There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Peter`, `Arnold`
 - Each person has a unique favorite drink: `water`, `milk`, `tea`
 - They all have a unique favorite flower: `carnations`, `lilies`, `daffodils`
 - Everyone has a unique favorite cigar: `pall mall`, `prince`, `blue master`

## Clues:
1. The person who smokes Blue Master is the person who loves a bouquet of daffodils.
2. Arnold is somewhere to the right of the person who loves a carnations arrangement.
3. Eric is somewhere to the right of the Prince smoker.
4. The one who only drinks water is the person partial to Pall Mall.
5. The person who loves the boquet of lilies is Peter.
6. The tea drinker is not in the third house.
 What is the value of attribute House for the person whose attribute Flower is lilies? Please reason step by step, and put your final answer within \boxed{}

Answer: 1


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Bob`, `Arnold`, `Peter`, `Alice`
 - People use unique phone models: `oneplus 9`, `google pixel 6`, `iphone 13`, `samsung galaxy s21`, `huawei p50`
 - People have unique hair colors: `gray`, `blonde`, `black`, `brown`, `red`
 - People own unique car models: `bmw 3 series`, `tesla model 3`, `ford f150`, `toyota camry`, `honda civic`

## Clues:
1. The person who uses an iPhone 13 is in the third house.
2. The person who owns a Tesla Model 3 is the person who uses a OnePlus 9.
3. The person who has blonde hair is not in the second house.
4. The person who uses a Samsung Galaxy S21 is not in the second house.
5. The person who owns a BMW 3 Series is not in the fourth house.
6. The person who owns a Honda Civic and the person who has brown hair are next to each other.
7. The person who owns a Honda Civic and Arnold are next to each other.
8. The person who owns a Ford F-150 is somewhere to the left of the person who has black hair.
9. Eric is not in the first house.
10. Eric is somewhere to the left of the person who has brown hair.
11. The person who owns a Toyota Camry is Peter.
12. The person who has blonde hair is somewhere to the right of Alice.
13. The person who uses an iPhone 13 is Bob.
14. Peter is the person who uses a Google Pixel 6.
15. The person who owns a Toyota Camry is somewhere to the right of the person who owns a Honda Civic.
16. Peter is the person who has gray hair.
 What is the value of attribute House for the person whose attribute Name is Peter? Please reason step by step, and put your final answer within \boxed{}

Answer: 5


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Carol`, `Alice`, `Arnold`, `Peter`, `Bob`
 - People have unique favorite book genres: `historical fiction`, `science fiction`, `fantasy`, `romance`, `mystery`, `biography`
 - People use unique phone models: `iphone 13`, `google pixel 6`, `samsung galaxy s21`, `oneplus 9`, `huawei p50`, `xiaomi mi 11`
 - Each person has a unique favorite drink: `tea`, `water`, `boba tea`, `milk`, `root beer`, `coffee`

## Clues:
1. The boba tea drinker is the person who uses a Xiaomi Mi 11.
2. Eric is the root beer lover.
3. The person who loves science fiction books is the person who likes milk.
4. The person who likes milk is somewhere to the right of Eric.
5. Arnold is the person who uses a Huawei P50.
6. The person who loves romance books is not in the fourth house.
7. Alice is the person who uses an iPhone 13.
8. The person who loves fantasy books is somewhere to the right of Bob.
9. There are two houses between the person who uses an iPhone 13 and the person who loves fantasy books.
10. The root beer lover is directly left of Peter.
11. The person who uses an iPhone 13 and the person who loves science fiction books are next to each other.
12. The person who uses a Samsung Galaxy S21 is the person who loves romance books.
13. The tea drinker is the person who uses a Huawei P50.
14. There is one house between the person who loves romance books and the person who uses a OnePlus 9.
15. There are two houses between the person who loves biography books and the boba tea drinker.
16. The person who loves historical fiction books is somewhere to the right of the root beer lover.
17. There is one house between the one who only drinks water and the person who uses a Huawei P50.
 What is the value of attribute PhoneModel for the person whose attribute House is 6? Please reason step by step, and put your final answer within \boxed{}

Answer: xiaomi mi 11


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Bob`, `Peter`, `Eric`, `Carol`, `Alice`
 - Each person lives in a unique style of house: `craftsman`, `colonial`, `mediterranean`, `modern`, `victorian`, `ranch`
 - Each person has an occupation: `nurse`, `doctor`, `artist`, `engineer`, `lawyer`, `teacher`
 - People have unique favorite sports: `swimming`, `basketball`, `baseball`, `tennis`, `soccer`, `volleyball`

## Clues:
1. There are two houses between the person who loves baseball and the person in a modern-style house.
2. Arnold and the person who loves baseball are next to each other.
3. The person living in a colonial-style house is somewhere to the right of the person who is a lawyer.
4. The person who is an artist is Peter.
5. The person who is a nurse is not in the sixth house.
6. There is one house between Carol and Bob.
7. The person who is an engineer is directly left of the person who is an artist.
8. There are two houses between the person who loves swimming and the person who is a doctor.
9. Bob is in the sixth house.
10. The person residing in a Victorian house is not in the third house.
11. The person residing in a Victorian house is not in the second house.
12. The person who loves basketball is Eric.
13. The person living in a colonial-style house is in the fourth house.
14. There are two houses between the person in a ranch-style home and the person who loves tennis.
15. The person in a modern-style house is somewhere to the right of the person in a Mediterranean-style villa.
16. The person living in a colonial-style house is somewhere to the left of the person in a Mediterranean-style villa.
17. The person who loves volleyball is the person who is an engineer.
18. The person living in a colonial-style house is somewhere to the left of the person who loves basketball.
 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: victorian


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Arnold`, `Peter`, `Alice`, `Eric`, `Carol`
 - The people are of nationalities: `swede`, `dane`, `german`, `norwegian`, `chinese`, `brit`
 - Each person has a favorite color: `red`, `green`, `purple`, `blue`, `white`, `yellow`

## Clues:
1. Alice is directly left of Bob.
2. The person whose favorite color is green is the British person.
3. Alice is somewhere to the left of the German.
4. The person who loves blue is the Dane.
5. Eric is the person who loves purple.
6. Carol is in the second house.
7. The Swedish person is not in the second house.
8. The Swedish person is not in the first house.
9. The person who loves white is directly left of the Norwegian.
10. There are two houses between Alice and the person who loves yellow.
11. The Dane is in the fifth house.
12. The person who loves blue is Peter.
 What is the value of attribute House for the person whose attribute Color is blue? Please reason step by step, and put your final answer within \boxed{}

Answer: 5


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Alice`, `Peter`, `Eric`, `Arnold`, `Carol`
 - Each mother is accompanied by their child: `Fred`, `Timothy`, `Samantha`, `Alice`, `Meredith`, `Bella`
 - People have unique favorite music genres: `pop`, `hip hop`, `classical`, `jazz`, `rock`, `country`
 - People have unique heights: `average`, `very tall`, `tall`, `super tall`, `very short`, `short`

## Clues:
1. There is one house between the person's child is named Samantha and the person who is short.
2. The person's child is named Alice is Bob.
3. The person who loves country music is directly left of Arnold.
4. Alice is the person who is tall.
5. The person who loves pop music is Eric.
6. Bob is somewhere to the right of the person who is super tall.
7. The person's child is named Fred is Peter.
8. The person's child is named Bella is the person who loves hip-hop music.
9. The person who is the mother of Timothy is not in the sixth house.
10. The person who is super tall is somewhere to the right of the person who has an average height.
11. The person's child is named Alice is somewhere to the right of Arnold.
12. There is one house between the person who is short and the person who is very short.
13. The person who is very short is in the fifth house.
14. The person who loves jazz music is not in the fifth house.
15. Carol is somewhere to the left of the person who is the mother of Timothy.
16. The person who is very tall is not in the sixth house.
17. The person who loves classical music is in the sixth house.
18. The person who loves rock music is in the first house.
 What is the value of attribute Children for the person whose attribute MusicGenre is rock? Please reason step by step, and put your final answer within \boxed{}

Answer: Samantha


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Eric`, `Bob`, `Arnold`, `Carol`, `Alice`
 - Each person has a unique birthday month: `may`, `sept`, `jan`, `mar`, `feb`, `april`
 - People have unique hair colors: `brown`, `blonde`, `red`, `auburn`, `black`, `gray`
 - Each mother is accompanied by their child: `Alice`, `Samantha`, `Fred`, `Meredith`, `Timothy`, `Bella`
 - Everyone has a favorite smoothie: `blueberry`, `cherry`, `watermelon`, `lime`, `dragonfruit`, `desert`
 - The mothers' names in different houses are unique: `Sarah`, `Aniya`, `Janelle`, `Holly`, `Kailyn`, `Penny`

## Clues:
1. The person whose birthday is in September and the person's child is named Alice are next to each other.
2. The person whose birthday is in April is the person who is the mother of Timothy.
3. The person who likes Cherry smoothies is the person who has gray hair.
4. Alice is somewhere to the right of Arnold.
5. The person's child is named Alice is somewhere to the right of The person whose mother's name is Sarah.
6. Peter is the person who drinks Blueberry smoothies.
7. The person whose mother's name is Janelle is the person whose birthday is in May.
8. The person whose birthday is in January is somewhere to the left of The person whose mother's name is Sarah.
9. The Watermelon smoothie lover is the person who has brown hair.
10. The person who has red hair is somewhere to the right of the person who has gray hair.
11. The person who drinks Lime smoothies is in the fifth house.
12. The person who has black hair and the person's child is named Meredith are next to each other.
13. The person whose birthday is in March is somewhere to the right of the person who has red hair.
14. The person whose mother's name is Sarah is somewhere to the left of the person who has auburn hair.
15. The person who has black hair is directly left of The person whose mother's name is Penny.
16. The person's child is named Samantha is The person whose mother's name is Holly.
17. The person who has blonde hair and Bob are next to each other.
18. The person who has gray hair is the person whose birthday is in January.
19. The person who has black hair is in the first house.
20. The person whose mother's name is Sarah is the person who has blonde hair.
21. The person whose mother's name is Kailyn is Arnold.
22. The person whose birthday is in January is The person whose mother's name is Kailyn.
23. The person whose birthday is in September is Carol.
24. The person who drinks Lime smoothies is somewhere to the right of the Desert smoothie lover.
25. The person's child is named Fred is not in the third house.
 What is the value of attribute Birthday for the person whose attribute Children is Meredith? Please reason step by step, and put your final answer within \boxed{}

Answer: feb


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Arnold`, `Bob`, `Carol`, `Alice`, `Eric`
 - People have unique heights: `very tall`, `very short`, `super tall`, `average`, `tall`, `short`
 - Everyone has a favorite smoothie: `watermelon`, `cherry`, `blueberry`, `desert`, `dragonfruit`, `lime`
 - Everyone has a unique favorite cigar: `prince`, `blue master`, `yellow monster`, `dunhill`, `blends`, `pall mall`
 - Each person prefers a unique type of vacation: `camping`, `city`, `beach`, `cultural`, `mountain`, `cruise`

## Clues:
1. The person who is very short is directly left of the person who drinks Lime smoothies.
2. The person who smokes Yellow Monster is directly left of the person who drinks Lime smoothies.
3. The person who is short and the person who goes on cultural tours are next to each other.
4. The person who smokes many unique blends is the person who loves beach vacations.
5. There are two houses between the person who smokes Blue Master and the person who drinks Blueberry smoothies.
6. The Prince smoker is not in the fourth house.
7. The person who is tall is not in the first house.
8. The person who smokes Yellow Monster is the person who enjoys mountain retreats.
9. The person who is very tall is Alice.
10. The person who likes Cherry smoothies is the person who smokes Blue Master.
11. Bob is in the first house.
12. The person who smokes many unique blends is Alice.
13. The person who goes on cultural tours is the person partial to Pall Mall.
14. Eric is the person who drinks Blueberry smoothies.
15. The person who prefers city breaks is not in the sixth house.
16. The Prince smoker is the person who prefers city breaks.
17. The person who enjoys camping trips is somewhere to the left of Carol.
18. The person who has an average height is the person who prefers city breaks.
19. The Desert smoothie lover is in the fifth house.
20. The Watermelon smoothie lover is Arnold.
21. The person who is very tall is the person who drinks Lime smoothies.
22. Carol is the Dunhill smoker.
 What is the value of attribute Height for the person whose attribute Vacation is cruise? Please reason step by step, and put your final answer within \boxed{}

Answer: short


4. Let $A B C$ be a triangle. An interior point $P$ of $A B C$ is said to be good if we can find exactly 27 rays emanating from $P$ intersecting the sides of the triangle $A B C$ such that the triangle is divided by these rays into 27 smaller triangles of equal area. Determine the number of good points for a given triangle $A B C$.

Answer: 325


I4.2 In Figure 1, the shaded area is formed by two concentric circles and has area $96 \pi \mathrm{cm}^{2}$. If the two radii differ by $2 P \mathrm{~cm}$ and the large circle has area $Q \mathrm{~cm}^{2}$, find the value of $Q$. (Take $\pi=3$ )

I4.3 Let $R$ be the largest integer such that $R^{Q}<5^{200}$, find the value of $R$.

Answer: 300


8. A and B take turns rolling a die, with A starting first. The rule is: if A rolls a 1, A continues to roll; otherwise, B rolls. If B rolls a 3, B continues to roll; otherwise, A rolls. The two always follow this rule. Then the probability that the $n$-th roll is made by A is
$$
p_{n}=
$$
$\qquad$

Answer: \dfrac{1}{2} \left( 1 + \left( -\dfrac{2}{3} \right)^{n-1} \right)


5. The sum of the squares of the lengths of the sides of triangle $ABC$ is 36, and its area is $5 \sqrt{3}$.

Points $P, Q$, and $R$ are the vertices of isosceles triangles with an angle of $120^{\circ}$, constructed on the sides of triangle $ABC$ as bases (externally). Find the lengths of the sides of triangle $PQR$.

## Answers and Solutions

Problem 1 Answer: $s=2 \sqrt{3}+6 \sqrt{7}$

Answer: 2\sqrt{3} +6\sqrt{7}


On a plane several straight lines are drawn in such a way that each of them intersects exactly $15$ other lines. How many lines are drawn on the plane? Find all possibilities and justify your answer.

Poland

Answer: 16


2. Cut an $8 \times 8$ chessboard into $p$ rectangles without cutting any square, and this cutting method must satisfy (1) each rectangle has an equal number of black and white squares; (2) let $a_{i}$ be the number of white squares in the $i$-th rectangle, then $a_{1}<a_{2}<\cdots<a_{p}$. Find the maximum value of $P$ for which the above cutting method exists, and for this $P$, find all possible $a_{1}, a_{2}, \cdots, a_{p}$.

Answer: 7


Known that there is a regular 200-gon $A_{1}A_{2} \ldots A_{200}$, connecting the diagonals $A_{i}A_{i+9}(\mathrm{i}=1,2, \ldots, 200)$, where $A_{i+200}=A_{i}(i=1,2, \ldots, 9)$. Then there are a total of \_\_\_\_\_ distinct intersection points inside the regular 200-gon.

Answer: 1600


Two distinct squares of the $8\times8$ chessboard $C$ are said to be adjacent if they have a vertex or side in common.
Also, $g$ is called a $C$-gap if for every numbering of the squares of $C$ with all the integers $1, 2, \ldots, 64$ there exist twoadjacent squares whose numbers differ by at least $g$. Determine the largest $C$-gap $g$.

Answer: 9


Four, (50 points) During the Teachers' Day at a certain school, 200 teachers sent text messages to greet each other, with each person sending exactly one message to another teacher. Now, from them, the maximum number of $k$ representatives can be selected to attend a commendation conference, such that none of them has sent a message to any of the others. Try to find the minimum value of $k$.

---

The above text has been translated into English, preserving the original text's line breaks and format.

Answer: 67


11. (20 points) The ellipse $C_{1}$ and the hyperbola $C_{2}$ have common foci $( \pm c, 0)(c>0), C_{1}$ and $C_{2}$ have an eccentricity difference of no more than 1, and $C_{2}$ has an asymptote with a slope of no less than $\frac{4}{3}$. $C_{1}$ and $C_{2}$ intersect the positive x-axis at points $A$ and $B$, respectively, and the intersection point of the two curves in the first quadrant is $D$. Question: Does $\triangle A B D$ have a maximum area? If so, find the maximum area and provide the equations of $C_{1}$ and $C_{2}$; if not, explain the reason.

Answer: \dfrac{9\sqrt{5}}{50}c^2


7. Highway (from 7th grade, 3 points). A highway running from west to east intersects with $n$ equal roads, numbered from 1 to $n$ in order. Cars travel on these roads from south to north and from north to south. The probability that a car will approach the highway from each of these roads is $\frac{1}{n}$. Similarly, the probability that a car will turn off the highway onto each of these roads is $\frac{1}{n}$. The road by which a car leaves the highway is independent of the road by which it entered the highway.

![](https://cdn.mathpix.com/cropped/2024_05_06_142b35c881fc974fcd25g-05.jpg?height=411&width=532&top_left_y=654&top_left_x=1393)
which the car leaves the highway, is independent of the road by which it entered the highway.

Find the probability that a random car that has entered the highway will pass the $k$-th intersection (either drive through it or turn at it).

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


5. We will call a number greater than 25 semi-prime if it is the sum of some two different prime numbers. What is the maximum number of consecutive semi-prime numbers that can be semi-prime?

Answer: 5


## Task 1 - 330521

On a street in a small town, there are a total of 47 street lamps on the left and right side of the road. On each side of the street, the distance between any two adjacent lamps is $35 \mathrm{~m}$. On the left side of the road, there is a lamp exactly at the beginning and at the end of the street. How long is this street?

Answer: 805


11. Let $S(k)$ denote the sum of all the digits in the decimal representation of a positive integer $k$. Let $n$ be the smallest positive integer satisfying the condition $S(n)+S(n+1)=$ param1. As the answer to the problem, write down a five-digit number such that its first two digits coincide with the first two digits of $n$ and its last three digits coincide with the last three digits of $n$. For example, if $\mathrm{n}=1234567890$, then the answer must be 12890.

| param1 | Answer |
| :---: | :---: |
| 2016 | 59989 |
| 664 | 49989 |
| 1580 | 39989 |
| 4000 | 79989 |

## Condition

Let $S(k)$ denote the sum of all the digits in the decimal representation of a positive integer $k$. Let $n$ be the smallest positive integer such that $S(n)+S(n+1)=2016$. As the answer to the problem, write down a five-digit number such that its first two digits coincide with the first two digits of $n$ and its last three digits coincide with the last three digits of $n$. For example, if $\mathrm{n}=1234567890$, then the answer must be 12890.

## Answer.

59989

## Solution

If the number $n$ does not end in 9, then $S(n+1)=S(n)+1$, and the sum $S(n)+S(n+1)$ is odd, which means it cannot equal 2016. Therefore, the number $n$ must end in 9. If it ends in one 9, then $S(n+1)=S(n)-8$, and thus $S(n)=1012$. The smallest number with such a sum of digits is 599...989 (the total number of 9s is 111). If the number of 9s at the end of $n$ is not less than two, then by similar reasoning, the sum $S(n)$ will be larger, and the number $n$ will also be larger. Therefore, the answer is 59989.

12. Let $A C$ be the largest side of triangle $A B C$. Points $K$ and $M$ are chosen on side $A C$ such that $A M=A B$ and $C K=C B$. It is given that the radius of the circumcircle of triangle $K B M$ equals param1, and the radius of the incircle of triangle $A B C$ equals param2. Find the length of $B T$ squared if $T$ is the point where the incircle of $A B C$ touches its side $B C$.

| param1 | param2 | Answer |
| :---: | :---: | :---: |
| 7 | 5 | 24 |
| 8 | 5 | 39 |
| 9 | 7 | 32 |
| 11 | 8 | 57 |
| 13 | 9 | 88 |

## Condition

Let $A C$ be the largest side of triangle $A B C$. Points $K$ and $M$ are chosen on side $A C$ such that $A M=A B$ and $C K=C B$. It is given that the radius of the circumcircle of triangle $K B M$ equals 7, and the radius of the incircle of triangle $A B C$ equals 5. Find the length of $B T$ squared if $T$ is the point where the incircle of $A B C$ touches its side $B C$.

## Answer.

24

Answer: 24


In circle $\Omega$, let $\overline{AB}=65$ be the diameter and let points $C$ and $D$ lie on the same side of arc $\overarc{AB}$ such that $CD=16$, with $C$ closer to $B$ and $D$ closer to $A$. Moreover, let $AD, BC, AC,$ and $BD$ all have integer lengths. Two other circles, circles $\omega_1$ and $\omega_2$, have $\overline{AC}$ and $\overline{BD}$ as their diameters, respectively. Let circle $\omega_1$ intersect $AB$ at a point $E \neq A$ and let circle $\omega_2$ intersect $AB$ at a point $F \neq B$. Then $EF=\frac{m}{n}$, for relatively prime integers $m$ and $n$. Find $m+n$.
[asy]
size(7cm);
pair A=(0,0), B=(65,0), C=(117/5,156/5), D=(125/13,300/13), E=(23.4,0), F=(9.615,0);
draw(A--B--C--D--cycle);
draw(A--C);
draw(B--D);
dot("$A$", A, SW);
dot("$B$", B, SE);
dot("$C$", C, NE);
dot("$D$", D, NW);
dot("$E$", E, S);
dot("$F$", F, S);
draw(circle((A + C)/2, abs(A - C)/2));
draw(circle((B + D)/2, abs(B - D)/2));
draw(circle((A + B)/2, abs(A - B)/2));
label("$\mathcal P$", (A + B)/2 + abs(A - B)/2 * dir(-45), dir(-45));
label("$\mathcal Q$", (A + C)/2 + abs(A - C)/2 * dir(-210), dir(-210));
label("$\mathcal R$", (B + D)/2 + abs(B - D)/2 * dir(70), dir(70));
[/asy]
[i]Proposed by [b]AOPS12142015[/b][/i]

Answer: 961


5. For any $x \in\left[-\frac{\pi}{6}, \frac{\pi}{2}\right]$, the inequality
$$
\sin ^{2} x+a \sin x+a+3 \geqslant 0
$$

always holds. Then the range of the real number $a$ is $\qquad$

Answer: [-2, +\infty)


8. Let the sequence $a_{n}=\left[(\sqrt{2}+1)^{n}+\left(\frac{1}{2}\right)^{n}\right], n \geq 0$, where $[x]$ denotes the greatest integer less than or equal to $x$. Then
$$
\sum_{n=1}^{\infty} \frac{1}{a_{n-1} a_{n+1}}=
$$

Answer: \dfrac{1}{8}


Observing the temperatures recorded in Cesenatico during the December and January, Stefano noticed an interesting coincidence: in each day of this period, the low temperature is equal to the sum of the low temperatures the preceeding day and the succeeding day.
Given that the low temperatures in December $3$ and January $31$ were $5^\circ \text C$ and $2^\circ \text C$ respectively, find the low temperature in December $25$.

Answer: -3


3. In a convex 12-sided polygon, all angles are equal. It is known that the lengths of some ten of its sides are 1, and the length of one more is 2. What can the area of this 12-sided polygon be? (M. A. Evdokimov) Solution. Consider the 12-sided polygon $A_{1} A_{2} \ldots A_{12}$, satisfying the condition of the problem. It has ten sides of length 1 and one side of length 2. Let $x$ be the length of the remaining side. Consider the vectors $\overline{A_{1} A_{2}}, \overline{A_{2} A_{3}}, \ldots, \overline{A_{12} A_{1}}$, as well as the unit vectors $\vec{e}_{1}, \vec{e}_{2}, \ldots, \vec{e}_{12}$ collinear with them. Then for some $i$ and $j$ the equality $\vec{e}_{1}+\ldots+2 \vec{e}_{i}+\ldots+x \vec{e}_{j}+\ldots+\vec{e}_{12}=\overrightarrow{0}$ holds. In addition, $\vec{e}_{1}+\vec{e}_{7}=\vec{e}_{2}+\vec{e}_{8}=\ldots=\vec{e}_{6}+\vec{e}_{12}=\overrightarrow{0}$, so $\vec{e}_{1}+\vec{e}_{2}+\ldots+\vec{e}_{12}=\overrightarrow{0}$. Subtracting the second obtained equality from the first, we get $\vec{e}_{i}+(x-1) \vec{e}_{j}=\overrightarrow{0}$. This is possible only if $\vec{e}_{i}=-\vec{e}_{j}$ and $x=2$. Therefore, in the original 12-sided polygon, there is a pair of parallel sides of length 2.

Due to the equality of all angles and corresponding sides, this 12-sided polygon has an axis of symmetry (see figure). To find the area, we will divide it into 4 trapezoids and a rectangle. We find $A_{3} A_{12}=A_{6} A_{9}=1+\sqrt{3}, A_{4} A_{11}=A_{5} A_{10}=2+\sqrt{3}$, so the desired area is

$$
S=2 \cdot(2+\sqrt{3})+\frac{\sqrt{3} \cdot(2+\sqrt{3}+1+\sqrt{3})}{2}+\frac{1+\sqrt{3}+1}{2}=8+4 \sqrt{3}
$$

Answer: 8 + 4\sqrt{3}


# Problem 3.

B-1

At the time when a lion cub, located 6 minutes away, set off for a drink, the second, having already quenched its thirst, headed back along the same path at 1.5 times the speed of the first. At the same time, a tortoise, located 32 minutes away, set off along the same path for a drink. After some time, the first lion cub stepped on the tortoise, and some time later, the second lion cub did the same. 28 minutes and 48 seconds after the second incident, the tortoise reached the water hole. How many minutes passed between the two incidents, given that all three moved at constant speeds?

Answer: \dfrac{12}{5}


A servant is sent to a spring with two vessels of 7 pints and 11 pints capacity. She needs to bring back exactly 2 pints of water.

What is the minimum number of operations required in this case? An "operation" is defined as filling a vessel, emptying a vessel, or pouring water from one vessel to another.

Answer: 14


Evan's analog clock displays the time 12:13; the number of seconds is not shown. After 10 seconds elapse, it is still 12:13. What is the expected number of seconds until 12:14?

Answer: 25


Bob is coloring lattice points in the coordinate plane. Find the number of ways Bob can color five points in \(\{(x, y) \mid 1 \leq x, y \leq 5\}\) blue such that the distance between any two blue points is not an integer.

Answer: 80


On the radius $AO$ of a circle with center $O$, a point $M$ is chosen. On one side of $AO$, points $B$ and $C$ are chosen on the circle such that $\angle AMB = \angle OMC = \alpha$. Find the length of $BC$ if the radius of the circle is 21 and $\sin \alpha = \frac{\sqrt{40}}{7}$.

Answer: 18


An $8 \times 8$ chessboard is divided into $p$ rectangles such that no single square is cut in half. Each such division must satisfy the following conditions:

(a) Each rectangle must contain an equal number of white and black squares.

(b) If $a_{i}$ denotes the number of white squares in the $i$-th rectangle, the inequality $a_{1} < a_{2} < \ldots < a_{p}$ must hold.

Determine the largest possible value of $p$ for which such a division exists.

Additionally, generate the sequences $a_{1}, a_{2}, \ldots, a_{p}$ for this value of $p$.

Answer: 7


A cone has a height of $12 \mathrm{~cm}$. When the lateral surface is unwrapped, we obtain a circular sector with a central angle of $100^{\circ} 48^{\prime}$. What is the surface area and the volume of the cone?

Answer: 49\pi^3


Consider the line \( 12x - 4y + (Q - 305) = 0 \). If the area of the triangle formed by the \( x \)-axis, the \( y \)-axis, and this line is \( R \) square units, what is the value of \( R \)?

Answer: 6


A circle with a radius of \(1 + \sqrt{2}\) is circumscribed around an isosceles right triangle. Find the radius of a circle that is tangent to the legs of this triangle and internally tangent to the circumscribed circle.

Answer: 2


Arrange the leftmost numbers of Pascal's triangle in the following manner:

Diagonals are a sequence of numbers connected by arrows in a slanted direction. For example, the first diagonal is " $1 \leftarrow 1$", the second diagonal is " $1 \longleftrightarrow \rightarrow 2$", and the third diagonal is " $1 \longleftrightarrow 3 \longleftrightarrow \rightarrow 1$".

Question: What is the sum of all the numbers in the 13th diagonal?

Answer: 610


Natural numbers \(m\) and \(n\) are such that the fraction \(\frac{m}{n}\) is in its simplest form, but the fraction \(\frac{4m + 3n}{5m + 2n}\) can be simplified. To which natural numbers does it simplify?

Answer: 7


Let $\Sigma(n)$ denote the sum of the digits of the number $n$. Find the smallest three-digit $n$ such that $\Sigma(n) = \Sigma(2n) = \Sigma(3n) = \ldots = \Sigma(n^2)$.

Answer: 999


A group of friends is sharing a bag of candy:

1. On the first day, they eat \(\frac{1}{2}\) of the candies in the bag.
2. On the second day, they eat \(\frac{2}{3}\) of the remaining candies.
3. On the third day, they eat \(\frac{3}{4}\) of the remaining candies.
4. On the fourth day, they eat \(\frac{4}{5}\) of the remaining candies.
5. On the fifth day, they eat \(\frac{5}{6}\) of the remaining candies.

At the end of the fifth day, there is 1 candy remaining in the bag. How many candies were in the bag before the first day?
(A) 512
(B) 720
(C) 1024
(D) 1440
(E) 2048

Answer: 720


What is the maximum number of rooks that can be placed on a $4 \times 4$ chessboard such that each one attacks exactly one other rook? What is the answer for an $8 \times 8$ chessboard?

Answer: 10


The rectangular region \( A \) is in the first quadrant. The bottom side and top side of \( A \) are formed by the lines with equations \( y = 0.5 \) and \( y = 99.5 \), respectively. The left side and right side of \( A \) are formed by the lines with equations \( x = 0.5 \) and \( x = 99.5 \), respectively.

(a) Determine the number of lattice points that are on the line with equation \( y = 2x + 5 \) and are inside the region \( A \). (A point with coordinates \((r, s)\) is called a lattice point if \( r \) and \( s \) are both integers.)

(b) For some integer \( b \), the number of lattice points on the line with equation \( y = \frac{5}{3}x + b \) and inside the region \( A \) is at least 15. Determine the greatest possible value of \( b \).

(c) For some real numbers \( m \), there are no lattice points that lie on the line with equation \( y = mx + 1 \) and inside the region \( A \). Determine the greatest possible real number \( n \) that has the property that, for all real numbers \( m \) with \( \frac{2}{7} < m < n \), there are no lattice points on the line with equation \( y = mx + 1 \) and inside the region \( A \).

Answer: \frac{27}{94}


A 10 x 10 grid is filled with non-negative numbers. It is known that in every (vertical or horizontal) strip of 1 x 3 the sum of the numbers is equal to 9. What is the maximum possible value of the sum of the numbers in the entire grid?

Answer: 306


In a class of 45 students, all students participate in a tug-of-war. Among the remaining three events, each student participates in at least one event. It is known that 39 students participate in the shuttlecock kicking event and 28 students participate in the basketball shooting event. How many students participate in all three events?

Answer: 22


Answer: -k


Example 3 In a regular $\triangle A B C$ with side length 8, take a moving point $P$ on side $B C$. Find the maximum and minimum values of the segment $M N$ intercepted by the perpendicular bisector of $A P$ on $\triangle A B C$.

Answer: 4\sqrt{3}


Chords \(AB\) and \(CD\) of a circle \(\Gamma\) with center \(O\) each have a length of 4. The extensions of segments \(BA\) and \(CD\) beyond points \(A\) and \(D\) respectively intersect at point \(P\). Line \(PO\) intersects segment \(AC\) at point \(L\) such that \(AL:LC = 1:4\).

a) Find \(AP\).

b) Additionally, suppose the radius of circle \(\Gamma\) is 3 and point \(T\) is the center of the circle inscribed in triangle \(ACP\). Find the length of segment \(PT\) and the area of triangle \(ACP\).

Answer: \dfrac{4}{3}


There are $n \geq 2$ classes organized $m \geq 1$ extracurricular groups for students. Every class has students participating in at least one extracurricular group. Every extracurricular group has exactly $a$ classes that the students  in this group participate in. For any two extracurricular groups, there are no more than $b$ classes with students participating in both groups simultaneously.
a) Find $m$ when $n = 8, a = 4 , b = 1$ .
b) Prove that $n \geq 20$ when $m = 6 , a = 10 , b = 4$.
c) Find the minimum value of $n$ when $m = 20 , a = 4 , b = 1$.

Answer: 14


10,11

All dihedral angles of a trihedral angle are equal to $60^{\circ}$. Find the angles formed by the edges of this trihedral angle with the planes of the opposite faces.

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


Task 9. The Federation of Wrestling has assigned each participant in the competition a qualification number. It is known that in matches between wrestlers whose qualification numbers differ by more than 2, the wrestler with the lower number always wins. The tournament for 256 wrestlers is held on an Olympic system: at the beginning of each day, the wrestlers are paired, the loser is eliminated from the competition (there are no draws). What is the highest qualification number that the winner can have?

Answer: 16


4. Let positive integers $a, b$ make $15a + 16b$ and $16a - 15b$ both squares of positive integers. Find the smallest value that the smaller of these two squares can take.

Answer: 231361


B2 For each positive integer $n$, define $\varphi(n)$ to be the number of positive divisors of $n$. For example, $\varphi(10)=4$, since 10 has 4 positive divisors, namely $\{1,2,5,10\}$.
Suppose $n$ is a positive integer such that $\varphi(2 n)=6$. Determine the minimum possible value of $\varphi(6 n)$.

Answer: 8


For the function \(f(x)\), if \(f(x) = x\), then \(x\) is called the "fixed point" of \(f(x)\); if \(f(f(x)) = x\), then \(x\) is called the "stable point" of \(f(x)\). The sets of the "fixed points" and the "stable points" of the function \(f(x)\) are denoted by \(A\) and \(B\), respectively. Given that \(f(x) = a x^{2} - 1\) (where \(a \in \mathbb{R}\), \(x \in \mathbb{R}\)) and \(A = B \neq \varnothing\), find the range of the real number \(a\).

Answer: [-\frac{1}{4}, \frac{3}{4}]


[Example 1.3.8] Let $l, m$ be two skew lines, and on $l$ there are three points $A, B, C$ such that $AB = BC$. Through $A, B, C$ respectively, draw perpendiculars to $m$ at points $D, E, F$. It is known that $AD = \sqrt{15}, BE = \frac{7}{2}, CF = \sqrt{10}$. Find the distance between $l$ and $m$.

Answer: \sqrt{6}


We colored the faces of a regular dodecahedron with red, blue, yellow, and green such that any two adjacent faces have different colors. How many edges does the dodecahedron have where one of the two adjacent faces is blue and the other is green?

Answer: 5


10.276. A right triangle $ABC$ is divided by the altitude $CD$, drawn to the hypotenuse, into two triangles $BCD$ and $ACD$. The radii of the circles inscribed in triangles $BCD$ and $ACD$ are 4 and $3 \text{ cm}$, respectively. Find the distance between their centers.

Answer: 5\sqrt{2}


6. [5] Let $f(x)=x^{3}-x^{2}$. For a given value of $c$, the graph of $f(x)$, together with the graph of the line $c+x$, split the plane up into regions. Suppose that $c$ is such that exactly two of these regions have finite area. Find the value of $c$ that minimizes the sum of the areas of these two regions.

Answer: -\dfrac{11}{27}


4. In $\triangle A B C$, the sides opposite to $\angle A$, $\angle B$, and $\angle C$ are denoted as $a$, $b$, and $c$ respectively. If $b^{2}=a^{2}+c^{2}-a c$, and $c-a$ equals the height $h$ from vertex $A$ to side $A C$, then $\sin \frac{C-A}{2}=$ $\qquad$

Answer: \dfrac{1}{2}


2. Let $n$ be a three-digit positive integer without the digit 0. If the digits of $n$ in the units, tens, and hundreds places are permuted arbitrarily, the resulting three-digit number is never a multiple of 4. Find the number of such $n$.
(54th Ukrainian Mathematical Olympiad)

Answer: 283


In trapezoid $A B C D$, diagonal $A C$ is perpendicular to side $C D$, and diagonal $D B$ is perpendicular to side $A B$. The extensions of the lateral sides $A B$ and $D C$ intersect at point $K$, forming triangle $A K D$ with an angle of $45^{\circ}$ at vertex $K$. The area of trapezoid $A B C D$ is $P$. Find the area of triangle $A K D$.

Answer: 2P


5. Given a pyramid $S-ABCD$ inscribed in a sphere with the base being a rectangle $ABCD$, and $SA=4, SB=8, SD=7$, $\angle SAC=\angle SBC=\angle SDC$. Then the length of $BD$ is

Answer: 9


1. In a tennis tournament, 1152 schoolchildren are participating. 1 point is awarded for a win, and 0 points for a loss. Before each round, pairs are formed by drawing lots among participants with the same number of points (those who do not find a pair are awarded a point without playing). A player is eliminated after the second loss. The tournament continues as long as at least one pair of opponents can be formed. How many rounds will have to be played?

Answer: 14


1. Determine all natural numbers of the form $\overline{20 a 2 b 2}$ that are divisible by 72.

Answer: 200232


As shown in the figure, $\triangle ABC$ is divided into six smaller triangles by lines drawn from the vertices through a common interior point. The areas of four of these triangles are as indicated. Find the area of $\triangle ABC$ . AIME 1985 Problem 6.png
Please give the answer directly without any intermediate steps.

Answer: 315


3. On the extensions of sides $\boldsymbol{A B}, \boldsymbol{B C}, \boldsymbol{C D}$ and $\boldsymbol{A}$ of the convex quadrilateral $\boldsymbol{A} \boldsymbol{B C D}$, points $\boldsymbol{B}_{1}, \boldsymbol{C}_{1}, \boldsymbol{D}_{1}$ and $\boldsymbol{A}_{1}$ are taken such that $\boldsymbol{B} \boldsymbol{B}_{1}=\boldsymbol{A B}, \boldsymbol{C} \boldsymbol{C}_{1}=\boldsymbol{B C}, \boldsymbol{D D}_{1}=\boldsymbol{C D}$ and $\boldsymbol{B} \boldsymbol{B}_{1}=\boldsymbol{A B}$ and $\boldsymbol{A} \boldsymbol{A}_{1}=\boldsymbol{A}$. How many times smaller is the area of quadrilateral $\boldsymbol{A} \boldsymbol{B} \boldsymbol{C D}$ compared to the area of quadrilateral $\boldsymbol{A}_{1} \boldsymbol{B}_{1} C_{1} \boldsymbol{D}_{1}$. (10 points)

Answer: 5


In a certain triangle, the difference between two sides: $b-c$ is twice the distance from the angle bisector $f_{\alpha}$, which starts from the common vertex $A$ of the two sides, to the foot $M$ of the altitude. What is the angle $\alpha$ formed by the two sides?

Answer: 60^\circ


Twelve students are given two versions of a test (a total of six identical tickets of each version). In how many ways can the students be seated in two rows such that no students sitting next to each other have the same ticket, and students sitting one behind the other have the same version?

Answer: 1036800


4. Arrange $n(n \geqslant 2)$ coins in a row. If there is a coin with the head side up, then one can select one of them, and flip the consecutive odd number of coins (which can be one) starting from this coin from left to right simultaneously (flipping means turning a head-up coin to head-down, or a head-down coin to head-up), which is called one operation. When all coins are head-down, the operation stops. If all coins are head-up at the beginning, is there a way to perform $\left[\frac{2^{n+1}}{3}\right]([x]$ denotes the greatest integer not exceeding the real number $x$) operations?
（Gu Bin provided the problem）

Answer: \left[\frac{2^{n+1}}{3}\right]


In an acute triangle $A B C$ with angle $C$ equal to $30^{\circ}$, the altitudes intersect at point $M$. Find the area of triangle $A M B$, if the distances from the center of the circle circumscribed around triangle $A B C$ to sides $B C$ and $A C$ are $\sqrt{2}$ and $\frac{\sqrt{3}}{3}$, respectively.

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


5. The first four members of a sequence are: $2,0,1,8$. Each subsequent member of the sequence is the unit digit of the sum of the previous four members. (e.g., the fifth member is 1). Is the 2018th member of the sequence an even or odd number? Justify your answer!

Answer: \text{Odd}


5. In an isosceles triangle $M N K$, the sides $M N = N K = 8$, and $M K = 4$. A point $F$ is chosen on side $N K$ such that the circles inscribed in triangles $M N F$ and $M K F$ touch each other. Find the areas of triangles $M N F$ and $M K F$.

Answer: 3\sqrt{15}


26. (GBR 2) ${ }^{\mathrm{IMO} 3} \mathrm{~A}$ function $f$ defined on the positive integers (and taking positive integer values) is given by  $$ \begin{aligned} f(1) & =1, \quad f(3)=3 \\ f(2 n) & =f(n) \\ f(4 n+1) & =2 f(2 n+1)-f(n) \\ f(4 n+3) & =3 f(2 n+1)-2 f(n) \end{aligned} $$  for all positive integers $n$. Determine with proof the number of positive integers less than or equal to 1988 for which $f(n)=n$.

Answer: 92


20. The five numbers $a, b, c, d, e$ are all different. The products of each pair of these numbers, arranged in ascending order, are $3, 6, 15, 18, 20, 50, 60, 100, 120, 300$. Then, the five numbers arranged in ascending order, the square of the 2nd number is $\qquad$ .

Answer: 10


Let $ABC$ be a triangle with $\angle A = 60^o$. The points $M,N,K$ lie on $BC,AC,AB$ respectively such that $BK = KM = MN = NC$. If $AN = 2AK$, find the values of $\angle B$ and $\angle C$.

by Mahdi Etesami Fard

Answer: 75^\circ


8.5. In the city of Omsk, a metro has been built, represented by a straight line. On this same line is the house where Nikita and Egor live. Every morning they leave the house for classes at the same time, after which Egor runs to the nearest metro station at a speed of 12 km/h, while Nikita walks along the metro line to another station at a speed of 6 km/h. Despite this, every day Nikita manages to arrive on time for the first lesson, while Egor does not, even though he is not delayed anywhere. Find the greatest possible speed of the metro trains, given that it is constant and equal to an integer. (Assume that the school is located directly on a certain metro station, different from the given ones).

Answer: 23


13.366. On a river with a current speed of 5 km/h, there are piers $A, B$, and $C$ in the direction of the current, with $B$ located halfway between $A$ and $C$. From pier $B$, a raft and a boat depart simultaneously in the direction of the current towards pier $C$, and the boat heads towards pier $A$, with the boat's speed in still water being $V$ km/h. Upon reaching pier $A$, the boat turns around and heads towards pier $C$. Find all values of $v$ for which the boat arrives at $C$ later than the raft.

Answer: (5, 15)


In an isosceles trapezoid $A B C D$ with bases $B C$ and $A D$, the diagonals intersect at point $O$. Find the perimeter of the trapezoid if $B O=\frac{7}{8}, O D=\frac{25}{8}, \angle A B D=90^{\circ}$.

Answer: \dfrac{62}{5}


The last three digits of $1978^{n}$ and $1978^{m}$ are equal. Try to find positive integers $n, m$, such that $m+n$ takes the minimum value. Here $n>m \geqslant 1$.

Answer: 106


8-1. Find all common points of the graphs

$$
y=8 \cos \pi x \cdot \cos ^{2} 2 \pi x \cdot \cos 4 \pi x \quad \text { and } \quad y=\cos 9 \pi x
$$

with abscissas belonging to the segment $x \in[0 ; 1]$. In your answer, specify the sum of the abscissas of the found points.

Answer: \dfrac{7}{2}


6. A circle can be circumscribed around quadrilateral $A B C D$. The lengths of sides $A B$ and $A D$ are equal.

On side $C D$ is point $Q$ such that $D Q=1$, and on side $B C$ is point $P$ such that $B P=5$.

Furthermore, $\measuredangle D A B=2 \measuredangle Q A P$. Find the length of segment $P Q$.

## Solutions and answers

## Variant 1

Problem 1 Answer: $7-\frac{\sqrt{2}}{2}$

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


1. [15] Four standard six-sided dice are rolled. Find the probability that, for each pair of dice, the product of the two numbers rolled on those dice is a multiple of 4 .

Answer: \dfrac{31}{432}


9.11. The numbers $a_{1}, a_{2}, \ldots, a_{n}$ are such that the sum of any seven consecutive numbers is negative, and the sum of any eleven consecutive numbers is positive. For what largest $n$ is this possible?

118 Chapter 9. Computation of Sums and Products

$$
\text { 9.3. Sums } S_{k}(n)=1^{k}+2^{k}+\ldots+n^{k}
$$

The sum $1+2+3+\ldots+n$ can be computed as follows. Add the equations $(k+1)^{2}=k^{2}+2 k+1$ for $k=1,2, \ldots, n$. After simplification, we get $(n+1)^{2}=1+2 S_{1}(n)+n$, where $S_{1}(n)$ is the desired sum. Therefore, $S_{1}(n)=\frac{n(n+1)}{2}$.

Answer: 16


## Task 1 - 220831

On a day in 1981, Cathrin asks her grandfather about his birth year. The grandfather, a friend of puzzle questions, replied:

"I am older than 65 years, but younger than 100 years. The year of my birth is not divisible by 2, 3, or 5. The remainder when this year is divided by 60 is not a prime number."

Investigate whether these statements can all be true for a birth year and whether they uniquely determine the birth year! What is the birth year of the grandfather?

Hint: The year should be given in full, e.g., not 11 but 1911.

Answer: 1909


1. In a given triangle, the radius of the inscribed circle and the radii of the excircles form (in some order) consecutive terms of a geometric progression. Determine the largest angle of the triangle.

Answer: 90^\circ


Let $ABCD$ be a tetrahedron. Let $a$ be the length of $AB$ and let $S$ be the area of the projection of the tetrahedron onto a plane perpendicular to $AB$. Determine the volume of the tetrahedron in terms of $a$ and $S$.

Answer: \dfrac{aS}{3}


7.5. Each cell of a $5 \times 5$ table is painted in one of several colors. Lada shuffled the rows of this table so that no row remained in its original position. Then Lera shuffled the columns so that no column remained in its original position. To their surprise, the girls noticed that the resulting table was the same as the original one. What is the maximum number of different colors this table can be painted with?

Answer: 7


Let $f(x) = \log_{10} \left(\sin(\pi x) \cdot \sin(2 \pi x) \cdot \sin (3 \pi x) \cdots \sin(8 \pi x)\right)$ . The intersection of the domain of $f(x)$ with the interval $[0,1]$ is a union of $n$ disjoint open intervals. What is $n$

Answer: 12


4. Given that the ratio of the areas of the upper base, lower base, and the four isosceles trapezoidal sides of a regular frustum is $2: 5: 8$. Then the angle between the side and the base is $\qquad$ .

Answer: \arctan\left( \dfrac{\sqrt{55}}{3} \right)


ニ、(Full score 25 points) Write 1995 equations of the form $A_{i} x^{2}+B_{i} x+C_{i}=0(i=1,2, \cdots, 1995)$ on the blackboard. Two people, A and B, take turns to play a game. Each time, only one non-zero real number is allowed to replace one of the $A_{i}, B$ or $C$ in the $n$ equations $(i=1,2, \cdots, 1995)$. Once a coefficient in the equation is filled, it cannot be changed. After $3 \times 1995$ steps, 1995 quadratic equations with specific real coefficients are obtained. A always tries to make as many quadratic equations as possible have no roots, while B always tries to prevent A from achieving his goal. If A fills first, how many rootless equations can A get at most, regardless of what B does?

Answer: 998


A freight train leaves the town of Jenkinsville at 1:00 PM traveling due east at constant speed. Jim, a hobo, sneaks onto the train and falls asleep. At the same time, Julie leaves Jenkinsville on her bicycle, traveling along a straight road in a northeasterly direction (but not due northeast) at 10 miles per hour. At 1:12 PM, Jim rolls over in his sleep and falls from the train onto the side of the tracks. He wakes up and immediately begins walking at 3.5 miles per hour directly towards the road on which Julie is riding. Jim reaches the road at 2:12 PM, just as Julie is riding by. What is the speed of the train in miles per hour?

Answer: 62.5


Answer: \dfrac{4}{3}


3. Ducks were flying. One in front, two behind, one behind and two in front, one between two and three in a row. How many ducks were flying in total?

Answer: 3


4. At a rectangular table, eight chairs are placed, four on one side and four opposite them on the other side. In how many ways can eight friends be seated at this table so that Ana and Bane sit opposite each other, and Vesna and Goran sit next to each other? (It is known that all friends have distinct names.)

Answer: 1152


1. Each of the 4 colors has 6 pencils. Each of the six children received 4 pencils. For what smallest $k$ can we always choose $k$ children and decorate their pencils so that we have at least one in each color?

Answer: 3


13. Draw two perpendicular chords $O A, O B$ through the vertex of the parabola $y^{2}=4 p x(p>0)$, find the locus of the projection $M$ of the vertex $O$ of the parabola on the line $A B$.

Answer: (x - 2p)^2 + y^2 = (2p)^2


Through an internal point $P$ of an equilateral triangle $ABC$, perpendiculars are drawn to the three sides, with the feet of the perpendiculars being $D, E, F$ in sequence. Connect $AP, BP, CP$. If the area of the equilateral triangle $ABC$ is 2028 square centimeters, and the areas of triangles $PAD$ and $PBE$ are both 192 square centimeters, then the area of triangle $PCF$ is $\qquad$ square centimeters.

【Answer】 630
【Problem Type】Geometry: Half Model
【Solution】

Answer: 630


4. Given that $n$ is a positive integer, if $\frac{n^{2}+3 n-10}{n^{2}+6 n-16}$ is a reduced fraction, then the value of this fraction is $\qquad$ .

Answer: \dfrac{8}{11}


7. Let $m, n, p, l$ be natural numbers. If the following system:

$$
\left\{\begin{array}{l}
n p^{2}+m l^{2} \leq 23456 \\
m n+p^{7} l=1626 \\
m(l+1)^{p}+n p l=2197+m
\end{array}\right.
$$

has solutions, then present the one for which the sum $m+n+p+l$ is minimal. Otherwise, prove that there are no solutions.

Answer: \text{No solutions}


The sequence of positive integers $a_1, a_2, \dots$ has the property that $\gcd(a_m, a_n) > 1$ if and only if $|m - n| = 1$. Find the sum of the four smallest possible values of $a_2$.

Answer: 42


Putnam 1994 Problem A2 Let R be the region in the first quadrant bounded by the x-axis, the line 2y = x, and the ellipse x 2 /9 + y 2 = 1. Let R' be the region in the first quadrant bounded by the y-axis, the line y = mx, and the ellipse. Find m such that R and R' have the same area.

Answer: \dfrac{2}{9}


$f(x)=\sum\limits_{i=1}^{2013}\left[\dfrac{x}{i!}\right]$. A integer $n$ is called [i]good[/i] if $f(x)=n$ has real root. How many good numbers are  in $\{1,3,5,\dotsc,2013\}$?

Answer: 587


In triangle \(ABC\), points \(M\), \(N\), and \(P\) are chosen such that \(M\) and \(N\) lie on sides \(AC\) and \(BC\) respectively, and \(P\) lies on segment \(MN\). The following conditions are satisfied:

\[
\frac{|AM|}{|MC|} = \frac{|CN|}{|NB|} = \frac{|MP|}{|PN|}
\]

Find the area of triangle \(ABC\) given that the areas of triangles \(AMP\) and \(BNP\) are \(T\) and \(Q\) respectively.

Answer: \left( \sqrt[3]{T} + \sqrt[3]{Q} \right)^3


A circle with integer radius $r$ is centered at $(r, r)$ . Distinct line segments of length $c_i$ connect points $(0, a_i)$ to $(b_i, 0)$ for $1 \le i \le 14$ and are tangent to the circle, where $a_i$ $b_i$ , and $c_i$ are all positive integers and $c_1 \le c_2 \le \cdots \le c_{14}$ . What is the ratio $\frac{c_{14}}{c_1}$ for the least possible value of $r$

Answer: 17


4. Given that $\odot O_{1}$ and $\odot O_{2}$ are externally tangent, their radii are $112$ and $63$, respectively. The segment $A B$ is intercepted by their two external common tangents on their internal common tangent. Then, the length of $A B$ is $\qquad$ .

Answer: 168


8.5. What is the maximum number of members that can be in a sequence of non-zero integers, for which the sum of any seven consecutive numbers is positive, and the sum of any eleven consecutive numbers is negative?

Answer: 16


2. In $\triangle A B C$, segment $A D$ intersects $B C$ at $D$, and $A D$ divides $\triangle A B C$ into two similar triangles, with a similarity ratio of $\sqrt{3}: 3$. Then the interior angles of $\triangle A B C$ are $\qquad$ .

Answer: 90^\circ


2. Oleg and Sergey take turns writing down one digit from left to right until a nine-digit number is formed. At the same time, they cannot write down digits that have already been written. Oleg starts (and finishes). Oleg wins if the resulting number is divisible by 4; otherwise, Sergey wins. Who will win with correct play? (O. Dmitriev, R. Zhenodarov)

Answer: Sergey


There is a new type of problem-solving robot with a limited IQ balance. Every time it solves a problem, its IQ balance decreases by the value of the score for that problem. If the robot answers a problem correctly, its IQ balance increases by 1. When the robot's IQ balance is less than the score of the problem it is solving, it fails to solve the problem. If Xiao Peng uses a beginner-level robot with an initial IQ limit of 25 to solve a set of problems with scores ranging from 1 to 10, what is the maximum score that can be achieved?

Answer: 31


There were 14 participants sitting at a round table during a conference. During the break, some of them (but not all) went to drink coffee. It turned out that each participant remaining at the table had exactly one neighbor who left. How many participants could have gone for coffee? Provide all possible answers.

Answer: 6


5. Given is a triangle $\triangle A B C$ such that $|\angle B A C|+|\angle C B A|=135^{\circ}$. Point $N$ is the foot of the altitude from vertex $A$ such that $|N C|=6 \cdot|N B|$. Calculate the length of side $\overline{B C}$ if the area of triangle $\triangle A B C$ is $3024 \mathrm{~mm}^{2}$.

## COUNTY COMPETITION IN MATHEMATICS

Answer: 84


9.6. Ten-digit natural numbers $a, b, c$ are such that $a+b=c$. What is the maximum number of their 30 digits that can be odd?

(I. Bogdanov)

#

Answer: 29


The worm consists of a white head and several segments, as shown in the figure.

When the worm is born, it has a head and one white segment. Each day, a new segment is added to the worm in one of the following ways:

- Either a white segment splits into a white and a gray segment.
- Or a gray segment splits into a gray and a white segment.

(On the fourth day, the worm reaches maturity and stops growing - its body consists of a head and four segments.)

How many different color variants of mature worms of this species can exist?

Hint: What could a two-day-old worm look like?

Answer: 4


A porter needs to transport 200 buns from the kitchen to the construction site (he is currently in the kitchen). Each time, he can carry 40 buns. However, he eats 1 bun each way, whether traveling from the kitchen to the construction site or from the construction site to the kitchen. How many buns can the porter transport to the construction site at most?

Answer: 191


Let \( f(n) \) and \( g(n) \) be polynomials of degree 2014 such that \( f(n) + (-1)^n g(n) = 2^n \) for \( n = 1, 2, \ldots, 4030 \). Find the coefficient of \( x^{2014} \) in \( g(x) \).

Answer: \frac{3^{2014}}{2^{2014}\cdot2014!}


Several siblings are sharing an inheritance. According to the will, $A$ receives 2000 crowns and 1/10 of the remainder; $B$ receives 4000 crowns and 1/10 of the remainder; $C$ receives 6000 crowns and 1/10 of the remainder, and so on. It was observed that each sibling received an equal amount. How many siblings were there, how much did each receive, and what was the total inheritance?

Answer: 18000


Consider all functions \( f: \mathbb{Z} \rightarrow \mathbb{Z} \) satisfying

\[ f(f(x)+2x+20) = 15. \]

Call an integer \( n \) good if \( f(n) \) can take any integer value. In other words, if we fix \( n \), for any integer \( m \), there exists a function \( f \) such that \( f(n) = m \). Find the sum of all good integers \( x \).

Answer: -35


Let $[x]$ denote the greatest integer less than or equal to the real number $x$. Find the number of elements in the set $\left\{n \left|\, n=\left[\frac{k^{2}}{2005}\right], 1 \leqslant k \leqslant 2004, k \in \mathbf{N}\right\}$.

Answer: 1503


There are 2012 dwarves, each of whom is either good or bad. Every day they attend a meeting in groups of 3 or 5. During each meeting, if the majority of the attendees are good, all attendees of that meeting become good; if the majority are bad, all attendees become bad. If after the third day's meetings all 2012 dwarves have become good, what is the minimum number of good dwarves before the first day's meetings?

Answer: 435


Answer: 1149852


Teresa the bunny has a fair 8-sided die. Seven of its sides have fixed labels \(1, 2, \ldots, 7\), and the label on the eighth side can be changed and begins as \(1\). She rolls it several times, until each of \(1, 2, \ldots, 7\) appears at least once. After each roll, if \(k\) is the smallest positive integer that she has not rolled so far, she relabels the eighth side with \(k\). The probability that 7 is the last number she rolls is \(\frac{a}{b}\), where \(a\) and \(b\) are relatively prime positive integers. Compute \(100a + b\).

Answer: 104


The edges of \( K_{2017} \) are each labeled with 1, 2, or 3 such that any triangle has a sum of labels of at least 5. Determine the minimum possible average of all \( \binom{2017}{2} \) labels.

Answer: 2-\frac{1}{2017}


As shown in the diagram, there are two cubes of different sizes. The edge length of the larger cube is 6 times the edge length of the smaller cube. All 6 faces of the larger cube are painted red, and all 6 faces of the smaller cube are painted yellow. Then, the two cubes are glued together. What is the ratio of the red surface area to the yellow surface area on the resulting combined figure?

Answer: 43


Suppose that \( n \) is a positive integer and that the set \( S \) contains exactly \( n \) distinct positive integers. If the mean of the elements of \( S \) is equal to \( \frac{2}{5} \) of the largest element of \( S \) and is also equal to \( \frac{7}{4} \) of the smallest element of \( S \), determine the minimum possible value of \( n \).

Answer: 5


2021 people are sitting around a circular table. In one move, you may swap the positions of two people sitting next to each other. Determine the minimum number of moves necessary to make each person end up 1000 positions to the left of their original position.

Answer: 1021000


As shown in the figure, in the right triangular prism $ABC-A_{1}B_{1}C_{1}$, $\angle ACB=90^\circ$, $AC=6$, $BC=CC_{1}=\sqrt{2}$, let $P$ be a moving point on the segment $B_{1}C$. Find the minimum value of $AP + PC_{1}$.

Answer: 5\sqrt{2}


A race track for car racing consists of three sections: paved, dirt, and muddy. The speeds of two cars participating in the race on each section of the track are the same, being 100, 70, and 15 km/h, respectively. The timer started when the red car was on the paved section, 500 meters ahead of the white car, and the white car was just crossing the starting line at the beginning of the paved section. Find the distance between the cars when both of them are on the dirt section. Find the distance between the cars when both of them are on the muddy section.

Answer: 75\,


Given the set \( A = \{1, 2, 3, 4\} \), Anders randomly picks a number from \( A \) every second (numbers can be repeated). The picking stops when the sum of the last two picked numbers is a prime number. What is the probability that the last number picked is "1"?

Answer: \frac{15}{44}


143. Find the general solution of the equation $y^{\prime \prime}-6 y^{\prime}+9 y=0$.

Answer: y = (C_1 + C_2 x) e^{3x}


## Task B-4.4.

The digits of the nine-digit number are mutually different and different from 0. Any two adjacent digits form a two-digit number that is divisible by 7 or 13. Determine that nine-digit number.

Answer: 784913526


Trains $A$ and $B$ are on the same track a distance $100$ miles apart heading towards one another, each at a speed of $50$ miles per hour. A fly starting out at the front of train $A$ flies towards train $B$ at a speed of $75$ miles per hour. Upon reaching train $B$, the fly turns around and flies towards train $A$, again at $75$ miles per hour. The fly continues flying back and forth between the two trains at $75$ miles per hour until the two trains hit each other. How many minutes does the fly spend closer to train $A$ than to train $B$ before getting squashed?

Answer: 43\frac{1}{3}


Problem 3. A carried path (one square wide) is infinite in both directions. Two squares on the path are traps. There are $N$ squares between the traps, and one of them is where the hopper stands. Before each hop, we choose a random natural number $X$, after which the hopper jumps $X$ squares to the left or right (at its own choice). During the game, we can see where the hopper is. For which $N$ can we choose the numbers so that we can definitely trap the hopper, no matter where it initially stood and regardless of how it chooses the directions for jumping?

Answer: N = 2^k - 1


Jerry starts at $0$ on the real number line. He tosses a fair coin $8$ times. When he gets heads, he moves $1$ unit in the positive direction; when he gets tails, he moves $1$ unit in the negative direction. The probability that he reaches $4$ at some time during this process $\frac{a}{b},$ where $a$ and $b$ are relatively prime positive integers. What is $a + b?$ (For example, he succeeds if his sequence of tosses is $HTHHHHHH.$

Answer: 151


1. As shown in Figure 1, let the area of $\triangle A B C$ be $1, A F=\frac{1}{m} A B, B D=\frac{1}{n} B C, C E=\frac{1}{p} C A$. Then the area of $\triangle D E F$ is . (1984, Xi'an City Junior High School Competition)

Answer: 1 - \frac{1}{m} - \frac{1}{n} - \frac{1}{p} + \frac{1}{mn} + \frac{1}{mp} + \frac{1}{np}


In a square with a side of 1, there is a broken line (polygonal chain) of length $L$. It is known that each point of the square is at a distance less than $\varepsilon$ from some point of this broken line. Prove that $L \geq \frac{1}{2 \varepsilon} - \frac{\pi \varepsilon}{2}$.

Answer: L \geq \frac{1}{2\varepsilon} - \frac{\pi \varepsilon}{2}


1st Swedish 1961 Problem 3 For which n is it possible to put n identical candles in a candlestick and to light them as follows. For i = 1, 2, ... , n, exactly i candles are lit on day i and burn for exactly one hour. At the end of day n, all n candles must be burnt out. State a possible rule for deciding which candles to light on day i.

Answer: n \text{ is an odd positive integer}


A large equilateral triangle is constructed by using toothpicks to create rows of small equilateral triangles. For example, in the figure, we have $3$ rows of small congruent equilateral triangles, with $5$ small triangles in the base row.  How many toothpicks would be needed to construct a large equilateral triangle if the base row of the triangle consists of $2003$ small equilateral triangles?
[asy] unitsize(15mm); defaultpen(linewidth(.8pt)+fontsize(8pt)); pair Ap=(0,0), Bp=(1,0), Cp=(2,0), Dp=(3,0), Gp=dir(60); pair Fp=shift(Gp)*Bp, Ep=shift(Gp)*Cp; pair Hp=shift(Gp)*Gp, Ip=shift(Gp)*Fp; pair Jp=shift(Gp)*Hp; pair[] points={Ap,Bp,Cp,Dp,Ep,Fp,Gp,Hp,Ip,Jp}; draw(Ap--Dp--Jp--cycle); draw(Gp--Bp--Ip--Hp--Cp--Ep--cycle); for(pair p : points) { fill(circle(p, 0.07),white); } pair[] Cn=new pair[5]; Cn[0]=centroid(Ap,Bp,Gp); Cn[1]=centroid(Gp,Bp,Fp); Cn[2]=centroid(Bp,Fp,Cp); Cn[3]=centroid(Cp,Fp,Ep); Cn[4]=centroid(Cp,Ep,Dp); label("$1$",Cn[0]); label("$2$",Cn[1]); label("$3$",Cn[2]); label("$4$",Cn[3]); label("$5$",Cn[4]); for (pair p : Cn) { draw(circle(p,0.1)); }[/asy]

Answer: 1507509


B1. Circles $\mathcal{K}_{1}$ and $\mathcal{K}_{2}$ in the figure are the circumcircle and incircle of an equilateral triangle $A B C$. Circle $\mathcal{K}_{2}$ has a square $D E F G$ inscribed in it, with point $D$ lying on side $A B$. Circles $\mathcal{K}_{3}$ and $\mathcal{K}_{4}$ are of equal size and touch each other, and each touches two sides of the square $D E F G$. Determine the ratio of the radii of circles $\mathcal{K}_{1}$ and $\mathcal{K}_{4}$.

(6 points)

![](https://cdn.mathpix.com/cropped/2024_06_07_58dcb28c79ea4cc315e1g-07.jpg?height=643&width=603&top_left_y=198&top_left_x=1252)

Answer: 2(\sqrt{2} + 1)


A square was cut into four equal rectangles and used to form a large letter П, as shown in the figure, with a perimeter of 56. What is the perimeter of the original square?

Answer: 32


1. Determine all prime numbers of the form $1+2^{p}+3^{p}+\ldots+p^{p}$ where $p$ is a prime number.

Answer: 5


13. As shown in the figure, $ABCD$ is a trapezoid, and the intersection point of its diagonals is $O$. Extend $AC$ to point $E$, such that $CE = AO$. Extend $DB$ to point $F$, such that $BF = DO$. If the area of $\triangle BFG$ is 2015 square centimeters, find: the area of $\triangle CGE$.

Answer: 2015


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

Answer: 660


9.48 A pile of stones has a total weight of 100 kilograms, with each stone weighing no more than 2 kilograms. By taking some of these stones in various ways and calculating the difference between the sum of their weights and 10 kilograms. Among all these differences, the minimum absolute value is denoted as $d$. Among all piles of stones that meet the above conditions, find the maximum value of $d$.

Answer: \dfrac{10}{11}


In scalene $\triangle ABC$, $I$ is the incenter, $I_a$ is the $A$-excenter, $D$ is the midpoint of arc $BC$ of the circumcircle of $ABC$ not containing $A$, and $M$ is the midpoint of side $BC$. Extend ray $IM$ past $M$ to point $P$ such that $IM = MP$. Let $Q$ be the intersection of $DP$ and $MI_a$, and $R$ be the point on the line $MI_a$ such that $AR\parallel DP$. Given that $\frac{AI_a}{AI}=9$, the ratio $\frac{QM} {RI_a}$ can be expressed in the form $\frac{m}{n}$ for two relatively prime positive integers $m,n$. Compute $m+n$.

[i]Ray Li.[/i]

[hide="Clarifications"][list=1][*]"Arc $BC$ of the circumcircle" means "the arc with endpoints $B$ and $C$ not containing $A$".[/list][/hide]

Answer: 11


On a night march, the Wolf and Fox patrols each received one identical candle. They lit the candles together at the start and set off. Each member of the patrol carried the candle for the duration it took for its length to be halved. The Wolves reached the finish line at the moment when the sixth carrier was passing the candle to the seventh. After that, their candle burned for another three minutes. The Foxes reached the finish line in two hours and 57 minutes.

- Which patrol reached the finish line first?
- By how many minutes did the winning patrol finish earlier than the other patrol?

(The candle burns evenly: it always burns the same amount in the same time.)

(M. Dillingerová)

Answer: 12


We have a rectangular piece of paper. We want to produce 20-sided polygons using the following procedure. In each step, we select a piece of paper (initially, only the starting piece of paper is available), and cut it into two along a straight line. Continuing this process, what is the minimum number of cuts needed to obtain at least one hundred 20-sided polygons?

Answer: 1699


$1 \cdot 12$ Consider a 100-digit number consisting only of the digits 1 and 2. You are allowed to pick any 10 consecutive digits and swap the positions of the first 5 and the last 5 digits. If one 100-digit number can be obtained from another by performing this operation several times, then these two numbers are called congruent. How many 100-digit numbers, consisting only of the digits 1 and 2, can be selected such that no two of them are congruent?

Answer: 4084101


## Task A-4.4.

A natural number is said to be quirky if in its decimal representation it has 100 digits and if by removing any of its digits, a 99-digit number divisible by 7 is formed. How many quirky natural numbers are there?

Answer: 2^{98}


1. Given that the function $f(x)$ is a monotonically increasing function defined on $[-4,+\infty)$, to ensure that for all real numbers $x$,
$$
f\left(\cos x-b^{2}\right) \geqslant f\left(\sin ^{2} x-b-3\right)
$$

holds true. Find the range of real numbers $b$.

Answer: \left[ \dfrac{1 - 2\sqrt{2}}{2}, 1 \right]


Answer: 508032


IMO 1975 Problem B1 Let A be the sum of the decimal digits of 4444 4444 , and B be the sum of the decimal digits of A. Find the sum of the decimal digits of B.

Answer: 7


2.1. The seller has weights of $1, 2, 4, 8, 16, 32$ grams (one of each) and a balance scale. On the first pan, a candy weighing 25 grams and some three weights were placed, and on the second pan, the remaining three weights, with the scales coming into balance. Indicate the weights of all three weights on the second pan.

Answer: 4


In a $3 \times 3$ grid, there are four $2 \times 2$ subgrids, each of which is bordered by a thick line in one of the four grids below.
![](https://cdn.mathpix.com/cropped/2024_04_17_d13f9eaf1d614231018ag-2.jpg?height=254&width=1134&top_left_y=1346&top_left_x=554)

Kasun wants to place an integer from 1 to 4 inclusive in each cell of a $3 \times 3$ grid so that every $2 \times 2$ subgrid contains each integer exactly once. For example, the grid below and on the left satisfies the condition, but the grid below and on the right does not. In how many ways can Kasun place integers in the grid so that they satisfy the condition?

| 1 | 2 | 3 |
| :--- | :--- | :--- |
| 3 | 4 | 1 |
| 1 | 2 | 3 |

| 1 | 3 | 2 |
| :--- | :--- | :--- |
| 2 | 4 | 1 |
| 1 | 3 | 3 |

Answer: 72


Task 1. Gjorgja's phone number consists of two three-digit numbers written side by side. Each of them is divisible by 45, and the middle digit is 8. Determine the phone number, if the three-digit number written from left in the phone number is smaller than the three-digit number from the right.

Answer: 180585


13.278. A passenger on a train knows that the speed of this train on this section of the track is 40 km/h. As soon as a passing train started to go by the window, the passenger started a stopwatch and noticed that the passing train took $3 \mathrm{s}$ to pass by the window. Determine the speed of the passing train, given that its length is 75 m.

Answer: 50


Answer: \dfrac{x^{2^{2019}}}{x^{2^{2019}} - (x - 1)^{2^{2019}}}


Let $n$ be a positive integer. There are $n$ ants walking along a line at constant nonzero speeds. Different ants need not walk at the same speed or walk in the same direction. Whenever two or more ants collide, all the ants involved in this collision instantly change directions. (Different ants need not be moving in opposite directions when they collide, since a faster ant may catch up with a slower one that is moving in the same direction.) The ants keep walking indefinitely.

Assuming that the total number of collisions is finite, determine the largest possible number of collisions in terms of $n$.

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


# 1. Option 1.

Vasya strikes the strings of a 6-string guitar from 1 to 6 and back. Each subsequent strike hits the adjacent string. On which string number will the 2000th strike fall? (The order of striking the strings: $1-2-3-4-5-6-5-4-3-2-1-2-\ldots$)

Answer: 2


7. A test paper consists of four multiple-choice questions, each with three options $\mathrm{A}$, $\mathrm{B}$, and $\mathrm{C}$. Several students take the exam, and after grading, it is found that any three students have different answers to at least one question, and no student leaves any question unanswered. Then the maximum number of students who can take the exam is $\qquad$ people.

Answer: 9


Given an acute triangle $\triangle ABC$ with midpoints $A_1$, $B_1$, and $C_1$ on sides $BC$, $CA$, and $AB$ respectively. From points $A_1$, $B_1$, and $C_1$, perpendiculars are drawn to the other two sides of $\triangle ABC$, meeting the sides at points $A_2$, $B_2$, and $C_2$ respectively. Prove that the area of hexagon $A_1C_2B_1A_2C_1B_2$ is equal to half of the area of $\triangle ABC$.

Answer: \frac{1}{2}


# 2.1. Condition:

In the campus, rooms are numbered consecutively $1,2,3,4 \ldots, 10,11, \ldots$ For room numbering, stickers with digits were purchased, with the digits 1, 2, and 3 being purchased in equal quantities, and the digit 5 being purchased three more than the digit 6. How many rooms are there in the campus, given that there are fewer than 100?

Answer: 66


3. A football invitational tournament involves sixteen cities, each sending two teams, Team A and Team B. According to the competition rules, after several days of matches, it was found that except for Team A of City A, all other teams had played a different number of matches. How many matches has Team B of City A played? Please prove your conclusion.

Answer: 15


Three distinct positive integers \( a, b, c \) form a geometric sequence, and their sum is 111. Determine the set \( \{a, b, c\} \).

Answer: \{1, 10, 100\}


9. (10 points) In a certain exam, 153 people scored no more than 30 points, with an average of 24 points; 59 people scored no less than 80 points, with an average of 92 points; those who scored more than 30 points had an average of 62 points, and those who scored less than 80 points had an average of 54 points. How many people in total participated in this exam?

Answer: 1007


10. (14 points) In $\triangle A B C$, $B C=a, C A=$ $b$, a square $A B E F$ is constructed outward with side $A B$, $O$ is the center of the square $A B E F$, $M$ and $N$ are the midpoints of sides $B C$ and $C A$, respectively. When $\angle B C A$ varies, find the maximum value of $O M+O N$.

---

Please note that the translation preserves the original text's formatting and structure.

Answer: \dfrac{(a + b)(1 + \sqrt{2})}{2}


Let $I$ be the set of points $(x,y)$ in the Cartesian plane such that $$x>\left(\frac{y^4}{9}+2015\right)^{1/4}$$ Let $f(r)$ denote the area of the intersection of $I$ and the disk $x^2+y^2\le r^2$ of radius $r>0$ centered at the origin $(0,0)$. Determine the minimum possible real number $L$ such that $f(r)<Lr^2$ for all $r>0$.

Answer: \dfrac{\pi}{3}


Richard likes to solve problems from the IMO Shortlist. In 2013, Richard solves $5$ problems each Saturday and $7$ problems each Sunday. He has school on weekdays, so he ``only'' solves $2$, $1$, $2$, $1$, $2$ problems on each Monday, Tuesday, Wednesday, Thursday, and Friday, respectively -- with the exception of December 3, 2013, where he solved $60$ problems out of boredom. Altogether, how many problems does Richard solve in 2013?

[i]Proposed by Evan Chen[/i]

Answer: 1099


A rook has traveled across an \( n \times n \) board, visiting each cell exactly once, where each of its moves was exactly one cell. The cells are numbered from 1 to \( n^2 \) in the order of the rook's path. Let \( M \) be the maximum difference between the numbers of adjacent (by side) cells. What is the smallest possible value of \( M \)?

Answer: 2n - 1


30th IMO 1989 shortlist Problem 15 Given a convex n-gon A 1 A 2 ... A n with area A and a point P, we rotate P through an angle x about A i to get the point P i . Find the area of the polygon P 1 P 2 ... P n .

Answer: 2A(1 - \cos x)


2. In quadrilateral $ABCD$, the diagonals intersect at point $P$, diagonal $AC$ is divided into four equal segments by points $P, N$, and $M$ $(|AP|=|PN|=|NM|=|MC|)$, and diagonal $BD$ is divided into four equal segments by points $L, K$, and $P$ $(|BL|=|LK|=|KP|=|PD|)$. Determine the ratio of the area of quadrilateral KLMN to the area of quadrilateral ABCD.

Answer: \dfrac{3}{16}


15. Given the sequence $\left\{a_{n}\right\}, a_{n}+a_{n+1}=n \cdot(-1)^{\frac{n(n+1)}{2}}$, the sum of the first $n$ terms is $S_{n}$, and $m+S_{2015}=-1007, a_{1} m>0$. Then the minimum value of $\frac{1}{a_{1}}+\frac{4}{m}$ is . $\qquad$

Answer: 9


365. The lengths of two parallel sides of a rectangle are 1 cm. In addition, it is known that it can be divided into four rectangles by two perpendicular lines, three of which have an area of no less than $1 \mathrm{~cm}^{2}$, and the fourth no less than $2 \mathrm{~cm}^{2}$. What is the minimum length of the other two sides of the rectangle for this to be possible?

Answer: 3 + 2\sqrt{2}


87.3. Let $f$ be a strictly increasing function defined in the set of natural numbers satisfying the conditions $f(2)=a>2$ and $f(m n)=f(m) f(n)$ for all natural numbers $m$ and $n$. Determine the smallest possible value of $a$.

Answer: 4


1. Can we obtain the triplet of numbers $2,6,9$ in some order from the triplet with numbers $2,4,7$?

Answer: \text{No}


3. How many five-digit numbers exist where the sum of the first two digits is twice the sum of the last two digits?

Answer: 2600


2. In triangle $ABC$, a point $M$ is given. Through $M$, three lines $p, q$, and $r$ are drawn parallel to $AB, BC$, and $CA$ respectively. Let $R=p \cap BC, L=p \cap AC$, $P=q \cap AB, T=q \cap AC, Q=r \cap AB, N=r \cap BC$. If the areas $P_{1}=P_{P Q M}, P_{2}=P_{M R N}$, and $P_{3}=P_{L M T}$ are known, find the area $P$ of triangle $ABC$.

Answer: (\sqrt{P_1} + \sqrt{P_2} + \sqrt{P_3})^2


Question 39, Let $a, b$ be real numbers, and the function $f(x)=x^{3}+a x^{2}+b x$. If there exist three real numbers $x_{1}, x_{2}, x_{3}$ satisfying $x_{1}+1 \leq x_{2} \leq x_{3}-1$, and $f\left(x_{1}\right)=f\left(x_{2}\right)=f\left(x_{3}\right)$, and the minimum value of $|a|+2|b|$.

Answer: \sqrt{3}


$\underline{\text { Khamtsov }}$ D:

A cube with side $n$ ( $n \geq 3$ ) is divided by partitions into unit cubes. What is the minimum number of partitions between unit cubes that need to be removed so that from each cube it is possible to reach the boundary of the cube?

Answer: (n-2)^3


18.2. A segment 20 cm long is divided into two segments, and a square is constructed on each of them as a side. Find the lengths of the sides of these squares, if the difference in the areas of the squares is $120 \mathrm{~cm}^{2}$.

$$
\text { (7-8 grades) }
$$

Answer: 7


1. $\arctan x+\arctan \frac{1-x}{1+x}=(x>-1)$.

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

1. $\arctan x+\arctan \frac{1-x}{1+x}=(x>-1)$.

Answer: \frac{\pi}{4}


## Task B-2.5.

A bus left from place $A$ to place $B$. 50 minutes later, a car left from place $A$ and arrived at place $B$ 10 minutes before the bus. If they had left simultaneously, one from place $A$ and the other from place $B$ (one heading towards the other), they would have met after one hour and 12 minutes. If traveling the same route and at the same speed, how long will it take for the bus to return from place $B$ to place $A$?

Answer: 3


7.5. In Nastya's room, 16 people gathered, each pair of whom either are friends or enemies. Upon entering the room, each of them wrote down the number of friends who had already arrived, and upon leaving - the number of enemies remaining in the room. What can the sum of all the numbers written down be, after everyone has first arrived and then left?

Answer: 120


11.069. The lateral edge of a regular triangular prism is equal to the height of the base, and the area of the section made through this lateral edge and the height of the base is $Q$. Determine the volume of the prism.

Answer: \dfrac{\sqrt{3}}{3} Q^{\frac{3}{2}}


One, (15 points) Given $T=\{1,2, \cdots, 8\}$. For $A \subseteq T, A \neq \varnothing$, define $S(A)$ as the sum of all elements in $A$. Question: How many non-empty subsets $A$ of $T$ are there such that $S(A)$ is a multiple of 3 but not a multiple of 5?
(Chen Yonggao provided)

Answer: 70


A unit side regular hexagonal lattice point is reflected to the midpoint of the six sides. Calculate the area of the resulting hexagon.

Consider an arbitrary internal point of a regular hexagon with a side length of one unit. This point is reflected to the midpoint of each of the six sides. Calculate the area of the resulting hexagon.

Answer: \dfrac{9\sqrt{3}}{2}


A list of $11$ positive integers has a mean of $10$ , a median of $9$ , and a unique mode of $8$ . What is the largest possible value of an integer in the list?

Answer: 35


20. Xiaoming sets off from home to his grandmother's house, cycling at a speed of 12 kilometers per hour. 2.5 hours after he leaves, his father realizes that Xiaoming has forgotten his homework, so he sets off on a motorcycle at a speed of 36 kilometers per hour to catch up. As a result, half an hour after Xiaoming arrives at his grandmother's house, his father also arrives. The distance from Xiaoming's home to his grandmother's house is $\qquad$ kilometers.

Answer: 36


For a positive integer $n$, $\sigma(n)$ denotes the sum of the positive divisors of $n$. Determine $$\limsup\limits_{n\rightarrow \infty} \frac{\sigma(n^{2023})}{(\sigma(n))^{2023}}$$

[b]Note:[/b] Given a sequence ($a_n$) of real numbers, we say that $\limsup\limits_{n\rightarrow \infty} a_n = +\infty$ if ($a_n$) is not upper bounded, and, otherwise, $\limsup\limits_{n\rightarrow \infty} a_n$ is the smallest constant $C$ such that, for every real $K > C$, there is a positive integer $N$ with $a_n < K$ for every $n > N$.

Answer: 1


Task B-3.3. Determine all integers $k$ for which the equation $x^{2014} + k x^{2} + 36 = 0$ has at least one integer solution.

Answer: -37


3.6. Znayka cut out a semicircle from paper. Neznaika marked a point $M$ on the diameter $A B$ of this semicircle and cut off two semicircles from Znayka's semicircle with diameters $A M$ and $M B$. The area of the remaining figure turned out to be $16 \pi^{3}$. Find the length of the part of the chord lying inside it, passing through point $M$ perpendicular to $A B$. If necessary, round the answer to two decimal places.

Answer: 8\pi


Task 1. Which of the numbers 723, 732, and 273 will decrease the most and by how much if in each of them the digits 7 and 3 swap places?

Answer: 723


Two players are engaged in a game: they alternately name prime numbers not exceeding 100 such that the last digit of the number named by one player is equal to the first digit of the number named by the next player (except for the very first prime number named in the game). Already named numbers cannot be repeated. The player who cannot name the next prime number according to these rules loses. Prove that one of the players can act in such a way as to guarantee a win, and find the smallest possible number of primes that will be used by both players in such a game.

Answer: 3


2. Let $n$ be a positive integer. If $n$ is divisible by 2010 and exactly one of the digits of $n$ is even, find the smallest possible value of $n$.
(1 mark)
Let $n$ be a positive integer. If $n$ is divisible by 2010 and exactly one of the digits of $n$ is even, find the smallest possible value of $n$.

Answer: 311550


9.A, B, C, D, E, F six people are each assigned a number, and these six numbers are all different.

A says: “Our six people's numbers form an arithmetic sequence”
B says: “The smallest number in this arithmetic sequence is 2"
C says: “The sum of our six people's numbers is 42"
D says: “The sum of A, C, E's numbers is twice the sum of B, D, F's numbers”
E says: “The sum of B, F's numbers is twice A's number"
Then D's number is ( ).

Answer: 2


a) A castle has the shape of an equilateral triangle with a side length of 100 meters. It is divided into 100 triangular halls. All the walls of the halls are of the same length, 10 meters. In the middle of each wall between the halls, there is a door. Prove that if a person wants to tour the castle, visiting each hall no more than once, they will be able to see no more than 91 halls.

b) Each side of an equilateral triangle is divided into $k$ equal parts. Lines are drawn through the division points parallel to the sides. As a result, the triangle is divided into $k^{2}$ small triangles. A "chain" is defined as a sequence of small triangles where no triangle appears more than once, and each subsequent triangle shares a side with the previous one. What is the maximum possible number of small triangles in a "chain"?

Answer: 91


Example 8 On the plane: there is an $n$-sided polygon $A_{1} A_{2} \cdots A_{n}$. The area is $S$, and there is another point $M$. $M$ rotates around $A$ by an angle $\alpha$ to get points $M_{i}(i=1,2, \cdots, n)$. Find the area of the $n$-sided polygon $M_{1} M_{2} \cdots M_{n}$.
(30th IMO Preliminary Problem)

Answer: 4S\sin^2\frac{\alpha}{2}


1. As shown in the figure, there is a sequence of curves $P_{0}, P_{1}, P_{2}, \cdots$. It is known that the area of the equilateral triangle enclosed by $P_{0}$ is 1, and $P_{k+1}$ is obtained from $P_{k}$ by the following operation: divide each side of $P_{k}$ into three equal parts, construct an equilateral triangle outward using the middle segment of each side, and then remove the middle segment $(k=0,1,2, \cdots)$. Let $S_{n}$ be the area of the figure enclosed by the curve $P_{n}$.
Find the general term formula for the sequence $\left\{S_{n}\right\}$.
... ...

Answer: S_n = \dfrac{8}{5} - \dfrac{3}{5} \left( \dfrac{4}{9} \right)^n


7.5. Misha and Masha had the same multi-digit integer written in their notebooks, ending in 9876. Masha placed a plus sign between the third and fourth digits from the right, while Misha placed a plus sign between the fourth and fifth digits from the right. To the surprise of the schoolchildren, both resulting sums turned out to be the same. What number was originally written by the schoolchildren? Provide all possible answers and prove that there are no others.

Answer: 9999876


Simão needs to discover a number that is the code of the Treasure Ark hidden in the table.

| 5 | 9 | 4 | 9 | 4 | 1 |
| :--- | :--- | :--- | :--- | :--- | :--- |
| 6 | 3 | 7 | 3 | 4 | 8 |
| 8 | 2 | 4 | 2 | 5 | 5 |
| 7 | 4 | 5 | 7 | 5 | 2 |
| 2 | 7 | 6 | 1 | 2 | 8 |
| 5 | 2 | 3 | 6 | 7 | 1 |

To discover the code, he has to form groups of 3 digits that are in consecutive cells, either horizontally or vertically, whose sum is 14. After removing these groups, the code is the sum of the numbers that do not appear in these groups. What is this code?

## Solutions from List 6

Answer: 29


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Arnold`, `Bob`, `Carol`, `Alice`, `Eric`
 - People have unique heights: `very tall`, `very short`, `super tall`, `average`, `tall`, `short`
 - Everyone has a favorite smoothie: `watermelon`, `cherry`, `blueberry`, `desert`, `dragonfruit`, `lime`
 - Everyone has a unique favorite cigar: `prince`, `blue master`, `yellow monster`, `dunhill`, `blends`, `pall mall`
 - Each person prefers a unique type of vacation: `camping`, `city`, `beach`, `cultural`, `mountain`, `cruise`

## Clues:
1. The person who is very short is directly left of the person who drinks Lime smoothies.
2. The person who smokes Yellow Monster is directly left of the person who drinks Lime smoothies.
3. The person who is short and the person who goes on cultural tours are next to each other.
4. The person who smokes many unique blends is the person who loves beach vacations.
5. There are two houses between the person who smokes Blue Master and the person who drinks Blueberry smoothies.
6. The Prince smoker is not in the fourth house.
7. The person who is tall is not in the first house.
8. The person who smokes Yellow Monster is the person who enjoys mountain retreats.
9. The person who is very tall is Alice.
10. The person who likes Cherry smoothies is the person who smokes Blue Master.
11. Bob is in the first house.
12. The person who smokes many unique blends is Alice.
13. The person who goes on cultural tours is the person partial to Pall Mall.
14. Eric is the person who drinks Blueberry smoothies.
15. The person who prefers city breaks is not in the sixth house.
16. The Prince smoker is the person who prefers city breaks.
17. The person who enjoys camping trips is somewhere to the left of Carol.
18. The person who has an average height is the person who prefers city breaks.
19. The Desert smoothie lover is in the fifth house.
20. The Watermelon smoothie lover is Arnold.
21. The person who is very tall is the person who drinks Lime smoothies.
22. Carol is the Dunhill smoker.
 What is the value of attribute Cigar for the person whose attribute Height is average? Please reason step by step, and put your final answer within \boxed{}

Answer: prince


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

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

Answer: 4


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Peter`, `Eric`, `Alice`
 - Each person lives in a unique style of house: `craftsman`, `colonial`, `victorian`, `ranch`
 - People have unique hair colors: `red`, `blonde`, `black`, `brown`
 - Each mother is accompanied by their child: `Bella`, `Fred`, `Meredith`, `Samantha`
 - People have unique favorite book genres: `mystery`, `fantasy`, `romance`, `science fiction`

## Clues:
1. The person in a Craftsman-style house is in the third house.
2. Alice is the person who loves romance books.
3. The person who has brown hair is in the fourth house.
4. The person's child is named Samantha is in the fourth house.
5. The person in a ranch-style home is somewhere to the right of the person who has red hair.
6. Peter is the person's child is named Bella.
7. Arnold is the person who has red hair.
8. Alice is the person living in a colonial-style house.
9. The person who has black hair is in the second house.
10. The person who loves fantasy books is Peter.
11. Arnold is the person's child is named Meredith.
12. The person who has black hair is Eric.
13. The person who loves science fiction books is Arnold.
 What is the value of attribute BookGenre for the person whose attribute HouseStyle is victorian? 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: `Alice`, `Arnold`, `Eric`, `Peter`
 - They all have a unique favorite flower: `daffodils`, `roses`, `lilies`, `carnations`
 - Everyone has a favorite smoothie: `desert`, `watermelon`, `cherry`, `dragonfruit`
 - Each person lives in a unique style of house: `craftsman`, `colonial`, `ranch`, `victorian`
 - Each person has a unique hobby: `cooking`, `photography`, `gardening`, `painting`
 - Each person has a unique favorite drink: `coffee`, `tea`, `milk`, `water`

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

Answer: watermelon


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

## Clues:
1. Alice is in the fifth house.
2. Arnold is the person who loves white.
3. Bob is not in the fourth house.
4. The person who loves blue is Bob.
5. The person whose favorite color is red is not in the third house.
6. The person who loves a bouquet of daffodils and Eric are next to each other.
7. The person whose favorite color is green is Alice.
8. Peter is directly left of the person who loves a carnations arrangement.
9. The person who loves white is the person who loves the vase of tulips.
10. The person who loves the rose bouquet is Alice.
11. The person who loves the vase of tulips and Peter are next to each other.
 What is the value of attribute House for the person whose attribute Name is Arnold? Please reason step by step, and put your final answer within \boxed{}

Answer: 2


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Bob`, `Alice`, `Eric`, `Peter`
 - People have unique heights: `very tall`, `average`, `tall`, `very short`, `short`
 - Everyone has something unique for lunch: `stew`, `grilled cheese`, `spaghetti`, `pizza`, `stir fry`

## Clues:
1. Alice is the person who is short.
2. The person who is tall is in the third house.
3. The person who has an average height is not in the second house.
4. The person who has an average height is somewhere to the left of the person who loves the stew.
5. The person who loves stir fry is Arnold.
6. The person who is a pizza lover is the person who is tall.
7. Eric is the person who is tall.
8. Bob is somewhere to the right of Arnold.
9. The person who loves eating grilled cheese is somewhere to the right of Eric.
10. The person who is very short is somewhere to the left of Arnold.
 What is the value of attribute Name for the person whose attribute House is 1? Please reason step by step, and put your final answer within \boxed{}

Answer: Peter