodils is the person living in a colonial-style house.
4. The person living in a colonial-style house is the person who loves white.
5. The person who prefers city breaks and the person whose favorite color is red are next to each other.
6. The person who loves the rose bouquet is not in the fourth house.
7. The person who likes going on cruises is in the second house.
8. The person who prefers city breaks is the person who loves rock music.
9. The person who loves pop music is Alice.
10. The person who loves the rose bouquet is the person who prefers city breaks.
11. Eric is the person who enjoys mountain retreats.
12. The person who enjoys mountain retreats is directly left of the person who loves a bouquet of daffodils.
13. The person who loves yellow is somewhere to the left of the person who likes going on cruises.
14. The person who enjoys mountain retreats is the person residing in a Victorian house.
15. The person who loves classical music is the person who likes going on cruises.
16. Peter is somewhere to the right of the person who loves the boquet of lilies.
 What is the value of attribute MusicGenre for the person whose attribute Flower is lilies? Please reason step by step, and put your final answer within \boxed{}

Answer: jazz


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`
 - People own unique car models: `toyota camry`, `tesla model 3`, `ford f150`
 - People have unique favorite book genres: `mystery`, `romance`, `science fiction`
 - Each person has an occupation: `doctor`, `teacher`, `engineer`

## Clues:
1. The person who loves science fiction books is directly left of the person who is a teacher.
2. Arnold is in the third house.
3. The person who owns a Ford F-150 is the person who is a teacher.
4. The person who is an engineer is the person who owns a Toyota Camry.
5. The person who is an engineer is somewhere to the right of the person who is a teacher.
6. The person who loves mystery books is the person who is a teacher.
7. The person who loves science fiction books is Eric.
 What is the value of attribute CarModel for the person whose attribute BookGenre is mystery? Please reason step by step, and put your final answer within \boxed{}

Answer: ford f150


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`, `Carol`, `Alice`, `Bob`, `Eric`
 - Each person has a unique level of education: `bachelor`, `high school`, `doctorate`, `trade school`, `associate`, `master`
 - Each person has an occupation: `teacher`, `artist`, `nurse`, `lawyer`, `engineer`, `doctor`
 - People own unique car models: `ford f150`, `toyota camry`, `honda civic`, `chevrolet silverado`, `bmw 3 series`, `tesla model 3`
 - Each person has a unique birthday month: `sept`, `jan`, `mar`, `april`, `feb`, `may`

## Clues:
1. The person with a bachelor's degree is somewhere to the right of the person who owns a Toyota Camry.
2. The person whose birthday is in February is the person with an associate's degree.
3. The person whose birthday is in September is not in the fifth house.
4. The person who owns a BMW 3 Series is somewhere to the right of the person who is an engineer.
5. The person whose birthday is in January is Arnold.
6. The person who is an artist is directly left of the person with a doctorate.
7. The person who owns a Ford F-150 is the person who is an engineer.
8. The person who attended trade school and Eric are next to each other.
9. There is one house between the person who attended trade school and the person with a bachelor's degree.
10. The person whose birthday is in May is the person who owns a BMW 3 Series.
11. The person who owns a Chevrolet Silverado is somewhere to the left of Alice.
12. The person who is an engineer is directly left of Bob.
13. The person who owns a Honda Civic is the person whose birthday is in April.
14. The person who is an artist is the person with a high school diploma.
15. Arnold is the person who owns a Toyota Camry.
16. The person who owns a Tesla Model 3 is in the first house.
17. The person who is a doctor is somewhere to the left of the person whose birthday is in April.
18. The person who is a nurse is directly left of the person whose birthday is in September.
19. The person who is a teacher is the person with a doctorate.
20. There is one house between the person who owns a Toyota Camry and the person who owns a Honda Civic.
21. The person who owns a Tesla Model 3 is Peter.
 What is the value of attribute Birthday for the person whose attribute Education is doctorate? Please reason step by step, and put your final answer within \boxed{}

Answer: may


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`, `Eric`, `Bob`, `Arnold`, `Peter`, `Carol`
 - People have unique favorite music genres: `classical`, `rock`, `jazz`, `pop`, `country`, `hip hop`
 - Each person has an occupation: `teacher`, `artist`, `nurse`, `engineer`, `lawyer`, `doctor`
 - Each person prefers a unique type of vacation: `cruise`, `camping`, `city`, `cultural`, `mountain`, `beach`
 - Each person has a favorite color: `blue`, `white`, `red`, `yellow`, `purple`, `green`

## Clues:
1. The person who loves beach vacations is Arnold.
2. The person who is an engineer is the person who enjoys mountain retreats.
3. Peter is not in the second house.
4. The person whose favorite color is red is in the third house.
5. The person who loves beach vacations and the person who is a teacher are next to each other.
6. The person who is a nurse is Carol.
7. The person who goes on cultural tours is the person who loves pop music.
8. The person who loves rock music and the person who loves country music are next to each other.
9. The person whose favorite color is green is in the second house.
10. The person whose favorite color is red is the person who enjoys camping trips.
11. Peter is the person who loves pop music.
12. The person who loves classical music is the person who loves purple.
13. The person who loves blue is directly left of Alice.
14. The person who loves rock music is in the fourth house.
15. The person who loves white is the person who is a doctor.
16. The person who loves jazz music is the person whose favorite color is red.
17. Carol is somewhere to the right of the person who prefers city breaks.
18. The person who loves rock music is the person who is an artist.
19. The person who enjoys mountain retreats is in the first house.
20. Bob is the person who loves classical music.
 What is the value of attribute House for the person whose attribute Vacation is city? Please reason step by step, and put your final answer within \boxed{}

Answer: 2


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Bob`, `Peter`, `Alice`, `Carol`, `Eric`
 - Everyone has something unique for lunch: `stew`, `grilled cheese`, `stir fry`, `soup`, `pizza`, `spaghetti`
 - People have unique heights: `tall`, `average`, `super tall`, `very short`, `very tall`, `short`
 - Each person has a unique favorite drink: `root beer`, `boba tea`, `coffee`, `water`, `tea`, `milk`
 - Each person has a unique type of pet: `hamster`, `fish`, `cat`, `dog`, `bird`, `rabbit`
 - People use unique phone models: `samsung galaxy s21`, `xiaomi mi 11`, `google pixel 6`, `iphone 13`, `huawei p50`, `oneplus 9`

## Clues:
1. The person who uses an iPhone 13 is in the third house.
2. Bob is the person who is tall.
3. The person who loves the soup is in the second house.
4. The root beer lover is directly left of the person who uses a Xiaomi Mi 11.
5. The person who uses a Huawei P50 is directly left of the person who loves eating grilled cheese.
6. The person who loves stir fry is the person who likes milk.
7. The person who loves eating grilled cheese is the person who is tall.
8. The person who uses a Xiaomi Mi 11 is the coffee drinker.
9. The person who uses a OnePlus 9 is Arnold.
10. The person who owns a rabbit is not in the fifth house.
11. The person with a pet hamster is somewhere to the right of the person who uses a Google Pixel 6.
12. The person who is super tall is the person with an aquarium of fish.
13. The person with an aquarium of fish is Alice.
14. The tea drinker is directly left of the person who is a pizza lover.
15. The person who uses a Samsung Galaxy S21 is Carol.
16. The person who is a pizza lover is the person who is short.
17. Arnold is the person who is very tall.
18. The person who loves the spaghetti eater is the person who uses a Google Pixel 6.
19. The boba tea drinker is somewhere to the right of the person who loves the soup.
20. The person with a pet hamster is not in the fifth house.
21. The person who is very tall is not in the second house.
22. The person who is super tall is somewhere to the left of Peter.
23. The person who is very short is the person who loves the spaghetti eater.
24. The person who keeps a pet bird is somewhere to the left of the person who loves the spaghetti eater.
25. The person with an aquarium of fish is directly left of Eric.
26. The person who owns a dog is the person who likes milk.
 What is the value of attribute Drink for the person whose attribute Food is stir fry? Please reason step by step, and put your final answer within \boxed{}

Answer: milk


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Arnold`, `Peter`
 - People use unique phone models: `google pixel 6`, `iphone 13`, `samsung galaxy s21`
 - The people keep unique animals: `bird`, `horse`, `cat`
 - Each person has an occupation: `doctor`, `engineer`, `teacher`
 - Each person has a unique birthday month: `jan`, `sept`, `april`
 - Everyone has something unique for lunch: `grilled cheese`, `pizza`, `spaghetti`

## Clues:
1. The person who keeps horses is directly left of the person who is a teacher.
2. The person who loves the spaghetti eater is the person who is an engineer.
3. The person who uses a Samsung Galaxy S21 is not in the third house.
4. The person who is a doctor is the cat lover.
5. The person who uses an iPhone 13 is Peter.
6. The person who loves the spaghetti eater is the person whose birthday is in January.
7. The person who uses a Samsung Galaxy S21 is the person who loves eating grilled cheese.
8. The person who uses a Samsung Galaxy S21 and Eric are next to each other.
9. The person whose birthday is in January is somewhere to the right of the person whose birthday is in September.
 What is the value of attribute Occupation for the person whose attribute Animal is bird? Please reason step by step, and put your final answer within \boxed{}

Answer: teacher


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`
 - Each person has a unique hobby: `gardening`, `photography`, `cooking`
 - Everyone has a unique favorite cigar: `pall mall`, `prince`, `blue master`
 - Each person has a unique favorite drink: `water`, `tea`, `milk`

## Clues:
1. Eric is directly left of the Prince smoker.
2. There is one house between the person who likes milk and the person who enjoys gardening.
3. Arnold is the Prince smoker.
4. Peter is not in the third house.
5. The tea drinker is the person who loves cooking.
6. The person partial to Pall Mall is the photography enthusiast.
 What is the value of attribute Name for the person whose attribute Cigar is pall mall? Please reason step by step, and put your final answer within \boxed{}

Answer: Peter


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Alice`, `Eric`, `Bob`, `Arnold`
 - Each person has a favorite color: `white`, `red`, `green`, `blue`, `yellow`
 - Each person lives in a unique style of house: `craftsman`, `modern`, `colonial`, `victorian`, `ranch`
 - People have unique favorite book genres: `fantasy`, `science fiction`, `biography`, `romance`, `mystery`

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

Answer: yellow


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Peter`, `Alice`, `Eric`, `Arnold`
 - The mothers' names in different houses are unique: `Penny`, `Kailyn`, `Aniya`, `Janelle`, `Holly`
 - Everyone has a unique favorite cigar: `blue master`, `pall mall`, `prince`, `dunhill`, `blends`
 - The people keep unique animals: `bird`, `cat`, `horse`, `fish`, `dog`
 - Everyone has something unique for lunch: `grilled cheese`, `stir fry`, `pizza`, `spaghetti`, `stew`

## Clues:
1. The person who loves the stew is somewhere to the left of the fish enthusiast.
2. The person who is a pizza lover is somewhere to the right of the Dunhill smoker.
3. Peter is the cat lover.
4. The dog owner is directly left of the person who loves the spaghetti eater.
5. The person who loves stir fry is The person whose mother's name is Aniya.
6. There are two houses between the fish enthusiast and Eric.
7. The person who smokes many unique blends is The person whose mother's name is Kailyn.
8. The fish enthusiast is the person who loves the spaghetti eater.
9. The fish enthusiast is in the second house.
10. The person whose mother's name is Kailyn is the person who loves the stew.
11. The person whose mother's name is Janelle is not in the fifth house.
12. The person whose mother's name is Holly is the person who smokes Blue Master.
13. Arnold is The person whose mother's name is Aniya.
14. The person who loves the spaghetti eater and the person partial to Pall Mall are next to each other.
15. Peter is not in the third house.
16. Arnold is the person who keeps horses.
17. The person whose mother's name is Holly is Bob.
 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: `Eric`, `Peter`, `Bob`, `Arnold`, `Carol`, `Alice`
 - Everyone has a favorite smoothie: `cherry`, `blueberry`, `dragonfruit`, `watermelon`, `lime`, `desert`
 - The people keep unique animals: `bird`, `cat`, `dog`, `rabbit`, `horse`, `fish`
 - The mothers' names in different houses are unique: `Sarah`, `Holly`, `Janelle`, `Aniya`, `Kailyn`, `Penny`
 - They all have a unique favorite flower: `lilies`, `iris`, `carnations`, `tulips`, `roses`, `daffodils`
 - Each person has an occupation: `lawyer`, `doctor`, `teacher`, `engineer`, `artist`, `nurse`

## Clues:
1. The person who loves a bouquet of daffodils is The person whose mother's name is Holly.
2. The person who likes Cherry smoothies is not in the third house.
3. The person who is a nurse is directly left of Peter.
4. The person who is an artist is the person who loves the rose bouquet.
5. The person who drinks Lime smoothies is not in the fifth house.
6. The cat lover is The person whose mother's name is Sarah.
7. Bob is the Desert smoothie lover.
8. The Dragonfruit smoothie lover is in the sixth house.
9. The person who loves the boquet of lilies and The person whose mother's name is Penny are next to each other.
10. Peter is the bird keeper.
11. The person whose mother's name is Kailyn is the person who loves the boquet of iris.
12. Carol is somewhere to the left of the person who likes Cherry smoothies.
13. The person who is a doctor is directly left of The person whose mother's name is Holly.
14. Bob is The person whose mother's name is Janelle.
15. Arnold is in the third house.
16. Alice is the person who is an engineer.
17. The Dragonfruit smoothie lover is the person who is an artist.
18. The dog owner is directly left of the person who is a teacher.
19. The person who keeps horses is The person whose mother's name is Aniya.
20. The person who loves the boquet of lilies is directly left of the person who loves a carnations arrangement.
21. There is one house between the Desert smoothie lover and the person who loves a carnations arrangement.
22. The person who drinks Lime smoothies is somewhere to the right of The person whose mother's name is Aniya.
23. The person whose mother's name is Kailyn is directly left of the fish enthusiast.
24. The Watermelon smoothie lover is the rabbit owner.
 What is the value of attribute Animal for the person whose attribute Name is Alice? Please reason step by step, and put your final answer within \boxed{}

Answer: fish


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Arnold`, `Bob`, `Eric`, `Peter`
 - Each person has a unique hobby: `painting`, `photography`, `cooking`, `gardening`, `knitting`
 - Each person has a unique birthday month: `jan`, `sept`, `mar`, `feb`, `april`
 - People have unique favorite sports: `swimming`, `basketball`, `tennis`, `baseball`, `soccer`
 - Each person has an occupation: `engineer`, `lawyer`, `teacher`, `doctor`, `artist`
 - Each person has a unique level of education: `bachelor`, `master`, `high school`, `doctorate`, `associate`

## Clues:
1. The person with a high school diploma is directly left of the person whose birthday is in February.
2. The person who loves soccer is directly left of the photography enthusiast.
3. Arnold is the person whose birthday is in January.
4. The person who is a doctor is the person who loves baseball.
5. The person who is an artist is somewhere to the left of Bob.
6. The photography enthusiast is somewhere to the right of the person with an associate's degree.
7. The photography enthusiast is directly left of the person who paints as a hobby.
8. Eric is in the fifth house.
9. The person whose birthday is in February is directly left of the person who loves tennis.
10. The person who loves baseball is somewhere to the right of the person whose birthday is in January.
11. The person who is an engineer is not in the first house.
12. Alice is not in the second house.
13. The person who enjoys gardening is somewhere to the left of the person whose birthday is in September.
14. The person who loves swimming is not in the first house.
15. The person who is a teacher is Bob.
16. Arnold is the person who enjoys knitting.
17. The person who is an artist is the person whose birthday is in March.
18. Arnold is the person with a master's degree.
19. The person whose birthday is in March is directly left of the person who is a doctor.
20. The person whose birthday is in September is the person with a bachelor's degree.
 What is the value of attribute FavoriteSport for the person whose attribute Occupation is artist? Please reason step by step, and put your final answer within \boxed{}

Answer: soccer


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

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

Answer: short


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

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

Answer: white


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

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

Answer: Peter


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Eric`, `Alice`, `Peter`
 - People own unique car models: `honda civic`, `tesla model 3`, `toyota camry`, `ford f150`
 - They all have a unique favorite flower: `roses`, `daffodils`, `lilies`, `carnations`
 - Each person has a unique favorite drink: `milk`, `water`, `tea`, `coffee`
 - People have unique heights: `very short`, `average`, `tall`, `short`
 - Each person has a favorite color: `red`, `white`, `yellow`, `green`

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

Answer: water


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

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

Answer: 5


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Eric`, `Peter`, `Alice`
 - Each person has a unique level of education: `master`, `high school`, `associate`, `bachelor`
 - Everyone has something unique for lunch: `stew`, `grilled cheese`, `pizza`, `spaghetti`
 - People have unique favorite sports: `soccer`, `swimming`, `tennis`, `basketball`
 - People have unique hair colors: `blonde`, `black`, `brown`, `red`
 - People have unique favorite book genres: `mystery`, `fantasy`, `science fiction`, `romance`

## Clues:
1. The person who loves the stew is in the first house.
2. The person who loves science fiction books is the person who has black hair.
3. The person who has blonde hair is not in the second house.
4. The person with a master's degree is in the first house.
5. Eric is the person who loves soccer.
6. The person with a master's degree and the person who loves mystery books are next to each other.
7. The person who loves tennis is the person with a high school diploma.
8. The person who loves the spaghetti eater is the person who loves science fiction books.
9. The person who is a pizza lover is the person with an associate's degree.
10. The person who has blonde hair is not in the fourth house.
11. The person with a master's degree is directly left of Eric.
12. The person with a bachelor's degree is directly left of the person who loves fantasy books.
13. The person who has red hair is Peter.
14. The person who loves swimming is directly left of the person who is a pizza lover.
15. Alice is not in the first house.
 What is the value of attribute House for the person whose attribute BookGenre is romance? Please reason step by step, and put your final answer within \boxed{}

Answer: 1


18. Let $f(x), g(x)$ be odd and even functions defined on $R$, respectively, and $f(x)+g(x)=2^{x}$. If for $x \in\left[\frac{1}{2}, 2\right]$, the inequality $a f(x)-f(3 x) \leqslant 2 g(2 x)$ always holds, find the range of the real number $a$.

Answer: (-\infty, 10]


1. In the village of Big Vasyuki, there is a riverside promenade 50 meters long, running along the river. A boat 10 meters long passes by it in 5 seconds when traveling downstream and in 4 seconds when traveling upstream. How many seconds will it take for a paper boat to float from one end of the promenade to the other?

Answer: \dfrac{100}{3}


One square is inscribed in a circle, and another square is circumscribed around the same circle such that its vertices lie on the extensions of the sides of the first square. Find the angle between the sides of these squares.

Answer: 15^\circ


Let \( p \) be an odd prime and \( S_{q} = \frac{1}{2 \cdot 3 \cdot 4} + \frac{1}{5 \cdot 6 \cdot 7} + \ldots + \frac{1}{q(q+1)(q+2)} \), where \( q = \frac{3p - 5}{2} \). We express \( \frac{1}{p} - 2 S_{q} \) in the form \( \frac{m}{n} \), where \( m \) and \( n \) are integers. Show that \( m \equiv n \pmod{p} \); that is, \( m \) and \( n \) leave the same remainder when divided by \( p \).

Answer: m \equiv n \pmod{p}


The numbers $1,2,\dots,10$ are written on a board. Every minute, one can select three numbers $a$, $b$, $c$ on the board, erase them, and write $\sqrt{a^2+b^2+c^2}$ in their place. This process continues until no more numbers can be erased. What is the largest possible number that can remain on the board at this point?

[i]Proposed by Evan Chen[/i]

Answer: 8\sqrt{6}


24.3.11  For a competition with 4 multiple-choice questions, each with $A, B, C$ as options, for any three participants, there is at least one question where their answers are all different. How many people can participate in the competition at most?

Answer: 9


13. (15 points) There are 60 students in the class, and their birthdays are recorded as a certain month and day. Ask each student the same two questions: How many students in the class have the same birth month as you? How many students in the class have the same birth day as you (for example, the birth day of January 12 and December 12 is the same). It is found that the answers obtained include all integers from 0 to 14. Then, what is the minimum number of students who have the same birthday?

Answer: 2


14. Let $\left\{a_{n}\right\}$ be a sequence of positive numbers, with the sum of its first $n$ terms being $S_{n}$, and for all natural numbers $n$, we have
$$
S_{n}=\frac{3 n+1}{2}-\frac{n}{2} a_{n} \text {. }
$$
(1) Write down the first three terms of the sequence $\left\{a_{n}\right\}$;
(2) Find the general term formula for the sequence $\left\{a_{n}\right\}$ (include the derivation process);
(3) Let $b_{n}=\frac{1}{a_{n}-1}\left(n \in \mathbf{N}_{+}\right)$, find the sum of the first $n$ terms of the sequence $\left\{b_{n}\right\}$.

Answer: \dfrac{n(n +1)(n +2 )(n +3 )}{8}


10. In a row of wooden cells containing several balls, two players, A and B, take turns moving the balls. The movement rules are: each person can choose any number of balls from a certain cell and move them to the cell immediately to the right; when all the balls are moved to the far-right cell, the game ends, and the player who moves the last ball wins.
Given the situation shown in the figure (the numbers in the cells represent the number of balls, and the numbers below the cells indicate the cell numbers), the first player has a winning strategy. To ensure victory, the first player should move $\qquad$ balls from cell $\qquad$ in the first step.

Answer: 1


Let \(x, y, z \in \mathbf{R}^{+}\) and \(\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 1\). Prove that \(\sqrt{x + yz} + \sqrt{y + zx} + \sqrt{z + xy} \geq \sqrt{xyz} + \sqrt{x} + \sqrt{y} + \sqrt{z}\).

Answer: \sqrt{x + yz} + \sqrt{y + zx} + \sqrt{z + xy} \geq \sqrt{xyz} + \sqrt{x} + \sqrt{y} + \sqrt{z}


Find the least positive integer $m$ such that $m^2 - m + 11$ is a product of at least four not necessarily distinct primes.
Please give the answer directly without any intermediate steps.

Answer: 132


One, (Full marks 30 points) Let $f(x)=x^{2}-(4 a-2) - x-6 a^{2}$ have the minimum value $m$ on the interval $[0,1]$. Try to write the expression for $m$ in terms of $a$, $m=F(a)$. And answer: For what value of $a$ does $m$ achieve its maximum value? What is this maximum value?

---

The translation maintains the original text's format and line breaks as requested.

Answer: 0


9. In rectangle $A B C D$, $A D=a, A B=b(b>a)$, fold the rectangle so that point $A$ coincides with point $C$, resulting in the crease $M N$. Then open the half-plane $D A M N$ so that it forms a $57^{\circ}$ dihedral angle with the half-plane $M N C B$. The angle formed between line $A C$ and line $M N$ is $\qquad$

Answer: 90^\circ


2. Define the sequence $\left\{a_{n}\right\}: a_{1}=0, a_{2}=1$,
$$
\begin{aligned}
a_{n}= & \frac{1}{2} n a_{n-1}+\frac{1}{2} n(n-1) a_{n-2}+ \\
& (-1)^{n}\left(1-\frac{n}{2}\right)(n \geqslant 3) .
\end{aligned}
$$

Find the simplest expression for $f_{n}=\sum_{k=1}^{n} k \mathrm{C}_{n}^{k-1} a_{n+1-k}$.

Answer: 2n! - n - 1


4. In a regular tetrahedron $P-ABC$, it is known that the angle between the lateral edge and the base is $45^{\circ}$. Then the cosine value of the dihedral angle between two adjacent lateral faces is $\qquad$

Answer: \dfrac{1}{5}


5. Given a convex quadrilateral $ABCD$ with area $S$, $O$ is a point outside the quadrilateral, $M_{1}, M_{2}, M_{3}, M_{4}$ are the centroids of $\triangle OAB$, $\triangle OBC$, $\triangle OCD$, $\triangle ODA$ respectively. Then the area of quadrilateral $M_{1} M_{2} M_{3} M_{4}$ is $\qquad$

Answer: \dfrac{2}{9}S


3. Two circles with equal radii touch at point $C$ and touch externally a third circle with radius $r=5$ cm at points $A$ and $B$. Calculate the area of triangle $ABC$, if $\overline{AB}=6$ cm.

Answer: 18


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 $DP^2$.

[i]2020 CCA Math Bonanza Lightning Round #4.3[/i]

Answer: 169


Problem 10.7. Olya drew $N$ different lines on the plane, any two of which intersect. It turned out that among any 15 lines, there are always two lines such that the angle between them is $60^{\circ}$. What is the largest $N$ for which this is possible?

Answer: 42


Let an integer $ n > 3$. Denote the set $ T\equal{}\{1,2, \ldots,n\}.$ A subset S of T is called [i]wanting set[/i] if S has the property: There exists a positive integer $ c$ which is not greater than $ \frac {n}{2}$ such that $ |s_1 \minus{} s_2|\ne c$ for every pairs of arbitrary elements $ s_1,s_2\in S$. How many does a [i]wanting set[/i] have at most are there ?

Answer: \left\lfloor \dfrac{2n}{3} \right\rfloor


8. For a regular triangular prism $A B C-A_{1} B_{1} C_{1}$ with all edges of length 3, a line segment $M N$ of length 2 has one endpoint $M$ moving on $A A_{1}$ and the other endpoint $N$ moving on the base $A B C$. Then, the trajectory (surface) of the midpoint $P$ of $M N$ and the three faces of the regular triangular prism sharing vertex $A$ enclose a geometric body with a volume of $\qquad$

Answer: \dfrac{\pi}{9}


Natural numbers \(a\) and \(b\) are such that \(a^3 + b^3 + ab\) is divisible by \(ab(a - b)\). Prove that \(\operatorname{lcm}(a, b)\) is a perfect square. (LCM - least common multiple).

Answer: \operatorname{lcm}(a, b) \text{ is a perfect square}


3. In how many ways can the set $\{1,2, \ldots, 12\}$ be divided into six disjoint two-element subsets such that each of them contains coprime numbers (i.e., numbers that do not have a common divisor greater than one)?

Answer: 252


7. A row of 100 coins is laid out. All the coins look the same, but somewhere among them are 50 counterfeit ones (and the rest are genuine). All genuine coins weigh the same, and the counterfeit ones may weigh differently, but each counterfeit is lighter than a genuine one. Can at least 34 genuine coins be found with a single weighing on a balance scale without weights? (O. Dmitriev, R. Zhenodarov)

Answer: 34


4. Tomi dropped a rubber ball from a certain height onto a hard floor. After each bounce, it reached $\frac{3}{4}$ of the previous height. The ball bounced four times from the ground. At the moment it fell for the fifth time, his neighbor Jani caught it. From what height did Tomi drop the ball if it traveled a total distance of $653 \mathrm{~cm}$? Write down your answer.

Answer: 128


3. In triangle $ABC$, the median $AM$ is drawn. Circle $\alpha$ passes through point $A$, touches line $BC$ at point $M$, and intersects sides $AB$ and $AC$ at points $D$ and $E$ respectively. On the arc $AD$ that does not contain point $E$, a point $F$ is chosen such that $\angle BFE = 72^{\circ}$. It turns out that $\angle DEF = \angle ABC$. Find the angle $\angle CME$.

Answer: 36^\circ


Let $X$ be a $5\times 5$ matrix with each entry be $0$ or $1$. Let $x_{i,j}$ be the $(i,j)$-th entry of $X$ ($i,j=1,2,\hdots,5$). Consider all the $24$ ordered sequence in the rows, columns and diagonals of $X$ in the following:
\begin{align*}
&(x_{i,1}, x_{i,2},\hdots,x_{i,5}),\ (x_{i,5},x_{i,4},\hdots,x_{i,1}),\ (i=1,2,\hdots,5) \\
&(x_{1,j}, x_{2,j},\hdots,x_{5,j}),\ (x_{5,j},x_{4,j},\hdots,x_{1,j}),\ (j=1,2,\hdots,5) \\
&(x_{1,1},x_{2,2},\hdots,x_{5,5}),\ (x_{5,5},x_{4,4},\hdots,x_{1,1}) \\
&(x_{1,5},x_{2,4},\hdots,x_{5,1}),\ (x_{5,1},x_{4,2},\hdots,x_{1,5})
\end{align*}
Suppose that all of the sequences are different. Find all the possible values of the sum of all entries in $X$.

Answer: 12


3. In the convex quadrilateral $ABCD$, $\angle ADB=70^{\circ}, \angle CDB=$ $40^{\circ}, \angle BAC=20^{\circ}, \angle BCA=35^{\circ}$. Find the degree measure of the acute angle between $AC$ and $BD$.

Answer: 75


4. Elisa makes so-called fake dice. On each side of a fake die, one of the numbers 1 through 6 is written, but not every number has to appear, and some numbers may appear more often. However, from all sides, it must look like a real die. This means: at each corner, three different numbers come together, and no two of them add up to 7 (on a real die, such pairs of numbers are always opposite each other). The numbers 1, 2, and 4 can come together at a corner, for example, but 1, 2, and 5 cannot because $2+5=7$. A regular die, of course, also counts as a fake die.

Elisa is interested in the eye sum of the fake die: the sum of the six numbers on the fake die.

How many different possibilities are there for the eye sum of a fake die?

Answer: 19


Example 5 As shown in Figure 5, in $\triangle A B C$, $\angle C=90^{\circ}$, $I$ is the intersection of the angle bisectors $A D$ and $B E$ of $\angle A$ and $\angle B$. Given that the area of $\triangle A B I$ is 12. Then the area of quadrilateral $A B D E$ is $\qquad$
(2004, Beijing Middle School Mathematics Competition (Grade 8))

Answer: 24


Ninety-eight apples who always lie and one banana who always tells the truth are randomly arranged along a line. The first fruit says "One of the first forty fruit is the banana!'' The last fruit responds "No, one of the $\emph{last}$ forty fruit is the banana!'' The fruit in the middle yells "I'm the banana!'' In how many positions could the banana be?

Answer: 21


Place 2004 bookmarks around a circle, each with one white side and one black side. Define a move as selecting a bookmark with the black side facing up and flipping it along with the two adjacent bookmarks. Assuming there is only one bookmark with the black side facing up initially, can all bookmarks be turned white side up after a series of moves? What about the case with 2003 bookmarks?

Answer: \text{No}


Kolya and Zhenya agreed to meet in the metro sometime between noon and 1 PM. Kolya arrives at the meeting place at some random time between noon and 1 PM, waits for 10 minutes, and then leaves. Zhenya does the same.

a) What is the probability that they meet?

b) How does the probability of meeting change if Zhenya decides to arrive before 12:30 PM, while Kolya still arrives between noon and 1 PM?

c) How does the probability of meeting change if Zhenya decides to arrive at a random time between 12:00 and 12:50, while Kolya still arrives between 12:00 and 1:00?

Answer: \dfrac{11}{36}


4. Given $n(n \geqslant 3)$ lines where exactly $m(m \geqslant 2)$ lines are parallel, and no three lines intersect at the same point. The maximum number of regions these $n$ lines can divide the plane into is $\qquad$

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


1. The oldest can be either Boris or Viktor. But Viktor could not take second place, as he lost to both Anton and Grigory. Therefore, Boris is second. This means Grigory is first, Anton is third, and Viktor is fourth.

Answer: \text{Grigory}, \text{Boris}, \text{Anton}, \text{Viktor}


Triangle $ABC$ lies in a regular decagon as shown in the figure.
What is the ratio of the area of the triangle to the area of the entire decagon?
Write the answer as a fraction of integers.
[img]https://1.bp.blogspot.com/-Ld_-4u-VQ5o/Xzb-KxPX0wI/AAAAAAAAMWg/-qPtaI_04CQ3vvVc1wDTj3SoonocpAzBQCLcBGAsYHQ/s0/2007%2BMohr%2Bp1.png[/img]

Answer: \dfrac{1}{5}


In the figure, many identical matchsticks form a rectangle. The length of this rectangle is composed of 60 matchsticks, and the width is composed of 10 matchsticks. Finally, these matchsticks are distributed equally among more than 100 children, with each child receiving the same number of matchsticks and no matchsticks left over. How many children are there in total?

Answer: 127


Problem 37. The base of an isosceles triangle relates to its lateral side as 4:3 (Fig. 28). In what ratio does the height, drawn to the lateral side, divide it?

Answer: 1:8


12: Use 6 equal-length thin iron rods to weld into a regular tetrahedral frame, ignoring the thickness of the iron rods and welding errors. Let the radius of the largest sphere that this frame can contain be $R_{1}$, and the radius of the smallest sphere that can enclose this frame be $R_{2}$. Then $\frac{R_{1}}{R_{2}}$ equals $\qquad$

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


4. Find the minimum value of the function $f(x)=\sum_{i=1}^{2011}|i x-1|$.

 untranslated text remains unchanged.

Answer: \dfrac{592043}{711}


5. The set of integer points on the plane
$$
S=\{(a, b) \mid 1 \leqslant a, b \leqslant 5(a, b \in \mathbf{Z})\},
$$
$T$ is a set of integer points on the plane, such that for any point $P$ in $S$, there exists a point $Q$ in $T$ different from $P$, such that the line segment $P Q$ contains no other integer points except $P$ and $Q$. What is the minimum number of elements in $T$?
(Supplied by Chen Yonggao)

Answer: 2


We call a monic polynomial $P(x) \in \mathbb{Z}[x]$ [i]square-free mod n[/i] if there [u]dose not[/u] exist polynomials $Q(x),R(x) \in \mathbb{Z}[x]$ with $Q$ being non-constant and $P(x) \equiv Q(x)^2 R(x) \mod n$. Given a prime $p$ and integer $m \geq 2$. Find the number of monic [i]square-free mod p[/i] $P(x)$ with degree $m$ and coeeficients in $\{0,1,2,3,...,p-1\}$.

[i]Proposed by Masud Shafaie[/i]

Answer: p^m - p^{m-1}


Eve randomly chooses two $\textbf{distinct}$ points on the coordinate plane from the set of all $11^2$ lattice points $(x, y)$ with $0 \le x \le 10$, $0 \le y \le 10$. Then, Anne the ant walks from the point $(0,0)$ to the point $(10, 10)$ using a sequence of one-unit right steps and one-unit up steps. Let $P$ be the number of paths Anne could take that pass through both of the points that Eve chose. The expected value of $P$ is $\dbinom{20}{10} \cdot \dfrac{a}{b}$ for relatively prime positive integers $a$ and $b$. Compute $100a+b$.

[i]Proposed by Michael Tang[/i]

Answer: 942


8. Let $D$ be a point inside $\triangle A B C$ such that $\angle B A D=$ $\angle B C D$, and $\angle B D C=90^{\circ}$. Given $A B=5, B C=$ $6, M$ is the midpoint of $A C$. Then $D M=$ $\qquad$

Answer: \dfrac{\sqrt{11}}{2}


IMO 1990 Problem A2 Take n ≥ 3 and consider a set E of 2n-1 distinct points on a circle. Suppose that exactly k of these points are to be colored black. Such a coloring is "good" if there is at least one pair of black points such that the interior of one of the arcs between them contains exactly n points from E. Find the smallest value of k so that every such coloring of k points of E is good. Solution

Answer: n


96. Processing a batch of parts, if A works for 5 hours first, then B joins to work together, A will have made 400 more parts than B when completed; if B works for 5 hours first, then A joins to work together, B will have made 40 more parts than A when completed. If A and B start working together from the beginning, how many more parts will A have made than B when completed?

Answer: 180


4. Xiao Pang, Xiao Dingding, Xiao Ya, and Xiao Qiao's four families, a total of 8 parents and 4 children, went to the amusement park together. The amusement park's ticket pricing is: 100 yuan per adult; 50 yuan per child; for 10 people or more, a group ticket is available at 70 yuan per person. They need to spend at least yuan to buy the tickets.

Answer: 800


Exercise 2. A permutation of the set $\{1, \ldots, 2021\}$ is a sequence $\sigma=\left(\sigma_{1}, \ldots, \sigma_{2021}\right)$ such that each element of the set $\{1, \ldots, 2021\}$ is equal to exactly one term $\sigma_{i}$. We define the weight of such a permutation $\sigma$ as the sum

$$
\sum_{i=1}^{2020}\left|\sigma_{i+1}-\sigma_{i}\right|
$$

What is the greatest possible weight of permutations of $\{1, \ldots, 2021\}$?

Answer: 2042219


Let $m>1$ be an odd integer. Let $n=2m$ and $\theta=e^{2\pi i\slash n}$. Find integers $a_{1},\ldots,a_{k}$ such that
$\sum_{i=1}^{k}a_{i}\theta^{i}=\frac{1}{1-\theta}$.

Answer: 1


3. In a right triangle $\triangle A B C\left(\measuredangle C=90^{\circ}\right)$, the line passing through the midpoint of the hypotenuse and the center of the inscribed circle intersects the leg $A C$ at point $N$ at an angle of $75^{\circ}$. Determine the acute angles in $\triangle A B C$.

Answer: 30^\circ


## Task 4 - 300824

Imagine the numbers 1, 2, 3, 4, ..., 9999 written consecutively to form a number $z$.

The beginning of this representation is $z=123456789101112131415 \ldots$; for example, the digit 0 appears at the 11th position, the digit 2 appears at the 2nd position, the 15th position, and at several other positions in $z$.

Which digit stands at the 206788th position of $z$?

Answer: 7


4. As shown in Figure $4, B D$ and $C E$ are altitudes of acute $\triangle A B C$, $\angle A B C=60^{\circ}$, $A E=8, B E=4$, and $P$ is a moving point on side $B C$. Then the minimum value of $P D+P E$ is $\qquad$.

Answer: \dfrac{20\sqrt{7}}{7}


\section*{Problem 3 - 291013}

Two players have invented the following game: On a game board, 14 fields are arranged in a circle, one of these fields is designated as the starting field \(A\). Each player has a game piece and places it on field \(A\).

Then each player rolls a die. If both players roll different numbers, the player with the higher number moves their piece 4 steps clockwise, the other player moves their piece 2 steps. If they roll the same number, each player moves their piece 3 steps forward.

This rolling and moving of both pieces is considered one turn. Due to the circular arrangement of the fields, it is possible for a piece to reach or pass field \(A\) (and thus start a new lap) during its movement.

The game is over as soon as after performing a turn, the piece of at least one player lands exactly on field \(A\). This player wins if the other player's piece is not on \(A\). However, if both pieces are on \(A\), the game ends in a draw.

What is the smallest possible number of turns in which a game can end in a draw? Justify your answer!

Answer: 7


C2. Let $m, n \geq 2$ be positive integers. Each entry of an $m \times n$ grid contains a real number in the range $[-1,1]$, i.e. between -1 and 1 inclusively. The grid also has the property that the sum of the four entries in every $2 \times 2$ subgrid is equal to 0 . (A $2 \times 2$ subgrid is the intersection of two adjacent rows and two adjacent columns of the original grid.) Let $S$ be the sum of all of the entries in the grid.
a. Suppose $m=6$ and $n=6$. Explain why $S=0$.
b. Suppose $m=3$ and $n=3$. If the elements of the grid are
\begin{tabular}{|l|l|l|}
\hline$a$ & $b$ & $c$ \\
\hline$d$ & $e$ & $f$ \\
\hline$g$ & $h$ & $i$ \\
\hline
\end{tabular}
show that $S+e=a+i=c+g$.
c. Suppose $m=7$ and $n=7$. Determine the maximum possible value of $S$.

Answer: 7


5. [25] Thaddeus is given a $2013 \times 2013$ array of integers each between 1 and 2013 , inclusive. He is allowed two operations:
1. Choose a row, and subtract 1 from each entry.
2. Chooses a column, and add 1 to each entry.
He would like to get an array where all integers are divisible by 2013. On how many arrays is this possible?

Answer: 2013^{4025}


The Fruit and Vegetable Kingdom is holding a king election, with each of the 205 people nationwide casting 1 vote, choosing 1 from Tomato Warrior, Banana Superman, and Carrot Doctor, and the one with the most votes will be elected. As of now, Tomato Warrior has 54 votes, Banana Superman has 46 votes, and Carrot Doctor has 36 votes. So, how many more votes does Tomato Warrior need to ensure he can be elected king.

Answer: 31


3. As shown in Figure 4, the diagonals $AC$ and $BD$ of quadrilateral $ABCD$ intersect at point $O$, and $E$, $F$, and $G$ are the midpoints of $AB$, $OC$, and $OD$ respectively. Given that $OA = AD$, $OB = BC$, and $CD = \sqrt{3} AB$. Then the degree measure of $\angle FEG$ is $\qquad$

Answer: 120


Let's determine how many shuffles it takes to return 52 cards to their original order if we shuffle them as described in the article about Faro Shuffle. Solve this problem in the case where we start the shuffle with the bottom card of the right-hand deck, meaning the card originally in the 26th position ends up at the bottom.

Answer: 52


8.82 Suppose there are 128 ones written on the blackboard. In each step, you can erase any two numbers $a$ and $b$ on the blackboard, and write $ab+1$. After 127 such steps, only one number remains. Let the maximum possible value of this remaining number be $A$. Find the last digit of $A$.

Answer: 2


10th Irish 1997 Problem B2 The quadrilateral ABCD has an inscribed circle. ∠A = ∠B = 120 o , ∠C = 30 o and BC = 1. Find AD.

Answer: \dfrac{\sqrt{3} - 1}{2}


6. Let the two foci of the ellipse $\Gamma$ be $F_{1}, F_{2}$, and a line passing through point $F_{1}$ intersects $\Gamma$ at points $P, Q$. If $\left|P F_{2}\right|=\left|F_{1} F_{2}\right|$, and $3\left|P F_{1}\right|=4\left|Q F_{1}\right|$, then the ratio of the minor axis to the major axis of the ellipse $\Gamma$ is $\qquad$.

Answer: \dfrac{2\sqrt{6}}{7}


Podlipsky o.k.

Oleg drew an empty $50 \times 50$ table and wrote a number above each column and to the left of each row. It turned out that all 100 written numbers are distinct, with 50 of them being rational and the other 50 being irrational. Then, in each cell of the table, he wrote the product of the numbers written next to its row and its column ("multiplication table"). What is the maximum number of products in this table that could be rational numbers?

Answer: 1275


4. Given $S_{\triangle M B C}=4, 3 A B=2 B C$, draw the perpendicular from point $C$ to the angle bisector $B E$ of $\angle A B C$, and let the foot of the perpendicular be $D$. Then $S_{\triangle B D C}=$ $\qquad$

Answer: 3


40th Putnam 1979 Problem B4 Find a non-trivial solution of the differential equation F(y) ≡ (3x 2 + x - 1)y'' - (9x 2 + 9x - 2)y' + (18x + 3)y = 0. y = f(x) is the solution of F(y) = 6(6x + 1) such that f(0) = 1, and ( f(-1) - 2)( f(1) - 6) = 1. Find a relation of the form ( f(-2) - a)( f(2) - b) = c. Solution

Answer: (f(-2) - 6)(f(2) - 14) = 1


3. (7 points) A traveler was riding in a bus and saw a two-digit number on a kilometer post. He fell asleep and woke up an hour later to see a three-digit number on the kilometer post, where the first digit was the same as the second digit an hour ago, the second digit was zero, and the third digit was the same as the first digit an hour ago. Two hours later, he looked out the window and saw a kilometer post with the same number as two hours ago, except the zero was replaced by another digit. Find the speed of the bus (assuming it moved at a constant speed).

Answer: 45


Example 3.3.2 (League 2004) In an acute triangle $ABC$, the altitude $CE$ from $C$ to $AB$ intersects the altitude $BD$ from $B$ to $AC$ at point $H$. The circle with diameter $DE$ intersects $AB$ and $AC$ at points $F$ and $G$, respectively. $FG$ intersects $AH$ at point $K$. Given $BC=25, BD=20, BE=7$, find the length of $AK$.

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The translation maintains the original text's formatting and structure.

Answer: \dfrac{216}{25}


$A, B, C, D$ four people attend a meeting, and they all receive the same positive integer. Next, each person describes this positive integer. Each person says three sentences, of which at least one is true and at least one is false. What they say is as follows:
$A:\left(A_{1}\right)$ This number is less than 12;
$\left(A_{2}\right) 7$ does not divide this number;
$\left(A_{3}\right) 5$ multiplied by this number is less than 70.
$B:\left(B_{1}\right) 12$ multiplied by this number is greater than 1000;
$\left(B_{2}\right) 10$ divides this number;
$\left(B_{3}\right)$ This number is greater than 100.
$C:\left(C_{1}\right) 4$ divides this number;
$\left(C_{2}\right) 11$ multiplied by this number is less than 1000;
$\left(C_{3}\right) 9$ divides this number.
$D:\left(D_{1}\right)$ This number is less than 20;
$\left(D_{2}\right)$ This number is a prime number;
$\left(D_{3}\right)$ 7 divides this number.

This number is - . $\qquad$

Answer: 89


8. Place 10 numbers on a given circle such that their total sum is 200, and the sum of any three consecutive numbers is not less than 58. Then the maximum value of the largest number among all sets of 10 numbers that satisfy the above requirements is $\qquad$

Answer: 26


Example 2 In $\triangle A B C$, it is known that the coordinates of vertex $A$ are $A(1,4), \angle A B C$ and $\angle A C B$'s bisector line equations are $x-2 y=0$ and $x+y-1=0$, respectively. Find the equation of line $B C$.

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

Answer: 4x + 17y + 12 = 0


The electronic clock shown can display time from 00:00:00 to 23:59:59. How many times in a day (24 hours) does the clock display a time that consists of the digits 1, 2, 3, 4, 5, and 6?

Answer: 96


A sequence $(z_n)$ of complex numbers satisfies the following properties:

$z_1$ and $z_2$ are not real.
$z_{n+2}=z_{n+1}^2z_n$ for all integers $n\geq 1$.
$\dfrac{z_{n+3}}{z_n^2}$ is real for all integers $n\geq 1$.
$\left|\dfrac{z_3}{z_4}\right|=\left|\dfrac{z_4}{z_5}\right|=2$.

Find the product of all possible values of $z_1$.

Answer: 65536


Link is coloring a triforce, which consists of four equilateral triangles and is depicted below. He has three colors to use: gold, black, and green. So that it remains recognizable, he doesn't want to color any two triangles the same color if they share a side. How many different ways can he color the triforce? (Two colorings that differ by rotation are considered distinct.) [asy]
size(50);
draw((0,0)--(2,0)--(1,1.732)--cycle);
draw((1,0)--(1.5,.866)--(.5,.866)--cycle);
[/asy]

Answer: 24


A belt is drawn tightly around three circles of radius $10$ cm each, as shown. The length of the belt, in cm, can be written in the form $a + b\pi$ for rational numbers $a$ and $b$. What is the value of $a + b$? [asy]

size(120); defaultpen(linewidth(0.8));
picture p; real r = 3^.5/2;
pair A = expi(-pi/6); draw(p, circle(A, r)); draw(p, (A.x,A.y - r) -- (-A.x,A.y - r));
add(p); add(rotate(120)*p); add(rotate(240)*p);
[/asy]

Answer: 80


Shapovalov A.V.

There are three piles of 40 stones each. Petya and Vasya take turns, Petya starts. On a turn, one must combine two piles, then divide these stones into four piles. The player who cannot make a move loses. Which of the players (Petya or Vasya) can win, regardless of how the opponent plays?

Answer: Vasya


Four boys, $A, B, C$, and $D$, made 3 statements each about the same number $x$. We know that each of them has at least one correct statement, but also that at least one of their statements is false. Can $x$ be determined? The statements are:

A: (1) The reciprocal of $x$ is not less than 1.

(2) The decimal representation of $x$ does not contain the digit 6.

(3) The cube of $x$ is less than 221.

$B$: (4) $x$ is an even number.

(5) $x$ is a prime number.

(6) $x$ is an integer multiple of 5.

$C$: (7) $x$ cannot be expressed as the quotient of two integers,

(8) $x$ is less than 6.

(9) $x$ is the square of a natural number.

$D$: (10) $x$ is greater than 20.

(11) $x$ is positive and its base-10 logarithm is at least 2.

(12) $x$ is not less than 10.

Answer: 25


On the side $BC$ of the equilateral triangle $ABC$, choose any point $D$, and on the line $AD$, take the point $E$ such that $| B A | = | BE |$. Prove that the size of the angle $AEC$ is  of does not depend on the choice of point $D$, and find its size.

Answer: 30^\circ


5. Calculate $f(2)$, if $25 f\left(\frac{x}{1580}\right)+(3-\sqrt{34}) f\left(\frac{1580}{x}\right)=2017 x$. Round the answer to the nearest integer.

Answer: 265572


136. How many times should a die be thrown so that the probability of the inequality

$$
|m / n-1 / 6| \leqslant 0.01
$$

is not less than the probability of the opposite inequality, where $m$ is the number of times one dot appears in $n$ throws of a die?

Answer: 632


At the base of a quadrilateral pyramid lies a rhombus $A B C D$, in which $\angle B A D=60^{\circ}$. It is known that $S A=S C$, $S D=S B=A B$. A point $E$ is taken on the edge $D C$ such that the area of triangle $B S E$ is the smallest among the areas of all sections of the pyramid containing the segment $B S$ and intersecting the segment $D C$. Find the ratio $D E: E C$.

Answer: \dfrac{2}{3}


To build the paper kite represented in the figure, Eduardo started by painting a rectangle ABCD on a sheet of paper. Then, he extended each side of the rectangle, tripling its length, and obtained the quadrilateral $A^{\prime} B^{\prime} C^{\prime} D^{\prime}$.

![](https://cdn.mathpix.com/cropped/2024_05_01_4e05cbad380df980abcfg-7.jpg?height=452&width=782&top_left_y=542&top_left_x=474)

Figure 20.1

Knowing that the area of rectangle $A B C D$ is $200 \mathrm{~cm}^{2}$, what is the area of the kite constructed by Eduardo?

Answer: 2600


Anna and Béla agree to meet between 5 and $\frac{1}{2} 6$ in the afternoon. What is the probability that the one who arrives first does not wait more than 10 minutes for the other, if both adhere to what they agreed upon?

Answer: \dfrac{5}{9}


7. In a chess tournament, 8 players participated. They scored $7, 6, 4, 4, 3, 2, 1, 5$ and 0.5 points respectively. How many points did the players who took the first four places lose in matches against the others?

Answer: 1


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

Answer: 95


100. There are 5 different lines on a plane, these 5 lines form $n$ intersection points, then $n$ has $\qquad$ different values.

Answer: 9


For example, the vertices $B$ and $A$ of a regular $\triangle A B P$ with side length $a$ slide on the $x$-axis and $y$-axis respectively $(A, B, P$ are arranged clockwise). Find the trajectory of vertex $P$ in the first quadrant.

Answer: x^2 + y^2 - \sqrt{3}xy = \frac{a^2}{4}


Example 2 In a simple graph with eight vertices, what is the maximum number of edges in a graph without a quadrilateral?

The maximum number of edges in a graph without a quadrilateral in a simple graph with eight vertices is what?

Answer: 11


6. In the Cartesian coordinate system $x O y$, the line $l_{1}$: $4 x+5 y=20$ intersects the $x$-axis and $y$-axis at points $A$ and $B$, respectively. The line $l_{2}$ intersects the line segment $A B$ and $O A$ at points $C$ and $D$, respectively, and bisects the area of $\triangle A O B$. Then the minimum value of $C D^{2}$ is

Answer: 5\sqrt{41} -25


16. (6 points) As shown in the figure, the book "Classic Fairy Tales" has a total of 382 pages. Then, the number of digit 0 in the page numbers of this book is $\qquad$ .

Answer: 68


Three, 18 football teams are participating in a single round-robin tournament, meaning each round the 18 teams are divided into 9 groups, with each group's two teams playing one match. In the next round, the teams are regrouped to play, for a total of 17 rounds, ensuring that each team plays one match against each of the other 17 teams. After $n$ rounds of the tournament according to any feasible schedule, there always exist 4 teams that have played a total of only 1 match among themselves. Find the maximum possible value of $n$.
(Li Chengzhang, contributor)

Answer: 7


7. For $x \in \mathbf{R}, n \in \mathbf{N}^{*}$, define $C_{x}^{n}=\frac{x(x-1) \cdots(x-n+1)}{n!}$. Let $P(x)$ be a 6th degree polynomial that satisfies $P(0)=1, P(k)=2^{k-1}, k=1,2, \cdots, 6$, express $P(x)=$ using $C_{x}^{k}(k=1,2, \cdots, 6)$.

Answer: 1 + C_{x}^{2} + C_{x}^{4} + C_{x}^{6}


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Arnold`, `Peter`, `Alice`, `Eric`
 - Each person has a unique favorite drink: `milk`, `root beer`, `coffee`, `tea`, `water`
 - Each person has a favorite color: `blue`, `green`, `white`, `yellow`, `red`
 - They all have a unique favorite flower: `daffodils`, `roses`, `lilies`, `tulips`, `carnations`
 - Each person has a unique hobby: `painting`, `cooking`, `photography`, `gardening`, `knitting`

## Clues:
1. Alice is not in the fourth house.
2. The root beer lover is the person who enjoys gardening.
3. The person whose favorite color is green is the coffee drinker.
4. The person whose favorite color is green is the person who loves the boquet of lilies.
5. The person who loves blue is somewhere to the right of the person who loves a bouquet of daffodils.
6. The person who loves cooking is the person who loves blue.
7. Eric is directly left of the tea drinker.
8. The one who only drinks water is Peter.
9. Arnold is the photography enthusiast.
10. The person who loves white is the person who loves the rose bouquet.
11. There is one house between the person who loves a carnations arrangement and the person whose favorite color is red.
12. The person who loves cooking is somewhere to the left of the person who paints as a hobby.
13. The one who only drinks water is in the third house.
14. The person who loves a carnations arrangement is the root beer lover.
15. The person who loves white is in the second house.
 What is the value of attribute Hobby for the person whose attribute Name is Eric? Please reason step by step, and put your final answer within \boxed{}

Answer: gardening