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`, `Peter`, `Eric`, `Carol`, `Alice`, `Bob`
 - The people are of nationalities: `swede`, `brit`, `chinese`, `dane`, `german`, `norwegian`
 - People have unique favorite book genres: `science fiction`, `mystery`, `fantasy`, `biography`, `historical fiction`, `romance`
 - Everyone has a favorite smoothie: `blueberry`, `dragonfruit`, `desert`, `watermelon`, `cherry`, `lime`
 - People use unique phone models: `oneplus 9`, `samsung galaxy s21`, `huawei p50`, `xiaomi mi 11`, `iphone 13`, `google pixel 6`

## Clues:
1. Alice is the British person.
2. There is one house between Eric and the Dragonfruit smoothie lover.
3. The Chinese is somewhere to the left of Alice.
4. The person who loves historical fiction books is directly left of the person who loves biography books.
5. The person who loves biography books is somewhere to the right of Carol.
6. The person who uses an iPhone 13 is somewhere to the right of the Desert smoothie lover.
7. The person who loves mystery books is the British person.
8. Bob is somewhere to the right of the person who uses a OnePlus 9.
9. Carol is the person who drinks Blueberry smoothies.
10. There are two houses between the person who loves romance books and the Watermelon smoothie lover.
11. The person who likes Cherry smoothies is the Swedish person.
12. The person who loves science fiction books is the person who uses a Google Pixel 6.
13. The person who drinks Lime smoothies is Eric.
14. Bob is in the fifth house.
15. Carol is in the second house.
16. The Norwegian and Alice are next to each other.
17. The Swedish person is not in the fifth house.
18. The German and the Norwegian are next to each other.
19. The person who loves fantasy books is the Dragonfruit smoothie lover.
20. The British person is somewhere to the left of Peter.
21. The person who loves romance books is the person who uses a Samsung Galaxy S21.
22. Alice is the person who uses a Xiaomi Mi 11.
23. The person who loves romance books is Eric.
 What is the value of attribute House for the person whose attribute Nationality is german? Please reason step by step, and put your final answer within \boxed{}

Answer: 2


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Eric`, `Arnold`, `Alice`
 - People have unique favorite music genres: `rock`, `jazz`, `classical`, `pop`
 - The people are of nationalities: `dane`, `brit`, `swede`, `norwegian`
 - Each person has a favorite color: `green`, `red`, `yellow`, `white`

## Clues:
1. The British person and Peter are next to each other.
2. Eric is somewhere to the left of the British person.
3. The Swedish person is directly left of the person whose favorite color is red.
4. The person who loves classical music is in the second house.
5. There is one house between the British person and the person whose favorite color is red.
6. The person who loves rock music is the Norwegian.
7. The person who loves yellow is not in the first house.
8. The Norwegian is Arnold.
9. The Dane and the person who loves white are next to each other.
10. There are two houses between the person who loves pop music and the person who loves rock music.
 What is the value of attribute Name for the person whose attribute MusicGenre is jazz? Please reason step by step, and put your final answer within \boxed{}

Answer: Peter


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Eric`, `Peter`, `Carol`, `Bob`, `Arnold`
 - Everyone has a favorite smoothie: `watermelon`, `blueberry`, `desert`, `cherry`, `dragonfruit`, `lime`
 - Everyone has something unique for lunch: `stew`, `pizza`, `grilled cheese`, `stir fry`, `soup`, `spaghetti`
 - People use unique phone models: `google pixel 6`, `iphone 13`, `xiaomi mi 11`, `huawei p50`, `samsung galaxy s21`, `oneplus 9`
 - People own unique car models: `tesla model 3`, `honda civic`, `toyota camry`, `ford f150`, `chevrolet silverado`, `bmw 3 series`
 - Each person lives in a unique style of house: `craftsman`, `ranch`, `modern`, `victorian`, `mediterranean`, `colonial`

## Clues:
1. Bob is the person who uses a Xiaomi Mi 11.
2. The person who loves the soup is in the fourth house.
3. The Dragonfruit smoothie lover is somewhere to the left of the person in a ranch-style home.
4. There is one house between the person who owns a Chevrolet Silverado and the person residing in a Victorian house.
5. The person in a Mediterranean-style villa is the person who drinks Lime smoothies.
6. Eric is in the sixth house.
7. The Desert smoothie lover is the person who is a pizza lover.
8. The person living in a colonial-style house is the person who drinks Blueberry smoothies.
9. The Dragonfruit smoothie lover and the person who uses a Google Pixel 6 are next to each other.
10. The person who loves the soup is Peter.
11. Alice is somewhere to the right of the person who owns a BMW 3 Series.
12. The person who loves stir fry is the person in a ranch-style home.
13. The person who owns a Ford F-150 is the person living in a colonial-style house.
14. The person in a Craftsman-style house is somewhere to the right of the person in a modern-style house.
15. The person who loves the stew is directly left of the person in a ranch-style home.
16. The person who owns a Tesla Model 3 is directly left of the person who loves stir fry.
17. The person who loves eating grilled cheese is the person who owns a Honda Civic.
18. The person in a Mediterranean-style villa is the person who uses a Google Pixel 6.
19. The person in a Craftsman-style house is the Watermelon smoothie lover.
20. The person in a modern-style house is Carol.
21. The person who uses an iPhone 13 is in the first house.
22. The person who owns a Honda Civic is somewhere to the left of the person who owns a BMW 3 Series.
23. The person who uses a Xiaomi Mi 11 is the person who is a pizza lover.
24. The person who uses a Samsung Galaxy S21 is the person who loves stir fry.
25. The person residing in a Victorian house is directly left of the person who uses a OnePlus 9.
 What is the value of attribute PhoneModel for the person whose attribute Food is grilled cheese? Please reason step by step, and put your final answer within \boxed{}

Answer: google pixel 6


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Carol`, `Eric`, `Peter`, `Arnold`, `Alice`, `Bob`
 - Everyone has a favorite smoothie: `watermelon`, `lime`, `dragonfruit`, `desert`, `blueberry`, `cherry`

## Clues:
1. Peter is somewhere to the right of the person who drinks Lime smoothies.
2. The person who likes Cherry smoothies is not in the fourth house.
3. Eric is not in the sixth house.
4. The Watermelon smoothie lover is somewhere to the left of Eric.
5. The Desert smoothie lover is somewhere to the right of Peter.
6. The person who drinks Blueberry smoothies is directly left of the person who drinks Lime smoothies.
7. The Desert smoothie lover is somewhere to the right of Alice.
8. Eric is not in the fifth house.
9. The person who drinks Lime smoothies is Arnold.
10. Bob is directly left of Peter.
 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: 3


Three, (25 points) On the first day of operation, the factory's production does not exceed 20 units, and the daily production increases thereafter, but the increase in production each time does not exceed 20 units. When the daily production reaches 1996 units, find the minimum value of the total production of the factory.

When the daily production reaches 1996 units, find the minimum value of the total production of the factory.

Answer: 100600


5. Divide the positive integers $1,2, \cdots, 7$ into two groups in any way, such that each group has at least one number. Then the probability that the sum of the numbers in the first group is equal to the sum of the numbers in the second group is $\qquad$

Answer: \dfrac{4}{63}


Opgave 4. Vind alle functies $f: \mathbb{Z} \rightarrow \mathbb{Z}$ die voldoen aan

- $f(p)>0$ voor alle priemgetallen $p$,
- $p \mid(f(x)+f(p))^{f(p)}-x$ voor alle $x \in \mathbb{Z}$ en alle priemgetallen $p$.

Answer: f(x) = x


5. Solve the equation $\sin \frac{\pi n}{12} \cdot \sin \frac{\pi k}{12} \cdot \sin \frac{\pi m}{12}=\frac{1}{8}$. Here $k, m, n-$ are natural numbers not exceeding 5.

Answer: (2,2,2)


## Task A-1.4. (4 points)

A regular pentagon $A B C D E$ is given. The lines $B C$ and $D E$ intersect at point $F$. Determine the angles of triangle $B E F$.

Answer: 36^\circ


3. On the planet Sharp Teeth, animals reproduce in a special way, specifically: every hamster gives birth to four hamsters every 4 months; every groundhog gives birth to one small groundhog every 4 months; and rabbits, mysterious creatures, reproduce faster the more time passes, specifically, if a person does not give away or buy new rabbits, then after $2k$ months, the number of their rabbits increases by a factor of $k!$ (where $k!$ is the product of numbers from 1 to $k$).

Boy Ilya bought rodents from a pet store. It is known that after a year, Ilya had 1040 domestic animals. How many rodents did Ilya buy from the pet store?

If there are several possible answers, list them in ascending order separated by a semicolon.

Answer: 13


(treated during the session)

Let $ABCD$ be a quadrilateral, $P$ the intersection of $(AC)$ and $(BD)$. Suppose that $\widehat{CAD}=50^{\circ}$, $\widehat{BAC}=70^{\circ}$, $\widehat{DCA}=40^{\circ}$, and $\widehat{ACB}=20^{\circ}$. Calculate the angle $\widehat{CPD}$.

Answer: 70^\circ


Given that \( n \) is a positive even number and graph \( G \) has \( n \) vertices and \(\frac{n^2}{4}\) edges, without loops or multiple edges (any two vertices are either connected by one edge or not connected at all). For an unordered pair of distinct vertices \( (x, y) \), if they are adjacent to the same vertex (there exists a vertex \( z \) such that both \( xz \) and \( yz \) are edges), then the pair \( (x, y) \) is called "friendly". Prove that graph \( G \) has at least \( 2 \binom{\frac{n}{2}}{2} \) friendly unordered pairs.

Answer: 2 \binom{\frac{n}{2}}{2}


Exercise 8. In a tournament organized between 6 teams, each team plays against each other team exactly once. When a team wins, it gets 3 points, and the losing team receives 0 points. If the game is a draw, both teams receive 1 point. Determine the values of \( a \) for which it is possible that the final scores of the teams can be the six consecutive numbers \( a, a+1, \ldots, a+5 \)?

Answer: 4


In the quadratic equation \(x^{2} + px + q = 0\), the coefficients \(p\) and \(q\) are increased by one. This operation is repeated four times. Provide an example of such an initial equation so that the roots of each of the five resulting equations are integers.

Answer: x^2 + 3x + 2 = 0


12. Arrange the numbers of Pascal's Triangle aligned to the far left as follows:

The numbers in the diagonal direction connected by arrows are referred to as a diagonal, for example, the first diagonal is “ $1 \leftarrow 1$ ”, the second diagonal is “ $1 \longleftrightarrow \rightarrow 2$ ”, and the third diagonal is “ $1 \longleftrightarrow 3 \longleftrightarrow \rightarrow 1$ ”.
What is the sum of all the numbers on the 13th diagonal?

Answer: 610


4. Let's highlight the complete squares: $\left\{\begin{array}{l}(x-2)^{2}+(y+1)^{2}=5, \\ (x-2)^{2}+(z-3)^{2}=13, \\ (y+1)^{2}+(z-3)^{2}=10 .\end{array}\right.$

Add all the equations $\left\{\begin{array}{c}(x-2)^{2}+(y+1)^{2}=5, \\ (x-2)^{2}+(z-3)^{2}=13, \\ (x-2)^{2}+(y+1)^{2}+(z-3)^{2}=14 .\end{array}\right.$

And, by subtracting the first and second from the third, we get:

$$
\left\{\begin{array}{l}
(x-2)^{2}=4 \\
(z-3)^{2}=9 \\
(y+1)^{2}=1
\end{array}\right.
$$

Solve the quadratic equations and write down all possible solutions. The system has eight solutions.

Answer: (4, 0, 6)


6. If the three sides of $\triangle A_{1} A_{2} A_{3}$ are $8, 10, 12$ respectively, and the midpoints of the three sides are $B, C, D$, connecting the midpoints in pairs to form three midlines, the resulting figure is the surface development of the tetrahedron $A-BCD$. Then, the surface area of the circumscribed sphere of this tetrahedron is

Answer: \dfrac{77}{2} \pi


Example 5. Five monkeys found Wang Yiguan's peaches. How can they be fairly divided when one monkey secretly gets up, eats one peach, and the remaining peaches are exactly divided into five equal parts, after which it hides its share and goes back to sleep. The second monkey gets up, also eats one peach, and the remaining peaches are again exactly divided into five equal parts, after which it also takes its share and goes back to sleep. The third, fourth, and fifth monkeys all do the same. How many peaches were there at the very beginning?

Answer: 3121


Let \( f(n) \) be the degree of the lowest order polynomial \( p(x) \) with integer coefficients and leading coefficient 1, such that \( n \) divides \( p(m) \) for all integral \( m \). Describe \( f(n) \). Evaluate \( f(1000000) \).

Answer: 25


10. Let the focus of the parabola $y^{2}=4 x$ be $F$, and draw any line through point $F$ intersecting the parabola at points $A$ and $B$. Does there exist a line $l$ such that for any point $M$ on line $l$, the slopes of lines $M A$, $M F$, and $M B$ always form an arithmetic sequence? If it exists, find the equation of line $l$; if not, explain the reason.

Answer: x = -1


Two equal cones have a common vertex and touch along a common generatrix. The angle in the axial section of each cone is $60^{\circ}$. Find the angle between two planes, each of which touches the cones but does not pass through the common generatrix.

Answer: \arccos \left( \dfrac{1}{3} \right)


Billy and Bobby are located at points $A$ and $B$, respectively. They each walk directly toward the other point at a constant rate; once the opposite point is reached, they immediately turn around and walk back at the same rate. The first time they meet, they are located 3 units from point $A$; the second time they meet, they are located 10 units from point $B$. Find all possible values for the distance between $A$ and $B$.

[i]Proposed by Isabella Grabski[/i]

Answer: 15


Given 100 distinct prime numbers \( p_{1}, p_{2}, \cdots, p_{100} \), let \( N=p_{1}^{2}+p_{2}^{2}+\cdots+p_{100}^{2} \). What is the remainder when \( N \) is divided by 3?

Answer: 1


The angle bisector of angle \( A \) of triangle \( ABC \) intersects side \( BC \) at point \( D \). A circle with radius 35, whose center lies on the line \( BC \), passes through points \( A \) and \( D \). It is known that \( AB^2 - AC^2 = 216 \), and the area of triangle \( ABC \) is \( 90 \sqrt{3} \). Find the radius of the circumcircle of triangle \( ABC \).

Answer: 7\sqrt{3}


A chocolate bar is divided into 24 sections by 3 horizontal and 5 vertical notches for easier splitting. We want to cut it into pieces along the notches. Each cut can only split one already formed piece into two. What is the minimum number of cuts required?

Answer: 23


As shown in the figure, $A$ and $B$ are endpoints of a diameter of a circular track. Three micro-robots, Alpha, Beta, and Gamma, start simultaneously on the circular track and move uniformly in a circular motion. Alpha and Beta start from point $A$, and Gamma starts from point $B$. Beta moves clockwise, while Alpha and Gamma move counterclockwise. After 12 seconds, Alpha reaches point $B$. Nine seconds later, Alpha catches up with Gamma for the first time and also coincides with Beta for the first time. When Gamma reaches point $A$ for the first time, how many seconds will it take for Beta to reach point $B$ for the first time?

Answer: 56


Through the focus \( F \) of the parabola \( y^2 = 8(x+2) \), a line with an inclination angle of \( 60^\circ \) is drawn. If this line intersects the parabola at points \( A \) and \( B \), and the perpendicular bisector of chord \( AB \) intersects the \( x \)-axis at point \( P \), then the length of segment \( PF \) is equal to:

A. \( \frac{16}{3} \)

B. \( \frac{8}{3} \)

C. \( \frac{16}{3} \sqrt{3} \)

D. \( 8 \sqrt{3} \)

Answer: \frac{16}{3}


In a classroom, 34 students are seated in 5 rows of 7 chairs. The place at the center of the room is unoccupied. A teacher decides to reassign the seats such that each student will occupy a chair adjacent to his/her present one (i.e., move one desk forward, back, left, or right). In how many ways can this reassignment be made?

Answer: 0


Each Kinder Surprise contains exactly 3 different Smurfs, and there are a total of 11 types of Smurfs. The box contains a sufficient number of Kinder Surprises, where no two of them have identical sets of three Smurfs. What is the minimum number of Kinder Surprises that need to be purchased so that after opening them, there is at least one of each of the 11 types of Smurfs?

Answer: 121


As shown in the diagram, two lines intersect with two circles at 9 points. Choose 4 points from these 9 points such that any 3 points among these 4 are neither collinear nor lie on the same circle. How many different selections can be made?

Answer: 18


Petya bought a construction set containing 100 rods of various lengths. The instructions for the construction set state that any three rods from the set can form a triangle. Petya decided to verify this statement by forming triangles from the rods. The rods in the set are arranged in increasing order of length. What is the minimum number of checks (in the worst-case scenario) that Petya must perform to either prove or disprove the statement in the instructions?

Answer: 1


Calculate the force with which water presses on a dam, whose cross-section has the shape of an isosceles trapezoid. The density of water is $\rho=1000 \ \text{kg/m}^3$ and the acceleration due to gravity $g$ is $10 \ \text{m/s}^2$.

Hint: Pressure at a depth $x$ is $\rho g x$.

Given:
- \( a = 7.2 \ \text{m} \)
- \( b = 12.0 \ \text{m} \)
- \( h = 5.0 \ \text{m} \)

Answer: 1100000\,


One day in October, Nikita looked at the wall clock and saw that four flies had fallen asleep exactly on the numbers 12, 3, 6, and 9.

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

Answer: \frac{1}{3}


In a square room, each wall has a light bulb that can emit one of the seven colors of the rainbow. No two bulbs in the room emit the same color at the same time. At each turn, a person can change the color of one of the light bulbs to a color that is not currently being used by any other light bulb in the room at that moment, and they must also change the colors to two other remaining unused colors. (After this, there will still be no two bulbs with the same color in the room). What is the minimum number of moves needed so that each bulb has emitted each of the seven colors at least once?

Answer: 8


In the trapezoid \(ABCD\), a circle is inscribed, touching the side \(AD\) at point \(K\). Find the area of the trapezoid if \(AK=16\), \(DK=4\), and \(CD=6\).

Answer: 432


In a school with classes from I to VIII, there is only one class per grade. There are a total of 130 students in grades I-IV. In grade V, there are 7 more students than in grade II; in grade VI, there are 5 fewer students than in grade I; in grade VII, there are 10 more students than in grade IV; and in grade VIII, there are 4 fewer students than in grade I. How many students are there in total from grades I-VIII?

Answer: 268


An unreliable typist can guarantee that when they try to type a word with different letters, every letter of the word will appear exactly once in what they type, and each letter will occur at most one letter late (though it may occur more than one letter early). Thus, when trying to type MATHS, the typist may type MATHS, MTAHS, or TMASH, but not ATMSH. Determine, with proof, the number of possible spellings of OLYMPIADS that might be typed.

Answer: 256


A non-self-intersecting broken line is situated in a given half-plane, with the ends of the broken line lying on the boundary of this half-plane. The length of the broken line is $L$, and the area of the polygon bounded by the broken line and the boundary of the half-plane is $S$. Prove that $S \leq \frac{L^{2}}{2\pi}$.

Answer: S \leq \frac{L^{2}}{2\pi}


Let $ABC$ be an equilateral triangle with side length $1$.  Say that a point $X$ on side $\overline{BC}$ is [i]balanced[/i] if there exists a point $Y$ on side $\overline{AC}$ and a point $Z$ on side $\overline{AB}$ such that the triangle $XYZ$ is a right isosceles triangle with $XY = XZ$.  Find with proof the length of the set of all balanced points on side $\overline{BC}$.

Answer: 2 - \sqrt{3}


The ﬁgure below has only two sizes for its internal angles. The larger angles are three times the size of the
smaller angles. Find the degree measure of one of the larger angles. For figure here: http://www.purplecomet.org/welcome/practice

P.S The figure has 9 sides.

Answer: 252


13. (3 points) The judging panel of the Young Singers Grand Prix consists of several people. Each judge gives a score to the singer, with a maximum of 10 points. After the first singer performs, the scoring situation is as follows: the average score given by all judges is 9.64 points; if the highest score is removed, the average score of the remaining judges is 9.60 points; if the lowest score is removed, the average score of the remaining judges is 9.68 points. Therefore, the lowest score given by all judges can be at least $\qquad$ points, and the number of judges in this Grand Prix is $\qquad$.

Answer: 9.28


# 6. Variant 1

A doll maker makes one doll in 1 hour 45 minutes. After every three dolls made, the master has to rest for half an hour. Ten dolls are needed for gifts. At what time the next day (specify hours and minutes) will the order be completed if the master started making dolls at 10:00 AM and worked through the night?

Answer: 5:00


10.4. On the cells of an 8 by 8 board, chips are placed such that for each chip, the row or column of the board in which it lies contains only one chip. What is the maximum possible number of chips on the board?

Answer: 14


For a finite set  $X$  define \[
S(X) = \sum_{x \in X} x \text{ and }
P(x) = \prod_{x \in X} x.
\] Let  $A$  and  $B$  be two finite sets of positive integers such that  $\left\lvert A \right\rvert = \left\lvert B \right\rvert$ ,  $P(A) = P(B)$  and  $S(A) \neq S(B)$ .  Suppose for any  $n \in A \cup B$  and prime  $p$  dividing  $n$ , we have  $p^{36} \mid n$  and  $p^{37} \nmid n$ .  Prove that \[ \left\lvert S(A) - S(B) \right\rvert > 1.9 \cdot 10^{6}. \]*Proposed by Evan Chen*

Answer: \left\lvert S(A) - S(B) \right\rvert > 1.9 \cdot 10^{6}


The argument of the complex numbers \( z_{1}, z_{2}, z_{3} \) are \( \alpha, \beta, \gamma \) respectively, and \( \left|z_{1}\right|=1,\left|z_{2}\right|=k \), \( \left|z_{3}\right|=2-k \). Given that \( z_{1} + z_{2} + z_{3} = 0 \), determine the value of \( k \) for which \( \cos (\beta - \gamma) \) achieves its maximum or minimum value. Also, find the maximum and minimum values.

Answer: 1


5. Find all values of the parameter $b$, for which for any value of the parameter $a \in[-1 ; 2]$ the inequality $\operatorname{tg}^{2} x+4(a+b) \operatorname{tg} x+a^{2}+b^{2}-18<0$ is satisfied for each $x \in[-\pi / 4 ; \pi / 4]$.

Answer: (-2, 1)


3. Let the integer $k$ be such that the quadratic equation in $x$
$$
2 x^{2}+(11-5 k) x+3 k^{2}-13 k-2003=0
$$

has at least one integer root. Then all such integers $k$ are
$\qquad$

Answer: -4030


# Problem 7. Option 1.

Given a triangle with sides 6, 8, and 10. Find the length of the shortest segment connecting points on the sides of the triangle and dividing it into two equal areas.

Answer: 4


$\underline{\text { Berlov S.L. }}$

A five-digit number is called non-decomposable if it cannot be factored into the product of two three-digit numbers.

What is the maximum number of consecutive non-decomposable five-digit numbers?

Answer: 99


Consider the numbers from $1$ to $16$. The "solitar" game consists in the arbitrary grouping of the numbers in pairs and replacing each pair with the great prime divisor of the sum of the two numbers (i.e from $(1,2); (3,4); (5,6);...;(15,16)$ the numbers which result are $3,7,11,5,19,23,3,31$). The next step follows from the same procedure and the games continues untill we obtain only one number. Which is the maximum numbers with which the game ends.

Answer: 19


Let $ABCD$ be an isosceles trapezoid with $AB=5$, $CD = 8$, and $BC = DA = 6$. There exists an angle $\theta$ such that there is only one point $X$ satisfying $\angle AXD = 180^{\circ} - \angle BXC = \theta$. Find $\sin(\theta)^2$.

Answer: \dfrac{27}{32}


In a tournament, each participant plays a match against every other participant. The winner of a match earns 1 point, the loser 0 points, and if the match is a draw, both players earn half a point. At the end of the tournament, the participants are ranked according to their score (in case of a tie, the order is arbitrary). It is then noticed that each participant has won half of their points against the bottom ten in the ranking. How many people participated in the tournament

Answer: 25


3. The number of real solutions to the equation $x=10 \sin x$ is

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

Answer: 7


Let $M=\left\{1,2, \cdots, 2^{m} n\right\}$ (where $m$ and $n$ are positive integers) be a set consisting of continuous $2^{m}n$ positive integers. Find the minimum value of the positive integer $k$ such that any $k$-element subset of $M$ contains $m+1$ numbers $a_1, a_2, \cdots, a_{m+1}$ that satisfy $a_i$ divides $a_{i+1}$ for $i=1,2, \cdots, m$.

Answer: (2^m -1)n +1


1. Let set $A=\{0,1,2, \cdots, 9\},\left\{B_{1}, B_{2}, \cdots, B_{k}\right\}$ be a collection of non-empty subsets of $A$, and when $i \neq j$, $B_{i} \cap B_{j}$ has at most two elements. Then the maximum value of $k$ is $\qquad$
(1999, National High School Mathematics League Guangxi Preliminary Contest (High) Three))

Answer: 175


5. For any two points $P, Q$ on the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$, if $O P \perp O Q$, then the minimum value of $|O P| \cdot|O Q|$ is $\qquad$.

Answer: \dfrac{2a^2b^2}{a^2 + b^2}


\section*{Problem 5 - 031225}

Two shepherds sell a number of animals, each of which brings in as many groschen as the number of animals. They distribute the proceeds as follows:

The first shepherd receives 10 groschen, the second 10 groschen, then again the first 10 groschen, the second 10 groschen, and so on. After the first shepherd receives 10 groschen for the last time, a remainder remains that is less than 10 groschen.

With this remainder, they buy a knife.

How much does the knife cost?

Answer: 6


7. As shown in Figure 2, the radii of circles $A$ and $B$ are both 1, and they are externally tangent to each other. The radii of circles $P, Q, R, S$ are all $r$, and circle $P$ is externally tangent to circles $A, B, Q, S$, circle $Q$ is externally tangent to circles $P, B, R$, circle $R$ is externally tangent to circles $A, B, Q, S$, and circle $S$ is externally tangent to circles $P, A, R$. Then $r=$ $\qquad$

Answer: \dfrac{3 + \sqrt{17}}{2}


1. Given a triangular pyramid $S A B C$, the base of which is an equilateral triangle $A B C$, and all plane angles at vertex $S$ are equal to $\alpha$. For what least $\alpha$ can we assert that this pyramid is regular?

M. Malkin

Answer: 60^\circ


10. A sphere touches all the edges of the pyramid $S A B C$, and the lateral edges $S A, S B$, and $S C$ at points $A^{\prime}, B^{\prime}$, and $C^{\prime}$. Find the volume of the pyramid $S A^{\prime} B^{\prime} C^{\prime}$, if $A B=B C=S B=5$ and $A C=4$.

Answer: \dfrac{2\sqrt{59}}{15}


Zaslavsky A.A.

Given a triangle $ABC$. On side $AB$ as the base, an isosceles triangle $ABC'$ is constructed externally with the angle at the vertex being $120^{\circ}$, and on side $AC$ an equilateral triangle $ACB'$ is constructed internally. Point $K$ is the midpoint of segment $BB'$. Find the angles of triangle $KCC'$.

Answer: 90^\circ


3. For $n$ positive numbers $x_{1}, x_{2}, \cdots, x_{n}$ whose sum equals 1, let $S$ be the largest of the following numbers: $\frac{x_{1}}{1+x_{1}}, \frac{x_{2}}{1+x_{1}+x_{2}}$, $\cdots, \frac{x_{n}}{1+x_{1}+x_{2}+\cdots+x_{n}}$. Find the minimum possible value of $S$, and determine the values of $x_{1}, x_{2}, \cdots, x_{n}$ that achieve this minimum value.
（6th All-Soviet Union Olympiad problem）

Answer: 1 - \frac{1}{\sqrt[n]{2}}


A motorcade of cars with astronauts is moving uniformly along an avenue at a speed of $v$ km/h. The length of the motorcade is constantly maintained at $m$ meters. A bouquet of flowers, thrown from a window of a house, landed in the sidecar of a motorcyclist riding behind the motorcade. The motorcyclist drove forward, passed the bouquet to the astronaut in the first car, and immediately went back. It took the motorcyclist $t$ minutes to travel there and back along the moving motorcade. Calculate the motorcyclist's speed, assuming it was constant throughout the entire journey.

Answer: \dfrac{3m + \sqrt{9m^2 + 2500t^2v^2}}{50t}


[ Arithmetic. Mental calculation, etc. ]  [ Problems on percentages and ratios ]

## Author: Klepcyn V.A.

Vanya had a certain number of cookies; he ate some, and then Tanya came to visit, and they divided the remaining cookies equally. It turned out that Vanya ate five times more cookies than Tanya. What fraction of all the cookies did Vanya eat by the time Tanya arrived?

Answer: \dfrac{2}{3}


1. The older brother noticed that in 10 years, the younger brother will be as old as he is now, and his age will be twice the current age of the younger brother. How old is the younger brother now?

Answer: 20


Example 7 (1992 National High School League Question) Let the sequence $a_{1}, a_{2}, \cdots, a_{n}, \cdots$ satisfy $a_{1}=a_{2}=1, a_{3}=2$, and for any positive integer $n$, $a_{n} a_{n+1} a_{n+2} \neq 1$. Also, $a_{n} a_{n+1} a_{n+2} a_{n+3}=a_{n}+a_{n-1}+a_{n+2}+a_{n+3}$, then the value of $a_{1}+a_{2}+\cdots+a_{100}$ is $\qquad$.

Answer: 200


Let $a_{1}, a_{2}, \ldots, a_{2018}$ be non-negative real numbers whose sum is 1. Determine the maximum possible value of

$$
S=\sum_{i \neq j, i \mid j} a_{i} a_{j}
$$

where $i \mid j$ denotes that $i$ divides $j$.

Answer: \dfrac{5}{11}


21. Let $P$ be a 30 -sided polygon inscribed in a circle. Find the number of triangles whose vertices are the vertices of $P$ such that any two vertices of each triangle are separated by at least three other vertices of $P$.

Answer: 1900


For example, (2003 Japan Mathematical Olympiad) $P$ is a point inside $\triangle A B C$, the line $A C$ and $B P$ intersect at $Q$, the line $A B$ and $C P$ intersect at $R$. Given that $A R=R B=C P, O Q=P Q$. Try to find the size of $\angle B R C$.

---

The translation is provided as requested, maintaining the original text's line breaks and format.

Answer: 120^\circ


27.11. (Belgium, 77). Three shooters $A, B, C$ decided to duel simultaneously. They positioned themselves at the vertices of an equilateral triangle and agreed on the following: the first shot is taken by $A$, the second by $B$, the third by $C$, and so on in a circle; if one of the shooters is eliminated, the duel continues between the two remaining. It is known that shooter $A$ hits the target with a probability of 0.3, shooter $C$ with a probability of 0.5, and shooter $B$ never misses. Each shooter aims at one of the two others or at the air in such a way as to maximize their probability of winning the duel. Where should shooter $A$ direct his first shot: 1) at shooter $C$; 2) at shooter $B$; 3) at the air?

Answer: 3


Let $A B C$ be an equilateral triangle with side length 1. Points $A_{1}$ and $A_{2}$ are chosen on side $B C$, points $B_{1}$ and $B_{2}$ are chosen on side $C A$, and points $C_{1}$ and $C_{2}$ are chosen on side $A B$ such that $B A_{1}<B A_{2}, C B_{1}<C B_{2}$, and $A C_{1}<A C_{2}$. Suppose that the three line segments $B_{1} C_{2}, C_{1} A_{2}$, and $A_{1} B_{2}$ are concurrent, and the perimeters of triangles $A B_{2} C_{1}, B C_{2} A_{1}$, and $C A_{2} B_{1}$ are all equal. Find all possible values of this common perimeter.

Answer: 1


In an acute triangle $ABC$, the segment $CD$ is an altitude and $H$ is the orthocentre. Given that the circumcentre of the triangle lies on the line containing the bisector of the angle $DHB$, determine all possible values of $\angle CAB$.

Answer: 60^\circ


10.3. Vasya has $n$ candies of several types, where $n \geqslant 145$. It is known that if any group of at least 145 candies is chosen from these $n$ candies (in particular, the group can consist of all $n$ candies), then there exists a type of candy such that the chosen group contains exactly 10 candies of this type. Find the largest possible value of $n$.

(A. Antropov)

Answer: 160


The positive integers $a, b, c$ are such that
$$gcd \,\,\, (a, b, c) = 1,$$
$$gcd \,\,\,(a, b + c) > 1,$$
$$gcd \,\,\,(b, c + a) > 1,$$
$$gcd \,\,\,(c, a + b) > 1.$$
Determine the smallest possible value of $a + b + c$.

Clarification: gcd stands for greatest common divisor.

Answer: 30


16. Lego Explorer Guguji arrives at a medieval kingdom, which is ruled by an evil wizard. With the help of Guguji and his friends, the prince defeats the wizard and returns to his castle. The castle is built with many cubes with an edge length of 1 meter, arranged from top to bottom in a pattern as shown in the figure. The castle has a total of 25 layers. What is the surface area of the castle in square meters (excluding the bottom base area)?

Answer: 4901


4. Let M be a set of six distinct positive integers whose sum is 60. We will write all of them on the faces of a cube, with exactly one on each face. In one step, we choose any three faces of the cube that share a common vertex and increase each of the numbers on these three faces by 1. Determine the number of all such sets M whose numbers can be written on the faces of the cube in such a way that after a finite number of appropriate steps, the numbers on all faces will be the same.

The regional round of category A takes place

## on Tuesday, January 18, 2011

so that it starts in the morning and the contestants have 4 hours of clean time to solve the problems. For each problem, a contestant can earn 6 points, and a successful solver is a student who scores 10 points or more. Allowed aids are writing and drawing tools, school MF tables, and calculators without a graphical display. These details will be communicated to the students before the start of the competition.

Answer: 84


## [ Coordinate method in space ] Vectors (other).

Find the angle between the line passing through points $A(-3 ; 0 ; 1)$ and $B(2 ; 1 ;-1)$, and the line passing through points $C(-2 ; 2 ; 0)$ and $D(1 ; 3 ; 2)$.

Answer: \arccos\left( \dfrac{2\sqrt{105}}{35} \right)


[The Pigeonhole Principle (continued).]

In a photo studio, 20 birds flew in - 8 sparrows, 7 wagtails, and 5 woodpeckers. Each time the photographer clicks the camera shutter, one of the birds flies away (permanently). How many shots can the photographer take to be sure: he will have at least four birds of one species left, and at least three of another?

#

Answer: 7


327. Find $\left(\cos ^{3} x\right)^{\prime}$.

Answer: -3 \sin x \cos^2 x


[ Tangent circles [ Opposite the larger side lies the larger angle ]

Two circles of radii $r$ and $p (r < p)$ touch each other externally and both touch internally a circle of radius $R$. It is known that the triangle with vertices at the centers of the circles is isosceles, and the angle between the lateral sides is greater than $\frac{\pi}{3}$. Find the length of the base of this triangle.

Answer: R - r


9. (3 points) Using red, yellow, blue, black, white, and green to paint each face of a cube (each face is painted with only one color), now four small cubes with exactly the same painting method are combined to form a rectangular prism, as shown in the figure. Try to answer: The opposite face of the red face on each small cube is $\qquad$ color, the opposite face of the yellow face is

color, and the opposite face of the black face is $\qquad$ color.

Answer: white


8. (10 points) A certain exam consists of 7 questions, each of which only concerns the answers to these 7 questions, and the answers can only be one of $1, 2, 3, 4$. It is known that the questions are as follows:
(1) How many questions have the answer 4?
(2) How many questions do not have the answer 2 or 3?
(3) What is the average of the answers to questions (5) and (6)?
(4) What is the difference between the answers to questions (1) and (2)?
(5) What is the sum of the answers to questions (1) and (7)?
(6) Which question is the first one with the answer 2?
(7) How many answers are unique to only one question?
Then, the total sum of the answers to the 7 questions is $\qquad$

Answer: 16


11. (This question is worth 20 points) Let the ellipse $C_{1}: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$ have left and right foci $F_{1}$ and $F_{2}$, respectively, and the right vertex be $A$. Let $P$ be any point on the ellipse $C_{1}$, and the maximum value of $\overrightarrow{P F_{1}} \cdot \overrightarrow{P F_{2}}$ is in the range $\left[c^{2}, 3 c^{2}\right]$, where $c=\sqrt{a^{2}-b^{2}}$.
(1) Find the range of the eccentricity $e$ of the ellipse $C_{1}$:
(2) Let the hyperbola $C_{2}$ have the foci of the ellipse $C_{1}$ as its vertices and the vertices of the ellipse $C_{1}$ as its foci. Let $B$ be any point on the hyperbola $C_{2}$ in the first quadrant. When $e$ takes its minimum value, does there exist a constant $\lambda(\lambda>0)$ such that $\angle B A F_{1}=\lambda \angle B F_{1} A$ always holds? If it exists, find the value of $\lambda$; if not, explain the reason.

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


7. Let non-zero distinct complex numbers $x, y$ satisfy $x^{2}+x y+$ $y^{2}=0$. Then the value of the expression
$$
\left[\frac{x y}{(x+y)(x-y)^{2}}\right]^{2000}\left(x^{2006}+y^{2006}\right)
$$

is $\qquad$ .

Answer: -\dfrac{1}{3^{2000}}


Example 3 Solve the inequality $(\lg x+3)^{7}+\lg ^{7} x+\lg x^{2}+3 \geqslant 0$.

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

Answer: [10^{-\frac{3}{2}}, \infty)


3. In a $999 \times 999$ grid, some cells are white, and the others are red. Let $T$ be the number of cell groups $\left(C_{1}, C_{2}, C_{3}\right)$ such that $C_{1}$ and $C_{2}$ are in the same row, $C_{2}$ and $C_{3}$ are in the same column, and $C_{1}$ and $C_{3}$ are white, while $C_{2}$ is red. Find the maximum value of $T$.

Answer: 147556443852


5. If the function $f(x)=\frac{a+\sin x}{2+\cos x}+b \tan x$ has a sum of its maximum and minimum values equal to 4, then $a+b=$ $\qquad$ .

Answer: 3


15. Xiaohua wrote a computer program. One minute after the program runs, some bubbles first appear on the computer screen, and at each subsequent integer minute, the same number of new bubbles appear. Half a minute after the 11th appearance of bubbles, one bubble bursts. Thereafter, every minute, more bubbles burst, with the number increasing by 1 each minute (i.e., half a minute after the 12th appearance of bubbles, 2 bubbles burst...). At a certain moment, the total number of burst bubbles is exactly equal to the total number of bubbles that have appeared on the screen, meaning all bubbles have disappeared at this moment. How many bubbles were there at most on the screen at any given time during the entire process of the program running?

Answer: 1026


10.3. In a row from left to right, all natural numbers from 1 to 37 are written in such an order that each number, starting from the second to the 37th, divides the sum of all numbers to its left: the second divides the first, the third divides the sum of the first and second, and so on, the last divides the sum of the first thirty-six. The number 37 is on the first place from the left, what number is on the third place?

Answer: 2


30.6. Let $A B C$ be an isosceles triangle with $A B=A C$. Let $D$ and $E$ be the feet of the perpendiculars from $B$ and $C$ to $\overline{A C}$ and $\overline{A B}$, respectively. Suppose that $\overline{C E}$ and $\overline{B D}$ intersect at point $H$. If $E H=1$ and $A D=4$, find $D E$.

Answer: \dfrac{8\sqrt{17}}{17}


The population of a certain city on January 1, 1950, was 117,751. The average annual increase due to migration (in-migration and out-migration) is 640 people, while the natural growth rate is $0.87 \%$ per year. a) What will the population be on January 1, 1960? - b) What is the percentage of annual growth? - c) Under the same conditions, in which year will the city reach a population of 200,000?

Answer: 135065


Example 5. Find the derivative of the function $y=\arcsin \frac{1}{x}$.

Answer: -\dfrac{1}{|x| \sqrt{x^2 - 1}}


59. Solve the equation $x-y=x^{2}+x y+y^{2}$ in integers.

Answer: (0, 0)


10. (20 points) Given points $M(-1,0), N(1,0)$, the perimeter of $\triangle M N Q$ is 6, and the trajectory of the moving point $Q$ is the curve $C$. $P$ is any point on the circle $x^{2}+y^{2}=4$ (not on the x-axis), and $P A, P B$ are tangent to the curve $C$ at points $A, B$ respectively. Find the maximum value of the area of $\triangle O A B$ (where $O$ is the origin).

Answer: \dfrac{3}{2}


Let $ABC$ be a triangle with $\angle C = 90^o$ and $AC = 1$. The median $AM$ intersects the incircle at the points $P$ and $Q$, with $P$ between $A$ and $Q$, such that $AP = QM$. Find the length of $PQ$.

Answer: \sqrt{2\sqrt{5} - 4}


Answer: \dfrac{x^{64}}{(x + 1)^{64} - x^{64}}


Problem 6.3. Vitya and his mother left home at the same time and walked in opposite directions at the same speed: Vitya - to school, and his mother - to work. After 10 minutes, Vitya realized he didn't have the keys to the house, and he would return from school earlier than his mother, so he started to catch up with her, increasing his speed fivefold. How many minutes after he realized he needed to get the keys will Vitya catch up with his mother?

Answer: 5


4. Given the dihedral angle $\alpha-l-\beta$ is $60^{\circ}$, moving points $P, Q$ are in planes $\alpha, \beta$ respectively, the distance from $P$ to $\beta$ is $\sqrt{3}$, and the distance from $Q$ to $\alpha$ is $2 \sqrt{3}$, then the minimum distance between points $P, Q$ is $\qquad$ .

Answer: 2\sqrt{3}


We placed 8 balls of radius $r$ on a horizontal plane such that their centers are at the vertices of a regular octagon, and adjacent balls touch each other. What is the radius of the sphere that touches the plane and the balls?

Answer: \dfrac{(2 + \sqrt{2})}{2} r


Eve and Martti have a whole number of euros.
Martti said to Eve: ''If you give give me three euros, so I have $n$ times the money compared to you. '' Eve in turn said to Martti: ''If you give me $n$ euros then I have triple the amount of money compared to you'' . Suppose, that both claims are valid.
What values can a positive integer $n$ get?

Answer: 1


16. Two cars are driving on a highway, 100 meters apart, both traveling at 60 kilometers per hour. The highway has different speed points (the speed points are far apart). After each car passes the first speed point, their speed immediately increases to 80 kilometers per hour; after passing the second speed point, their speed immediately increases to 100 kilometers per hour; after passing the third speed point, their speed immediately increases to 120 kilometers per hour. After both cars have passed the third speed point, how far apart are the two cars in meters?

Answer: 200


In the picture, there is a cube with an edge of $2 \mathrm{~cm}$ made up of eight smaller cubes with an edge of $1 \mathrm{~cm}$. Eight faces of the smaller cubes are painted black, and the rest are white. From them, a cube can be assembled whose surface is white. In how many ways can the smaller cubes be painted? Assume that identically painted smaller cubes cannot be distinguished and can therefore be interchanged.

![](https://cdn.mathpix.com/cropped/2024_04_17_2a0500418bf8c778174dg-4.jpg?height=177&width=203&top_left_y=551&top_left_x=961)

(K. Pazourek)

Answer: 10


10. In the complex plane, take any three different roots of the equation $z^{100}-1=0$ as vertices to form a triangle. Then the number of different acute triangles is $\qquad$.

Translating the text into English while preserving the original formatting and line breaks, the result is as follows:
```
10. In the complex plane, take any three different roots of the equation z^100 - 1 = 0 as vertices to form a triangle. Then the number of different acute triangles is      .
```

Answer: 39200


14. In a $3 \times 3$ grid filled with the numbers $1 \sim 9$, the largest number in each row is colored red, and the smallest number in each row is colored green. Let $M$ be the smallest number among the three red squares, and $m$ be the largest number among the three green squares. Then $M-m$ can have $\qquad$ different values.

Answer: 8


Nikolai with his son and Peter with his son went fishing. Nikolai caught as many fish as his son, and Peter caught three times as many as his son. In total, 25 fish were caught. What is the name of Peter's son?

#

Answer: Nikolai


4. A table of dimensions $8 \times 8$ is divided into 64 unit squares. We place dominoes in the shape of a + (see image) on the table such that their interiors do not overlap, the edges of the dominoes overlap with the edges of the corresponding unit squares, and the dominoes themselves lie entirely on the table. What is the maximum number of dominoes that can be placed on the table in this way?

![](https://cdn.mathpix.com/cropped/2024_06_07_44798e3c82dc47fe22e9g-4.jpg?height=168&width=165&top_left_y=2052&top_left_x=1645)

Solve the problems independently. Time for solving: $3 \frac{1}{2}$ hours.

The use of notes, literature, or a pocket calculator is not allowed.

45th Mathematical Competition for High School Students in Slovenia

Idrija, May 12, 2001

## PROBLEMS FOR THE FOURTH YEAR

Answer: 8


Answer: 0


On the hypotenuse $A B$ of the right triangle $A B C$, a square $A B D E$ is constructed outward. It is known that $A C=1, \quad B C=3$.

In what ratio does the angle bisector of angle $C$ divide the side $D E$?

Answer: \dfrac{1}{3}


2. In how many ways can the fraction $\frac{2020}{2021}$ be written as a product of two fractions of the form $\frac{n}{n+1}, n \in \mathbb{N}$.

Answer: 48


G1.2 In figure $1, A B$ is parallel to $D C, \angle A C B$ is a right angle, $A C=C B$ and $A B=B D$. If $\angle C B D=b^{\circ}$, find the value of $b$.

Answer: 15


8. In a commercial football tournament, five teams participated. Each was supposed to play exactly one match against each other. Due to financial difficulties, the organizers canceled some games. In the end, it turned out that all teams had scored a different number of points, and no team had a zero in the points scored column. What is the minimum number of games that could have been played in the tournament, if three points were awarded for a win, one for a draw, and zero for a loss?

Answer: 6


Example 4. Solve the equation $(x+y+1) d x+(2 x+2 y-1) d y=0$.

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

Answer: x + 2y + 3 \ln|x + y - 2| = C


6.50 In an exam, there are 4 multiple-choice questions, each with 3 options. A group of students take the exam, and the result is that for any 3 people, there is 1 question where their answers are all different. How many students can participate in the exam at most?

Answer: 9


Two capacitors with capacitance values $C_{1}=2000 \pm 10 \mathrm{pF}$ and $C_{2}=3000 \pm 15 \mathrm{pF}$ are connected in series. The voltage applied across this combination is $V=5.00 \pm 0.02 \mathrm{~V}$. What is the percentage error in the calculation of the energy stored in this combination of capacitors?
Please give the answer directly without any intermediate steps.

Answer: 1.3


2. There are ten cards, each with one of the ten digits $0-9$. They are arranged in a row to display the number 9072543681. If swapping the positions of two adjacent cards is considered one operation, then the minimum number of operations required to change the original number into one that is divisible by 99 is $\qquad$.

Answer: 2


4. The numbers $1,2,3,4$ and 5 are used once each in some order substituting for the letters in the series of powers $\left.M^{\left(A^{\left(T^{S}\right)}\right)}\right)$. In how many of the arrangements is the units digit of the value of this expression equal to 1 ?

Answer: 34


12.113. The base of the pyramid $ABCD$ is a right triangle $ABC\left(\angle C=90^{\circ}\right)$. The lateral edge $AD$ is perpendicular to the base. Find the acute angles of triangle $ABC$, if $\angle DBA=\alpha$ and $\angle DBC=\beta(\alpha<\beta)$.

Answer: \arcsin\left(\frac{\cos \beta}{\cos \alpha}\right)


A $5 \mathrm{~cm}$ by $5 \mathrm{~cm}$ pegboard and a $10 \mathrm{~cm}$ by $10 \mathrm{~cm}$ pegboard each have holes at the intersection of invisible horizontal and vertical lines that occur in $1 \mathrm{~cm}$ intervals from each edge. Pegs are placed into the holes on the two main diagonals of both pegboards. The $5 \mathrm{~cm}$ by $5 \mathrm{~cm}$ pegboard is shown; it has 16 holes. The 8 shaded holes have pegs, and the 8 unshaded holes do not. How many empty holes does the $10 \mathrm{~cm}$ by $10 \mathrm{~cm}$ pegboard have?

![](https://cdn.mathpix.com/cropped/2024_04_20_6ed09463f225f8ba1f07g-011.jpg?height=355&width=357&top_left_y=581&top_left_x=1315)

Answer: 64


20. Let the sequence $\left\{a_{n}\right\}$ be defined as
$$
a_{1}=a, a_{n+1}=1+\frac{1}{a_{1}+a_{2}+\cdots+a_{n}-1}(n \geqslant 1)
$$

Find all real numbers $a$ such that $0<a_{n}<1(n \geqslant 2)$.

Answer: (-\infty, 0)


II. (25 points) As shown in Figure 3, $P Q$ is tangent to $\odot O$ at point $Q$, and $Q$ is the midpoint of $\widehat{A B}$. Chord $A C = B C$, the extension of $C A$ intersects $P Q$ at point $D$, $O E \perp B C$ at point $E$, connect $D E$, and $F$ is the midpoint of $D E$. Given that $Q F = 2$, $A C = 6$. Find the length of $O E$.

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


8. What is the least number of odd numbers that can be chosen from the set of all odd numbers lying between 16 and 2016, so that no one of the chosen numbers is divisible by any other?

Answer: 672


$ABC$ is an isosceles triangle where $E$ is the midpoint of leg $AB$, and $F$ is the midpoint of the base $BC$. Points $H$ and $G$ are symmetric to point $F$ on the base $BC$. Point $K$ is the intersection of line $EH$ and the angle bisector of $\angle EAH$, and point $L$ is the intersection of line $EG$ and the angle bisector of $\angle EAG$. Prove that $KL \| BC$. What is the measure of $\angle KAL$ in terms of $\angle HAG$?

Answer: \frac{1}{2} \angle HAG


Four. (20 points) Given the ellipse $\frac{x^{2}}{4}+\frac{y^{2}}{3}=1$ with an inscribed $\triangle A B C$ where sides $A B$ and $A C$ pass through the left and right foci $F_{1}$ and $F_{2}$ respectively, and the left and right vertices of the ellipse are $D$ and $E$ respectively. The lines $D B$ and $C E$ intersect at point $P$. When point $A$ varies on the ellipse, find the locus of point $P$.

Answer: \dfrac{x^2}{4} + \dfrac{y^2}{27} = 1


In the city of Ikozapolisz, the road network graph matches the graph of the edge network of an icosahedron. Jorgosz's lodging is located at one vertex of the icosahedron, while his favorite theater is at the opposite vertex. After dark, when leaving the theater and heading home, he becomes uncertain at each vertex about which direction to take next. Suppose that at each vertex, the probability $p$ is that he encounters someone who points him in the direction that will take him to his lodging with the fewest edges. Otherwise, he continues in a random direction, with no direction being favored, meaning it is even possible for him to turn back. For what value of $p$ will there be a $50 \%$ probability that he reaches his lodging before returning to the theater?

Answer: \dfrac{5\sqrt{2} - 6}{14}


4. $(5-7,8)$ The phone PIN code consists of 4 digits (and can start with zero, for example, 0951). Petya calls "lucky" those PIN codes where the sum of the outer digits equals the sum of the middle digits, for example $1357: 1+7=3+5$. In his phone, he uses only "lucky" PIN codes. Petya says that even if he forgets one digit (but remembers its position), he can easily restore it. And if he forgets two digits (but remembers their positions), he will have to try only a small number of PIN codes.

a) How many PIN codes will Petya have to try in the worst case?

b) How many "lucky" PIN codes exist in total?

Answer: 10


A subtraction pyramid is a pyramid formed by non-negative integers, each of which is the difference between the two nearest numbers from the previous level (read from bottom to top). Here is an example of a subtraction pyramid:

## 1

2

2

5 7
4

3
5

8

A significant number is the largest number in the subtraction pyramid. An excellent pyramid is a subtraction pyramid that has a 0 at the top and at least one level consisting of distinct numbers.

1. How many levels must an excellent pyramid have at a minimum?
2. What is the smallest significant number that can be contained in an excellent pyramid with the minimum number of levels?

(K. Jasenčáková)

Hint. What does the level below the apex look like?

Answer: 3


Example 8 (2004 National Girls' Mathematical Olympiad) Let $a, b, c$ be positive real numbers. Find the minimum value of $\frac{a+3 c}{a+2 b+c}+\frac{4 b}{a+b+2 c}-$ $\frac{8 c}{a+b+3 c}$.

Answer: 12\sqrt{2} - 17


Given an ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) where \(a > b\) and a line \(x = -\frac{a^2}{c}\), where \(c = \sqrt{a^2 - b^2}\). On the ellipse, place \(n\) points \( (n > 1) \) such that the angle between lines connecting each pair of adjacent points with the left focus is equal: \(\angle P_{1} F P_{2} = \angle P_{2} F P_{3} = \cdots = \angle P_{n} F P_{1} = \frac{2\pi}{n}\). Prove: The sum of the reciprocals of the distances from these \(n\) points to the line \(x = -\frac{a^2}{c}\) is a constant related to \(n\).

Answer: \frac{n c}{b^2}


1. Let $f$ be the function such that
$$
f(x)=\left\{\begin{array}{ll}
2 x & \text { if } x \leq \frac{1}{2} \\
2-2 x & \text { if } x>\frac{1}{2}
\end{array}\right.
$$

What is the total length of the graph of $\underbrace{f(f(\ldots f}_{2012 f^{\prime} s}(x) \ldots))$ from $x=0$ to $x=1$ ?

Answer: \sqrt{4^{2012} + 1}


10.6. Circles $\omega_{1}$ and $\omega_{2}$ intersect at points $A$ and $B$. Let $PQ$ and $RS$ be the external common tangents of these two circles (points $P$ and $R$ are on circle $\omega_{1}$, points $Q$ and $S$ are on circle $\omega_{2}$). Given that $RB \parallel PQ$, the ray $RB$ intersects circle $\omega_{2}$ again at point $W$. Find the ratio $\frac{RB}{BW}$.

Answer: \dfrac{1}{3}


6. In a square $ABCD$ with side length 1, points $M$, $N$, $O$, $P$ are on sides $AB$, $BC$, $CD$, $DA$ respectively. If $AM=BM$, $DP=3AP$, then the minimum value of $MN+NO+OP$ is $\qquad$.

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

Answer: \dfrac{\sqrt{85}}{4}


# Problem 8. (5 points)

For each pair of numbers $\overline{a b b}$ and $\overline{a b b}$, where $a$ and $b$ are different digits, the GCD of these numbers was calculated. Find the greatest of these GCDs.

$\overline{a b b}$ is the standard notation for a number consisting of the digits $a, b$, and $b$ in that exact order.

Answer: 45

Answer: 45


Four consecutive three-digit numbers are divided respectively by four consecutive two-digit numbers. What minimum number of different remainders can be obtained?

[i](A. Golovanov)[/i]

Answer: 1


3. In Rt $\triangle A B C$, $\angle B=90^{\circ}$, $P$ is a point on the angle bisector of $\angle A$ inside $\triangle A B C$, $M$ (different from $A, B$) is a point on side $A B$, and lines $A P, C P, M P$ intersect side $B C, A B, A C$ at points $D, E, N$ respectively. If $\angle M P B = \angle P C N, \angle N P C = \angle M B P$, find $\frac{S_{\triangle A P C}}{S_{A C D E}}$.

Answer: \dfrac{1}{2}


\left.\begin{array}{l}\text { The Law of Sines } \\ \text { [Midline of a Triangle] }\end{array}\right]

Two chords, equal to 10 and 12, are drawn from one point on a circle. Find the radius of the circle if the distance from the midpoint of the smaller chord to the larger one is 4.

Answer: \dfrac{25}{4}


Circles $\mathcal{P}$ and $\mathcal{Q}$ have radii $1$ and $4$ , respectively, and are externally tangent at point $A$ . Point $B$ is on $\mathcal{P}$ and point $C$ is on $\mathcal{Q}$ so that line $BC$ is a common external tangent of the two circles. A line $\ell$ through $A$ intersects $\mathcal{P}$ again at $D$ and intersects $\mathcal{Q}$ again at $E$ . Points $B$ and $C$ lie on the same side of $\ell$ , and the areas of $\triangle DBA$ and $\triangle ACE$ are equal. This common area is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m+n$
[asy] import cse5; pathpen=black; pointpen=black; size(6cm);  pair E = IP(L((-.2476,1.9689),(0.8,1.6),-3,5.5),CR((4,4),4)), D = (-.2476,1.9689);  filldraw(D--(0.8,1.6)--(0,0)--cycle,gray(0.7)); filldraw(E--(0.8,1.6)--(4,0)--cycle,gray(0.7)); D(CR((0,1),1)); D(CR((4,4),4,150,390)); D(L(MP("D",D(D),N),MP("A",D((0.8,1.6)),NE),1,5.5)); D((-1.2,0)--MP("B",D((0,0)),S)--MP("C",D((4,0)),S)--(8,0)); D(MP("E",E,N)); [/asy]

Answer: 129


B3. Find all real solutions of the equation $\left(x^{2}+x+3\right)^{2}+3\left(x^{2}+x-1\right)=28$.

Answer: -2


Nathan has discovered a new way to construct chocolate bars, but it’s expensive! He starts with a single $1\times1$ square of chocolate and then adds more rows and columns from there. If his current bar has dimensions $w\times h$ ($w$ columns and $h$ rows), then it costs $w^2$ dollars to add another row and $h^2$ dollars to add another column. What is the minimum cost to get his chocolate bar to size $20\times20$?

Answer: 5339


Let $ S$ be the set of natural numbers $ n$ satisfying the following conditions:

$ (i)$ $ n$ has $ 1000$ digits,
$ (ii)$ all the digits of $ n$ are odd, and
$ (iii)$ any two adjacent digits of $ n$ differ by $ 2$.

Determine the number of elements of $ S$.

Answer: 8 \cdot 3^{499}


A bus leaves from city $A$ to city $B$, which is 240 km away from $A$, at a speed of 40 km/h. At the same time, a car leaves from $B$ heading towards $A$ at a speed of $v$ km/h. Half an hour after meeting the bus, the car turns back towards $B$ without reaching city $A$ and continues moving towards $B$ at the same speed. Determine all values of $v$ for which the car arrives in city $B$ before the bus.

Answer: (56, 120)


. We color the non-zero rational numbers in two colors: white and black. We assume that 1 is colored white, that $x$ and $x+1$ are never colored the same color, and that $x$ and $\frac{1}{x}$ always have the same color. What is the color of $\frac{1543}{275}$?

Answer: white


15. Given that $m, n$ are integers, $m, n \in\{1,2, \cdots, 1981\}$, and $\left(n^{2}-m n-\right.$ $\left.m^{2}\right)^{2}=1$, determine the maximum value of $m^{2}+n^{2}$.

Answer: 3524578


[ Processes and operations  [ Examples and counterexamples. Constructions ]

Several boxes together weigh 10 tons, and each of them weighs no more than one ton. How many three-ton trucks are definitely enough to haul this cargo?

#

Answer: 5


## Task A-1.1.

Marko drew a rectangle with two blue sides of length 24 and two red sides of length 36. He colored each point inside the rectangle with the color of the nearest side. Points that are equally distant from a blue and a red side were colored black. Determine the area of the red part of the rectangle.

Answer: 576


In the following figure-not drawn to scale!- $E$ is the midpoint of $B C$, triangle $F E C$ has area 7, and quadrilateral $D B E G$ has area 27. Triangles $A D G$ and $G E F$ have the same area, $x$. Find $x$.

![](https://cdn.mathpix.com/cropped/2024_04_17_6cb05c45fe7c141f3316g-4.jpg?height=661&width=700&top_left_y=1209&top_left_x=688)

Answer: 8


The distance between Coco da Selva and Quixajuba is $24 \mathrm{~km}$. Two friends need to go from Quixajuba to Coco da Selva and a third friend needs to go from Coco da Selva to Quixajuba. They have a bicycle that is initially in Quixajuba. Each of them can walk at a speed of $6 \mathrm{~km} / \mathrm{h}$, or ride the bicycle at a speed of $18 \mathrm{~km} / \mathrm{h}$. Additionally, they can leave the bicycle at any point along the route.

![](https://cdn.mathpix.com/cropped/2024_05_01_9f6ce5ffa52bbf7500dfg-2.jpg?height=177&width=1012&top_left_y=2210&top_left_x=702)

Show how they can proceed to reach their destinations in no more than $2 \mathrm{~h} 40 \mathrm{~min}$.

Answer: 2 \text{ h } 40 \text{ min}


9. Given the sequence $\left\{a_{n}\right\}$ satisfies $a_{1}=3$, and for any positive integers $m, n$, it holds that $a_{m+n}=a_{m}+a_{n}+2 m n$.
(1) Find the general term formula for the sequence $\left\{a_{n}\right\}$;
(2) If a real number $c$ makes $\sum_{i=1}^{k} \frac{1}{a_{i}}<c$ hold for all positive integers $k$, find the range of values for $c$.

Answer: [\dfrac{3}{4}, +\infty)


# Task 2.

B-1.

At the time when a lion cub, who was 5 minutes away, set off for the watering hole, the second cub, having already quenched its thirst, started heading back along the same road at 1.5 times the speed of the first. At the same time, a tortoise, which was half an hour away, set off for the watering hole along the same road. After some time, all three met at one point and then continued on their way. How many minutes after the meeting did the tortoise reach the watering hole, given that all three moved at constant speeds?

Answer: 28


We write integers into an $n \times k$ table in such a way that every integer from 1 to $k$ appears in each of the $n$ rows. Let $S$ denote the largest of the $k$ column sums. For every $n$ and $k$, determine the smallest possible value of $S$!

Answer: \left\lceil \frac{n(k+1)}{2} \right\rceil


We define a sequence with $a_{1}=850$ and

$$
a_{n+1}=\frac{a_{n}^{2}}{a_{n}-1}
$$

for $n \geq 1$. Determine all values of $n$ for which $\left\lfloor a_{n}\right\rfloor=2024$.
Here, the floor $\lfloor a\rfloor$ of a real number $a$ denotes the greatest integer less than or equal to $a$.

Answer: 1175


4. In triangle $A B C$, medians $B D$ and $C E$ intersect at point $K$. The circle constructed on segment $C K$ as a diameter passes through vertex $B$ and is tangent to line $D E$. It is known that $D K=3$. Find the height $A H$ of triangle $A B C$, the radius of the circle, and the area of triangle $A B C$.

Answer: 18


## Task 2 - 250522

From the Mathcity train station, a bus departs at every full quarter hour and arrives in Knobelhausen after 2 hours.

From there, buses also depart every quarter hour on the same road to Mathcity, where they arrive after 2 hours of travel.

In the morning, the first bus from Mathcity departs at 5:00 AM and the first bus from Knobelhausen departs at 7:10 AM. The bus drivers nod to each other every time they meet on the road.

How many oncoming colleagues does bus driver Franz Freundlich greet on a trip from Mathcity to Knobelhausen, if this trip starts at 10:00 AM?

Answer: 16


Example 3 There are ten birds on the ground pecking at food, and among any five birds, at least four birds are on the same circumference. Question: What is the minimum number of birds on the circumference that contains the most birds?

Answer: 9


Let \( f(n) \in \mathbb{N} \) be the smallest number such that the sum \( \sum_{k=1}^{f(n)} k \) is divisible by \( n \). Prove that \( f(n) = 2n - 1 \) if and only if \( n = 2^m \) where \( m \in \mathbb{Z}^+ \).

Answer: f(n) = 2n - 1 \text{ if and only if } n = 2^m \text{ for some } m \in \mathbb{Z}^+


Let $f(x)$ be an odd function on $\mathbb{R}$, such that $f(x)=x^2$ when $x\ge 0$. Knowing that for all $x\in [a,a+2]$, the inequality $f(x+a)\ge 2f(x)$ holds, find the range of real number $a$.

Answer: [\sqrt{2}, \infty)


Let there be \( n \) lamps. The number of ways to light up \( k \) lamps is denoted by \( C_{n}^{k} \). Prove that
\[ C_{n}^{0} + C_{n}^{1} + \ldots + C_{n}^{k} + \ldots + C_{n}^{n} = 2^{n}. \]
Is the notation \( C_{n}^{k} \) clear? What does \( C_{5}^{1} \) mean? What is the value of \( C_{5}^{4} \)?

Answer: 2^n


Answer: 3


Prince Zrychlený invited Princess Zpomalená to his castle. When she took too long, he set out to meet her. After two days of travel, he met her at one fifth of her journey. Together, they continued their journey twice as fast as when the princess was traveling alone. They arrived at Prince's castle the second Saturday after their meeting. Which day did they meet, if the princess set out from her castle on Friday?

(M. Dillingerová)

Answer: Wednesday


11. (20 points) Given that $O$ is the origin, points $A$ and $B$ lie on the ellipse $\frac{x^{2}}{4}+\frac{y^{2}}{3}=1$, and $|A B|=\frac{3}{2}$. The tangents to the ellipse at points $A$ and $B$ intersect at point $C$. Find the range of the area of quadrilateral $O A C B$.

Answer: \left[ \dfrac{6\sqrt{165}}{55}, \dfrac{6\sqrt{13}}{13} \right]


Problem 7.6. Natural numbers 13, 14, 15, 16, 17, 18, 19, 20 are placed in the circles on the diagram (each number in one circle) such that the sums of the three numbers along each line are equal. What number can be in the circle $X$? List all possible options.

![](https://cdn.mathpix.com/cropped/2024_05_06_4d36ea080cd7b65cb574g-5.jpg?height=287&width=422&top_left_y=1447&top_left_x=516)

Answer: 15


Problem 4.7. Denis threw darts at four identical dartboards: he threw exactly three darts at each board, where they landed is shown in the figure. On the first board, he scored 30 points, on the second - 38 points, on the third - 41 points. How many points did he score on the fourth board? (For hitting each specific zone - ring or central field - a certain number of points is awarded.)
![](https://cdn.mathpix.com/cropped/2024_05_06_c49ecf27446af41a8bcdg-06.jpg?height=260&width=1136&top_left_y=960&top_left_x=156)

Answer: 34


5.3. In a seven-story building, domovoi (Russian house spirits) live. The elevator travels between the first and the last floors, stopping at every floor. On each floor, starting from the first, one domovoi entered the elevator, but no one exited. When the thousandth domovoi entered the elevator, it stopped. On which floor did this happen? Explain your answer.

Answer: 4


The rational numbers \( a, -b, c \) are positioned on the number line as shown in the diagram. Determine the largest value among \( -\frac{1}{a b}, \frac{1}{b}, |a c|, \frac{1}{b^{2}}, \frac{1}{a^{2}} \).

Answer: \frac{1}{a^2}


39th IMO 1998 shortlist Problem N8 The sequence 0 ≤ a 0 < a 1 < a 2 < ... is such that every non-negative integer can be uniquely expressed as a i + 2a j + 4a k (where i, j, k are not necessarily distinct). Find a 1998 . Solution

Answer: 1227096648


For a nonempty set $\, S \,$ of integers, let $\, \sigma(S) \,$ be the sum of the elements of $\, S$.  Suppose that $\, A = \{a_1, a_2, \ldots, a_{11} \} \,$ is a set of positive integers with $\, a_1 < a_2 < \cdots < a_{11} \,$ and that, for each positive integer $\, n\leq 1500, \,$ there is a subset  $\, S \,$ of $\, A \,$ for which $\, \sigma(S) = n$.  What is the smallest possible value of $\, a_{10}$?

Answer: 248


Problem 2. Pentagon $A B C D E$ is inscribed in circle $\omega$. Diagonal $A C$ is the diameter of circle $\omega$. Find $\angle B E C$, if $\angle A D B=20^{\circ}$.

Answer: 70^\circ


Fedya wrote down all the natural divisors of the number \( n \) that lie in the interval \(\left[2^{100}, 3^{100}\right]\). Can it turn out that he wrote exactly 100 numbers, and exactly 35 of them are cubes of natural numbers?

Answer: \text{No}


Problem 3. Find the smallest positive integer $n$, such that there exist $n$ integers $x_{1}, x_{2}, \ldots, x_{n}$ (not necessarily different), with $1 \leq x_{k} \leq n, 1 \leq k \leq n$, and such that

$$
x_{1}+x_{2}+\cdots+x_{n}=\frac{n(n+1)}{2}, \quad \text { and } \quad x_{1} x_{2} \cdots x_{n}=n!
$$

but $\left\{x_{1}, x_{2}, \ldots, x_{n}\right\} \neq\{1,2, \ldots, n\}$.

Answer: 9


甽 2. As shown in the figure, in $\triangle ABC$, it is known that $M, N$ are on sides $AC, BC$ respectively, and $BM$ intersects $AN$ at $O$. If $S_{\triangle OMA}=3 \text{~cm}^{2}, S_{\triangle OAB}$ $=2 \text{~cm}^{2}, S_{\triangle OBN}=1 \text{~cm}^{2}$.

Find $S_{\triangle CMN}$. (1990, Shanghai Junior High School Mathematics Competition)

Answer: \dfrac{45}{2}


4. Solve the equation $\sqrt{5-x}=x^{2}-5$.

---

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

Answer: \dfrac{-1 - \sqrt{21}}{2}


Let $ABC$ be a triangle where $AC\neq BC$. Let $P$ be the foot of the altitude taken from $C$ to $AB$; and let $V$ be the orthocentre, $O$ the circumcentre of $ABC$, and $D$ the point of intersection between the radius $OC$ and the side $AB$. The midpoint of $CD$ is $E$.

a) Prove that the reflection $V'$ of $V$ in $AB$ is on the circumcircle of the triangle $ABC$.

b) In what ratio does the segment $EP$ divide the segment $OV$?

Answer: 1


Task 14. (8 points)

Elena decided to get a pet - a budgerigar. She faced the question of where to buy a cage and a bath more cost-effectively.

On the official websites of the stores, product catalogs are posted. Elena studied the offers and compiled a list of stores where the items she is interested in are available:

| Item | Store |  |
| :--- | :---: | :---: |
|  | ZooWorld | ZooIdea |
| Cage | 4500 rub. | 3700 rub. |
| Bath | 510 rub. | 680 rub. |

Elena plans to use delivery. The delivery cost from the store "ZooWorld" is 500 rub., and from the store "ZooIdea" it is 400 rub. If the purchase amount exceeds 5000 rub., delivery from the store "ZooWorld" is free.

Elena has a discount card from the store "ZooIdea," which provides a 5% discount on the purchase amount.

Determine the minimum total cost for purchasing a cage and a bath for the budgerigar, including delivery costs.

In your answer, provide only the number without units of measurement!

Answer: 4561


A penny in a corner. A circle moves so that it is continually in contact with all three coordinate planes of an ordinary rectangular system. Find the locus of the center of the circle. Please provide the num of the possible condition that $x_{i}^{2} \leq r^{2}$.

Answer: 3


The flag of the United Nations consists of a polar map of the world, with the North Pole as center, extending approximately to $45^{\circ}$ South Latitude. The parallels of latitude are concentric circles with radii proportional to their co-latitudes. Australia is near the periphery of the map and is intersected by the parallel of latitude $30^{\circ} \mathrm{S}$. In the very close vicinity of this parallel, how much are East and West distances exaggerated as compared to North and South distances? The original answer is in the format $m\pi/n\sqrt{j}\sim2.42$, please provide the value of m + n + j.

Answer: 10


Let $f(x)$ be a cubic polynomial with roots $x_{1}, x_{2}$, and $x_{3}$. Assume that $f(2 x)$ is divisible by $f^{\prime}(x)$ and compute the sum of the absolute values of the ratios $x_{1}: x_{2}: x_{3}$.

Answer: 9


A horizontal disc of diameter 3 inches is rotating at 4 revolutions per minute. A light is shining at a distant point in the plane of the disc. An insect is placed at the edge of the disc furthest from the light, facing the light. It at once starts crawling, and crawls so as always to face the light, at 1 inch per second. Set up the differential equation of motion, and find at what point the insect again reaches the edge of the disc. The original answer indicates that the insect leaves the disc at a point with a coordinate in the format \frac{m}{n},0. Please provide the value of m + n.

Answer: 5


Two uniform solid spheres of equal radii are so placed that one is directly above the other. The bottom sphere is fixed, and the top sphere, initially at rest, rolls off. At what point will contact between the two spheres be "lost"? Assume the coefficient of friction is such that no slipping occurs. The original answer is in the format \theta=\arccos (m / n), please give the value of m + n.

Answer: 27


In assigning dormitory rooms, a college gives preference to pairs of students in this order: $$ A A, A B, A C, B B, B C, A D, C C, B D, C D, D D $$ in which $A A$ means two seniors, $A B$ means a senior and a junior, etc. Determine numerical values to assign to $A, B, C, D$ so that the set of numbers $A + A, A + B, A + C, B + B$, etc., corresponding to the order above will be in descending magnitude. Please provide the sum of A + B + C + D for the least positive integers that satisfy this condition.

Answer: 23


Suppose that $G$ is a group generated by elements $A$ and $B$, that is, every element of $G$ can be written as a finite "word" $A^{n_1}B^{n_2}A^{n_3}\cdots B^{n_k}$, where $n_{1},\cdots, n_k$ are any integers, and $A^0=B^0=1$ as usual. Also, suppose that $A^4=B^7= ABA^{-1}B=1$, $A^2\neq1$, and $B\neq1$. How many elements of $G$ are of the form $C^2$ with $C$ in $G$?

Answer: 8


In how many ways can the integers from 1 to n be ordered subject to the condition that, except for the first integer on the left, every integer differs by 1 from some integer to the left of it? Please provide the number of ways for $n = 6$.

Answer: 32


Let $n$ be a positive integer. Let $a,b,x$ be real numbers, with $a \neq b$. and let $M_{n}$ denote the $2n \times 2n$ matrix whose $(i,j)$ entry $m_{ij}$ is given by $$\left.m_{ij}=\left\{\begin{array}{llll}x&\text{if}\quad i=j,\\a&\text{if}\quad i\neq j&\text{and}&i+j\text{is even},\\b&\text{if}\quad i\neq j&\text{and}&i+j\text{is odd}.\end{array}\right.\right.$$ Thus, for example, $M_2=\begin{pmatrix}x&b&a&b\\b&x&b&a\\a&b&x&b\\b&a&b&x\end{pmatrix}.$ Express $\lim_{x\rightarrow a}\det M_n/(x-a)^{2n-2}$ as a polynomial in $a,b$, and $n.$ where $\det M_n$ denotes the determinant of $M_n$. Please provide the value of this expression when $n = 3$, $a = 2$, and $b = 1$. The answer should be a single integer.

Answer: 27


Evaluate$$\lim _{n \rightarrow \infty} \frac{1}{n} \sum_{k=1}^{n}\left(\left[\frac{2 n}{k}\right]-2\left[\frac{n}{k}\right]\right)$$and express your answer in the form $\ln a-b$, with $a$ and $b$ positive integers. Please provide the value of $a + b$.

Answer: 5


Express the sum \( \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{1}{m^{2} n+m n^{2}+2 m n} \) as a rational number. The original answer is in the form of \( \frac{m}{n} \). Please provide the value of \( m + n \).

Answer: 11


There are four weights and a two-pan balance scale without an indicator. How many different weights can be exactly measured with these weights, if

a) the weights can only be placed on one pan of the scale;

b) the weights can be placed on both pans of the scale.

Answer: 15


9-8. All digits in the notation of 6-digit natural numbers $a$ and $b$ are even, and in the notation of any number between them, there is an odd digit. Find the largest possible value of the difference $b-a$.

---

Translation:
9-8. All digits in the notation of 6-digit natural numbers $a$ and $b$ are even, and in the notation of any number between them, there is an odd digit. Find the largest possible value of the difference $b-a$.

Answer: 111112


[ Fundamental Theorem of Arithmetic. Prime Factorization]

Author: Blinkov A. d:

The product of five different integers is 2022. What can their sum be? If there are multiple answers, list them all.

#

Answer: -342


Three, (30 points) In $\triangle A B C$, $A B=A C$, point $M$ is on $A B$ and $M A=M C$, point $N$ is on $A C$ and $C N=C B, \angle A: \angle N B A=2: 3$. Find the degree measure of $\angle N M C$.

Answer: 30


A sequence of integers $ a_{1},a_{2},a_{3},\ldots$ is defined as follows: $ a_{1} \equal{} 1$ and for $ n\geq 1$, $ a_{n \plus{} 1}$ is the smallest integer greater than $ a_{n}$ such that $ a_{i} \plus{} a_{j}\neq 3a_{k}$ for any $ i,j$ and $ k$ in $ \{1,2,3,\ldots ,n \plus{} 1\}$, not necessarily distinct. Determine $ a_{1998}$.

Answer: 4494


There are $5$ accents in French, each applicable to only specific letters as follows:

[list]
[*] The cédille: ç
[*] The accent aigu: é
[*] The accent circonflexe: â, ê, î, ô, û
[*] The accent grave: à, è, ù
[*] The accent tréma: ë, ö, ü
[/list]

Cédric needs to write down a phrase in French. He knows that there are $3$ words in the phrase and that the letters appear in the order: \[cesontoiseaux.\] He does not remember what the words are and which letters have what accents in the phrase. If $n$ is the number of possible phrases that he could write down, then determine the number of distinct primes in the prime factorization of $n$.

Answer: 4


4. Marko and Luka met at the beginning of a 1800 m long trail. Marko is riding a bicycle, and Luka is running, and both move at constant speeds. When they reach the end of the trail, they turn around and continue in the opposite direction without stopping. In 30 minutes, Marko has traveled 9 km, and Luka 4.5 km. How far apart will they be 30 minutes after the start of the training, and at what distance from the start of the trail did they first meet?

Answer: 900


[ Case Analysis $]$  [ Proof by Contradiction ]

In the cells of a $3 \times 3$ table, numbers are arranged such that the sum of the numbers in each column and each row is zero. What is the smallest number of non-zero numbers that can be in this table, given that this number is odd?

Answer: 7


Find the number of different sequences with $2 n+1$ terms, where each term is $\pm 1$ and the absolute value of the sum of any odd number of consecutive terms does not exceed 1.

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

Answer: 2^{n+2} - 2


19. As shown in the figure, two ants, Jia and Yi, are moving on the following circle. $AC$ is the diameter of the large circle, and point $B$ is on $AC$. $AB$ and $BC$ are the diameters of the two smaller circles, respectively. Ant Jia crawls clockwise on the large circle, while Ant Yi crawls in the direction indicated by the arrows on the two smaller circles, forming a figure "8" $(A \rightarrow B \rightarrow C \rightarrow B \rightarrow A)$. Jia and Yi start from point $A$ at the same moment and continue to crawl. The speed ratio is $v_{\text {Jia }}: v_{\text {Yi}}=3: 2$. After $T_{1}$ minutes, the two ants meet. Then, Jia increases its speed by $\frac{1}{3}$, while Yi's speed remains unchanged, and they continue to crawl on their original paths. After $T_{2}$ minutes, the two ants meet again. It is known that $T_{1}+T_{2}=100^{3}-99^{3}+98^{3}-97^{3}+\cdots+2^{3}-1^{3}$, then the time it takes for Jia to crawl around the large circle at its original speed is $\qquad$ minutes (the answer should be written as an improper fraction).

Answer: \dfrac{1015000}{9}


## Task 1.

Determine the smallest natural number $n$ for which there exist positive real numbers $x, y, z$ such that

$$
n=\lfloor x y\rfloor \cdot\lfloor y z\rfloor=(x+y+z)^{2}+2
$$

Note. For a real number $t$, we denote by $\lfloor t\rfloor$ the greatest integer less than or equal to $t$.

Answer: 70


3. Barbara is at the railway station and reports to her mother: "The train arrived 10 minutes ago, and we traveled exactly four hours and twenty-two minutes!" At that moment, her mother looked at the clock, and it showed 13 h and 5 min.

a) At what time did the train Barbara arrived on depart?

b) What type of angle did the clock hands form at the time of the train's departure?

Answer: 8:33


5.79 To guard a certain target, someone needs to be on duty day and night, so there are day shifts and night shifts. Personnel can be scheduled for a single day shift or a single night shift, or they can be scheduled for a continuous day and night shift. After working a day shift, a night shift, or a day and night shift, the personnel should rest for no less than 1 day and night, 1.5 day and night, and 2.5 day and night, respectively. If each shift requires 1 person, how many people are needed at a minimum to participate in the shifts?

Answer: 4


3. Find the number of positive integer solutions to the equation $x_{1}+x_{2}+x_{3}+x_{4}=23$, with the conditions $x_{1} \leqslant 9, x_{2} \leqslant 8, x_{3} \leqslant 7$, $x_{4} \leqslant 6$.

Answer: 115


6. On an island, there live chameleons of five colors. When one chameleon bites another, the color of the bitten chameleon changes according to some rule, and the new color depends only on the color of the biter and the color of the bitten. It is known that 2023 red chameleons can agree on a sequence of bites, after which all of them will turn blue. For what smallest $k$ can it be guaranteed that $k$ red chameleons can agree to turn blue?

For example, the rules might be: if a red chameleon bites a green one, the bitten one changes color to blue; if a green chameleon bites a red one, the bitten one remains red, i.e., "changes color to red"; if a red chameleon bites a red one, the bitten one changes color to yellow, and so on. (The specific rules for color changes may be different.)

(2022-60, M. Raskin)

Answer: 5


5. A line in which all natural numbers from 1 to $n$ are written exactly once will be called evident if for any $k$ one of the following conditions is met:

(a) $k$ is written first;

(b) among the numbers written to the left of $k$, there is either $(k-1)$ or $(k+1)$.

Find the number of all evident strings of length $n$.

Answer: 2^{n-1}


## Task 2 - 270912

In a domino game with the numbers $0,1, \ldots, 6$, each tile is divided into two halves, each half bearing one of the numbers. In a domino set, all combinations of two of the numbers $0,1, \ldots, 6$ appear exactly once (including those where the same number appears on both halves of a tile).

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

A chain is called "closed" if the two tile halves at the free ends of the chain also bear the same number (so that the chain, if it consists of enough tiles, can be led back to its beginning and closed there).

a) Determine the number of all tiles belonging to a domino set!

b) Determine the largest number of such tiles from a domino set that can form a closed chain!

Answer: 28


15. Red and Black (from 8th grade, 3 points). The dealer ${ }^{2}$ deals one card at a time to the player from a well-shuffled standard deck of cards. At any moment, while the dealer still has cards left, the player can say "Stop". After this, the dealer reveals one more card. If it is of a red suit, the player wins; if it is black, the player loses.

Does there exist a strategy for the player that makes the probability of winning greater than $0.5 ?$[^6]

Answer: No


20. (12 points) As shown in Figure 5, let $D$ be a point inside the acute triangle $\triangle A B C$ such that
$$
\begin{array}{l}
\angle A D B \\
=\angle A C B+90^{\circ},
\end{array}
$$

and $A C \cdot B D=A D \cdot B C$.
Find the value of $\frac{A B \cdot C D}{A C \cdot B D}$.

Answer: \sqrt{2}


In order to pass $B$ going $40$ mph on a two-lane highway, $A$, going $50$ mph, must gain $30$ feet. Meanwhile, $C$, who is $210$ feet from $A$, is headed toward him at $50$ mph. If $B$ and $C$ maintain their speeds, then, in order to pass safely, $A$ must increase his speed by how many mph?

Answer: 5


Asli will distribute $100$ candies among her brother and $18$ friends of his. Asli splits the friends of her brother into several groups and distributes all the candies into these groups. In each group, the candies are shared in a fair way such that each child in a group takes the same number of candies, and this number is the largest possible. Then, Asli's brother takes the remaining candies of each group. At most, how many candies can Asli's brother have?

Answer: 16


In Figure 1, if the sum of the interior angles is \(a^{\circ}\), find \(a\).

If the \(n^{\text{th}}\) term of the arithmetic progression \(80, 130, 180, 230, 280, \ldots\) is \(a\), find \(n\).

In Figure 2, \(AP: PB = 2: 1\).
If \(AC = 33 \text{ cm}\), \(BD = n \text{ cm}\), \(PQ = x \text{ cm}\), find \(x\).

If \(K = \frac{\sin 65^{\circ} \tan ^{2} 60^{\circ}}{\tan 30^{\circ} \cos 30^{\circ} \cos x^{\circ}}\), find \(K\).

Answer: 6


Volodya and Dima decided to start their own business producing still mineral water called "Dream" in July. They invested 1,500,000 rubles and purchased equipment for 500,000 rubles. The technical passport for this equipment indicates that the maximum production capacity is 100,000 bottles.

At the end of July, Dima and Volodya decided to launch a trial production batch and obtained 200 bottles, of which 5 were incomplete. In August, the equipment operated at full capacity and produced 15,000 bottles of water. In September, a breakdown occurred in the equipment, causing it to be idle for several days, but 12,300 bottles of water were produced over 20 days of that month.

In October, they decided to stop producing water, as the winter season would lead to lower demand, and decided to sell the equipment.

a) Determine the total depreciation of the equipment.

b) Determine the residual value of the equipment at the time of sale.

c) For what amount should the equipment be sold to achieve a profit of 10,000 rubles? (20 points)

Answer: 372,500\,


Let \( S = \{1, 2, \ldots, 50\} \). Determine the smallest natural number \( K \) such that in any subset of \( S \) with \( K \) elements, there exist two distinct numbers \( a \) and \( b \) such that \( (a+b) \mid a b \).

Answer: 39


Pierre says: "The day before yesterday I was 10 years old. Next year, I will celebrate my 13th birthday." What day is it today?

Answer: 1


Let \(ABC\) be an equilateral triangle with side length 2, and let \(\Gamma\) be a circle with radius \(\frac{1}{2}\) centered at the center of the equilateral triangle. Determine the length of the shortest path that starts somewhere on \(\Gamma\), visits all three sides of \(ABC\), and ends somewhere on \(\Gamma\) (not necessarily at the starting point). Express your answer in the form of \(\sqrt{p}-q\), where \(p\) and \(q\) are rational numbers written as reduced fractions.

Answer: \sqrt{\frac{28}{3}}-1


In an $n \times n$ grid of small squares, if a sequence of $k$ distinct small squares $\left(a_{1}, a_{2}, \ldots, a_{k}\right)$ satisfies: $k \geq 4$, and square $a_{i}$ shares a common edge with $a_{i+1}$ (for $i = 1, 2, \ldots, k$, and $a_{k+1} = a_{1}$), then this sequence $\left(a_{1}, a_{2}, \ldots, a_{k}\right)$ is called a cycle. If a set $X$ of some small squares in the grid satisfies that for every cycle on the grid, there is at least one small square in $X$, then set $X$ is called a "naughty set".

Find all real numbers $C$ such that for any integer $n \geq 2$, there exists a "naughty set" in an $n \times n$ grid containing at most $C n^{2}$ small squares.

Answer: \frac{1}{3}


In the cells of a $3 \times 3$ square are placed the numbers $0, 1, 2, \ldots, 8$. It is known that any two consecutive numbers are placed in adjacent (side-by-side) cells. What number can be in the central cell if the sum of the numbers in the corner cells is $18?$

Answer: 2


A quadrilateral \(ABCD\) is inscribed in a circle with radius 6. The rays \(AB\) and \(DC\) intersect at point \(P\), and the rays \(BC\) and \(AD\) intersect at point \(Q\). It is known that triangles \(ADP\) and \(QAB\) are similar (vertices are not necessarily given in the corresponding order).

a) Find \(AC\).

b) Additionally, it is known that the inscribed circles of triangles \(ABC\) and \(ACD\) touch the segment \(AC\) at points \(K\) and \(T\) respectively, and \(CK : KT : TA = 2 : 1 : 3\) (point \(T\) lies between \(K\) and \(A\)). Find \(\angle DAC\) and the area of the quadrilateral \(ABCD\).

Answer: 68


As shown in the figure, in the right triangle \(ABC\), the lengths of \(AB\) and \(BC\) are 15 and 20, respectively. The quadrilateral \(BDEF\) is a square. If the height \(EH\) of triangle \(EMN\) is 2, then the area of square \(BDEF\) is \(\qquad\).

Answer: 100


As shown in the figure, let points $E$ and $F$ be on the sides $AB$ and $AC$ of $\triangle ABC$, respectively. The line segments $CE$ and $BF$ intersect at point $D$. If the areas of $\triangle CDF$, $\triangle BCD$, and $\triangle BDE$ are 3, 7, and 7, respectively, what is the area of quadrilateral $AEDF$?

Answer: 18


We place a plane on each edge of a unit cube such that the plane forms a $45^{\circ}$ angle with each of the two faces adjacent to the edge and does not intersect the cube. What is the volume of the convex body bounded by these 12 planes?

Answer: 2


A cat ate half of the smallest fish in 30 seconds. In total, it ate 5 fish and took a whole number of minutes to do so (the cat eats fish at a constant 'cell' rate). The picture shows all the fish the cat had. Which fish did the cat not eat?

Answer: 6


The flea Kuzya can jump along a line $L$. The starting point for the jumps is the point $A$ on the line $L$. The length of each jump is $h$, and the direction of each jump is chosen randomly and equally likely. Find the probability that after making between five and ten random jumps, Kuzya will be at a distance of at least $5h$ from $A$ at least once.

Answer: \frac{9}{64}


Two cars are traveling on the highway at a speed of 80 km/h with a gap of 10 meters between them. At a speed limit sign, the cars instantly reduce their speed to 60 km/h. What will be the gap between them after the speed limit sign?

Answer: 7.5\,


## Zadatak A-2.6.

Trapez $A B C D$ s osnovicama $\overline{A B}$ i $\overline{C D}$ ima opisanu kružnicu $k$. Njegove dijagonale međusobno su okomite i sijeku se u točki $S$. Odredi omjer površine kruga omeđenog kružnicom $k$ i zbroja površina trokuta $A B S$ i $C D S$.

Answer: \pi


Gábor wanted to design a maze. He took a square-grid paper and marked a large square on it. In doing so, and subsequently, he always followed the grid lines, moving from grid point to grid point. Then, within the square, he drew some lines, totaling 400 units in length. These became the walls of the maze. When he was finished, he noticed that in his maze, from any unit square, one can reach any other unit square in exactly one way, excluding paths that pass through any unit square more than once. What is the side length of the first square he drew?

Answer: 21


A nine-member selection committee chooses from three candidates. Each of them ranks them, giving 3 points to their first choice, 2 points to their second choice, and 1 point to their third choice. Summing up the candidates' points, it turned out that the order was clear, with all three having different point totals. One member of the committee noticed that if the election had been conducted such that each of the nine only chose one candidate and gave that candidate 1 point, then the order of the candidates would have been reversed. How many points did the candidates originally receive?

Answer: 19


The altitudes of triangle \(ABC\), drawn from vertices \(B\) and \(C\), intersect at point \(M\). It is known that \(BM = CM\). Prove that triangle \(ABC\) is isosceles.

Answer: \triangle ABC \text{ is isosceles}


At a university in Budapest, the most successful students can apply for two types of scholarships. One of the conditions for the outstanding scholarship is that the average of the grades earned in the last semester must be above 4.5. For the republican scholarship, however, an average of at least 4.51 is required.

How many grades does someone need to earn to have an average above 4.5, but not reach 4.51?

Answer: 51


A round robin tournament is held with $2016$ participants. Each player plays each other player once and no games result in ties. We say a pair of players $A$ and $B$ is a [i]dominant pair[/i] if all other players either defeat $A$ and $B$ or are defeated by both $A$ and $B$. Find the maximum number dominant pairs.

[i]Proposed by Nathan Ramesh

Answer: 2015


Two bored millionaires, Bilion and Trilion, decide to play a game. They each have a sufficient supply of $\$ 1, \$ 2,\$ 5$, and $\$ 10$ bills. Starting with Bilion, they take turns putting one of the bills they have into a pile. The game ends when the bills in the pile total exactly $\$1{,}000{,}000$, and whoever makes the last move wins the $\$1{,}000{,}000$ in the pile (if the pile is worth more than $\$1{,}000{,}000$ after a move, then the person who made the last move loses instead, and the other person wins the amount of cash in the pile). Assuming optimal play, how many dollars will the winning player gain?

[i]Proposed by Yannick Yao[/i]

Answer: 1000000


2. Vasya constructed a robot, but he messed up the program, so the robot makes random moves that somehow always lead it to the right point.

A move is a one-kilometer displacement either north (N), east (E), south (S), or west (W).

The robot reached its goal in 22 moves:

# EENENNWNENESEEEEESSSWN.

How many moves would the robot need if it did not make any extra moves?

Answer: 10


11.175. A right parallelepiped is described around a sphere, with the diagonals of the base being $a$ and $b$. Determine the total surface area of the parallelepiped.

Answer: 3ab


3. In $\triangle A B C$, it is known that $\angle A=60^{\circ}, \angle C=$ $75^{\circ}, A B=10$, points $D, E, F$ are on sides $A B, B C, C A$ respectively. Then the minimum perimeter of $\triangle D E F$ is $\qquad$

Answer: 5\sqrt{6}


431 There is a card game called "Twelve Months" that is only played during the Spring Festival. The rules are as follows:

First, take a brand new deck of cards, remove the big joker, the small joker, and the 4 Ks, leaving 48 cards, which are then shuffled.

Second, lay out the shuffled cards face down in 12 columns, with 4 cards in each column.

Third, start by flipping the first card in the first column. If you flip over an $N (N=1,2, \cdots, 12$, where $\mathrm{J}$ and $\mathrm{Q}$ represent $11$ and $12$ respectively, without distinguishing suits), place this face-up card at the end of the $N$th column.

Fourth, continue to flip the first face-down card in the $N$th column, and operate according to the method in the third step.

Fifth, continue this process until it can no longer proceed. If all 12 columns are eventually flipped over, it signifies that everything will go smoothly for the next 12 months; otherwise, if some columns remain unflipped when the game ends, it indicates that there will be difficult events in the corresponding months.
Find the probability of the event "all flipped over" occurring.

Answer: \dfrac{1}{12}


For the digits of the decimal number $\overline{a b c d}$, it holds that $a>b>c>d$. These same digits, in some order, are also the digits of the difference $\overline{a b c d}-\overline{d c b a}$. Which is this four-digit number?

Answer: 7641


In triangle $ABC,$ $AB = 13,$ $BC = 14,$ $AC = 15,$ and point $G$ is the intersection of the medians. Points $A',$ $B',$ and $C',$ are the images of $A,$ $B,$ and $C,$ respectively, after a $180^\circ$ rotation about $G.$ What is the area of the union of the two regions enclosed by the triangles $ABC$ and $A'B'C'?$
Please give the answer directly without any intermediate steps.

Answer: 112


Task 1. Masha has a piggy bank where she puts a 50 or 100 ruble bill every week. At the end of every 4 weeks, she picks the bill of the smallest denomination from the piggy bank and gives it to her little sister. After a year, it turned out that she had given her sister 1250 rubles. What is the minimum amount of money that could have accumulated with her over this time?

Answer: 3750


4. Find all real numbers $x, y$ that satisfy the system of equations

$$
\begin{aligned}
& x \cdot\left\lfloor\frac{y}{x}\right\rfloor=5, \\
& y \cdot\left\lfloor\frac{x}{y}\right\rfloor=-6 .
\end{aligned}
$$

(The notation $\lfloor a\rfloor$ denotes the floor function of $a$, i.e., the greatest integer less than or equal to $a$.)

The regional round of category B is held

## on Tuesday, April 17, 2012

so that it starts in the morning and contestants have 4 hours of pure time to solve the problems. Each problem can earn a contestant 6 points, and a successful solver is a student who scores 10 points or more. Allowed aids are writing and drawing materials and school MF tables. Calculators, laptops, and any other electronic aids are not allowed. This information will be communicated to the students before the start of the competition.

Answer: (-5, 3)


### 1.26. Find the center of gravity of a hemisphere

$$
0 \leqslant z \leqslant \sqrt{R^{2}-x^{2}-y^{2}},
$$

where the volume density \( p \) of the material at each point is proportional to the distance from the origin.

Answer: \dfrac{2R}{5}


11. (12 points) Five students, Jia, Yi, Bing, Ding, and Wu, ranked in the top 5 (no ties) in a math competition and stood in a row for a photo. They each said the following:

Jia said: The two students next to me are both ranked behind me;
Yi said: The two students next to me are both ranked adjacent to me;
Bing said: All the students to my right (at least 1) are ranked ahead of me;
Ding said: All the students to my left (at least 1) are ranked behind me;
Wu said: I am standing in the second position from the right.

Given that they are all honest children, Jia, Yi, Bing, Ding, and Wu ranked $A, B, C, D, E$ respectively. What is the five-digit number $\overline{\mathrm{ABCDE}}$?

Answer: 23514


6. Let the function $f(x)=(2 a-b) x+2 b-8$ $(b \leqslant 6)$, when $x \in[-4,2]$, $f(x) \geqslant 0$ always holds. Then the range of $\frac{a^{4}-2 b^{4}}{a^{3} b}$ is $\qquad$ .

Answer: [-\frac{161}{3}, -\frac{39071}{4116}]


Given a triangular pyramid \( S-ABC \) with a regular triangular base, the projection of point \( A \) onto side face \( SBC \), denoted as \( H \), is the orthocenter of \( \triangle SBC \). If \( SA = a \), find the maximum volume of this triangular pyramid.

Answer: \dfrac{a^3}{6}


2. Determine the maximum number of points in a plane such that any three of them form the vertices of a right triangle.

Answer: 4