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 prefers a unique type of vacation: `city`, `beach`, `mountain`
 - Everyone has something unique for lunch: `pizza`, `spaghetti`, `grilled cheese`
 - Everyone has a favorite smoothie: `desert`, `watermelon`, `cherry`
 - People own unique car models: `toyota camry`, `tesla model 3`, `ford f150`
 - They all have a unique favorite flower: `daffodils`, `carnations`, `lilies`

## Clues:
1. The person who likes Cherry smoothies is directly left of the person who loves a bouquet of daffodils.
2. Eric is the person who loves beach vacations.
3. The person who loves a carnations arrangement is somewhere to the left of the person who loves a bouquet of daffodils.
4. The person who owns a Toyota Camry is not in the first house.
5. The Watermelon smoothie lover is directly left of the person who is a pizza lover.
6. Peter is somewhere to the left of Arnold.
7. Arnold is not in the third house.
8. The person who loves beach vacations is the person who owns a Ford F-150.
9. The person who loves the spaghetti eater is the person who prefers city breaks.
10. Peter and the person who loves the spaghetti eater are next to each other.
 What is the value of attribute House for the person whose attribute Vacation is mountain? Please reason step by step, and put your final answer within \boxed{}

Answer: 1


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Alice`, `Eric`, `Arnold`, `Bob`, `Carol`
 - The people are of nationalities: `dane`, `brit`, `norwegian`, `swede`, `german`, `chinese`
 - People have unique favorite book genres: `historical fiction`, `mystery`, `romance`, `fantasy`, `science fiction`, `biography`
 - The mothers' names in different houses are unique: `Holly`, `Aniya`, `Penny`, `Kailyn`, `Sarah`, `Janelle`

## Clues:
1. The person whose mother's name is Janelle is the Swedish person.
2. Alice is the person who loves science fiction books.
3. The person who loves romance books is not in the fifth house.
4. Carol and Peter are next to each other.
5. The person whose mother's name is Sarah is the Norwegian.
6. There is one house between Arnold and the British person.
7. The German and the person who loves historical fiction books are next to each other.
8. There is one house between the German and Carol.
9. The Swedish person is the person who loves fantasy books.
10. Arnold is in the second house.
11. The person whose mother's name is Sarah is in the third house.
12. The person who loves historical fiction books is somewhere to the right of The person whose mother's name is Penny.
13. The Chinese is not in the sixth house.
14. There are two houses between Alice and the person who loves mystery books.
15. Carol is somewhere to the right of Alice.
16. Arnold is the Dane.
17. The person whose mother's name is Aniya is in the fourth house.
18. The person whose mother's name is Penny is somewhere to the left of the Swedish person.
19. The person whose mother's name is Holly and Eric are next to each other.
 What is the value of attribute Nationality for the person whose attribute BookGenre is science fiction? Please reason step by step, and put your final answer within \boxed{}

Answer: german


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Bob`, `Arnold`, `Alice`, `Eric`, `Carol`
 - The people keep unique animals: `rabbit`, `bird`, `dog`, `horse`, `cat`, `fish`
 - Each person has an occupation: `doctor`, `teacher`, `lawyer`, `artist`, `engineer`, `nurse`

## Clues:
1. Arnold is not in the sixth house.
2. The person who is an engineer is directly left of Eric.
3. Carol and the rabbit owner are next to each other.
4. The person who is an artist is Peter.
5. The person who keeps horses is in the second house.
6. Eric is directly left of the person who is a lawyer.
7. The person who is an artist is somewhere to the left of the person who is a doctor.
8. The cat lover is not in the sixth house.
9. Peter is in the fifth house.
10. The dog owner is the person who is an artist.
11. The person who keeps horses is directly left of the rabbit owner.
12. The bird keeper is Alice.
13. The person who is a teacher and Alice are next to each other.
14. Alice is not in the first house.
 What is the value of attribute Animal for the person whose attribute Name is Carol? Please reason step by step, and put your final answer within \boxed{}

Answer: horse


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`
 - Each person has a unique hobby: `cooking`, `gardening`, `painting`, `photography`, `knitting`
 - People have unique favorite sports: `swimming`, `tennis`, `soccer`, `baseball`, `basketball`
 - Each person lives in a unique style of house: `ranch`, `craftsman`, `victorian`, `modern`, `colonial`
 - Each mother is accompanied by their child: `Timothy`, `Samantha`, `Bella`, `Meredith`, `Fred`
 - People have unique heights: `average`, `very tall`, `very short`, `short`, `tall`

## Clues:
1. The person who has an average height is the person's child is named Meredith.
2. The person who is tall is in the second house.
3. Peter is directly left of the person residing in a Victorian house.
4. Alice is the person who is tall.
5. The person who loves baseball is the person who is very tall.
6. The person's child is named Meredith and the person who is the mother of Timothy are next to each other.
7. Bob is the person who paints as a hobby.
8. The person who enjoys gardening is in the second house.
9. The person who is very short is somewhere to the right of Eric.
10. The person who loves tennis is the person's child is named Samantha.
11. The person who loves soccer is not in the first house.
12. The person's child is named Samantha is the person in a modern-style house.
13. The person in a Craftsman-style house is the person who has an average height.
14. The person's child is named Fred is the person residing in a Victorian house.
15. The person who is short is the person who loves basketball.
16. Peter is the person who is very tall.
17. The person in a ranch-style home is somewhere to the left of the person who loves cooking.
18. The person who enjoys knitting and the person who enjoys gardening are next to each other.
19. The person in a modern-style house is the person who loves cooking.
20. The person residing in a Victorian house is in the fifth house.
 What is the value of attribute HouseStyle for the person whose attribute Name is Alice? Please reason step by step, and put your final answer within \boxed{}

Answer: ranch


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 House for the person whose attribute Color is yellow? Please reason step by step, and put your final answer within \boxed{}

Answer: 1


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 BookGenre for the person whose attribute Drink is coffee? Please reason step by step, and put your final answer within \boxed{}

Answer: romance


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Arnold`, `Eric`
 - People have unique favorite book genres: `science fiction`, `mystery`, `romance`
 - Everyone has a favorite smoothie: `watermelon`, `desert`, `cherry`
 - Each person has a unique birthday month: `april`, `jan`, `sept`
 - People have unique heights: `average`, `very short`, `short`

## Clues:
1. The person who likes Cherry smoothies is not in the second house.
2. Arnold is the person who loves mystery books.
3. The person whose birthday is in January is not in the first house.
4. The person who is very short is the person who loves romance books.
5. The person who loves mystery books is the person whose birthday is in September.
6. The person who has an average height is the Desert smoothie lover.
7. Eric is in the first house.
8. The Watermelon smoothie lover is the person who is short.
9. The Watermelon smoothie lover is Eric.
 What is the value of attribute Name for the person whose attribute BookGenre is romance? 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`, `Arnold`, `Eric`, `Bob`, `Alice`
 - People have unique heights: `average`, `very tall`, `very short`, `short`, `tall`
 - Everyone has a unique favorite cigar: `prince`, `dunhill`, `blends`, `pall mall`, `blue master`
 - Everyone has a favorite smoothie: `lime`, `cherry`, `dragonfruit`, `watermelon`, `desert`
 - People use unique phone models: `oneplus 9`, `samsung galaxy s21`, `iphone 13`, `huawei p50`, `google pixel 6`

## Clues:
1. The Prince smoker is the Desert smoothie lover.
2. There is one house between Eric and Alice.
3. The person who is short is the person who smokes many unique blends.
4. The person who uses an iPhone 13 is directly left of the person who smokes Blue Master.
5. The person who has an average height is the Dunhill smoker.
6. Eric is the person who is very tall.
7. Arnold is directly left of the person who uses a Huawei P50.
8. Bob is not in the fourth house.
9. Eric is directly left of the person who likes Cherry smoothies.
10. Bob is the Dunhill smoker.
11. The Dragonfruit smoothie lover is Bob.
12. The person who uses an iPhone 13 and the person who uses a OnePlus 9 are next to each other.
13. The person who uses a Samsung Galaxy S21 is the person who is short.
14. There are two houses between the person who is very tall and the Dragonfruit smoothie lover.
15. The person who uses an iPhone 13 is Eric.
16. The Desert smoothie lover is somewhere to the left of the person who drinks Lime smoothies.
17. Arnold and the person who is very short are next to each other.
 What is the value of attribute Height for the person whose attribute Cigar is prince? Please reason step by step, and put your final answer within \boxed{}

Answer: very tall


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Bob`, `Alice`, `Peter`, `Arnold`
 - Each person has a favorite color: `yellow`, `green`, `red`, `white`, `blue`
 - Each mother is accompanied by their child: `Samantha`, `Fred`, `Meredith`, `Bella`, `Timothy`
 - People have unique favorite book genres: `biography`, `mystery`, `romance`, `science fiction`, `fantasy`
 - Each person has a unique birthday month: `april`, `feb`, `sept`, `jan`, `mar`
 - People have unique favorite sports: `soccer`, `basketball`, `swimming`, `baseball`, `tennis`

## Clues:
1. The person who loves swimming is the person who loves mystery books.
2. Eric is somewhere to the left of the person who is the mother of Timothy.
3. The person who loves basketball is somewhere to the left of the person who is the mother of Timothy.
4. The person who loves soccer is the person whose birthday is in September.
5. Eric is the person who loves biography books.
6. The person who loves baseball is the person who loves white.
7. The person who loves science fiction books is the person whose birthday is in March.
8. The person who loves swimming and the person whose favorite color is red are next to each other.
9. The person who loves biography books is not in the first house.
10. The person who loves baseball is somewhere to the left of the person whose favorite color is red.
11. The person who loves fantasy books is Bob.
12. The person's child is named Bella is the person whose birthday is in February.
13. The person who loves tennis and the person's child is named Samantha are next to each other.
14. The person's child is named Bella is not in the first house.
15. The person's child is named Meredith is the person whose birthday is in March.
16. The person who loves blue is directly left of the person who loves yellow.
17. There is one house between Alice and the person whose birthday is in March.
18. The person whose favorite color is red is directly left of the person whose birthday is in January.
19. The person's child is named Meredith is not in the first house.
20. Arnold is somewhere to the right of the person who loves swimming.
21. The person who loves fantasy books is the person who is the mother of Timothy.
 What is the value of attribute BookGenre for the person whose attribute Birthday is mar? Please reason step by step, and put your final answer within \boxed{}

Answer: science fiction


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Eric`, `Arnold`, `Alice`
 - Each person has a unique birthday month: `april`, `sept`, `jan`, `feb`
 - Everyone has a unique favorite cigar: `dunhill`, `prince`, `pall mall`, `blue master`
 - People use unique phone models: `google pixel 6`, `samsung galaxy s21`, `oneplus 9`, `iphone 13`
 - They all have a unique favorite flower: `lilies`, `roses`, `carnations`, `daffodils`
 - People have unique heights: `average`, `short`, `tall`, `very short`

## Clues:
1. The person who loves a bouquet of daffodils is Arnold.
2. The person whose birthday is in April is directly left of the person who smokes Blue Master.
3. The person who is short is the person who uses an iPhone 13.
4. The person who uses a Samsung Galaxy S21 is the person who is tall.
5. Eric is the person who uses a OnePlus 9.
6. The person who uses a Samsung Galaxy S21 is directly left of the person who uses a Google Pixel 6.
7. The person whose birthday is in February is Alice.
8. The Prince smoker is somewhere to the right of the person who is short.
9. The Prince smoker is the person whose birthday is in February.
10. The person who uses a OnePlus 9 is directly left of the person who loves the rose bouquet.
11. The person partial to Pall Mall is somewhere to the left of the person who loves the boquet of lilies.
12. The person who has an average height and the person whose birthday is in September are next to each other.
13. The person who loves a bouquet of daffodils is in the fourth house.
 What is the value of attribute Flower for the person whose attribute Name is Alice? Please reason step by step, and put your final answer within \boxed{}

Answer: lilies


7. In village $A$, there are 100 schoolchildren, and in village $B$, there are 50 schoolchildren. The distance between the villages is 3 kilometers. At what point on the road from $A$ to $B$ should a school be built to minimize the total distance traveled by all schoolchildren?

Answer: 0


6. Find all triples $(m, n, p)$ which satisfy the equation
$$
p^{n}+3600=m^{2}
$$
where $p$ is prime and $m, n$ are positive integers.

Answer: (61, 2, 11)


11. Given the ellipse $C_{1}: \frac{x^{2}}{24}+\frac{y^{2}}{b^{2}}=1(0<b<2 \sqrt{6})$ with its right focus $F_{1}$ coinciding with the focus of the parabola $C_{2}: y^{2}=4 p x\left(p \in \mathbf{Z}_{+}\right)$, a line $l$ passing through $F_{1}$ with a positive integer slope intersects $C_{1}$ at points $A, B$ and $C_{2}$ at points $C, D$. If $13|A B|=\sqrt{6}|C D|$, find the values of $b$ and $p$.

Answer: 2\sqrt{2}


Example 2. Solve the Euler equation $x^{2} y^{\prime \prime}-x y^{\prime}+2 y=x \ln x$.

Answer: y = C_1 x \cos(\ln x) + C_2 x \sin(\ln x) + x \ln x


Three. (50 points) Consider the 60 vertices of a regular 60-gon, where 10 vertices are colored red, 20 vertices are colored yellow, and the remaining 30 vertices are colored blue. Find the smallest positive integer $m$, such that it is always possible to find $m$ lines that do not pass through any vertices and divide the plane into several regions, each of which contains points of the same color.

Answer: 30


4. In an isosceles triangle $A B C$ the angle opposite the base $\overline{A B}$ is $80^{\circ}$. On the height to the base of this triangle, a point $D$ is chosen such that $|\angle D B A|=30^{\circ}$. On the segment $\overline{B D}$, a point $T$ is chosen such that $|\angle T A D|=20^{\circ}$. Determine the size of the angle $\angle T C A$.

Answer: 70^\circ


[ Decimal number system ] [ Divisibility of numbers. General properties ]

In a 100-digit number 12345678901234...7890, all digits standing at odd positions were erased; in the resulting 50-digit number, all digits standing at odd positions were erased again, and so on. The erasing continued until there was nothing left to erase. What was the last digit to be erased?

Answer: 4


2. On one website, there is a poll to determine the best hockey player of the last decade. The number of votes for each player is given as a percentage, rounded to the nearest whole number. After Mirek voted for Jaromír Jágr, his share of $7 \%$ did not change. What is the minimum number of people, including Mirek, who voted? We assume that each participant in the poll voted exactly once, and for only one player.

Answer: 107


2. A square with a side of 49 cm is given. Divide the given square into 2009 smaller squares of two types, which have integer sides (the side lengths of the divided squares can have one of two different integer values).

Answer: 1960


6. There are 8 English letters $K, Z, A, I, G, A, K$, and $\mathrm{U}$, each written on 8 cards. Ask:
(1) How many ways are there to arrange these cards in a row?
(2) How many ways are there to arrange 7 of these cards in a row?

Answer: 10080


2. Solve the equation $x+\frac{x}{\sqrt{x^{2}-1}}=\frac{35}{12}$.

Answer: \dfrac{5}{4}


## Task 10/75

Given is a regular pyramid with a square base. Each side face forms an angle $\alpha$ with the base. Two spheres are inscribed in this pyramid such that one sphere touches all five faces of the pyramid, while the other touches the four side faces of the pyramid and the surface of the first sphere.

In what ratio do the volumes of the two spheres stand to each other?

Answer: \tan^6\left(\dfrac{\alpha}{2}\right)


13.255. Three swimmers have to swim a lane 50 m long, immediately turn around, and return to the starting point. The first starts first, the second starts 5 s later, and the third starts another 5 s later. At some point in time, before reaching the end of the lane, the swimmers found themselves at the same distance from the start. The third swimmer, having reached the end of the lane and turned back, met the second 4 m from the end of the lane, and the first - 7 m from the end of the lane. Find the speed of the third swimmer.

Answer: \dfrac{22}{15}


1. What is the smallest number of digits that can be erased from the number 20162016 so that the result is divisible by 2016 (it is not allowed to erase nothing)? Please note that you need to not only provide an example but also explain why it is impossible to do with fewer digits.

Answer: 3


3B. Given a right triangle СВА with legs of length 3 and 4. Squares BCSR and $A B Q P$ are constructed outside the triangle on the smaller leg and the hypotenuse, respectively. Calculate the area of triangle $B R Q$.

Answer: 6


Example 1 A city has $n$ high schools, the $i$-th high school sends $c_{i}$ students $\left(1 \leqslant c_{i} \leqslant 39\right)$ to watch a ball game at the gymnasium, where $\sum_{i=1}^{n} c_{i}=1990$. Each row in the stand has 199 seats, and it is required that students from the same school sit in the same row. Question: What is the minimum number of rows that need to be arranged so that all students can definitely be seated?

Answer: 12


9.6. Inside an acute scalene triangle $A B C$, where $\angle A B C=60^{\circ}$, a point $T$ is marked such that $\angle A T B = \angle B T C = \angle A T C = 120^{\circ}$. The medians of the triangle intersect at point $M$. The line $T M$ intersects the circumcircle of triangle $A T C$ again at point $K$. Find $T M / M K$.

(A. Kuznetsov)

Answer: \dfrac{1}{2}


Several chips of two colors are arranged in a row (both colors are present). It is known that the chips that are 10 or 15 chips apart are the same.

What is the maximum number of chips possible?

Answer: 25


Thirty-six players are playing a game: they take turns drawing a random card from a 36-card deck. If a player draws the ace of diamonds, they win. If they draw any other card, they return it to the deck, and the next player takes their turn. This continues in a cycle: first player, then the second, and so on. If the ace of diamonds is not drawn in the first round, the players continue drawing cards in the same order in the second round, and so on. This process repeats until someone draws the ace of diamonds.

Assume that before the game starts, the players place bets, and the winner takes all. How should the players' bets be related so that the game is fair, meaning the expected winnings for all players are equal to each other (i.e., equal to zero taking into account the amount each player bet)?

Answer: \frac{35}{36}


In the figure alongside, the triangle \(ABC\) is equilateral and \(BD = CE = AF = \frac{AB}{3}\). The ratio \(\frac{EG}{GD}\) can be written in the form \(\frac{m}{n}\), where \(\operatorname{gcd}(m, n) = 1\). What is the value of \(m + n\)?

Answer: 5


Two circles are given. Their common internal tangents are mutually perpendicular. The chords connecting the points of tangency are 3 and 5. Find the distance between the centers of the circles.

Answer: 8


In the star $ABCDE$ shown in the figure, we know that $\measuredangle GBF=20^{\circ}$ and $\measuredangle GHI=130^{\circ}$. What is the value of the angle $\measuredangle JEI$?

Answer: 30^\circ


Let \( f(x) \) be an \( n \)-th degree polynomial such that \( f(k) = 2^k \) for \( k = 1, 2, \ldots, n+1 \). Find the values of \( f(n+2) \) and \( f(n+3) \) (express the answers as functions of \( n \) in their simplest form).

Answer: 2^{n+3} - 2n - 6


Let \( A, B, C, \) and \( D \) be four points lying in this order on a circle. Assume that there is a point \( K \) on the line \( AB \) such that \( BD \) bisects the line \( KC \) and \( AC \) bisects the line \( KD \). Determine the smallest possible value that \( \left|\frac{AB}{CD}\right| \) can take.

Answer: 2


Peter received a new jigsaw with a function for measuring the length of cuts for his birthday. To test the gift, Peter took a square piece of plywood with a side of 50 cm, and cut it into squares with sides of 10 cm and 20 cm. How many squares did he obtain in total if the jigsaw displayed a total cut length of \(2 \mathbf{m} 80 \mathrm{~cm}\)?

Answer: 16


Miguel and Sofia, a young couple who recently immigrated to a new country, love sharing stories of their homeland with their new friends. They have a collection of 120 traditional recipes from their homeland, which they plan to share at a cultural event. They decide to prepare 5 different dishes using these recipes.

For the event, they need to prepare enough servings for all the guests. Each dish from their homeland requires 4 servings. If they expect 60 guests at the event, how many total servings do Miguel and Sofia need to prepare to ensure each guest can try all 5 dishes?

Answer: 300


Ms. Carter is an English language teacher specialized in teaching business English. She has 5 business English classes each week, and each class has 12 students. She plans to provide each student with 3 business English vocabulary worksheets every month. How many worksheets will Ms. Carter need to prepare for all her students in one month?

Answer: 180


Jenna, whose spouse was a miner who tragically lost their life in a mining accident, is now committed to advocating for stricter safety regulations in the mining industry. She decides to organize a community meeting to raise awareness and gather support. Jenna plans to invite 85 people to the meeting. Each person who attends will be asked to sign a petition for stronger safety measures.

At the meeting, Jenna provides refreshments. She buys 3 bottles of water for every 5 attendees and 2 sandwiches for every 3 attendees. If each bottle of water costs $1.50 and each sandwich costs $3.00, how much will Jenna spend in total on refreshments if all 85 invited people attend the meeting?

Answer: 247.50


Maria is a young Brazilian lady who recently converted to Catholicism. She is passionate about social justice and fighting racial discrimination. She decided to organize a charity event at her local church to raise funds for a campaign promoting racial equality.

Maria plans to sell handmade bracelets at the event. She makes each bracelet for $3 using materials she bought with her own savings. She plans to sell each bracelet for $8. Maria's goal is to raise $500 for the campaign.

During the event, Maria sells 40 bracelets. Additionally, she receives a donation of $120 from a generous supporter who believes in her cause.

How much more money does Maria need to reach her fundraising goal of $500?

Answer: 180


Professor Zhang at Shanxi Technology and Business College has been guiding students through their academic journey for 10 years. Each year, he mentors 5 new students. By the end of each year, 3 of his previously mentored students graduate and he remains in contact with 2 of them. After 10 years, how many of Professor Zhang's former students does he still keep in contact with?

Answer: 20


The museum curator is organizing a special exhibition that features a portrait titled "The Vendor's Spirit," which symbolizes resilience and inclusion. The exhibition will have 5 different sections, each highlighting a different aspect of resilience or inclusion. The curator decides that each section will showcase 3 portraits, including "The Vendor's Spirit," which will be displayed in all sections as the centerpiece.

If each portrait, including "The Vendor's Spirit," costs $200 to frame, and the museum already has 10 frames available for free, how much will the museum spend on framing all the portraits needed for the exhibition?

Answer: 1000


Answer: -2


In the trapezoid $ABCD$, the sides $AB$ and $CD$ are parallel. Let $M$ be the midpoint of the diagonal $AC$, $N$ the midpoint of the diagonal $BD$, and $P$ the midpoint of the side $AB$. Given that $AB = 15 \text{ cm}$, $CD = 24 \text{ cm}$, and the height of the trapezoid is $h = 14 \text{ cm}$:

a) Calculate the length of the segment $MN$.

b) Calculate the area of the triangle $MNP$.

Answer: 15.75^2


Seven distinct balls are to be placed into four labeled boxes. It is required that boxes 1 and 2 must contain an even number of balls, and box 3 must contain an odd number of balls. Find the number of ways to do this.

Answer: 2080


In triangle $ABC$, the angle bisector $AL$ is drawn. Points $E$ and $D$ are marked on segments $AB$ and $BL$ respectively such that $DL = LC$ and $ED \parallel AC$. Find the length of segment $ED$ if it is known that $AE = 15$ and $AC = 12$.

Answer: 3


Six pirates - a captain and five crew members - are sitting around a campfire facing the center. They need to divide a treasure of 180 gold coins. The captain proposes a way to divide the coins (i.e., how many coins each pirate should receive: each pirate must receive a whole non-negative number of coins; different pirates may receive different amounts of coins). After this, the other five pirates vote on the captain's proposal. A pirate will vote "yes" only if he receives more coins than each of his two neighbors. The proposal is accepted if at least three out of the five crew members vote "yes".

What is the maximum number of coins the captain can receive under these rules?

Answer: 59


In a square with a side length of 1, a figure is placed such that the distance between any two points of the figure is not equal to 0.001. Prove that the area of this figure does not exceed: a) 0.34; b) 0.287.

Answer: 0.287


In how many ways can you place pieces on a chessboard so that they do not attack each other:
a) two rooks
b) two kings
c) two bishops
d) two knights
e) two queens?

All pieces are of the same color.

Answer: 1288


In the triangle \(ABC\), we draw the altitude \(AA_1\) from vertex \(A\). What is the length of \(BA_1\), if \(AB = 8 \text{ cm}\), \(AC = 10 \text{ cm}\), and \(BC = 12 \text{ cm}\)?

Answer: 4.5


There are 100 silver coins, ordered by weight, and 101 gold coins, also ordered by weight. It is known that all coins have different weights. We have a two-pan balance scale that allows us to determine which of any two coins is heavier. How can we find the coin that ranks 101st in weight among all the coins with the minimum number of weighings?

Answer: 8


In triangle \( ABC \), angle \( A \) is equal to \( 60^{\circ} \). The distances from vertices \( B \) and \( C \) to the center of the inscribed circle of triangle \( ABC \) are 3 and 4, respectively. Find the radius of the circumscribed circle around triangle \( ABC \).

Answer: \sqrt{\frac{37}{3}}


What is the minimum number of straight lines needed to construct exactly 100 squares? In the provided figure on the left, 20 squares are constructed with the help of nine straight lines (12 with side equal to $AB$, 6 with side equal to $AC$, and 2 with side equal to $AD$). On the same figure on the right, with one additional straight line, the number of squares increases to 17. Therefore, it is important how the lines are drawn, not just the total number of lines. Keep in mind that the goal is to obtain exactly 100 squares (not more and not less).

Answer: 15


In a plane, there is a point set \( M \) and seven distinct circles \( C_{1}, C_{2}, \ldots, C_{7} \). Circle \( C_{7} \) passes through exactly 7 points in \( M \), circle \( C_{6} \) passes through exactly 6 points in \( M \), and so on, until circle \( C_{1} \) passes through exactly 1 point in \( M \). What is the minimum number of points in \( M \)?

(A) 11 ;

(B) 12 ;

(C) 21 ;

(D) 28 .

Answer: 12


The square root of half of a swarm of bees flew to a jasmine bush. Eight-ninths of the swarm remained at home. One bee flew after a male bee, worried about its buzzing in a lotus flower, where it got trapped overnight, attracted by the pleasant aroma, and now cannot get out because the flower closed. Tell me the number of bees in the swarm.

Answer: 72


"Come, Juci, hurry to the market,

Fresh eggs gleam white in your basket.

Life is precious now... if you give wisely:

You'll get a crown for them, indeed, six of them!"

Said the lady. And off went the girl...

What and how it happened, still today is clear:

Once she stumbled, tripped over a stone...

Twenty white eggs turned yellow from it.

She would have burst into tears if many hadn't seen,

For who would take the side of a poor servant?

The girl was clever... quickly realizing

She quietly calculated, eggs into fillers...

"If I sell each for one filler more

I'll still get the six crowns!"

The fresh eggs had a great demand,

Juci sold them all... Singing, laughing

She took her money home... Who will guess:

How many eggs were in the basket?

Answer: 120


4. (13 points) In a dance ensemble, there are 8 boys and 16 girls. Some of them form mixed (boy and girl) dance pairs. It is known that in each pair, at least one of the partners does not belong to any other pair. What is the maximum number of dance pairs that can be formed in this ensemble?

Answer: 22


10. (20 points) As shown in Figure 1, in the Cartesian coordinate system $x C y$, the tangent line $l$ of a circle with center at the origin $C$ and radius 1 intersects the $x$-axis at point $N$ and the $y$-axis at point $M$. Point $A(3,4)$, and $\overrightarrow{A C}=$
$x \overrightarrow{A M}+y \overrightarrow{A N}$. Let
point $P(x, y)$.
(1) Find the equation of the trajectory of point $P$;
(2) Find the minimum value of $9 x^{2}+16 y^{2}$.

Answer: 4


8. (10 points) On a plane, use toothpicks of length 6 cm to form squares. To form a square with a side length of 6 cm, 4 toothpicks are needed. To form 5 such squares, at least $\qquad$ toothpicks are needed.

Answer: 15


## Task 5 - 211235

37 cards, each of which is colored red on one side and blue on the other, are laid on a table so that exactly 9 of them show their blue side on top.

Now, in work steps, cards are to be flipped, and in each individual work step, exactly 20 of the 37 cards are to be flipped.

Investigate whether it is possible to achieve with a finite number of work steps that all 37 cards

a) show their red side on top,

b) show their blue side on top.

If this is possible, determine the smallest number of work steps required for each case!

Answer: 2


On a $16 \times 16$ torus as shown all 512 edges are colored red or blue. A coloring is good if every vertex is an endpoint of an even number of red edges. A move consists of switching the color of each of the 4 edges of an arbitrary cell. What is the largest number of good colorings such that none of them can be converted to another by a sequence of moves?

Answer: 4


(5) Let two regular tetrahedra $P-ABC$ and $Q-ABC$ be inscribed in the same sphere. If the dihedral angle between a lateral face and the base of the regular tetrahedron $P-ABC$ is $45^{\circ}$, then the tangent value of the dihedral angle between a lateral face and the base of the regular tetrahedron $Q-ABC$ is $\qquad$.

Answer: 4


82. A vanguard unit of the PLA sets off from location $A$ via location $B$ to reach location $C$. Three hours later, the unit is urgently dispatched to send a communicator by car to make contact. It is known that the communicator's car speed is 20 kilometers per hour faster than the vanguard unit's speed. The vanguard unit reaches location $B$ 2 hours before the communicator arrives at $B$. When the communicator reaches $B$, the vanguard unit has already arrived at $C$. It is known that the distance between $B$ and $C$ is 120 kilometers. How far is it from $A$ to $B$ in kilometers?

Answer: 240


Define a \emph{crossword puzzle} to be a $15 \times 15$ grid of squares, each of which is either black or white. In a crossword puzzle, define a \emph{word} to be a sequence of one or more consecutive white squares in a row or column such that the squares immediately before and after the sequence both are either black or nonexistent. (The latter case would occur if an end of a word coincides with an end of a row or column of the grid.)  A crossword puzzle is \emph{tasty} if every word consists of an even number of white squares. Compute the sum of all nonnegative integers $n$ such that there exists a tasty crossword puzzle with exactly $n$ white squares.

[i]Proposed by Luke Robitaille[/i]

Answer: 4900


7. Xiao Le has some balls, and he puts them into 25 small squares, with at least 1 ball in each small square. No matter how he places them, there are always 4 small squares with the same number of balls. Xiao Le can have at most $\qquad$ balls.

Answer: 116


Problem 8.5. In a convex quadrilateral $A B C D$, the bisectors of angles $A$ and $C$ are parallel, and the bisectors of angles $B$ and $D$ intersect at an angle of $46^{\circ}$, as shown in the figure. How many degrees does the acute angle between the bisectors of angles $A$ and $B$ measure?

![](https://cdn.mathpix.com/cropped/2024_05_06_191a5994d6b917032c66g-03.jpg?height=490&width=616&top_left_y=94&top_left_x=419)

Answer: 67


## 11. Polar Expedition

A polar explorer set out from one point to another on a sled pulled by a team of five dogs. But after 24 hours, two of the dogs died, causing the explorer's speed to decrease to $3 / 5$ of the original speed, and he was delayed by two days.

- Oh! - exclaimed the polar explorer. - If the two dogs that died had run another 120 km, I would have been delayed by only one day.

What distance did the polar explorer travel?

Answer: 320


3. Initially, the numbers $1, 2, 3, \cdots, 2013$ are written on the blackboard. In each step, you can erase two numbers $a, b$ from the blackboard and write down $a+b$. What is the minimum number of steps required to ensure that the sum of any remaining numbers on the blackboard is not equal to 2014?
(Lin Chang provided the problem)

Answer: 503


## Task A-2.4.

The vertices of the pentagon $A B C D E$ lie on the same circle. If $\varangle C A D=50^{\circ}$, determine

$$
\varangle A B C+\varangle A E D \text {. }
$$

Answer: 230


4. (20 points) The figure below shows the setting of a "gesture password" on a mobile phone. In a $2 \times 2$ grid, there are 9 grid points. A "gesture password" starts from one of the grid points and connects several grid points with line segments. Each line segment must not have any unused grid points in between, and the two endpoints of the line segment cannot both be already used grid points. If a person's gesture password starts from the center grid point and only uses three grid points, how many connection methods are there?

---

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

Answer: 48


Problem 7.7. The numbers from 1 to 200 were arranged in a random order on a circle such that the distances between adjacent numbers on the circle are the same.

For any number, the following is true: if you consider 99 numbers standing clockwise from it and 99 numbers standing counterclockwise from it, there will be an equal number of numbers that are smaller than it in both groups. Which number stands opposite the number 113?

Answer: 114


75. The least common multiple of three different two-digit numbers can be divided by the 16 natural numbers from $1 \sim 16$. The sum of these three two-digit numbers is . $\qquad$

Answer: 270


9,10

Someone has arranged a 10-volume collection of works in a random order. Let's call a disorder a pair of volumes where the volume with the higher number is to the left. For this arrangement of volumes, the number $S$ of all disorders has been calculated. What values can $S$ take?

Answer: 0


Let  $n\ge 1$   be a fixed integer. Calculate the distance  $\inf_{p,f}\, \max_{0\le x\le 1} |f(x)-p(x)|$   , where  $p$   runs over polynomials of degree less than  $n$   with real coefficients and  $f$   runs over functions  $f(x)= \sum_{k=n}^{\infty} c_k x^k$    defined on the closed interval  $[0,1]$  , where  $c_k \ge 0$   and  $\sum_{k=n}^{\infty} c_k=1$ .

Answer: \dfrac{1}{2^{2n-1}}


33. A project is completed in whole days when worked on in the order of Jia, Yi, Bing each working for one day. If the work is done in the order of Bing, Jia, Yi each working for one day, it will take 0.5 days longer than the original plan; if the work is done in the order of Yi, Bing, Jia each working for one day, it will take 1 day longer than the original plan. Yi alone can complete the project in 30 days. Jia, Yi, and Bing working together need $\qquad$ days to complete the project.

Answer: \dfrac{15}{2}


2. Given the function $f(x)=\log _{2} \frac{x-3}{x-2}+\cos \pi x$. If $f(\alpha)=10, f(\beta)=-10$, then $\alpha+\beta=$ $\qquad$

Answer: 5


5. Given the set
$$
A=\{n|n \in \mathbf{N}, 11| S(n), 11 \mid S(n+1)\} \text {, }
$$

where $S(m)$ denotes the sum of the digits of the natural number $m$. Then the smallest number in set $A$ is $\qquad$ .

Answer: 2899999


15.1 A can of coffee is shared by person A and person B, and they finish it together in 10 days. If person A drinks it alone, it takes 12 days. 1 pound of tea is shared by person A and person B, and they finish it together in 12 days. If person B drinks it alone, it takes 20 days. Assuming that person A will never drink coffee when there is tea, and person B will never drink tea when there is coffee. How many days will it take for the two to finish 1 pound of tea and 1 can of coffee together?

Answer: 35


5. In $\triangle A B C$, suppose
$$
\frac{a+b}{a}=\frac{\sin B}{\sin B-\sin A},
$$

and $\cos (A-B)+\cos C=1-\cos 2 C$. Then the value of $\frac{a+c}{b}$ is

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


(solved by François Caddet, Marc Coiffier, and Jean-Alix David). Let $\left(a_{n}\right)$ be defined by $a_{1}=3$, $a_{2}=2$, and for $n \geqslant 1, a_{n+2}$ is the remainder of the Euclidean division of $a_{n}+a_{n+1}$ by 100. Calculate the remainder of the Euclidean division of:

$$
a_{1}^{2}+a_{2}^{2}+\cdots+a_{2007}^{2}
$$

by 8.

Answer: 6


1. Points $A, B, C, D$ lie on the circumference of a circle, and $B C=C D=4, E$ is the intersection of $A C$ and $B D$, and $A E=6$. The lengths of segments $B E$ and $D E$ are both integers. What is the length of $B D$?
(1988 National Junior High School League Question)

Answer: 7


7. The midline $E F$ of trapezoid $A B C D$ is 15 cm long (see figure), $\angle A B C=\angle A E F=90^{\circ}, G$ is a point on $E F$. If the area of triangle $A B G$ is $1 / 5$ of the area of trapezoid $A B C D$, what is the length of $E G$ in centimeters?

Answer: 6


8. In a unit square, if three of its four sides are colored with three different colors, then the unit square is called "colorful". As shown in the figure, a $1 \times 3$ grid table contains 10 unit line segments in its grid lines. Now, we need to color these 10 line segments, with each line segment being colored red, yellow, or blue, such that all three unit squares are colorful. The number of such coloring ways is $\qquad$ (answer in numerical form).

Answer: 5184


7. The function $f(x)$ satisfies for all real numbers $x$:
$$
f(2-x)=f(2+x), f(5+x)=f(5-x) \text {, and } f(0)=0
$$

Then $f(x)$ has at least $\qquad$ zeros in the interval $[-21,21]$.

Answer: 14


8. Priti is learning a new language called Tedio. During her one hour lesson, which started at midday, she looks at the clock and notices that the hour hand and the minute hand make exactly the same angle with the vertical, as shown in the diagram. How many whole seconds remain until the end of the lesson?

Answer: 276


Let $P(x) = 3\sqrt{x}$, and $Q(x) = x^3$. Consider functions $R(x) = x + 2$, and $S(x) = 2x$. Calculate $P(Q(R(S(P(Q(2))))))$.

Answer: 3\sqrt{(12\sqrt{2}+2)^3}


At Lincoln High School, there are 30 players on the soccer team. All players are participating in at least one of the three science courses: biology, chemistry, or physics. There are 15 players taking biology, 10 players taking chemistry, 5 players taking physics, and 2 players are taking all three sciences. How many unique players are taking chemistry, considering some might be taking more than one science subject?

Answer: 10


Jane shoots basketballs and makes 60% of her initial 30 shots. After taking an additional 10 shots, her shooting percentage increases to 62%. How many of these additional 10 shots did she make?

Answer: 7


A rectangle and two circles, each with a radius of 3 cm, are arranged as shown. What is the area, in square centimeters, of the rectangle? The circles are placed such that each touches two sides of the rectangle and each other.

Answer: 72 \text{ cm}^2


A QR-like code consists of a $5 \times 5$ grid with either black or white squares. Every valid code must contain at least one square of each color. A code is termed $\textit{symmetric}$ if it looks the same when rotated by multiples of $90^\circ$ around the grid center, or when reflected across a line joining opposite corners or midline of opposite sides. Determine the total number of possible symmetric codes.

Answer: 62


For a specific square in a Cartesian plane, two opposite vertices lie on the parabola $y = x^2$, and the other two opposite vertices lie on the parabola $y = -x^2 + 4$. Determine the smallest possible area of the square.

Answer: 4


Let $a$ and $b$ be the roots of the equation $x^2 - mx + 3 = 0.$ Suppose that $a + c\cdot(1/b)$ and $b + c\cdot(1/a)$ are the roots of the equation $x^2 - px + q = 0$. Determine $q$, where $c$ is a real number constant.

Answer: 3 + 2c + \frac{c^2}{3}


The current time is 6:00 a.m. What time will it be in 1876 minutes?

Answer: 1\!:\!16 \text{ p.m.}


A rectangular prism has dimensions 5 feet by 5 feet by 6 feet. A cylindrical section of radius 2.5 feet is removed from the solid, extending through the shortest dimension. What is the total remaining volume of the rectangular prism? Express your answer in cubic feet in terms of \(\pi\).

Answer: 150 - 31.25\pi \text{ cubic feet}


In a square, points $P$ and $R$ are midpoints of two adjacent sides. What fraction of the interior of the square is shaded if the triangle formed by the opposite vertex of $P$ and $R$ and point $P$ is unshaded? Express your answer as a common fraction.

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label("P",(1,0),S);
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Answer: \frac{7}{8}


Six circles of diameter 2 are packed in the first quadrant of the coordinate plane. Let region $\mathcal{S}$ be the union of these six circular regions. Line $l,$ with slope 2, divides $\mathcal{S}$ into two regions of equal area. The equation of line $l$ can be expressed in the form $ax=by+c,$ where $a, b,$ and $c$ are positive integers whose greatest common divisor is 1. Find $a^2+b^2+c^2.$

Answer: 6


In the town of Gridlock, the streets are all $25$ feet wide, and the blocks are rectangles with lengths of $450$ feet and widths of $350$ feet. Sarah runs around a block on the $450$-foot side of the street, while Sam runs on the opposite side of the street. Moreover, Sarah must also run through a $25$ feet wide alleyway that cuts through the center of the $450$ foot side, extending her path. How many more feet than Sarah does Sam run for each lap around the block?

Answer: 150


A right circular cone is inscribed in a right rectangular prism. The base of the prism measures $2a$ by $2b$ where $b < a$. The cone is tangent to the shorter sides of the rectangle and touches the midpoint of the longer sides. If the height of the cone is $h_c$ and the height of the prism is $h_p$, what is the ratio of the volume of the cone to the volume of the prism? Express your answer in terms of $\pi$.

Answer: \frac{\pi b h_c}{12a h_p}


Suppose we explore another number with similar properties, such as $999,999,999$. How many zeros are found in the square of $999,999,999$?

Answer: 8


Ella has designed a rectangular garden divided into four quadrants by paths, with a circular flower bed in each quadrant. Each flower bed is tangent to the paths and its adjacent flower beds. If the garden paths create a rectangle that is 30 inches long and 24 inches wide, how many square inches are not occupied by flower beds?

Answer: 720 - 144\pi \text{ square inches}


In a chess tournament each participant played exactly once against all others. The winner of each game received $1$ point, the loser $0$ points, and each scored $\frac{1}{2}$ point if the game was a draw. At the end of the tournament, it was found that exactly half of the points each player obtained came from playing against the twelve players with the lowest scores. Notably, each of these twelve lowest scoring participants got half of their points from games among themselves. Determine the total number of participants in the tournament.

Answer: 24


A stack contains three layers of square blocks, each block having sides of $10 \text{ cm}$. The stack is arranged in a stepped pyramid fashion where the top layer contains one block, the middle layer contains two blocks, and the bottom layer contains three blocks. Find the total height, $h$, of this pyramidal stack of blocks.

Answer: 30 \text{ cm}


Let $C$ be a point not on line $AE$, and $D$ a point on line $AE$ such that $CD \perp AE$. Also, $B$ is a point on line $CE$ such that $AB \perp CE$. Introduce a point $F$ on line $AE$ such that $BF \perp AE$. If $AB = 5$, $CD = 10$, $BF = 3$, and $AE = 6$, what is the length of $CE$?

Answer: 12


In the diagram provided, three semicircles are drawn on the straight line segment $AC$ of length 6 units, where $B$ is the midpoint of $AC$. Arc $ADB$ and arc $BEC$ have a radius of 2 units each. Point $D$, point $E$, and point $F$ are the midpoints of arc $ADB$, arc $BEC$, and arc $DFE$ respectively. If arc $DFE$ is also a semicircle with a radius of 1 unit, and a square $DPEF$ inscribes within arc $DFE$ such that $P$ lies at the intersection of lines $DE$ and $BF$, calculate the area of the square $DPEF$.

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Answer: 1


Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that
\[f(xf(y) + y) = xy^2 + f(x)\] for all $x,y \in \mathbb{R}.$

Find the number of possible values of $f(3)$, denote it by $n$. Let $s$ be the sum of all possible values of $f(3)$. Compute $n \times s.$

Answer: 0


Let \(A\) and \(B\) be two points on the parabola \(y = 4x^2\), such that the tangents at \(A\) and \(B\) intersect at an angle of \(45^\circ\). Determine the \(y\)-coordinate of their point of intersection, \(P\).

Answer: -\frac{1}{16}


For integers $0 \le m,n \le 64$, let $\alpha(m,n)$ be the number of nonnegative integers $k$ for which $\left\lfloor \frac{m}{2^k} \right\rfloor$ and $\left\lfloor \frac{n}{2^k} \right\rfloor$ are both odd integers. Consider a $65 \times 65$ matrix $M$ whose $(i,j)$th entry (for $1 \le i, j \le 65$) is:
\[ (-1)^{\alpha(i-1, j-1)}. \]
Compute the remainder when $\det M$ is divided by $1000$.

Answer: 208


Ana and Banana play a game. First, Ana picks a real number $p$ with $0 \le p \le 1$. Then, Banana picks an integer $h$ greater than $1$ and creates a spaceship with $h$ hit points. Every minute, Ana decreases the spaceship's hit points by $2$ with probability $1-p$, and by $3$ with probability $p$. Ana wins if and only if the number of hit points is reduced to exactly $0$ at some point (in particular, if the spaceship has a negative number of hit points at any time then Ana loses). Given that Ana and Banana select $p$ and $h$ optimally, compute the integer closest to $1000p$.

Answer: 382


Say that a function $f : \{1, 2, \ldots, 1001\} \to \mathbb{Z}$ is *almost* polynomial if there exists a polynomial $p(x) = a_{200}x^{200} + \ldots + a_1x + a_0$ where each $a_n$ is an integer with $|a_n| \le 201$, and $|f(x) - p(x)| \le 1$ for all $x \in \{1, 2, \ldots, 1001\}$. Let $N$ be the number of such almost polynomial functions. Compute the remainder when $N$ is divided by $199$.

Answer: 0


Triangle $ABC$ lies entirely in the first quadrant of the Cartesian plane, and its sides have slopes $63$, $73$, and $97$. Suppose the curve $\mathcal{V}$ with equation $y=(x+3)(x^2+3)$ passes through the vertices of $ABC$. Find the sum of the slopes of the three tangents to $\mathcal{V}$ at each of $A$, $B$, and $C$.

Answer: 237


A point is chosen uniformly at random from the interior of a unit square. Let $p$ be the probability that any circle centered at the point that intersects a diagonal of the square must also intersect a side of the square. Given that $p^2$ can be written as $m-\sqrt{n}$ for positive integers $m$ and $n$, what is $m+n$?

Answer: 11


How many subsets of the set \( \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\} \) have the property that no two of its elements differ by more than 5? For example, count the sets \( \{3\} \), \( \{2, 5, 7\} \), and \( \{5, 6, 7, 8, 9\} \) but not the set \( \{1, 3, 5, 7\} \).

Answer: 256


Yasha writes in the cells of a $99 \times 99$ table all positive integers from 1 to $99^2$, with each number appearing once. Grisha looks at the table and selects several cells such that no two selected cells share a common side. Grisha then sums up the numbers in all the selected cells. Determine the largest sum that Grisha can guarantee to achieve.

Answer: 24017351


Misha has accepted a job in the mines and will produce one ore each day. At the market, he can perform the following transactions:
- Buy or sell one ore for $3.
- Buy or sell bundles of three wheat for $12 each.
- Sell one wheat for one ore.

His ultimate goal is to build a city, which requires:
- Three ore
- Two wheat

How many dollars must Misha begin with in order to build a city after three days of working?

Answer: 9


The 16 fields of a $4 \times 4$ checkerboard can be arranged in 18 lines as follows: the four rows, the four columns, the five diagonals from northwest to southeast, and the five diagonals from northeast to southwest. These diagonals consist of 2, 3, or 4 edge-adjacent fields of the same color; the corner fields of the chessboard alone do not form a diagonal. Now, we place a token in 10 of the 16 fields. Each of the 18 lines containing an even number of tokens earns a point. What is the highest possible point total that can be achieved by optimally placing the 10 tokens? Explain your answer.

Answer: 16


Equilateral triangle $ABC$ has a side length of $2$. A semicircle is drawn with diameter $BC$ such that it lies outside the triangle, and the minor arc $BC$ is drawn so that it is part of a circle centered at $A$. The area of the "lune" that is inside the semicircle but outside sector $ABC$ can be expressed in the form $\sqrt{p} - \frac{q\pi}{r}$, where $p, q$, and $r$ are positive integers such that $q$ and $r$ are relatively prime. Compute $p + q + r$.

Answer: 10


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

Answer: 55


$A = 1 \cdot 4 \cdot 7 \cdots 2014$. Find the last non-zero digit of $A$ if it is known that $A \equiv 1 \pmod{3}$.

Answer: 2


The subnumbers of an integer $n$ are the numbers that can be formed by using a contiguous subsequence of the digits. For example, the subnumbers of 135 are 1, 3, 5, 13, 35, and 135. Compute the number of primes less than 1,000,000,000 that have no non-prime subnumbers. One such number is 37, because 3, 7, and 37 are prime, but 135 is not one, because the subnumbers 1, 35, and 135 are not prime.

Answer: 9


Let $ABCD$ and $AEFG$ be unit squares such that the area of their intersection is $\frac{20}{21}$. Given that $\angle BAE < 45^\circ$, $\tan \angle BAE$ can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a$ and $b$. Compute $100a + b$.

Answer: 4940


When Eva counts, she skips all numbers containing a digit divisible by 3. For example, the first ten numbers she counts are 1, 2, 4, 5, 7, 8, 11, 12, 14, 15. What is the $100^{\text{th}}$ number she counts?

Answer: 255


Let $x_1, x_2, \dots, x_{100}$ be real numbers such that $|x_1| = 63$ and $|x_{n+1}| = |x_n + 1|$ for $n = 1, 2, \dots, 99$. Find the largest possible value of $(-x_1 - x_2 - \cdots - x_{100})$.

Answer: 2034


From point \( T \), a tangent \( TA \) and a secant intersecting the circle at points \( B \) and \( C \) are drawn. The angle bisector of \(\angle ATC\) intersects chords \( AB \) and \( AC \) at points \( P \) and \( Q \) respectively. Prove that \( PA = \sqrt{PB \cdot QC} \).

Answer: PA = \sqrt{PB \cdot QC}


Example 3. There is a convex pentagon, the area of the triangles formed by any three consecutive vertices is 1. Find the area of this pentagon (1972, American Mathematics Competition)

Answer: \dfrac{5 + \sqrt{5}}{2}


10. (6 points) In stores A and B, the marked price of a certain commodity is the same. Store A sells this commodity at the marked price, with sales amounting to 7200 yuan; Store B sells it at an 80% discount of the marked price, selling 15 more items than Store A, with the same sales amount as Store A. Then, Store A sold $\qquad$ items of this commodity.

Answer: 60


Ali-Baba came to a cave where there is gold, diamonds, and a chest in which he can carry them away. A full chest of gold weighs 200 kg, a full chest of diamonds - 40 kg, an empty chest weighs nothing. A kilogram of gold is worth 20 dinars on the market, a kilogram of diamonds - 60 dinars. Ali-Baba can lift and carry no more than 100 kg. What is the maximum amount of money he can get for the treasures he brings out of the cave in one go?

Answer: 3000


[  [Theorem on the lengths of a tangent and a secant; the product of the entire secant and its external part. Pythagorean Theorem (direct and inverse). Area of a trapezoid

A circle passes through vertices $A$ and $B$ of rectangle $A B C D$ and is tangent to side $C D$ at its midpoint. A line is drawn through vertex $D$ that is tangent to the same circle at point $E$, and then intersects the extension of side $A B$ at point $K$.

Find the area of trapezoid $B C D K$, if $A B=10$ and $K E: K A=3: 2$.

#

Answer: 210


Given the rectangle $A B C D$. On the rays $A B, B C, C D$, and $D A$ starting from $A, B, C$, and $D$ respectively, we mark the points $A_{1}, B_{1}, C_{1}, D_{1}$ such that

$$
\frac{A A_{1}}{A B}=\frac{B B_{1}}{B C}=\frac{C C_{1}}{C D}=\frac{D D_{1}}{D A}=k>0
$$

For what value of $k$ will the area of the quadrilateral $A_{1} B_{1} C_{1} D_{1}$ be minimal?

Answer: \dfrac{1}{2}


A certain rectangle had its dimensions expressed in decimeters as whole numbers. Then it changed its dimensions three times. First, it doubled one of its dimensions and changed the other so that it had the same area as at the beginning. Then it increased one dimension by $1 \mathrm{dm}$ and decreased the other by $4 \mathrm{dm}$, while still having the same area as at the beginning. Finally, it decreased its shorter dimension by $1 \mathrm{dm}$ and left the longer one unchanged.

Determine the ratio of the lengths of the sides of the final rectangle.

(E. Novotná)

Answer: 4:1


5. Let $S$ denote the set of all 6-tuples $(a, b, c, d, e, f)$ of positive integers such that $a^{2}+b^{2}+c^{2}+d^{2}+e^{2}=$ $f^{2}$. Consider the set

$$
T=\{a b c d e f:(a, b, c, d, e, f) \in S\}
$$

Find the greatest common divisor of all the members of $T$.

Answer: 24


[Isosceles, Inscribed, and Circumscribed Trapezoids] [Pythagorean Theorem (Direct and Inverse)]

In an isosceles trapezoid \(ABCD\) with the larger base \(AD=12\) and \(AB=6\). Find the distance from point \(O\) where the diagonals intersect to point \(K\) where the extensions of the lateral sides intersect, if the extensions of the lateral sides intersect at a right angle.

Answer: \dfrac{12(3 - \sqrt{2})}{7}


A triangle's two vertices, the center of its inscribed circle, and its orthocenter lie on a circle. Calculate the angle at the third vertex of the triangle!

Answer: 60^\circ


Answer: -\dfrac{1}{2^{2015}}


1. Given that the function $f(x)$ is a decreasing function on $\mathbf{R}$ and is an odd function. If $m$ and $n$ satisfy
$$
\left\{\begin{array}{l}
f(m)+f(n-2) \leqslant 0, \\
f(m-n-1) \leqslant 0,
\end{array}\right.
$$

then the range of values for $5 m-n$ is

Answer: [7, +\infty)


7.4. Find the volume of a regular triangular pyramid, the midpoint of the height of which is at a distance of 2 and $\sqrt{7}$ from the lateral face and the lateral edge, respectively. Round your answer to two decimal places if necessary.

Answer: 112\sqrt{7}


The edges of a rectangular parallelepiped are \(a, b,\) and \(c\). Find the angle between the diagonal of the parallelepiped and the skew diagonal of the face with sides \(a\) and \(b\).

Answer: \arccos\left( \dfrac{\sqrt{a^2 + b^2}}{\sqrt{a^2 + b^2 + c^2}} \right)


Let \(ABCD\) be a convex cyclic quadrilateral with diagonals intersecting at \(S\). Let \(P\) be the circumcenter of triangle \(ABS\) and \(Q\) the circumcenter of triangle \(BCS\). The line parallel to \(AD\) through \(P\) and the line parallel to \(CD\) through \(Q\) intersect at point \(R\).

Prove that \(R\) lies on \(BD\).

Answer: R \text{ lies on } BD


## PROBLEM 13. TIC-TAC-TOE

Consider the game of "tic-tac-toe" on a three-dimensional cube $8 \times 8 \times 8$. How many straight lines can be indicated on which 8 symbols lie in a row?

Answer: 244


[ Arithmetic. Mental calculation, etc.]

In an example of adding two numbers, the first addend is less than the sum by 2000, and the sum is greater than the second addend by 6.

Restore the example.

#

Answer: 6 + 2000 = 2006


51. As shown in the figure, squares $A B C D$ and $B E F G$ with side lengths of 4 cm and 3 cm, respectively, are placed side by side. $A F$ intersects $B G$ at $P$, then the ratio of the area of $\triangle A P E$ to the area of $\triangle G F P$ is $\qquad$.

Answer: \dfrac{28}{9}


Kubarev A.M.

From a chessboard of 8x8, 10 cells have been cut out. It is known that among the cut-out cells, there are both black and white cells. What is the maximum number of two-cell rectangles that can be guaranteed to be cut out from this board after this?

#

Answer: 23


12.074. A sphere is inscribed in a cone. The ratio of the radius of the circle of contact between the spherical and conical surfaces to the radius of the base of the cone is $k$. Find the cosine of the angle between the slant height of the cone and the plane of the base.

Answer: 1 - k


5. A sign engraver makes signs with letters. He engraves identical letters in the same amount of time, and different letters possibly in different times. For two signs “ДОМ МОДЫ” and “ВХОД” together, he spent 50 minutes, and one sign “В ДЫМОХОД” he made in 35 minutes. How long will it take him to make the sign “ВЫХОД”?

Answer: 20


Answer: \sqrt[3]{a} + \sqrt[3]{b} + \sqrt[3]{c}


Problem 3. Six numbers are written in a row. It is known that among them there is a one and any three adjacent numbers have the same arithmetic mean. Find the maximum value of the geometric mean of any three adjacent numbers in this row, if the arithmetic mean of all 6 numbers is A.

Answer: \sqrt[3]{\left( \frac{3A - 1}{2} \right)^2}


35. NUMB3RS. The RSA Factoring Challenge, which ended in 2007, challenged computational mathematicians to factor extremely large numbers that were the product of two prime numbers. The largest number successfully factored in this challenge was RSA-640, which has 193 decimal digits and carried a prize of $\$ 20,000$. The next challenge number carried prize of $\$ 30,000$, and contains $N$ decimal digits. Your task is to submit a guess for $N$. Only the team(s) that have the closest guess(es) receives points. If $k$ teams all have the closest guesses, then each of them receives $\left\lceil\frac{20}{k}\right\rceil$ points.

Answer: 212


Example 1. Solve the equation $y^{IV}=\cos x$.

Answer: y = \cos x + C_1 x^3 + C_2 x^2 + C_3 x + C_4


2. Given $\left\{\begin{array}{l}4^{x}+y=4030, \\ 17 \times 2^{x}+2 y=2019 .\end{array}\right.$ Find all real roots of the system of equations.

Answer: \log_2 \left( \dfrac{17 + \sqrt{48617}}{4} \right)


Let $n \geqslant 3$ be integer. Given convex $n-$polygon $\mathcal{P}$. A $3-$coloring of the vertices of $\mathcal{P}$ is called [i]nice[/i] such that every interior point of $\mathcal{P}$ is inside or on the bound of a triangle formed by polygon vertices with pairwise distinct colors. Determine the number of different nice colorings.
([I]Two colorings are different as long as they differ at some vertices. [/i])

Answer: 2^n - 3 - (-1)^n


## 3. Before the Competition

A group of cyclists decided to train on a small road before the competition. All of them are riding at a speed of 35 km/h. Then one of the cyclists suddenly breaks away from the group and, moving at a speed of 45 km/h, covers 10 km, after which he turns back and, without slowing down, rejoins the others (who, ignoring him, continue to move at the same speed).

How much time has passed from the moment the cyclist broke away from the group until he rejoined them?

Answer: \dfrac{1}{4}


4. Petl wrote on the board in a row $n$ two-digit octal numbers $(n \geqslant 2)$, forming an arithmetic progression with a common difference of 8, and the first number does not contain the digit 2. Vasya thought that this was the octal representation of a natural number $x$, and he factored $x$ into prime factors. It turned out that there were only two of them, and they differ by 2. What is written on the board?

Answer: 3343


10. Let $n$ be a composite positive integer, $f(n)$ be the sum of the smallest three positive divisors of $n$, and $g(n)$ be the sum of the largest two positive divisors of $n$. If $g(n)=f^{3}(n)$, then all possible values of $n$ are $\qquad$

Answer: 144


4. As shown in the figure, on a semicircle with center $C$ and diameter $M N$, there are two different points $A$ and $B$. Point $P$ is on $C N$, and $\angle C A P = \angle C B P = 10^{\circ}$. If $\overparen{M A} = 40^{\circ}$, then $\overparen{B N}$ equals . $\qquad$

Answer: 20


[Example 2.2.2] Let $P(x)$ be an $n$-degree polynomial, and for $k=0,1,2, \cdots, n$, $P(k)=\frac{k}{k+1}$.
Find the value of $P(n+1)$.

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


7. There are 5 power poles, 2 of which will leak electricity. There are 5 birds each randomly choosing one of the poles to rest (if a bird touches a leaking pole, it will be stunned by the electric shock and fall to the ground). The probability that only 2 of the poles have birds on them is $\qquad$

Answer: \dfrac{342}{625}


A dart board is a regular octagon divided into regions as shown below. Suppose that a dart thrown at the board is equally likely to land anywhere on the board. Find the probability that the dart lands within the center square. The original answer is in the form \(\frac{k\sqrt{m} - n}{p}\), please find the value of k + m + n.

Answer: 4


A dart board is a regular octagon divided into regions as shown. Suppose that a dart thrown at the board is equally likely to land anywhere on the board. Find the probability that the dart lands within the center square. The original answer is in the form \(\frac{\sqrt{k} - 1}{m}\). Please give the value of k + m.

Answer: 4


Rhombus \(ABCD\) has side length \(2\) and \(\angle B = 120^\circ\). Region \(R\) consists of all points inside the rhombus that are closer to vertex \(B\) than any of the other three vertices. The area of \(R\) is given in the form \(\frac{k\sqrt{n}}{m}\). Please find the value of \(k + m + n\).

Answer: 8


Rhombus $ABCD$ has side length $2$ and $\angle B = 120^\circ$. Region $R$ consists of all points inside the rhombus that are closer to vertex $B$ than any of the other three vertices. The area of region $R$ is given in the form \(\frac{k\sqrt{m}}{n}\). Please find the value of \(k + m + n\).

Answer: 8


Let \(T_1\) be a triangle with side lengths \(2011\), \(2012\), and \(2013\). For \(n \geq 1\), if \(T_n = \triangle ABC\) and \(D, E\), and \(F\) are the points of tangency of the incircle of \(\triangle ABC\) to the sides \(AB\), \(BC\), and \(AC\), respectively, then \(T_{n+1}\) is a triangle with side lengths \(AD, BE\), and \(CF\), if it exists. Find the perimeter of the last triangle in the sequence \(\left(T_n\right)\). The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.

Answer: 1637


Let $P(z) = z^8 + \left(4\sqrt{3} + 6\right)z^4 - \left(4\sqrt{3} + 7\right)$. Find the minimum perimeter among all the $8$-sided polygons in the complex plane whose vertices are precisely the zeros of $P(z)$. The original answer is in the form of $k\sqrt{m}$, please give the value of k + m.

Answer: 10


II. (50 points) Let real numbers $a, b$ be such that the equation $a x^{3}-x^{2}+b x-1=0$ has three positive real roots. For all real numbers $a, b$ that satisfy the condition, find the minimum value of $P=\frac{5 a^{2}-3 a b+2}{a^{2}(b-a)}$.

---

The translation maintains the original text's formatting and line breaks.

Answer: 12\sqrt{3}


$\underline{\text { Cepreев } \text { I.n. }}$.

For given natural numbers $k_{0}<k_{1}<k_{2}$, determine the smallest number of roots on the interval $[0 ; 2 \pi)$ that the equation of the form

$$
\sin \left(k_{0} x\right)+A_{1} \cdot \sin \left(k_{1} x\right)+A_{2} \cdot \sin \left(k_{2} x\right)=0
$$

can have, where $A_{1}, A_{2}$ are real numbers.

Answer: 2k_0


## Task 3 - 331213

In a beauty contest for poodles, Asta, Benno, Cäsar, and Dolly face a jury of four members. Each jury member votes for one of the dogs by raising a card with the initial letter of the dog's name. As a rule for evaluating this voting result, it was stipulated: If two dogs clearly receive more votes than the other two dogs, they are considered qualified. However, if the voting result does not clearly show two dogs with more votes than the other two, a supplementary regulation will be made (e.g., a new vote will be scheduled).

Determine the number of all those different voting results that lead to this latter case! Voting results are considered the same only if they not only match in the number of votes for the dogs but also in which jury members voted for the respective dogs. For example, the voting results $A A B C$ and $C A B A$ are considered different.

Answer: 172


1. Petya came to the bus stop of the bus going to school with stops equally spaced from each other, and, not seeing the bus on the road, decided to run and get on the bus at the next stops along the way to school. Petya ran in such a way that at any moment he could notice the appearance of the bus on the road behind him. Seeing the bus, Petya can turn back or maintain his direction of movement. It is known that the speed of the bus is 4 times greater than Petya's running speed, and he can see the bus at a distance of no more than 1.5 km. Find the greatest value of the distance between stops, such that regardless of whether Petya turns back upon discovering the bus or not, he will be able to get on the bus at a stop.

Answer: \dfrac{4}{5}


Example 3 Let $P_{m}=\left\{A_{1}, A_{2}, \cdots, A_{m}\right\}$ be a partition of an $n$-element set $X$ (i.e., $A_{1}, A_{2}, \cdots, A_{m}$ is a partition of $X$). If one of the $A_{i}$ is further divided into two sets, this results in a partition $P_{m+1}$ of $X$, which consists of $m+1$ sets. $P_{m+1}$ is called a refinement of $P_{m}$. If
$$
P_{1}, P_{2}, \cdots, P_{n}
$$

are all partitions of the $n$-element set $X$, and each is a refinement of the previous one (clearly, in this case, $P_{n}$ consists of $n$ sets, while $P_{1}$ consists of only one set, which is $X$), then (7) is called a chain of length $n$. Find the number of chains of length $n$.

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


5. On a standard graph paper, an angle is drawn (see figure). Find its measure without using measuring instruments. Justify your answer.

Answer: 45^\circ


Find the number of zeros in which the decimal expansion of  $ 2007!$  ends. Also find its last non-zero digit.

Answer: 500


Let $\Gamma_{1}$ and $\Gamma_{2}$ be two circles such that $\Gamma_{1}$ is internally tangent to $\Gamma_{2}$ at point $A$. Let $P$ be a point on $\Gamma_{2}$. The tangents to $\Gamma_{1}$ from $P$ intersect $\Gamma_{1}$ at points $X$ and $Y$, and $\Gamma_{2}$ at points $Q$ and $R$.

Show that $\widehat{Q A R} = 2 \widehat{X A Y}$.

Answer: \widehat{QAR} = 2 \widehat{XAY}


3. On the side $B C$, resp. $C D$ of the parallelogram $A B C D$, determine points $E$, resp. $F$ such that segments $E F, B D$ are parallel and triangles $A B E, A E F$ and $A F D$ have the same areas.

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


6. When rolling a die twice, the numbers obtained are $m, n$, and the vector $a=(m, n)$ is formed. Then the probability that the angle between $a$ and $b=(1,-1)$ is an angle in a right triangle is $\qquad$
6. $\frac{7}{12}$.

Since $m, n$ can both take values from $1 \sim 6$, the vector $a$ has

Answer: \dfrac{7}{12}


5. In $\triangle A B C$, it is known that $A B=A C$, point $D$ is on side $A C$, $\angle A D B=60^{\circ}$, point $E$ is on side $B D$, $\angle E C B=30^{\circ}$. Find the degree measure of $\angle A E C$.

Answer: 150


3. During the shooting practice, each soldier fired 10 times. One of them completed the task successfully and scored 90 points. How many times did he score 9 points, if there were 4 tens, and the results of the hits were sevens, eights, and nines. There were no misses at all.

Answer: 3


Answer: \dfrac{1}{9}


4. In the country of Landia, which breeds an elite breed of horses, an annual festival is held to test their speed, in which only one-year-old, two-year-old, three-year-old, and four-year-old horses can participate. For each horse that meets the speed standard, the festival organizers pay a fixed amount of money to the stud farm where the horse was raised: 1 landric for a one-year-old, 2 landrics for a two-year-old, 3 landrics for a three-year-old, and 4 landrics for a four-year-old. Each stud farm participating in the festival annually enters four new horses (of any age combination as they wish) for testing, which have not previously participated in the tests, as well as personally all horses (not older than four years) that previously participated in the tests at a younger age and met the standard. What is the maximum amount of money a stud farm can earn in the first six years of its participation in the festival $\quad(12$ points)

Answer: 200


In a certain African village, there are 800 women, of which $3 \%$ wear only one earring. Of the remainder, half wear two earrings and the other half wear none. What is the total number of earrings worn by all the women in this village?
(a) 776
(b) 788
(c) 800
(d) 812
(e) 824

Answer: 800


12. B. As shown in Figure 5, given that the side length of square $A B C D$ is $1$, and $P, Q$ are two points inside it, and
$$
\angle P A Q=\angle P C Q=45^{\circ} \text {. }
$$

Find the value of $S_{\triangle P A B}+S_{\triangle P C Q}+S_{\triangle Q A D}$.

Answer: \dfrac{1}{2}


Given a positive integer $\mathrm{n}$, write the $\mathrm{n}$ positive integers $1, 2, \ldots, \mathrm{n}$ on paper, where some numbers are written in red and some in blue, with both colors used, and each number is only written in one color. For $1 \leq i \leq j \leq n$, if there are an odd number of blue numbers among $i, i+1, \ldots, j$, then the pair of numbers $(i, j)$ is called "good." Find the maximum possible number of good pairs.

Answer: \left\lfloor \frac{(n+1)^2}{4} \right\rfloor


8. Find all values of the parameter $a$ for which the system

$$
\left\{\begin{array}{l}
y=\frac{x+1}{|x|-1} \\
|x+y+a|+|x-y-a|=1
\end{array}\right.
$$

has a unique solution. In your answer, specify the smallest of all the obtained values of the parameter $a$.

Answer: \dfrac{7}{2}


Example 4. On a plane, there are $\mathrm{n}$ lines, among which no two are parallel, and no three or more pass through the same point. How many regions does the plane get divided into by these $\mathrm{n}$ lines?

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


Suppose that for some positive integer $n$, the first two digits of $5^n$ and $2^n$ are identical. Suppose the first two digits are $a$ and $b$ in this order. Find the two-digit number $\overline{ab}$.

Answer: 31


10. Given that $a$ and $b$ are real numbers, the system of inequalities about $x$
$$
\left\{\begin{array}{l}
20 x+a>0, \\
15 x-b \leqslant 0
\end{array}\right.
$$

has only the integer solutions $2, 3, 4$. Then the maximum value of $ab$ is $\qquad$

Answer: -1200


A circle \(O\) with radius \(R\) is drawn on a piece of paper, and a point \(A\) inside the circle is given such that \(O A = a\). The paper is folded so that a point \(A^{\prime}\) on the circumference of the circle coincides with \(A\). Every such fold leaves a crease line. Find the set of points on all crease lines as \(A^{\prime}\) varies over all points on the circumference of the circle.

Answer: \frac{(x - \frac{a}{2})^2}{(\frac{R}{2})^2} + \frac{y^2}{(\frac{\sqrt{R^2 - a^2}}{2})^2} = 1


9. On a rotating round table, there are 8 white teacups and 7 black teacups, and around the table sit 15 dwarfs, each wearing a hat, with 8 white hats and 7 black hats. Each dwarf places a teacup of the same color as their hat in front of them. After the table rotates randomly, what is the maximum number of teacups that can be guaranteed to match the color of the hats (the dwarfs can choose their seating, but they do not know how the table will rotate)?

Answer: 7


For a positive integer $n$, let $S(n)$ denote its digit sum. Find all positive integers $M$ such that for every positive integer $k$ not exceeding $M$, we have $S(M k)=S(M)$.

Answer: 9


In a $9 \times 9$ square table, 9 cells are marked at the intersections of the second, fifth, and eighth rows with the second, fifth, and eighth columns. How many ways are there to get from the bottom-left cell to the top-right cell, moving only through unmarked cells upwards or to the right?

Answer: 678


An array composed of certain nonzero natural numbers has the following properties:
1. Each number in the array (except for 1) is divisible by at least one of $2$, $3$, or $5$.
2. For any nonzero natural number $n$, if the array contains one of $2n$, $3n$, or $5n$, the array must also contain $n$, $2n$, $3n$, and $5n$.
If the number of elements in this array is between 300 and 400, then the array contains how many numbers?

Answer: 364


270. The Runner's Puzzle. $A B C D$ is a square field with an area of 19.36 hectares. $B E$ is a straight path, and $E$ is 110 meters from $D$. During the competition, Adams ran straight from $A$ to $D$, while Brown started running from $B$, reached $E$, and then continued towards $D$.

![](https://cdn.mathpix.com/cropped/2024_05_21_56f37ca27ac3d928838dg-080.jpg?height=591&width=529&top_left_y=321&top_left_x=772)

Each ran at a constant speed, and when Brown reached $E$, he saw Adams 30 meters ahead of him.

Who won the competition and by what margin?

Answer: 8


26. As shown in the figure, in tetrahedron $ABCD$, $CD$ is the common perpendicular to the opposite edges $AD$ and $BC$. Given that $CD=a$, $AB=b$, and the angle between $AD$ and $BC$ is $30^{\circ}$. Find the radius of the circumscribed sphere of tetrahedron $ABCD$.

Answer: \dfrac{\sqrt{4b^2 - 3a^2}}{2}


Problem 5.1. After a football match, the coach lined up the team as shown in the figure, and commanded: "Run to the locker room, those whose number is less than that of any of their neighbors." After several people ran away, he repeated his command. The coach continued until only one player was left. What is Igor's number, if it is known that after he ran away, 3 people remained in the line? (After each command, one or several players ran away, after which the line closed, and there were no empty spaces between the remaining players.)

![](https://cdn.mathpix.com/cropped/2024_05_06_6da73bfd3e09e8b55e3fg-08.jpg?height=386&width=1072&top_left_y=485&top_left_x=191)

Answer: 5


# 11.3. (7 points)

The base of the pyramid with vertex $P$ is a quadrilateral $A B C D$, where the sum of angles $A$ and $D$ is five times smaller than the sum of angles $B$ and $C$. Find the angle between the planes of the faces $P A B$ and $P C D$, if both are perpendicular to the base.

Answer: 60^\circ


(solved by Mathieu Aria, Jeanne Nguyen and Thomas Williams). Let $n \geqslant 3$ and $x_{1}, \ldots, x_{n-1}$ be non-negative integers. We assume:

$$
\begin{aligned}
x_{1}+x_{2}+\cdots+x_{n-1} & =n \\
x_{1}+2 x_{2}+\cdots+(n-1) x_{n-1} & =2 n-2
\end{aligned}
$$

Calculate the minimum value of:

$$
\sum_{k=1}^{n-1} k(2 n-k) x_{k}
$$

Answer: 3n(n-1)


About pentagon $ABCDE$ is known that angle $A$ and angle $C$ are right and that the sides $| AB | = 4$, $| BC | = 5$, $| CD | = 10$, $| DE | = 6$. Furthermore, the point $C'$ that appears by mirroring $C$ in the line $BD$, lies on the line segment $AE$. Find angle $E$.

Answer: 90^\circ


7.279. $\left\{\begin{array}{l}x^{y^{2}-7 y+10}=1, \\ x+y=8\end{array},(x>0)\right.$.

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

7.279. $\left\{\begin{array}{l}x^{y^{2}-7 y+10}=1, \\ x+y=8\end{array},(x>0)\right.$.

Answer: (1, 7)


Given \( c > 0 \). Let \( p \) : The function \( y=c' \) is monotonically decreasing on \( \mathbf{R} \), \( q \) : The solution set of the inequality \( x + |x - 2c| > 1 \) is \( \mathbf{R} \). If exactly one of \( p \) and \( q \) is correct, determine the range of values for \( c \).

Answer: (0, \frac{1}{2}] \cup [1, \infty)


7. Three rectangles are placed mutually adjacent and without gaps or overlaps to form a larger rectangle. One of the three rectangles has dimensions 70 by 110 . Another of the rectangles has dimensions 40 by 80 . What is the maximum perimeter of the third rectangle?

Answer: 300


2. Eight numbers stand in a row such that the sum of any three consecutive numbers equals 50. The first and last numbers of these eight are known. Fill in the six empty places:

11 12.

ANSWER: $11,12,27,11,12,27,11,12$

Answer: 11


4.9. The circumference of each of two equal circles of radius $R$ passes through the center of the other. Find the area of the common part of these circles.

Answer: \left( \dfrac{2\pi}{3} - \dfrac{\sqrt{3}}{2} \right) R^2


1st Centromerican 1999 Problem B1 ABCD is a trapezoid with AB parallel to CD. M is the midpoint of AD, ∠MCB = 150 o , BC = x and MC = y. Find area ABCD in terms of x and y.

Answer: \dfrac{xy}{2}


Vasiniev N. 5.

Can the numbers be arranged in a $4 \times 4$ table such that the following conditions are simultaneously satisfied:

1) the products of the numbers in each row are the same for all rows;
2) the products of the numbers in each column are the same for all columns;
3) there are no equal numbers among them;
4) all numbers are no greater than 100?

Answer: Yes


7. (15 points) The plant shooters include Peashooter, Twinshot, Triple-shot, Ice-shooter, Dual-shooter, and Pea Pod, requiring 100, 200, 300, 150, 125, 125 suns respectively to plant one of each type. Feifei planted 10 plant shooters, spending a total of 2500 suns. The number of different possible ways she could have planted them is $\qquad$. (For example, 7 Triple-shot + 1 Ice-shooter + 1 Pea Pod is one possible way that meets the requirements.)

Answer: 8


123. In reverse order. What nine-digit number, when multiplied by 123456 789, gives a product where the nine least significant digits are 9,8,7,6,5,4,3, 2,1 (in that exact order)?

Answer: 989010989


12. Chocolate Coupons. A chocolate candy factory places a coupon in each box of candies. For every ten coupons collected, a customer receives a free box of candies. What fraction of the cost of a box does one coupon represent?

Answer: \dfrac{1}{9}


12. Use 1 red bead, 2 blue beads, and 2 green beads to make a bracelet, you can make $\qquad$ different bracelets.

Answer: 4


1. Paul is filling the cells of a rectangular table alternately with crosses and circles (he starts with a cross). When the table is filled in completely, he determines his score as $X-O$ where $X$ is the sum of squares of the numbers of crosses in all the rows and columns, and $O$ is the sum of squares of the numbers of circles in all the rows and columns. Find all possible values of the score for a $67 \times 67$ table.

(Josef Tkadlec)

Answer: 134


8. Consider all three-element subsets $\{a, b, c\}$ of the set $\{1,2, \cdots, n\}$, where $a<b<c$, and call $b$ the middle element. Then the sum of all middle elements $S(n)=$ $\qquad$

Answer: \dfrac{(n-2)(n-1)n(n+1)}{12}


10.013. Given a triangle with sides 12, 15, and \(18 \, \text{cm}\). A circle is drawn that touches both smaller sides and has its center on the larger side. Find the segments into which the center of the circle divides the larger side of the triangle.

Answer: 10


For any positive integer $n$, let $D_n$ denote the set of all positive divisors of $n$, and let $f_i(n)$ denote the size of the set
$$F_i(n) = \{a \in D_n | a \equiv i \pmod{4} \}$$
where $i = 1, 2$.
Determine the smallest positive integer $m$ such that $2f_1(m) - f_2(m) = 2017$.

Answer: 2 \times 5^{2016}


1. Two sisters were picking currants: the elder one in a four-liter bucket, and the younger one in a bucket with a capacity of 3.5 liters. The elder sister always worked faster than the younger one. When the elder sister had collected three quarters of her bucket, and the younger one had collected more than half of hers, the girls switched buckets. Continuing to work at the same speed as before the switch, they finished their work simultaneously. How many times faster did the elder sister pick berries compared to the younger one?

Answer: \dfrac{3}{2}


3.20. In the planes of the faces of the tetrahedron \(ABCD\), points \(A_1, B_1, C_1\), and \(D_1\) are taken such that the lines \(AA_1, BB_1, CC_1\), and \(DD_1\) are parallel. Find the ratio of the volumes of the tetrahedra \(ABCD\) and \(A_1B_1C_1D_1\).

## § 4. Calculation of Volume

Answer: \dfrac{1}{3}


Answer: 4


Problem 7.8. Tanya and Vера are playing a game. Tanya has cards with numbers from 1 to 30. She arranges them in some order in a circle. For each pair of adjacent numbers, Vера calculates their difference, subtracting the smaller number from the larger one, and writes down the resulting 30 numbers in her notebook. After that, Vера gives Tanya a number of candies equal to the smallest number written in the notebook. Tanya arranges the cards so as to get as many candies as possible. What is the maximum number of candies she can get?

Answer: 14


12.407 A line perpendicular to the chord of a segment divides the chord in the ratio 1:4, and the arc - in the ratio $1: 2$. Find the cosine of the central angle subtended by this arc.

Answer: -\dfrac{23}{27}


Inside each of the circles that appear in the figure below, little Luana placed one of the numbers 1, 2, 3, 4, 5, 6, and 7.

![](https://cdn.mathpix.com/cropped/2024_05_01_cc59d970f26105cc4ca1g-13.jpg?height=416&width=483&top_left_y=503&top_left_x=889)

She did this in such a way that all the numbers were used. Her older brother, Pedro, looked at each trio of collinear circles and added the three numbers placed in them. Pedro observed that the sum was always the same.

a) Show that the number placed by Luana in the top circle is 4 and that the constant sum observed by Pedro is equal to 12.

b) Find a way in which Luana could have managed to achieve this feat.

Answer: 4


3. For a sequence of real numbers $x_{1}, x_{2}, \cdots, x_{n}$, define its "value" as $\max _{1 \leqslant i \leqslant n}\left\{\left|x_{1}+x_{2}+\cdots+x_{i}\right|\right\}$. Given $n$ real numbers, David and George want to arrange these $n$ numbers into a sequence with low value. On one hand, diligent David examines all possible ways to find the minimum possible value $D$. On the other hand, greedy George chooses $x_{1}$ such that $\left|x_{1}\right|$ is as small as possible. From the remaining numbers, he chooses $x_{2}$ such that $\left|x_{1}+x_{2}\right|$ is as small as possible, $\cdots \cdots$ At the $i$-th step, he chooses $x_{i}$ from the remaining numbers such that $\left|x_{1}+x_{2}+\cdots+x_{i}\right|$ is as small as possible. At each step, if there are multiple numbers that give the same minimum absolute sum, George arbitrarily chooses one. Finally, the value of the sequence he gets is $G$. Find the smallest constant $c$, such that for every positive integer $n$, every array of $n$ real numbers, and every sequence George can obtain, we have
$$
G \leqslant c D \text {. }
$$

Answer: 2


02.2. In two bowls there are in total $N$ balls, numbered from 1 to $N$. One ball is moved from one of the bowls to the other. The average of the numbers in the bowls is increased in both of the bowls by the same amount, $x$. Determine the largest possible value of $x$.

Answer: \dfrac{1}{2}


Sandy's daughter has a playhouse in the backyard, which includes one shaded exterior wall and two identical sloped roof faces. The exterior wall is 10 feet wide and 7 feet high, and each of the roof sections is 10 feet wide and 6 feet high at the peak. The special siding used to cover these areas is sold only in 10-foot by 15-foot sections, costing $35 each. If Sandy can cut the siding as needed, how much will she spend on the siding?

Answer: \$70


Suppose $x$ and $y$ are inversely proportional. If $x=40$ when $y=9$, find the value of $x$ when $y=20$. Additionally, find the value of $z$ directly proportional to $y$, with $z=45$ when $y=10$.

Answer: 90


On a triangle ABCD circling around a point, the diagram follows:
```
[point positions remain unchanged as well]
```
$\angle ACB = 60^\circ$ and $\angle CAD = 50^\circ$. Find the sum of the degrees in $\angle CAB + \angle ACD$.

Answer: 70^\circ


The Everton Youth Soccer Team consists of 25 players, including 4 goalies. In a training session, each goalie will defend penalty kicks from every other team member including the other goalies. How many penalty kicks are needed to ensure each player has a chance to shoot against each goalie?

Answer: 96


At Logical Learning, to pass the final exam which includes both algebra and geometry sections, a student must score at least \(85\%\) in each section separately to pass. If the algebra section consists of 40 problems and the geometry section has 30 problems, what is the greatest number of problems a student can miss in each section and still pass?

Answer: 4


Thirty gremlins and twenty imps attend the Annual Mischief Convention. Due to some alliances, exactly five imps are willing to shake hands with each other but refuse to shake hands with the remaining imps. All imps shake hands with all gremlins. Meanwhile, being sociable, all gremlins shake hands with each other as well as all the imps. Calculate the total number of handshakes at the convention.

Answer: 1045


In a bowling tournament, 6 bowlers are competing for the top prizes with a playoff system. First, #6 bowls against #5, and the winner enters the main playoff with bowlers #1 through #4. The winner of #6 vs. #5 then bowls #4, the winner of which bowls #3, and so on, until they finally bowl #1. The matches continue in a knockout format where the loser of each match receives the next available lowest prize starting from 5th prize down to 2nd, after all matches. How many different orders can the bowlers receive the prizes?

Answer: 32


A sequence $b_1, b_2, \ldots$ of non-negative integers is defined by $b_{n+2} = |b_{n+1} - b_n|$ for $n \geq 1$. If $b_1 = 1024$, $b_2 < 1024$, and $b_{1004} = 1$, how many different values of $b_2$ are possible?

Answer: 512


Consider the cubic polynomial \(x^3 + bx^2 + cx + d = 0\) where \(b\), \(c\), and \(d\) are integers. Determine all possible values of \(m\), where \(m\) is the exact number of distinct integer roots of the polynomial.

Answer: 0, 1, 2, 3


In a city represented by a Cartesian plane, Dennis is located at $(8, -15)$, Emily is at $(3, 20)$, and they plan to meet at a point that lies equidistant from both of them, then walk together to their friend Felix's location at $(5, 5)$. How many units do Dennis and Emily need to walk upwards to reach Felix?

Answer: 2.5


Marguerite drove 120 miles in 3 hours. She took a 30-minute stop during her journey. If Sam drove for 3.5 hours at the same average driving rate as Marguerite, but without any stops, how many miles did he drive?

Answer: 168 \text{ miles}


In \( \triangle ABC \) with \( AB = 15 \), \( BC = 14 \), and \( AC = 16 \), let \( N \) be a point on \( \overline{AC} \) such that the circumcircles of \( \triangle ABN \) and \( \triangle BNC \) have equal radii. If \( AN = x \) and \( NC = 16 - x \), determine \( \frac{x}{16-x} \).

Answer: \frac{15}{14}


The first $40$ numbers of an arrangement are described as follows:
- Row 1: $2$ numbers, each $2$
- Row 2: $4$ numbers, each $4$
- Row 3: $6$ numbers, each $6$
- Row 4: $8$ numbers, each $8$
- This pattern continues similarly in the subsequent rows. What is the value of the $80^{\mathrm{th}}$ number if the arrangement were continued?

Answer: 18


Max has drawn a pair of concentric circles. He begins to draw chords $\overline{PQ}$, $\overline{QR}$, and so on, of the larger circle, each chord being tangent to the smaller circle. Given that $\angle PQR = 60^\circ$, how many sections will Max draw before returning to his starting point at $P$?

Answer: 3


Evochkimov M.A.

Vasya thought of a two-digit number and told Petya the product of the digits in the number, and Sasha the sum of these digits. The boys had the following conversation:

Petya: "I can guess the number in three attempts, but two might not be enough".

Sasha: "If so, then I can guess it in four attempts, but three might not be enough".

What number was told to Sasha?

Answer: 10


6. From the set $\{1,2, \cdots, 2014\}$, randomly and without replacement, take three numbers $a_{1} 、 a_{2} 、 a_{3}$, and then from the remaining 2011 numbers, similarly randomly and without replacement, take three numbers $b_{1} 、 b_{2} 、 b_{3}$. The probability that a brick with dimensions $a_{1} \times a_{2} \times a_{3}$ can fit into a box with dimensions $b_{1} \times b_{2} \times b_{3}$ is $\qquad$

Answer: \dfrac{1}{4}


3. As shown in Figure 4, in quadrilateral $ABCD$, let $\angle BAD + \angle ADC = 270^{\circ}$, and $E, F$ are the midpoints of $AD, BC$ respectively, $EF=4$, the shaded parts are semicircles with diameters $AB, CD$ respectively. Then the sum of the areas of these two semicircles is $\qquad$ (the value of pi is $\pi$).

Answer: 8\pi


68. From Pickleminster to Quickville. Trains $A$ and $B$ depart from Pickleminster to Quickville at the same time as trains $C$ and $D$ depart from Quickville to Pickleminster. Train $A$ meets train $C$ 120 miles from Pickleminster and train $D$ 140 miles from Pickleminster. Train $B$ meets train $C$ 126 miles from Quickville, and train $D$ at the halfway point between Pickleminster and Quickville. What is the distance from Pickleminster to Quickville? All trains travel at constant speeds, not too different from the usual.

Answer: 210


Let $A B C D A^{\prime} B^{\prime} C^{\prime} D^{\prime}$ be a cube. The midpoint of edge $B^{\prime} C^{\prime}$ is $E$, and the midpoint of edge $C^{\prime} D^{\prime}$ is $F$. The plane $A E F$ divides the cube into two parts. Determine the ratio of the volumes of the two parts.

Answer: \dfrac{25}{47}


[ Tangent lines and tangent circles ] Described quadrilaterals $\quad]$

Two circles of radii 4 and 3 touch each other externally. Common external tangents $P Q$ and $R S$ are drawn to these circles such that points $P$ and $S$ lie on the circle of the larger radius, and points $Q$ and $R$ lie on the circle of the smaller radius. Find the radius of the circle that touches segments $R S$, $S P$, and $P Q$.

Answer: \dfrac{24}{7}


Problem 12.2. The excircle to the side $A B$ of a triangle $A B C$ is tangent to the circle with diameter $B C$. Find $\Varangle A C B$ if the lengths of the sides $B C, C A$ and $A B$ form (in this order) an arithmetic progression.

Oleg Mushkarov

Answer: 90^\circ