On an $n \times n$ chessboard, we write the number 1 in the first field of the first row, and in the remaining fields of the first row and the first column, we write the numbers $2,3, \ldots, n$ respectively. In every other field, we write the number that is one less than the sum of the numbers at the beginning of the row and the column. Then, we place a point mass of the written number grams at the center of each field. Where is the center of mass of this point system? (We consider the chessboard to be weightless.)

Where is the center of mass if we fill the inner fields of the table with the product of the two numbers instead of the reduced sum?

Answer: \left( \dfrac{(7n - 1)(n + 1)}{12n}, \dfrac{(7n - 1)(n + 1)}{12n} \right)


Let $x_1, x_2 \ldots , x_5$ be real numbers. Find the least positive integer $n$ with the following property: if some $n$ distinct sums of the form $x_p+x_q+x_r$ (with $1\le p<q<r\le 5$) are equal to $0$, then $x_1=x_2=\cdots=x_5=0$.

Answer: 7


We randomly select four points independently on the surface of a sphere. What is the probability that the tetrahedron defined by them contains the center of the sphere?

Answer: \dfrac{1}{8}


## Task A-1.3.

Two congruent squares with side lengths of $1+\sqrt{2}$ have the same center, and their intersection is a regular octagon. What is the area of this octagon?

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


5. Several children buy two types of goods, priced at 3 yuan and 5 yuan each. Each child buys at least one item, but the total amount of the items each child buys cannot exceed 15 yuan. Xiaomin says: Among the children, there must be at least three who bought the same quantities of the two types of goods. Question: How many children are there at least?

Answer: 25


10. Among the first 1000 positive integers, the number of positive integers that can be expressed as $[2 x]+[4 x]+[6 x]+[8 x]$ is $\qquad$ .

Answer: 600


2. (10 points) YY's family of 3, plus YY's cousin, have a total age of 95 years this year. Dad is 4 years older than Mom, and YY is 3 years older than the cousin. 8 years ago, the sum of their ages was 65 years. Then Dad is $\qquad$ years old this year.

Answer: 42


Example 9. Find the integral $\int \sin ^{2} \frac{x}{2} d x$.

Answer: \frac{1}{2}x - \frac{1}{2} \sin x + C


1. (10 points). Tourist Nikolai Petrovich was late by $\Delta t=5$ minutes for the departure of his riverboat, which had set off downstream. Fortunately, the owner of a fast motorboat agreed to help Nikolai Petrovich. Catching up with the riverboat and disembarking the unlucky tourist, the motorboat immediately set off on its return journey. How much time passed from the departure of the motorboat until its return? Assume that the speed of the riverboat relative to the water is $k=3$ times the speed of the river current, and the speed of the motorboat is $n=5$ times the speed of the river current.

Answer: 25


5. A train 280 meters long travels at a speed of 18 meters per second across a bridge. It takes 20 seconds from the moment the tail of the train gets on the bridge until the head of the train leaves the bridge. Therefore, the length of the bridge is $\qquad$ _.

Answer: 640


There are 64 points arranged in an 8-row by 8-column grid. What is the minimum number of straight lines required to separate all the points?

Answer: 14


Example 7.10 Let $A_{6}$ be a regular hexagon. Now we use red and blue to color the 6 vertices of $A_{6}$, with each vertex being one color. Find the number of type II patterns of the vertex-colored regular hexagon.

Answer: 13


We divided a chord $AB$ of a circle, which has the length of the radius, and the shorter arc $AB$ into three equal parts each. The line connecting the division points closer to $A$ intersects the line connecting the division points closer to $B$ at point $P$. What is the measure of angle $APB$?

Answer: 20^\circ


Problem 8.8. Masha wrote on the board in ascending order all natural divisors of some number $N$ (the very first divisor written is 1, the largest divisor written is the number $N$ itself). It turned out that the third from the end divisor is 21 times greater than the second from the beginning. What is the largest value that $N$ can take?

Answer: 441


A political analyst is studying the population growth in the semi-autonomous region of Puntland over a 5-year period. In the first year, the population was estimated to be 3,000,000 people. Each year, the population grew by 2%. Additionally, every year, 10,000 people migrated to Puntland from other regions. What will be the estimated population of Puntland at the end of the 5-year period?

Answer: 3364283


A Russian artist is planning an exhibition to showcase contemporary Russian art to gain more recognition. To organize the event, the artist has created 3 different series of paintings. Each series consists of 8 paintings. The artist also plans to invite 5 art critics to review the exhibition, and each critic will review 4 paintings. However, the artist wants to make sure that each painting is reviewed by exactly one critic.

If each critic writes a review for every painting they assess, how many total reviews will be written?

Answer: 24


An independent labor rights auditor is visiting a factory to conduct an inspection. The factory has three sections: manufacturing, packaging, and quality control. The auditor spends 3 hours in the manufacturing section, 2 hours in the packaging section, and 1 hour in the quality control section.

During the inspection, the auditor observes that there are 45 workers in the manufacturing section, 30 workers in the packaging section, and 15 workers in the quality control section. The auditor needs to ensure that each worker is following international labor standards, which takes approximately 10 minutes per worker.

If the auditor can only work for a total of 8 hours a day, how many additional hours will the auditor need to allocate to complete the inspection of all workers in the factory?

Answer: 7


In the Naruto franchise, let's say there are 720 episodes in the original series and 500 episodes in the sequel, "Naruto: Shippuden." Imagine you decide to rewatch both series. You plan to watch 3 episodes each day. On weekends, you double your pace and watch 6 episodes each day. If you start watching both series from the first episode on a Monday, how many full weeks will it take you to finish watching all the episodes?

Answer: 46


In the village of Halnad, an elderly resident named Mr. Naik has witnessed the village's growth over the years. When Mr. Naik was young, there were only 15 houses in the village. Over the decades, new houses were built at a steady rate of 3 houses every 5 years. Now, Mr. Naik is 80 years old, and he remembers that the last new house was built when he turned 75. How many houses are there in Halnad now?

Answer: 60


Liam lives in Larne, Northern Ireland, and frequently travels to the Isle of Man for work. The ferry from Larne to the Isle of Man takes 2 hours and 45 minutes one way. Liam makes this trip twice a week. Last month, Liam traveled to the Isle of Man every week and spent an additional 5 hours each week working on the island.

How many hours in total did Liam spend traveling to and working on the Isle of Man last month, assuming last month had 4 weeks?

Answer: 64


Alex is a train travel aficionado who is planning a railway circumnavigation trip around the world. He plans to visit 5 continents, spending an equal number of days on each continent. The total duration of his trip is 60 days. During his travels, Alex will use 10 different trains, with each train journey taking an average of 4 hours. If he travels every second day, how many hours in total will Alex spend on trains during his circumnavigation journey?

Answer: 40


Alexei is a big fan of Russian movies and works as customer support at a major tech company. He recently decided to organize a movie marathon weekend featuring his favorite Russian films. Alexei has a collection of 12 Russian movies, and he plans to watch as many movies as he can while also ensuring he gets enough sleep.

Each movie lasts approximately 2 hours. Since Alexei works on Saturday from 9 AM to 5 PM, he can only start watching movies on Saturday evening. He plans to watch movies until midnight and continue watching on Sunday from 10 AM to 10 PM.

If Alexei wants to make sure he gets at least 8 hours of sleep each night, how many movies can he watch over the weekend?

Answer: 9


Alex is an international development practitioner who frequently travels to different countries for work. To enhance their field experience, Alex is pursuing a degree that requires attending classes twice a week for 15 weeks. Each class lasts 3 hours. In addition to attending classes, Alex also spends 2 hours each week researching case studies related to international development.

If Alex has 10 weeks left in the semester, how many total hours will Alex spend on both attending classes and researching case studies by the end of the semester?

Answer: 80


An art dealer has a collection of 120 paintings from the Renaissance era, 85 from the Baroque period, and 95 modern art pieces. She decides to sell 15% of her Renaissance paintings, 20% of her Baroque paintings, and 10% of her modern art pieces to make space for new acquisitions. How many paintings will she have left in her collection after these sales?

Answer: 256


A screenwriter is collaborating with James Nesbitt on a new movie project. They need to write 120 pages of the script. The screenwriter wants to write 3 pages every day, while James suggests writing 5 pages on weekends. If the screenwriter follows this plan and writes every day, how many weekends will it take them to complete the script?

Answer: 5


Alex, a self-described nerd and an avid comic book collector, has been attending Comic-Con annually since 1995. To celebrate his love for comics, each year at Comic-Con, Alex buys a set of comic books equal to the number of years he's been attending. In 2023, how many comic books did Alex purchase in total at Comic-Con?

Answer: 435


a,b,c are positives  with  21ab+2bc+8ca \leq 12 . Find the least  possoble  value  of the  expresion 1/a + 2/b + 3/c.
Thanks

Answer: \dfrac{15}{2}


A small amount of air got into a certain barometer, so that at $15^{\circ}$, when a well-functioning barometer shows $762 \mathrm{~mm}$, it only shows $704 \mathrm{~mm}$. The height of the mercury-free space at this time is $143 \mathrm{~mm}$. With changing conditions, at $30^{\circ}$, the device shows $692 \mathrm{~mm}$. What is the true barometric pressure?

Answer: 748


4. One of the lateral sides of the trapezoid is perpendicular to the bases and equals $2 R$. A circle is constructed on this side as a diameter, which divides the other lateral side into three segments. The ratio of the lengths of these segments is 12:15:5 (counting from the lower base). Find the area of the trapezoid.

Answer. $S=\frac{7 R^{2}}{\sqrt{15}}$.

Answer: \dfrac{7 R^{2}}{\sqrt{15}}


4. Let quadrilateral $A B C D$ be inscribed. The rays $A B$ and $D C$ intersect at point K. It turns out that points $B, D$, and the midpoints $M$ and $N$ of segments $A C$ and $K C$ lie on the same circle. What values can the angle $A D C$ take?

![](https://cdn.mathpix.com/cropped/2024_05_06_cf77befabfd0e5d082b4g-1.jpg?height=388&width=457&top_left_y=1962&top_left_x=845)

Answer: 90^\circ


The lateral surface area of a cone is twice the area of its base. Find the angle in the lateral surface unfolding of the cone.

Answer: 180^\circ


12.102. The volume of a cone is $V$. A pyramid is inscribed in the cone, with an isosceles triangle as its base, where the angle between the lateral sides is $\alpha$. Find the volume of the pyramid.

Answer: \dfrac{2V}{\pi} \cos^2\left( \dfrac{\alpha}{2} \right) \sin \alpha


8th Chinese 1993 Problem B2 S = {1, 2, ... , 30}. A 1 , A 2 , ... , A 10 are subsets of S each containing 3 elements. A i and A j have at least one element in common. Let n i be the number of A j to which i belongs. What is the smallest possible value of max n i ? Solution

Answer: 5


2. In a computer game, a turtle moves across a grid on the computer screen, which contains 11 columns and 5 rows. Initially, it is located at the bottom-left corner of the screen - on the cell with coordinates $(0,0)$. If the program instructs the turtle to move off the screen, it reappears on the opposite side - for example, taking one step up from cell $(3,4)$, the turtle will end up in cell $(3,0)$. Where will the turtle be after executing the following program:
1) 1 step down; 2) 2 steps to the right; 3) 3 steps up; 4) 4 steps to the left; 5) 5 steps down; 6) 6 steps to the right; .. ; 2016) 2016 steps to the left; 2017) 2017 steps down?

Answer: (4,1)


6. How many ways are there to choose integers $a, b$ and $c$ with $a<b<c$ from the list $1,5,8,21,22,27,30,33,37,39,46,50$ so that the product abc is a multiple of 12 ?

Answer: 76


The medians divide a triangle into 6 smaller triangles. Four of the circles inscribed in the smaller triangles have equal radii. Prove that the original triangle is equilateral.

Answer: \text{The original triangle is equilateral.}


5. Let $Q_{+}$ be the set of all positive rational numbers. Find all functions $f: \mathbf{Q}_{+} \rightarrow \mathbf{Q}_{+}$ such that for all $x, y \in \mathbf{Q}_{+}$, we have
$$
f\left(f^{2}(x) y\right)=x^{3} f(x y) .
$$

Answer: \dfrac{1}{x}


Let \( a \) be a complex number lying on the unit circle \( S \) centered at the origin, and \( t \) be a real number (a point lying on the real axis). Let \( b \) be the point of intersection of the line \( at \) with the circle \( S \), distinct from \( a \). Prove that \(\bar{b} = (1 - ta)/(t - a)\).

Answer: \overline{b} = \dfrac{1 - ta}{t - a}


2. In quadrilateral $A B C D$, it is known that $A B=B C=$ $C D$, and $\angle A=80^{\circ}, \angle D=40^{\circ}$. Then the degree measure of $\angle B$ is $\qquad$.

Answer: 80


4. ( 1 mark) A multiple choice test consists of 100 questions. If a student answers a question correctly, he will get 4 marks; if he answers a question wrongly, he will get -1 mark. He will get 0 marks for an unanswered question. Determine the number of different total marks of the test. (A total mark can be negative.)
(1 分) A multiple choice test consists of 100 questions. If a student answers a question correctly, he will get 4 marks; if he answers a question wrongly, he will get -1 mark. He will get 0 marks for an unanswered question. Determine the number of different total marks of the test. (A total mark can be negative.)

Answer: 495


Answer: 23


Genevieve wants to further promote her school's annual Kite Olympics by making yet another large kite. This time, she designs it by quintupling both the height and width dimensions of the original small kite on a grid where points are one inch apart. Specifically, the quadrilateral extends for a base length of 35 inches (quintupling the original 7 inches) and a height of 15 inches from the mid-point (quintupling the original 3 inches). What is the area in square inches of this new large kite?

Answer: 525


In a survey of 1500 people, $25\%$ do not like television, and $15\%$ of the people who do not like television also do not like sports. How many people surveyed do not like both television and sports?

Answer: 56


Let $R = (10, 8)$. The lines whose equations are $9y = 18x$ and $15y = 6x$ contain points $P$ and $Q$, respectively, such that $R$ is the midpoint of $\overline{PQ}$. Find the length of $PQ$ expressed as a reduced fraction $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Then find $m + n$.

Answer: 31


Rectangle $ABCD$ has sides $\overline{AB}$ of length 5 and $\overline{CB}$ of length 4. Divide $\overline{AB}$ into 200 congruent segments with points $A=P_0, P_1, \ldots, P_{200}=B$, and divide $\overline{CB}$ into 200 congruent segments with points $C=Q_0, Q_1, \ldots, Q_{200}=B$. For $1 \leq k \leq 199$, draw the segments $\overline{P_kQ_k}$. Repeat this construction on the sides $\overline{AD}$ and $\overline{CD}$, and then draw the diagonal $\overline{AC}$. Compute the sum of the lengths of the 399 segments drawn in this way.

Answer: 200\sqrt{41}


Six numbers are given: 1877, 1999, 2039, 2045, 2119, and 2131. Four of these numbers have a mean of 2015. What is the mean of the other two numbers?

Answer: 2075


A student walks along a corridor of lockers numbered from $1$ to $2048$. Initially, all lockers are closed. He starts by opening every locker. On his return trip, he skips one locker and opens the next. When he reaches the beginning again, he skips two lockers and opens the next closed locker, and continues this pattern. He repeats this pattern, increasing the skip by one each time until all lockers are open. What is the number of the last locker he opens?

Answer: 2048


Answer: 55


The product of four different positive integers is equal to $11^4$. What is the sum of these four integers?

Answer: 144


On the Saturday of a weekend soccer tournament, Team A plays Team B, Team C plays Team D, and Team E plays Team F. On Sunday, the three Saturday winners play each other in a round-robin format to determine the first, second, and third places, while the three Saturday losers play another round-robin to determine the fourth, fifth, and sixth places. There are no ties in any matches. What is the total number of possible six-team ranking sequences at the end of the tournament?

Answer: 288


A new design involves a rectangle and four identical circles, each tangent to two adjacent sides of the rectangle and its neighboring circles. The design is such that the length of the rectangle is twice its width. If the rectangle has a width of 30 inches, determine the total shaded area when the circles are excluded.

Answer: 1800 - 225\pi \text{ square inches}


An artist wants to install a square glass pane (the shaded region) precisely within a square frame for an art installation. The outer perimeter of the frame measures 100 cm on each side. The width of each side of the frame is 15 cm. What is the area of the glass pane?

Answer: 4900 \text{ cm}^2


Find the total volume in cubic centimeters of a structure that consists of a cone on the bottom and a cylinder on the top. The entire structure has a consistent diameter of 8 cm. The cone's height is 10 cm, and the cylinder's height is 4 cm. Express your answer in terms of \(\pi\).

Answer: \frac{352}{3} \pi \text{ cm}^3


A new lattice pattern starts exactly as the previous one but has 7 columns per row instead of 5. The sequence is continued through 9 rows. What is the fifth number in the 9th row?

Answer: 61


44th Putnam 1983 Problem A6 Let T be the triangle with vertices (0, 0), (a, 0), and (0, a). Find lim a→∞ a 4 exp(-a 3 ) ∫ T exp(x 3 +y 3 ) dx dy. Solution

Answer: \dfrac{2}{9}


In how many ways can you transfer milk from a 12-liter can, filled with milk, to another empty 12-liter can using two empty jars with volumes of 1 liter and 2 liters? Pouring from jar to jar is not allowed.

Answer: 233


Let $n>2$ be a given positive integer. There are $n$ guests at Georg's bachelor party and each guest is friends with at least one other guest. Georg organizes a party game among the guests. Each guest receives a jug of water such that there are no two guests with the same amount of water in their jugs. All guests now proceed simultaneously as follows. Every guest takes one cup for each of his friends at the party and distributes all the water from his jug evenly in the cups. He then passes a cup to each of his friends. Each guest having received a cup of water from each of his friends pours the water he has received into his jug. What is the smallest possible number of guests that do not have the same amount of water as they started with?

Answer: 2


Consider polynomials $P$ of degree $2015$, all of whose coefficients are in the set $\{0,1,\dots,2010\}$. Call such a polynomial [i]good[/i] if for every integer $m$, one of the numbers $P(m)-20$, $P(m)-15$, $P(m)-1234$ is divisible by $2011$, and there exist integers $m_{20}, m_{15}, m_{1234}$ such that $P(m_{20})-20, P(m_{15})-15, P(m_{1234})-1234$ are all multiples of $2011$. Let $N$ be the number of good polynomials. Find the remainder when $N$ is divided by $1000$.

[i]Proposed by Yang Liu[/i]

Answer: 460


6. Suppose there are 5 cards, each with a different number written on it, and the numbers are $2,3,4,5,6$. When these cards are randomly placed in a row, the probability that for each $i(1 \leq i \leq 5)$, the number at the $\mathrm{i}$-th position from the left is greater than or equal to $\mathrm{i}$ is

Answer: \dfrac{2}{15}


5. (50 points) Vasya thought of a natural number not greater than 100. Fedya asks Vasya questions of the form “What is the quotient of the division of the thought number by $m$?”, where $m$ is a natural number not less than 2, and Fedya chooses it himself. What is the minimum number of such questions Fedya can guarantee to determine the number $B a$ that Vasya thought of?

Answer: 2


15. A and B take turns tossing a fair coin, and the one who tosses heads first wins. They play several rounds, and it is stipulated that the loser of the previous round tosses first in the next round. If A tosses first in the 1st round, what is the probability that A wins in the 6th round?

Answer: \dfrac{364}{729}


Problem 11. Denote by $d(a, b)$ the number of the divisors of a positive integer $a$, which are greater than or equal to $b$. Find all positive integers $n$ such that

$$
d(3 n+1,1)+d(3 n+2,2)+\cdots+d(4 n, n)=2006
$$

Ivan Landjev

Answer: 708


In triangle $ABC$, $AB = AC$. The distances from the two trisection points of the side $AB$ to the vertex $C$ are 17 cm and 20 cm, respectively. What could be the distance from vertex $A$ to the trisection points of the base?

Answer: 3\sqrt{65}


15. In the 6 circles below, fill in $0,1,2,3,4,5$ with each number used only once, so that the sum of the three numbers on each side is equal.

Answer: 6


Example 1. Solve the equation $\mathrm{x}^{2}-2 \mathrm{x}-3=0$ geometrically.

Answer: 3


Consider a regular $2 n+1$-gon $P$ in the plane, where $n$ is a positive integer. We say that a point $S$ on one of the sides of $P$ can be seen from a point $E$ that is external to $P$, if the line segment $S E$ contains no other points that lie on the sides of $P$ except $S$. We want to color the sides of $P$ in 3 colors, such that every side is colored in exactly one color, and each color must be used at least once. Moreover, from every point in the plane external to $P$, at most 2 different colors on $P$ can be seen (ignore the vertices of $P$, we consider them colorless). Find the largest positive integer for which such a coloring is possible.

Answer: 1


Sam spends his days walking around the following $2 \times 2$ grid of squares:
\[
\begin{array}{|c|c|}
\hline
1 & 2 \\
\hline
4 & 3 \\
\hline
\end{array}
\]
Two squares are considered adjacent if they share a side. Sam starts at the square labeled $1$ and every second, he walks to an adjacent square. How many paths can Sam take such that the sum of the numbers on every square he visits in his path is equal to $20$, excluding the square he started on?

Answer: 167


Vera has several identical matches, from which she forms a triangle. She wants any two sides of this triangle to differ in length by at least $10$ matches. However, it is impossible to create such a triangle with the available matches (without leaving any extra matches). What is the maximum number of matches Vera can have?

Answer: 62


Fedja used matches to form a parallelogram, ensuring that the sides are of equal length and the vertices do not lie on a single line. It is observed that exactly 7 or 9 matches fit into the diagonals. Determine the number of matches that make up the perimeter of the parallelogram.

Answer: 22


Given $1994$ points on a plane, it is known that any $100$ of them can be selected such that $98$ of these can be enclosed by a circle with a diameter of $1$. Determine the minimum number of circles with a radius of $1$ required to cover all $1994$ points.

Answer: 21


Consider all functions $f: \mathbb{N} \to \mathbb{N}$ satisfying $f(t^2 f(s)) = s(f(t))^2$ for all $s$ and $t$ in $\mathbb{N}$. Determine the least possible value of $f(1998)$.

Answer: 120


Let $r_1$, $r_2$, $\ldots$, $r_{20}$ be the roots of the polynomial $x^{20}-7x^3+1$.  If
\[\dfrac{1}{r_1^2+1}+\dfrac{1}{r_2^2+1}+\cdots+\dfrac{1}{r_{20}^2+1}\]
can be written in the form $\tfrac{m}{n}$ where $m$ and $n$ are positive coprime integers, find $m+n$.

Answer: 240


Ten test papers are to be prepared for the National Olympiad. Each paper has 4 problems, and no two papers have more than 1 problem in common. What is the minimum number of distinct problems needed?

Answer: 13


There are $2014$ students from high schools nationwide sitting around a round table in an arbitrary manner. The organizers want to rearrange the students so that those from the same school sit next to each other. This is done by performing swaps between two adjacent groups of students.

The task is to find the smallest number $k$ such that the desired arrangement can be achieved with no more than $k$ swaps.

Illustration of a swap:
$$...\underbrace{ABCD}_\text{1}\underbrace{EFG}_\text{2}... \to ...\underbrace{EFG}_\text{2}\underbrace{ABCD}_\text{1}...$$

Answer: 2013


Call a lattice point *visible* if the line segment connecting the point and the origin does not pass through another lattice point. Given a positive integer $k$, denote by $S_k$ the set of all visible lattice points $(x, y)$ such that $x^2 + y^2 = k^2$. Let $D$ denote the set of all positive divisors of $2021 \cdot 2025$. Compute the sum \[ \sum_{d \in D} |S_d| \]

Here, a lattice point is a point $(x, y)$ on the plane where both $x$ and $y$ are integers, and $|A|$ denotes the number of elements of the set $A$.

Answer: 20


There is a regular $17$-gon $\mathcal{P}$ and its circumcircle $\mathcal{Y}$ on the plane. The vertices of $\mathcal{P}$ are coloured in such a way that $A,B \in \mathcal{P}$ are of different colour if the shorter arc connecting $A$ and $B$ on $\mathcal{Y}$ has $2^k+1$ vertices, for some $k \in \mathbb{N}$, including $A$ and $B$. What is the least number of colours which suffices?

Answer: 4


It is given that the number $4^{11} + 1$ is divisible by some prime greater than $1000$. Determine this prime.

Answer: 2113


In triangle $ABC$, the lengths of all sides are integers, $\angle B = 2 \angle A$ and $\angle C > 90^\circ$. Find the smallest possible perimeter of this triangle.

Answer: 77


There exist complex numbers $z_1, z_2, \dots, z_{10}$ which satisfy the condition: $$|z_k i^k + z_{k+1} i^{k+1}| = |z_{k+1} i^k + z_k i^{k+1}|$$ for all integers $1 \leq k \leq 9$, where $i = \sqrt{-1}$. Given that $|z_1|=9$, $|z_2|=29$, and for all integers $3 \leq n \leq 10$, $|z_n|=|z_{n-1} + z_{n-2}|$, find the minimum value of $|z_1| + |z_2| + \cdots + |z_{10}|$.

Answer: 103


A permutation \((a_1, a_2, a_3, \dots, a_{2012})\) of \((1, 2, 3, \dots, 2012)\) is selected at random. If \(S\) is the expected value of
\[
\sum_{i = 1}^{2012} | a_i - i |
\],
then compute the sum of the prime factors of \(S\).

Answer: 2083


Let $D$ be a point on the side $[BC]$ of $\triangle ABC$ such that $|AB| + |BD| = |AC|$ and $m(\widehat{BAD}) = m(\widehat{DAC}) = 30^\circ$. If the answer is $x^\circ$, what is the value of $x$ for $m(\widehat{ACB})$? Please provide the value of $x$ as an integer.

Answer: 40


A unit cube is cut twice to form three triangular prisms, two of which are congruent, as shown in Figure 1. The cube is then cut in the same manner along the dashed lines shown in Figure 2. This creates nine pieces. What is the volume of the piece that contains vertex \(W\)? The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.

Answer: 13


A massless elastic cord (that obeys Hooke's Law) will break if the tension in the cord exceeds \(T_{max}\). One end of the cord is attached to a fixed point, the other is attached to an object of mass \(3m\). If a second, smaller object of mass \(m\) moving at an initial speed \(v_0\) strikes the larger mass and the two stick together, the cord will stretch and break, but the final kinetic energy of the two masses will be zero. If instead the two collide with a perfectly elastic one-dimensional collision, the cord will still break, and the larger mass will move off with a final speed of \(v_f\). All motion occurs on a horizontal, frictionless surface.

Find \(\frac{v_f}{v_0}\). The answer should be in the form \(\frac{k}{\sqrt{m}}\). Please calculate the value of k + m.

Answer: 7


The parabola $P$ has its focus at $(0,0)$ and passes through the points $(4,3)$ and $(-4,-3)$. Determine the number of points $(x,y) \in P$ with integer coefficients such that $|4x + 3y| \leq 1000$.

Answer: 40


Let $S$ be the set of points $(x, y)$ in the Cartesian plane that satisfy
\[\Big|\big| |x|-2\big|-1\Big|+\Big|\big| |y|-2\big|-1\Big|=1.\]What is the total length of all the lines that make up $S$?The answer is in the form k\sqrt{m}+n,. Please provide the value of k + m + n.

Answer: 66


There are $13$ cities in a certain kingdom. Between some pairs of the cities a two-way direct bus, train or plane connections are established. What is the least possible number of connections to be established so that choosing any two means of transportation one can go from any city to any other without using the third kind of vehicle?

Answer: 36


Let $x_1=1$ and $x_{n+1} =x_n+\left\lfloor \frac{x_n}{n}\right\rfloor +2$, for $n=1,2,3,\ldots $ where $x$ denotes the largest integer not greater than $x$. Determine $x_{1997}$.

Answer: 23913


We say that some positive integer $m$ covers the number $1998$, if $1,9,9,8$ appear in this order as digits of $m$. (For instance $1998$ is covered by $2\textbf{1}59\textbf{9}36\textbf{98}$ but not by $213326798$.) Let $k(n)$ be the number of positive integers that cover $1998$ and have exactly $n$ digits ($n\ge 5$), all different from $0$. What is the remainder of $k(n)$ on division by $8$?

Answer: 1


A photographer took some pictures at a party with $10$ people. Each of the $45$ possible pairs of people appears together on exactly one photo, and each photo depicts two or three people. What is the smallest possible number of photos taken?

Answer: 19


Bob’s Rice ID number has six digits, each a number from $1$ to $9$, and any digit can be used any number of times. The ID number satifies the following property: the first two digits is a number divisible by $2$, the first three digits is a number divisible by $3$, etc. so that the ID number itself is divisible by $6$. One ID number that satisfies this condition is $123252$. How many different possibilities are there for Bob’s ID number?

Answer: 324


A man named Juan has three rectangular solids, each having volume $128$. Two of the faces of one solid have areas $4$ and $32$. Two faces of another solid have areas $64$ and $16$.  Finally, two faces of the last solid have areas $8$ and $32$. What is the minimum possible exposed surface area of the tallest tower Juan can construct by stacking his solids one on top of the other, face to face? (Assume that the base of the tower is not exposed.)

Answer: 688


G. H. Hardy once went to visit Srinivasa Ramanujan in the hospital, and he started the conversation with: “I came here in taxi-cab number $1729$. That number seems dull to me, which I hope isn’t a bad omen.” “Nonsense,” said Ramanujan. “The number isn’t dull at all. It’s quite interesting. It’s the smallest number that can be expressed as the sum of two cubes in two different ways.” Ramujan had immediately seen that $1729=12^3+1^3=10^3+9^3$. What is the smallest positive integer representable as the sum of the cubes of [i]three[/i] positive integers in two different ways?

Answer: 251


Pyramid $EARLY$ has rectangular base $EARL$ and apex $Y$, and all of its edges are of integer length. The four edges from the apex have lengths $1, 4, 7, 8$ (in no particular order), and EY is perpendicular to $YR$. Find the area of rectangle $EARL$.

Answer: 28


1. A set of integers $T$ is orensan if there exist integers $a<b<c$ such that $a$ and $c$ belong to $T$ and $b$ does not belong to $T$. Find the number of subsets $T$ of $\{1,2, \ldots, 2019\}$ that are orensan.

Answer: 2^{2019} - \frac{2019 \times 2020}{2} - 1


24. There is a sequence of numbers: $1,4,7,10,13, \cdots \cdots, 2017,2020$, add a “+” or “-” sign in front of each number, then add all the numbers together. If the result is a non-negative number, what is the smallest this non-negative number can be? $\qquad$

Answer: 1


6. Let $A$ be a set of ten real-coefficient quadratic polynomials. It is known that there exist $k$ consecutive positive integers $n+1$, $n+2, \cdots, n+k$, and $f_{i}(x) \in A(1 \leqslant i \leqslant k)$, such that $f_{1}(n+1), f_{2}(n+2), \cdots, f_{k}(n+k)$ form an arithmetic sequence. Find the maximum possible value of $k$.

Answer: 20


As shown in Figure 3, from a point $P$ outside circle $\odot O$, two secants $PAB$ and $PCD$ are drawn, intersecting $\odot O$ at points $A, B$ and $C, D$ respectively. Chords $AD$ and $BC$ intersect at point $Q$, and $PQ$ intersects $\odot O$ at point $E$. If $PE = \frac{12}{5}$, the radius of $\odot O$ is 3, and the distance between points $O$ and $P$ is 5, find the length of $QE$.

Answer: \dfrac{96}{85}


The following figure shows a [i]walk[/i] of length 6:
[asy]
  unitsize(20);
  for (int x = -5; x <= 5; ++x)
    for (int y = 0; y <= 5; ++y)
      dot((x, y));
  label("$O$", (0, 0), S);
  draw((0, 0) -- (1, 0) -- (1, 1) -- (0, 1) -- (-1, 1) -- (-1, 2) -- (-1, 3));
[/asy]
This walk has three interesting properties:
[list]
[*]  It starts at the origin, labelled $O$.
[*]  Each step is 1 unit north, east, or west.  There are no south steps.
[*]  The walk never comes back to a point it has been to.[/list]

Let's call a walk with these three properties a [i]northern walk[/i].  There are 3 northern walks of length 1 and 7 northern walks of length 2.  How many northern walks of length 6 are there?

Answer: 239


A lock with three buttons opens if the buttons are pressed in a predetermined order, one after the other. What is the minimum number of button presses needed to ensure that the lock will open? (The correct sequence of three button presses is not affected by any preceding button presses.)

Translating the problem statement into English as requested, while preserving the original formatting and line breaks.

Answer: 9


9. (16 points) Given $a, b>0, a+b=1$. Find
$$
y=\left(a+\frac{1}{2015 a}\right)\left(b+\frac{1}{2015 b}\right)
$$

the minimum value.

Answer: \dfrac{2(\sqrt{2016} - 1)}{2015}


On the fields of a chesstable of dimensions $ n\times n$, where $ n\geq 4$ is a natural number, are being put coins. We shall consider a [i]diagonal[/i] of table each diagonal formed by at least $ 2$ fields. What is the minimum number of coins put on the table, s.t. on each column, row and diagonal there is at least one coin? Explain your answer.

Answer: 2n - 3


Given is an integer $k\geq 2$. Determine the smallest positive integer $n$, such that, among any $n$ points in the plane, there exist $k$ points among them, such that all distances between them are less than or equal to $2$, or all distances are strictly greater than $1$.

Answer: (k-1)^2 + 1


4.1.1. (12 points) From point $A$ to point $B$, a bus and a cyclist departed simultaneously at 13:00. After arriving at point $B$, the bus, without stopping, headed back and met the cyclist at point $C$ at 13:10. Upon returning to point $A$, the bus again, without stopping, headed to point $B$ and caught up with the cyclist at point $D$, which is located $\frac{2}{3}$ km from point $C$. Find the speed of the bus (in km/h), if the distance between points $A$ and $B$ is 4 km, and the speeds of the bus and the cyclist are constant.

Answer: 40


B2. Jure wrote down the natural numbers from 1 to 2015 on a board. Urška then reviewed the written numbers from the smallest to the largest and erased every number that was not divisible by 3. Among the numbers that remained on the board, she then erased every number that was not divisible by $3^{2}$ from the smallest to the largest. Among the remaining numbers, she then erased every number that was not divisible by $3^{3}$ from the smallest to the largest, and so on. Which number did Urška erase last?

Answer: 1458


## 51. Congratulations from Methuselah

Every New Year, starting from the first year of our era, Methuselah, who is still alive to this day, sends a greeting to his best friend, who, naturally, has changed many times over the centuries and decades. However, the formula for the greeting, on the contrary, has remained unchanged for almost two millennia. It is very simple: "Happy New Year 1", "Happy New Year 2", "Happy New Year 3", and so on, "Happy New Year 1978" and finally, "Happy New Year 1979".

Which digit has Methuselah used the least so far?

Answer: 0


At an international combinatorial conference, a hundred mathematicians are accommodated in a hotel where the rooms are numbered from one to a hundred. The receptionist plans to assign the mathematicians to the rooms corresponding to their arrival order. However, the first arriving guest is forgotten to be given the correct instruction, so they randomly choose one of the rooms. In the end, the receptionist instructs the rest to sequentially occupy the room corresponding to their arrival order; if that room is already occupied, they should randomly choose one of the available rooms. In how many ways can the guests be accommodated in the rooms?

Answer: 2^{99}


Some people know each other in a group of people, where "knowing" is a symmetric relation. For a person, we say that it is $social$ if it knows at least $20$ other persons and at least $2$ of those $20$ know each other. For a person, we say that it is $shy$ if it doesn't know at least $20$ other persons and at least $2$ of those $20$ don't know each other. Find the maximal number of people in that group, if we know that group doesn't have any $social$ nor $shy$ persons.

Answer: 40


The number 1 was written on the board. Every minute, restless Sasha obtains a new natural number by trying to subtract $2^{100}$ from the last recorded number. If it turns out that the result had already been recorded earlier, or the current number is less than $2^{100}$, then Sasha adds $3^{100}$ instead. Will the number $5^{100}$ ever appear on the board?

Answer: Yes


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 $\log a-b$, with $a$ and $b$ positive integers. Please provide the value of a-b. Here $[x]$ is defined to be the integer such that $[x] \leqq x<[x]+1$ and $\log x$ is the logarithm of $x$ to base $e$.

Answer: 3


For each $t\geqslant0$, let $S_{t}$ be the set of all nonnegative, increasing, convex, continuous, real-valued functions $f(x)$ defined on the closed interval $[0,1]$ for which $$ f(1)-2 f(2 / 3)+f(1 / 3)\geqslant t[f(2 / 3)-2 f(1 / 3)+f(0)] . $$ Develop necessary and sufficient conditions on $t$ for $S_{t}$ to be closed under multiplication. Please provide the sum of the endpoints of the interval for which the condition holds.

Answer: 1


For each $t\geqslant0$, let $S_t$ be the set of all nonnegative, increasing, convex, continuous, real-valued functions $f(x)$ defined on the closed interval $[0,1]$ for which $$f(1)-2f(2/3)+f(1/3)\geqslant[f(2/3)-2f(1/3)+f(0)]$$. Develop necessary and sufficient conditions on $t$ for $S_t$ to be closed under multiplication. Please provide the value of a + b if the original answer is in the format $a\leqslant t\leqslant b$.

Answer: 1


Let r and s be positive integers. Derive a formula for the number of ordered quadruples $(a, b, c, d)$ of positive integers such that
$$3^{r}·7^{s}=
\mathrm{lcm}[a,b,c]=\mathrm{lcm}[a,b,d]=\mathrm{lcm}[a,c,d]=\mathrm{lcm}[b,c,d]$$.
Please provide the value of the formula when r = 2 and s = 1. The answer should be the sum of the coefficients of the expanded expression of the original answer.

Answer: 1089


Find positive integers $n$ and $a_{1}, a_{2}, \ldots, a_{n}$ such that \n$$\na_{1}+a_{2}+\cdots+a_{n}=1979\n$$\nand the product $a_{1} a_{2} \cdots a_{n}$ is as large as possible. Please provide the value of $a + n$.

Answer: 662


Let $C$ be the class of all real valued continuously differentiable functions $f$ on the interval $0<x<1$ with $f(0)=0$ and $f(1)=1$. Determine the largest real number $u$ such that
$$
u \leqslant \int_{0}^{1}\left|f^{\prime}(x)-f(x)\right| d x$$
for all $f$ in $C$. The original answer is in \(\frac{m}{ne}\) format, please provide the value of m+ n.

Answer: 2


A point P moves inside a unit square in a straight line at unit speed. When it meets a corner it escapes. When it meets an edge its line of motion is reflected so that the angle of incidence equals the angle of reflection. Let $N(T)$ be the number of starting directions from a fixed interior point $P_{0}$ for which P escapes within T units of time. Find the least constant a for which constants b and c exist such that $$ N(T) \leq a T^{2}+b T+c $$ for all $T>0$ and all initial points $P_{0}$. The original answer is in \(\pi\) format, please give the integer part of the answer.

Answer: 1


Find the area of a convex octagon that is inscribed in a circle and has four consecutive sides of length 3 units and the remaining four sides of length 2 units. The answer is in the form $r+s \sqrt{t}$ with $r, s$, and $t$ being positive integers. Please provide the value of $r + s + t$.

Answer: 27


Three numbers are chosen independently at random, one from each of the three intervals $[0, L_i](i=1, 2,3)$. If the distribution of each random number is uniform with respect to length in the interval it is chosen from, determine the expected value of the smallest of the three numbers chosen. The original answer is in the format $\frac{a}{b} L_{1}^{2} L_{2} L_{3}-\frac{c}{d} L_{1}^{3}\left(L_{2}+L_{3}\right)+\frac{e}{f} L_{1}^{4}$, please give the value of a+b+c+d+e+f.

Answer: 23


(a) Find a solution that is not identically zero, of the homogeneous linear differential equation $$( 3x^2+x-1 ){y}''- ( 9x^2+9x-2 ){y}'+( 18x+3 )y=0$$ Intelligent guessing of the form of a solution may be helpful.
(b) Let $y=f(x)$ be the solution of the nonhomogeneous differential equation $$( 3x^2+x-1){y}''-( 9x^2+9x-2){y}'+( 18x+3)=6( 6x+1) $$ that has $f(0)=1$ and $(f(-1)-2)(f(1)-6)=1$. Find integers $a, b, c$ such that $(f(-2)-a)(f(2)-b)=c$. Please provide the value of a + b + c.

Answer: 21


A sequence of convex polygons $\{P_n\}$, $n\geqslant0$, is defined inductively as follows. $P_0$ is an equilateral triangle with sides of length 1. Once $P_n$ has been determined, its sides are trisected; the vertices of $P_{n+1}$ are the interior trisection points of the sides of $P_n$. Thus, $P_{n+1}$ is obtained by cutting corners off $P_n$, and $P_n$ has $3\cdot2^n$ sides. ($P_1$ is a regular hexagon with sides of length $1/3$.) Express $\lim_{n\rightarrow\infty}Area(P_n)$ in the form $\sqrt{a}/b$, where a and b are positive integers. Please provide the value of a + b.

Answer: 10


Express
$$\sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{1}{m^{2} n+m n^{2}+2 m n}$$
as a rational number. The original answer is in \(\frac{m}{n}\) format, please give the value of m + n.

Answer: 11


Let $S={(a, b) \mid a=1,2,..., n, b=1,2,3}$. A rook tour of $S$ is a polygonal path made up of line segments connecting points $p_1, p_2,\dots,p_{3n}$ in sequence such that  (i) $p_i \in S$,  (ii) $p_i$ and $p_{i+1}$ are a unit distance apart, for $1 \le i< 3n$,  (iii) for each $p\in S$ there is a unique i such that $p_i =p$. How many rook tours are there that begin at $(1,1)$ and end at $(n,1)$? Please provide the value of the number of rook tours when n = 5.

Answer: 8


Let $a_1, a_2, \ldots, a_n$ be real numbers, and let $b_1, b_2, \ldots, b_n$ be distinct positive integers. Suppose there is a polynomial $f(x)$ satisfying the identity $$(1-x)^n f(x)=1+\sum_{i=1}^n a_i x^{b_i}.$$ Find a simple expression (not involving any sums) for $f(1)$ in terms of $b_1, b_2, \ldots, b_n$ and $n$ (but independent of $a_1, a_2, \ldots, a_n$). Provide the value of $b_1 + b_n + n$ for $b_1 = 2$, $b_n = 3$, and $n = 2$.

Answer: 7


From a compact regular tetrahedron with unit edge length, we cut off a regular tetrahedron of edge length $\frac{1}{n}$ at each vertex, where $n>1$ is an integer. For what value of $n$ will the ratio of the surface area to the volume of the remaining solid be the smallest?

Answer: 3


Answer: 0


The numbers $p, q, r$, and $t$ satisfy $p<q<r<t$. When these numbers are paired, each pair has a different sum and the four largest sums are 19,22,25, and 28 . What is the the sum of the possible values for $p$ ?

Answer: \dfrac{17}{2}


Task 1 - 261241
500 candies are to be packed into a joke package using wrapping materials of suitable sizes, such that the following conditions (1), (2) are met.

Condition (2) should apply to every possible way of unpacking all the candies by gradually opening and removing an accessible wrapping (if multiple wrappings are accessible, in any order):

(1) There is exactly one wrapping that contains the entire package.

(2) When opening this and each subsequent wrapping, it is revealed that its content consists either of at least three individually wrapped sub-packages or of exactly one unwrapped candy.

Determine the maximum number of wrappings that such a package can have!

Answer: 749


Given that the dihedral angle between the lateral face and the base of a regular quadrilateral prism $P-ABCD$ is $\alpha$, and the dihedral angle between two adjacent lateral faces is $\beta$. Prove that: $\cos \beta = -\cos^2 \alpha$.

Answer: \cos \beta = -\cos^2 \alpha


5.103 Suppose there are 100 mutually hostile countries on Mars. To maintain peace, it is decided to form several alliances, with each alliance including at most 50 countries, and any two countries must belong to at least one alliance. Try to answer the following questions:
(1) What is the minimum number of alliances needed to meet the above requirements?
(2) If it is also required that the union of any two alliances does not exceed 80 countries, what would be the result?

Answer: 6


Four. (50 points) Take the subset $A_{i}=\left\{a_{i}, a_{i+1}, \cdots, a_{i+59}\right\}(i=1,2, \cdots$, 70 ) of the set $S=\left\{a_{1}, a_{2}, \cdots, a_{70}\right\}$, where $a_{70+i}=a_{i}$. If there exist $k$ sets among $A_{1}, A_{2}, \cdots, A_{70}$ such that the intersection of any seven of them is non-empty, find the maximum value of $k$.

---

The problem is asking to find the maximum number of sets $k$ from the collection $\{A_1, A_2, \ldots, A_{70}\}$ such that the intersection of any seven of these sets is non-empty. Each set $A_i$ is defined as a subset of $S$ containing 60 consecutive elements, with the indices wrapping around modulo 70.

Answer: 60


Example 5 In $\triangle A B C$, $E$ is the midpoint of $B C$, and $D$ is on $A C$. If $\angle B A C=60^{\circ}, \angle A C B=20^{\circ}$, $\angle D E C=80^{\circ}, S_{\triangle A B C}+2 S_{\triangle C D E}=\sqrt{3}$, find the length of $A C$.

Answer: 2\sqrt{2}


7. Let the three interior angles $\angle A, \angle B, \angle C$ of $\triangle ABC$ satisfy $\angle A=3 \angle B=9 \angle C$. Then
$$
\begin{array}{l}
\cos A \cdot \cos B+\cos B \cdot \cos C+\cos C \cdot \cos A \\
=\quad .
\end{array}
$$

Answer: -\dfrac{1}{4}


Adva van három egyenló sugarú, egymást érintő kör. Mekkora a körök között fekvő idom területe, ha mindegyik kör sugara $r$ ?

Answer: r^2 \left( \sqrt{3} - \frac{\pi}{2} \right)


Exercise 5. In a classroom, there are ten students. Aline writes ten consecutive integers on the board. Each student chooses one of the ten integers written on the board, such that any two students always choose two different integers. Each student then calculates the sum of the nine integers chosen by the other nine students. Each student whose result is a perfect square then receives a gift.

What is the maximum number of students who will receive a gift?

A perfect square is an integer of the form $n^{2}$, where $n$ is a natural number.

Answer: 4


1. Use the Euclidean algorithm from Theorem 4 in $\S 3$ to find the greatest common divisor (gcd) of the following sets of numbers, and express the gcd as an integer linear combination of these numbers:
(i) 1819, 3587;
(ii) 2947, 3997;
(iii) $-1109, 4999$.

Answer: 17


Let $n \geq 2$ be an integer. Lucia chooses $n$ real numbers $x_1,x_2,\ldots,x_n$ such that $\left| x_i-x_j \right|\geq 1$ for all $i\neq j$. Then, in each cell of an $n \times n$ grid, she writes one of these numbers, in such a way that no number is repeated in the same row or column. Finally, for each cell, she calculates the absolute value of the difference between the number in the cell and the number in the first cell of its same row. Determine the smallest value that the sum of the $n^2$ numbers that Lucia calculated can take.

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


[ Remarkable property of a trapezoid ] [ Transferring sides, diagonals, etc. ]

In quadrilateral $A B C D$, diagonals $A C$ and $B D$ intersect at point $K$. Points $L$ and $M$ are the midpoints of sides $B C$ and $A D$, respectively. Segment $L M$ contains point $K$. Quadrilateral $A B C D$ is such that a circle can be inscribed in it. Find the radius of this circle if $A B=3, A C=\sqrt{13}$, and $L K: K M=1: 3$.

Answer: \dfrac{3}{2}


In isosceles trapezoid $ABCD$ , parallel bases $\overline{AB}$ and $\overline{CD}$ have lengths $500$ and $650$ , respectively, and $AD=BC=333$ . The angle bisectors of $\angle{A}$ and $\angle{D}$ meet at $P$ , and the angle bisectors of $\angle{B}$ and $\angle{C}$ meet at $Q$ . Find $PQ$

Answer: 242


10. All edges of the regular quadrilateral pyramid $S A B C D$ are equal to $b$. The height of the regular quadrilateral prism $A B C D A_{1} B_{1} C_{1} D_{1}$ is equal to the height of the specified pyramid. The base $A B C D$ is common to both. (1) A plane $\alpha$ is drawn through points $A$ and $C_{1}$ parallel to the line $B D$. Find the area of the polygon formed by the intersection of plane $\alpha$ and the pyramid $S A B C D$.

(2) A plane $\beta$ is drawn through points $B$ and $D_{1}$. Find the minimum possible area of the polygon formed by the intersection of plane $\beta$ and the pyramid $S A B C D$.

- $b^{2} \sqrt{5} / 6$. This value is also equal to the smallest area of the polygon specified in the problem conditions.

Answer: \dfrac{b^2 \sqrt{5}}{6}


1. The digit at the 2007th position after the decimal point of the irrational number $0.2342343423434342343434342 \cdots$ is $\qquad$ .

Answer: 3


The number written on the board is allowed to be multiplied by 5 or to rearrange its digits (zero cannot be placed at the first position). Is it possible to obtain a 150-digit number $5222 \ldots 2223$ from the number 1 in this way?

Answer: \text{No}


13. In the sequence $\left\{a_{n}\right\}$, $a_{n}=a \cdot 2^{n}+b \cdot n-80$, where $a, b \in \mathbf{N}_{+}$. It is known that the sum of the first $n$ terms $S_{n}$ reaches its minimum value if and only if $n=6$, and $a_{36}$ is divisible by 7. Find the value of $\left|a_{1}\right|+\left|a_{2}\right|+\cdots+\left|a_{12}\right|$.

Answer: 8010


3. In parallelograms $A B C D$ and $A E F D$, heights $B P=5$ and $F Q=9$ are drawn to side $A D$. Find the length of the perpendicular from point $C$ to $E F$. If there are multiple possible values, list them in any order separated by a semicolon.

Answer: 4


## Task B-2.1.

A circle and two semicircles are inscribed in a square as shown in the figure. What is the area of the square if the area of the shaded part is equal to $48 \pi - 36 \sqrt{3}$?

![](https://cdn.mathpix.com/cropped/2024_05_30_ea3ef29009412d1e0d2bg-06.jpg?height=423&width=439&top_left_y=765&top_left_x=800)

Answer: 144


A competition involving $n\ge 2$ players was held over $k$ days. In each day, the players received scores of $1,2,3,\ldots , n$ points with no players receiving the same score. At the end of the $k$ days, it was found that each player had exactly $26$ points in total. Determine all pairs $(n,k)$ for which this is possible.

Answer: (3, 13)


2. The sum of all four-digit natural numbers divisible by 45, where the tens digit is three times the hundreds digit, minus the sum of all three-digit natural numbers divisible by 18, where the hundreds digit is the largest one-digit prime number.

Answer: 37098


10.2. What is the greatest number of consecutive natural numbers, each of which has exactly four natural divisors (including 1 and the number itself)?

Answer: 3


Let \(\{a_{1}, a_{2}, \cdots, a_{n}\}\) be a sequence of non-negative real numbers. Prove that for any positive integer \(n\), the inequality \(\sum_{i=1}^{n} x_{i}^{2} \leqslant 4 \sum_{i=1}^{n} a_{i}^{2}\) holds, where \(x_{i}\) is the arithmetic mean of \(a_{1}, a_{2}, \cdots, a_{i}\), i.e., \(x_{i} = \frac{a_{1} + a_{2} + \cdots + a_{i}}{i}\).

Answer: \sum_{i=1}^{n} x_{i}^{2} \leqslant 4 \sum_{i=1}^{n} a_{i}^{2}


1. In still water, the speed of boat A is twice the speed of boat B. Boats A and B start from points $A$ and $B$ respectively at the same time, moving towards each other, and meet at a point where the distances from $A$ and $B$ are in the ratio of 3:1. If boats A and B start from points $B$ and $A$ respectively at the same time, moving towards each other, the ratio of the distances from $A$ and $B$ when they meet is $\qquad$.

Answer: \dfrac{5}{7}


# 2. Clone 1

The teacher wanted to write an example for calculation on the board:

$$
1,05+1,15+1,25+1,4+1,5+1,6+1,75+1,85+1,95=?
$$

but accidentally forgot to write one comma. After this, Kolya went to the board and, correctly performing all the operations, obtained an integer result. What is it?

Answer: 27


5. The number 2015 was divided into 12 addends, after which all numbers that can be obtained by adding some of these addends (from one to nine) were written down. What is the minimum number of numbers that could have been written down?

Answer: 10


I4.1 Suppose the 9-digit number $\overline{32 x 35717 y}$ is a multiple of 72 , and $P=x y$, find the value of $P$.
I4.2 Given that the lines $4 x+y=\frac{P}{3}, m x+y=0$ and $2 x-3 m y=4$ cannot form a triangle. Suppose that $m>0$ and $Q$ is the minimum possible value of $m$, find $Q$.

I4.3 Given that $R, x, y, z$ are integers and $R>x>y>z$. If $R, x, y, z$ satisfy the equation $2^{R}+2^{x}+2^{y}+2^{z}=\frac{495 Q}{16}$, find the value of $R$.

I4.4 In Figure $1, Q$ is the interior point of $\triangle A B C$. Three straight lines passing through $Q$ are parallel to the sides of the triangle such that $F E / / A B, G K / / A C$ and $H J / / B C$. Given that the areas of $\triangle K Q E, \triangle J F Q$ and $\triangle Q G H$ are $R, 9$ and 49 respectively. If the area of $\triangle A B C$ is $S$, find the value of $S$.

Answer: 12


Sure, here is the translated text:

```
II. Fill-in-the-Blanks (9 points each, total 54 points)
1. Let the three interior angles of $\triangle A B C$ be $\angle A, \angle B, \angle C$ with the corresponding side lengths $a, b, c$. If $a < b < c$, and
$$
\left\{\begin{array}{l}
\frac{b}{a}=\frac{\left|b^{2}+c^{2}-a^{2}\right|}{b c}, \\
\frac{c}{b}=\frac{\left|c^{2}+a^{2}-b^{2}\right|}{c a}, \\
\frac{a}{c}=\frac{\left|a^{2}+b^{2}-c^{2}\right|}{a b},
\end{array}\right.
$$

then the radian measures of $\angle A, \angle B, \angle C$ are in the ratio $\qquad$ .
```

Answer: 1 : 2 : 4


Let $X$ be a point of intersection of circles $w_{1}$ and $w_{2}$ such that $X$ and $B$ lie on opposite sides of line $AC$. Prove that the point $X$ then lies on the altitude $BH$ of triangle $ABC$.

Answer: X \text{ lies on the altitude } BH


5、In the tetrahedron $A-BCD$, $\triangle BCD$ and $\triangle ACD$ are both equilateral triangles with side length 2, and $\triangle BCD$ lies in plane $\boldsymbol{\alpha}$. The side edge $AB=\sqrt{3}$. Now, four of the eight labels numbered 1 to 8 are randomly attached to its four vertices, and the corresponding labels are denoted as $\boldsymbol{f}(\boldsymbol{\eta}),(\boldsymbol{\eta}$ taking values $A, B, C, D)$. $E$ is a point on the side edge $AB$. If $|BE|:|EA|=f(B): f(A)$, then the probability that the dihedral angle $E-CD-A$ is greater than $\frac{\pi}{4}$ is $\qquad$

Answer: \dfrac{9}{56}


Consider the illustrated diagram below:

![](https://cdn.mathpix.com/cropped/2024_05_01_cc59d970f26105cc4ca1g-11.jpg?height=315&width=1174&top_left_y=445&top_left_x=544)

Augusto likes to count paths starting from some point, reaching point $A$ and never passing through the same vertex twice. To do this, he represents a path by the sequence of points that the path visits. For example, the dotted path in the figure below is represented by the sequence $D C B A$.

![](https://cdn.mathpix.com/cropped/2024_05_01_cc59d970f26105cc4ca1g-11.jpg?height=302&width=1168&top_left_y=1034&top_left_x=547)

Augusto calls a path unusual if the sequence representing this path is in reverse alphabetical order. In other words, the path is unusual if it never moves to the left, whether going up or down. For example, the path $D C B A$ is unusual. The path $D B C A$ is not unusual, since the letter $C$ appears before the letter $B$.

a) How many unusual paths exist starting from $D$ and ending at $A$?

b) Show that the number of unusual paths starting from $E$ is the sum of the number of unusual paths starting from $D$ and the number of unusual paths starting from $C$.

c) Augusto calculated the number of unusual paths starting from $K$ and ending at $A$. What is this number?

Answer: 3


In an isosceles triangle $A B C \quad \angle B=\operatorname{arctg} 8 / 15$. A circle of radius 1, inscribed in angle $C$, touches side $C B$ at point $M$ and cuts off a segment $K E$ from the base. It is known that $M B=15 / 8$. Find the area of triangle $K M B$, given that points $A, K, E, B$ follow on the base $A B$ in the specified order.

Answer: \dfrac{375}{272}


Example $10 f(x)$ is a continuous function defined on the interval $[0,2015]$, and $f(0)=f(2015)$. Find the minimum number of real number pairs $(x, y)$ that satisfy the following conditions.
(1) $f(x)=f(y)$;
(2) $x-y \in \mathbf{Z}_{+}$.

Answer: 2015


We build a tower of $2\times 1$ dominoes in the following way. First, we place $55$ dominoes on the table such that they cover a $10\times 11$ rectangle; this is the first story of the tower. We then build every new level with $55$ domioes above the exact same $10\times 11$ rectangle. The tower is called [i]stable[/i] if for every non-lattice point of the $10\times 11$ rectangle, we can find a domino that has an inner point above it. How many stories is the lowest [i]stable[/i] tower?

Answer: 5


3. Variant 1.

The height $A H$ and the bisector $C L$ of triangle $A B C$ intersect at point $O$. Find the angle $B A C$, if it is known that the difference between the angle $C O H$ and half the angle $A B C$ is $46^{\circ}$.

Answer: 92^\circ


12. Let the function $f(x)$ be a differentiable function defined on $(-\infty, 0)$, with its derivative being $f^{\prime}(x)$, and it satisfies $2 f(x)+x f^{\prime}(x)>x^{2}$. Then the solution set of the inequality $(x+2017)^{2} f(x+2017)-f(-1)>0$ is $\qquad$ .

Answer: (-\infty, -2018)


Let $z_1 = 18 + 83i$ , $z_2 = 18 + 39i,$ and $z_3 = 78 + 99i,$ where $i = \sqrt{-1}$ . Let $z$ be the unique complex number with the properties that $\frac{z_3 - z_1}{z_2 - z_1} \cdot \frac{z - z_2}{z - z_3}$ is a real number and the imaginary part of $z$ is the greatest possible. Find the real part of $z$ .
Please give the answer directly without any intermediate steps.

Answer: 56


6. On the table, there are 2021 piles of nuts, with one nut in each pile at the beginning. Petya and Vasya are playing the following game (Petya moves first). Each move consists of combining three piles that have the same number of nuts. The player who cannot make a move loses. Who will win with correct play, and how should they play?

Answer: Petya


1. Let integer $n \geqslant 2$, set
$$
A_{n}=\left\{2^{n}-2^{k} \mid k \in \mathbf{Z}, 0 \leqslant k<n\right\} \text {. }
$$

Find the largest positive integer such that it cannot be written as the sum of one or more numbers (repetition allowed) from the set $A_{n}$.

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


4. On a plane, there are 10 different points. We considered the midpoints of all segments connecting all pairs of points. What is the smallest number of midpoints that could have arisen?

Answer: 17


Let n be an odd positive integer.Let M be a set of  $n$  positive integers ,which are 2x2 different. A set  $T$   $\in$   $M$  is called "good" if the product of its elements is divisible by the sum of the elements in  $M$ , but is not divisible by the square of the same sum. Given that  $M$  is "good",how many "good" subsets of  $M$  can there be?

Answer: 2^{n-1}


4B. Find all values of the real parameter $a \neq 0,1$ for which the equations $x^{2}-(2 a+1) x+a=0$ and $x^{2}+(a-4) x+a-1=0$ have real roots $x_{1}, x_{2}$ and $x_{3}, x_{4}$, respectively, such that the equality

$$
\frac{x_{1}}{x_{3}}+\frac{x_{4}}{x_{2}}=\frac{x_{1} x_{4}\left(x_{1}+x_{2}+x_{3}+x_{4}\right)}{a}
$$

is satisfied.

Answer: -1 + \sqrt{5}


## Task A-4.4.

Let $\overline{B D}$ and $\overline{C E}$ be the altitudes of the acute triangle $A B C$. The circle with diameter $\overline{A C}$ intersects the segment $\overline{B D}$ at point $F$. The circle with diameter $\overline{A B}$ intersects the line $C E$ at points $G$ and $H$, with $G$ between $C$ and $E$. If $\varangle C H F=12^{\circ}$, determine $\varangle A G F$.

Answer: 78^\circ


96. Incorrect scales. When the pudding was placed on one pan of the scales, they showed 4 g more than $\frac{9}{11}$ of its true weight. When it was placed on the other pan, the scales showed 48 g more than in the first case. What is the true weight of the pudding?

![](https://cdn.mathpix.com/cropped/2024_05_21_56f37ca27ac3d928838dg-031.jpg?height=371&width=508&top_left_y=2079&top_left_x=774)

Answer: 143


(12) (22 points) Suppose a set of planar points $S$ has the properties: (i) no three points are collinear; (ii) the distances between any two points are all different. For two points $A$ and $B$ in $S$, if there exists $C \in S$ such that $|A C|<|A B|<|B C|$, then $A B$ is called a middle edge of $S$. For three points $A$, $B$, and $C$ in $S$, if $A B$, $A C$, and $B C$ are all middle edges of $S$, then $\triangle A B C$ is called a middle edge triangle of $S$. Find the smallest $n$ such that any $n$-element planar point set $S$ with properties (i) and (ii) must contain a middle edge triangle.

Answer: 6


It is given that the roots of the polynomial $P(z) = z^{2019} - 1$ can be written in the form $z_k = x_k + iy_k$ for $1\leq k\leq 2019$.  Let $Q$ denote the monic polynomial with roots equal to $2x_k + iy_k$ for $1\leq k\leq 2019$.  Compute $Q(-2)$.

Answer: -\dfrac{3^{2019} + 1}{2^{2018}}


12. Using the digits $1,2,3, \cdots 9$, without repetition, form 3 three-digit numbers such that the sum of the first two equals the third, for example: $245+736=981$. The minimum value of the sum of these three three-digit numbers is $\qquad$

Answer: 918


4. From the town "U ${ }^{\prime}$ " to the town " $A x$ ", Ivan set off on his bicycle at $11^{00}$ AM, having traveled two fifths of the distance, he passed the town " $O x$ ", from which at that moment Peter set off on foot towards the town "Ux". At the moment when Ivan arrived in the town " $A x$ ", from there in the opposite direction Nikolai set off on his bicycle and arrived in the town "Ux" at $12^{00}$ of the same day. How many kilometers from the town " $A x$ " did Nikolai catch up with Peter, if Peter arrived in the town "U $\$ " at $13^{30}$ of the same day, while the speed of each participant was constant, and the distance between the towns "U" and " $A x$ " is

7 km.

( 7 points)

Answer: 5


On sides $A B$ and $A C$ of triangle $A B C$, points $E$ and $F$ are taken, respectively. Segments $B F$ and $C E$ intersect at point $D$. Find the area of triangle $A E F$ if the areas of triangles $B C D$, $B D E$, and $C D F$ are $1$, $\frac{1}{2}$, and $\frac{1}{6}$, respectively.

Answer: \dfrac{7}{44}


49. A ship sails downstream from Port A to Port B, and upon arrival, immediately returns upstream to Port A, taking a total of 12 hours. It is known that the ship travels 16 kilometers more per hour downstream than upstream, and it also travels 80 kilometers more in the first 6 hours than in the last 6 hours. Therefore, the distance between Port A and Port B is $\qquad$ kilometers.

Answer: 280


3. Let $a, b, c$ be the three sides of a right triangle, with $c$ being the hypotenuse. The maximum value of $k$ such that the inequality $a^{2}(b+c)+b^{2}(c+a)$ $+c^{2}(a+b) \geqslant k a b c$ holds for all right triangles is $\qquad$.

Answer: 2 + 3\sqrt{2}


31. A black ant and a red ant start from two places 78 meters apart at the same time, moving towards each other. The black ant travels 6 meters per minute, and the red ant travels 7 meters per minute. After 1 minute, they both turn around and move away from each other, then after 3 minutes, they both turn around and move towards each other again, then after 5 minutes, they both turn around and move away from each other again, then after 7 minutes, they both turn around and move towards each other again, …‥ According to this pattern, they will first meet after $\qquad$ minutes.

Answer: 48


The bases of two cones, sharing a common vertex, lie in the same plane. The difference in their volumes is $V$. Find the volume of the smaller cone if the tangents drawn to the circle of its base from any point on the circle of the base of the larger cone form an angle $\alpha$.

Answer: V \tan^2 \frac{\alpha}{2}


5. In triangle $\mathrm{ABC}$, the sides $A B=4, B C=6$. Point $M$ lies on the perpendicular bisector of segment $A B$, and lines $A M$ and $A C$ are perpendicular. Find $M A$, if the radius of the circumscribed circle around triangle $A B C$ is 9.

Answer: 6


13.285. On a 10 km stretch of highway, devoid of intersections, the bus stops only for passengers to get on and off. It makes a total of 6 intermediate stops, spending 1 minute at each, and always moves at the same speed. If the bus were to travel without stopping, it would cover the same distance at a speed exceeding the average speed of its movement with stops by 5 km/h. How many minutes is the bus in motion on this stretch of highway?

Answer: 24


14.Can there be two lucky tickets among ten consecutive tickets? A ticket is considered lucky if the sums of its first three and last three digits are equal.

Answer: Yes


Point $O$ divides the segment $A B$ into segments $O A=6$ and $O B=4$. A circle is drawn with center at point $O$, and tangents are drawn from $A$ and $B$ to the circle, intersecting at point $M$, with the points of tangency lying on the same side of the line $A B$. Find the radius of the circle if $O M=12$.

Answer: \dfrac{6\sqrt{21}}{7}


26*. What kind of mapping is the composition of two homotheties in space? If this mapping is a homothety, what will be its coefficient? How to determine the position of its center?

Answer: k_1 k_2


Point $D$ is the midpoint of the lateral edge $B B 1$ of a regular triangular prism $A B C A 1 B 1 C 1$. On the lateral face $A A 1 C 1 C$, a point $E$ is taken, and on the base $A B C$, a point $F$ is taken such that the lines $E B 1$ and $F D$ are parallel. What is the maximum volume that the prism $A B C A 1 B 1 C 1$ can have if $E B 1=1, F D=\frac{3}{4}, E F=\frac{1}{2 \sqrt{3}}$?

Answer: \dfrac{4\sqrt{3}}{27}


Let S be the set of all real numbers greater than -1. Find all functions \( f : S \to S \) such that \( f(x + f(y) + x f(y)) = y + f(x) + y f(x) \) for all \( x \) and \( y \), and \( \frac{f(x)}{x} \) is strictly increasing on each of the intervals \( -1 < x < 0 \) and \( 0 < x \).

Answer: -\dfrac{x}{x + 1}


5. In $\triangle A B C$, it is known that $\angle A B C=44^{\circ}, D$ is a point on side $B C$, satisfying $D C=2 A B, \angle B A D=$ $24^{\circ}$. Then the size of $\angle A C B$ is $\qquad$.

Answer: 22


1. A line parallel to the selected side of a triangle with an area of 27 cuts off a smaller triangle with an area of 3. Find the area of the quadrilateral, three vertices of which coincide with the vertices of the smaller triangle, and the fourth lies on the selected side. Choose the answer option with the number closest to the one you found.

Choose the answer option with the number closest to the one you found.

Answer: 9


As shown in Figure 1, in triangle \( \triangle ABC \), \( AB > AC \), \( I \) is the incenter, and \( D \) is the foot of the perpendicular from \( I \) to side \( BC \). A perpendicular \( AH \) is drawn from point \( A \) to \( BC \), intersecting lines \( BI \) and \( CI \) at points \( P \) and \( Q \) respectively. Let \( O \) be the circumcenter of \( \triangle IPQ \). Extend \( AO \) to intersect side \( BC \) at point \( L \). Let \( N \) be the second intersection point of line \( BC \) and the circumcircle of \( \triangle AIL \). Prove that \(\frac{BD}{CD} = \frac{BN}{CN}\).

Answer: \frac{BD}{CD} = \frac{BN}{CN}


8. (10 points) In $\triangle A B C$, $B D=D E=E C$, $C F: A C=1: 3$. If the area of $\triangle A D H$ is 24 square centimeters more than the area of $\triangle H E F$, find the area of triangle $A B C$ in square centimeters?

Answer: 108


15. (15 points) Two cars, A and B, are traveling towards each other on a road parallel to a railway. A 180-meter-long train is moving in the same direction as car A at a speed of 60 kilometers/hour. The train takes 5 minutes from the moment it catches up with car A to the moment it meets car B. If the train takes 30 seconds to pass car A completely and 6 seconds to pass car B completely. How many minutes after car B meets the train will it meet car A?

Answer: \dfrac{5}{4}


14. (15 points) Let the sum of one hundred natural numbers $x, x+1, x+2, \cdots, x+99$ be $a$. If the sum of the digits of $a$ equals 50, what is the smallest value of $x$?

Answer: 99950


8.4. The volume of a regular pyramid is $8 \pi$. Can its lateral edge be equal to 3.96?

Answer: No


Suppose $X$ is a set with $|X| = 56$. Find the minimum value of $n$, so that for any 15 subsets of $X$, if the cardinality of the union of any 7 of them is greater or equal to $n$, then there exists 3 of them whose intersection is nonempty.

Answer: 41


1701. To determine the average burning duration of light bulbs in a batch of 200 identical boxes, one bulb was taken from each box as a sample. Estimate from below the probability that the average burning duration of the 200 selected light bulbs differs from the average burning duration of the entire batch by less than 5 hours in absolute value, given that the standard deviation of the burning duration of the bulbs in each box is less than 7 hours.

Answer: \dfrac{4951}{5000}


10. Fyodorovna's Tea Party. (7th grade. 3 points). Fyodorovna has 3 teacups. One day, she washed them and started a new life. Each time she wants to drink tea, Fyodorovna randomly picks one of the cups, and then puts it back with the others, but does not wash it again. Find the probability that during the fifth tea party since the start of her new life, Fyodorovna uses the last clean cup.

Answer: \dfrac{14}{81}


IMO 1976 Problem A1 A plane convex quadrilateral has area 32, and the sum of two opposite sides and a diagonal is 16. Determine all possible lengths for the other diagonal.

Answer: 8\sqrt{2}


Example 3. As shown in the figure, in $\triangle ABC$, $DE \parallel BC$, and $S_{\triangle ADE} : S_{\triangle CDE} = 1: 3$. Find the value of $S_{\triangle ADE} : S_{\triangle CDS}$. (1990, Sichuan Province Junior High School Mathematics Joint Competition)

Answer: 1:12


The world-renowned Marxist theorist [i]Joric[/i] is obsessed with both mathematics and social egalitarianism. Therefore, for any decimal representation of a positive integer $n$, he tries to partition its digits into two groups, such that the difference between the sums of the digits in each group be as small as possible. Joric calls this difference the [i]defect[/i] of the number $n$. Determine the average value of the defect (over all positive integers), that is, if we denote by $\delta(n)$ the defect of $n$, compute \[\lim_{n \rightarrow \infty}\frac{\sum_{k = 1}^{n}\delta(k)}{n}.\]
[i]Iurie Boreico[/i]

Answer: \dfrac{1}{2}


13. A7 (RUS) For a polynomial $P$ of degree 2000 with distinct real coefficients let $M(P)$ be the set of all polynomials that can be produced from $P$ by permutation of its coefficients. A polynomial $P$ will be called $n$-independent if $P(n)=0$ and we can get from any $Q$ in $M(P)$ a polynomial $Q_{1}$ such that $Q_{1}(n)=0$ by interchanging at most one pair of coefficients of $Q$. Find all integers $n$ for which $n$-independent polynomials exist.

Answer: 0


9.8 Dima calculated the reciprocal of the factorial of each integer from 80 to 100. He printed the resulting decimal fractions on 19 infinitely long strips of paper (for example,

Sasha cut a segment from one of the strips, which had exactly $n$ digits without a decimal point. If Sasha doesn't want Dima to guess which strip the $n$ digits were cut from, what is the maximum value of $n$?

Answer: 155


## 18. Seals

Once I entered a maritime museum and saw some seals there. There were not many of them - just seven eighths of all the seals plus another seven eighths of a seal that I counted in the pool. How many seals were there in the pool in total?

dili in two rounds). According to the French constitution, a second round of elections is not held if one candidate has already received an absolute majority of the votes in the first round; if no such candidate exists, only the two candidates who received the most votes in the first round participate in the second round.-Note by nepes.

${ }^{1}$ In French high schools, a 20-point grading system is used - from 1 (complete lack of knowledge of the subject) to 20 (the highest grade in French secondary education, which gives the right to continue education at a university). - Note by the editor.

Answer: 7


## Problem 5

Let $p(x)$ be the polynomial $(1-x)^{a}\left(1-x^{2}\right)^{b}\left(1-x^{3}\right)^{c} \ldots\left(1-x^{32}\right)^{k}$, where $a, b, \ldots, k$ are integers. When expanded in powers of $x$, the coefficient of $x^{1}$ is -2 and the coefficients of $x^{2}, x^{3}, \ldots, x^{32}$ are all zero. Find $\mathrm{k}$.

Answer: 134217728


2. Solve the numerical riddle

$$
\text { BLOW + BLOW = FIGHT }
$$

where different letters represent different digits.

Answer: 6354 + 6354 = 12708


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

Answer: 173


Problem 11-2. Mr. $A$ stood at the point with coordinates $(0,0)$ for an hour. During the same hour, moving uniformly and in a straight line, Mr. $B$ traveled from the point $(22,0)$ to the point $(2,20)$. During this same hour, Miss $C$, also moving uniformly and in a straight line, traveled from the point $(30,4)$ to the point $(0,24)$. How many times during the observation period did the area of triangle $A B C$ take on integer values? The initial and final moments are included.

Answer: 53


4. As shown in Figure 9, the side length of square $A B C D$ is 10, and points $P$ and $Q$ are on the line segment connecting the midpoints of $A D$ and $B C$. Connecting $P A$, $P C$, $Q A$, and $Q C$, the square is divided into three parts of equal area. Find the length of line segment $P Q$.

Answer: \dfrac{20}{3}


7. Let the quadratic function $f(x)=a x^{2}+b x+2(a \neq 0)$, and for any real number $x$, it always holds that $|f(x)| \geqslant 2$, then the coordinates of the focus of this parabola are $\qquad$ .

Answer: \left(0, \dfrac{9}{4}\right)


2. Let the side length of the regular octagon $A_{1} A_{2} \cdots A_{8}$ be 1. For any two points $A_{i} 、 A_{j}$, the maximum value of $\overrightarrow{A_{i} A_{j}} \cdot \overrightarrow{A_{1} A_{2}}$ is $\qquad$ .

Answer: 1 + \sqrt{2}


12. From 30 people with distinct ages, select two groups, the first group consisting of 12 people, and the second group consisting of 15 people, such that the oldest person in the first group is younger than the youngest person in the second group. The number of ways to select these groups is. $\qquad$ .

Answer: 4060


A set of real numbers is called "simple" if it contains elements such that \( x + y = z \). Given the set \( \{1, 2, \cdots, 2n+1\} \), find the maximum number of elements in one of its simple subsets.

Answer: n+1


[ Law of Sines ]

In a convex quadrilateral $A B C D$, the segment $C M$, connecting vertex $C$ with point $M$ located on side $A D$, intersects diagonal $B D$ at point $K$. It is known that $C K: K M = 2: 1, C D: D K = 5: 3$ and $\angle A B D + \angle A C D = 180^{\circ}$. Find the ratio of side $A B$ to diagonal $A C$.

Answer: \dfrac{5}{9}


8.2. Given an isosceles triangle $A B C$ with base $A C$. Points $K$ and $N$ are marked on the side $B C$ ( $K$ lies between $B$ and $N$ ). It turns out that $K N=A N$ and $\angle B A K=\angle N A C$. Find $\angle B A N$.

Answer: 60^\circ


3. Given that $a \cos x + b \cos 2x \geqslant -1 (x \in \mathbf{R})$ always holds. Then the range of $a + b$ is $\qquad$

Answer: [-1, 2]


Example 3 Given two quadratic functions $y_{1}$ and $y_{2}$, when $x$ $=\alpha(\alpha>0)$, $y_{1}$ reaches its maximum value of 5, and $y_{2}=25$; also, the minimum value of $y_{2}$ is $-2, y_{1}+y_{2}=x^{2}+16 x+$ 13. Find the value of $\alpha$ and the analytical expressions of the quadratic functions $y_{1}$ and $y_{2}$.

Answer: 1


We have $2021$ colors and $2021$ chips of each color. We place the $2021^2$ chips in a row. We say that a chip $F$ is [i]bad[/i] if there is an odd number of chips that have a different color to $F$ both to the left and to the right of $F$.

(a) Determine the minimum possible number of bad chips.

(b) If we impose the additional condition that each chip must have at least one adjacent chip of the same color, determine the minimum possible number of bad chips.

Answer: 1010


Point $A$ lies on the circumference of a circle $\Omega$ with radius $78$.  Point $B$ is placed such that $AB$ is tangent to the circle and $AB=65$, while point $C$ is located on $\Omega$ such that $BC=25$.  Compute the length of $\overline{AC}$.

Answer: 60


3. (1b, 8-11) The probability that a purchased light bulb will work is 0.95. How many light bulbs need to be bought to ensure that with a probability of 0.99, there are at least 5 working ones among them?

Answer: 7


16. Given positive integers $m, n, m<2001, n<2002$, there are $2001 \times 2002$ different real numbers. Place these numbers into a $2001 \times$ 2002 rectangular chessboard (the chessboard has 2001 rows and 2002 columns), so that each cell contains one number, and each number is placed in one cell. When the number in a cell is less than at least $m$ numbers in its column and also less than at least $n$ numbers in its row, the cell containing such a number is called a "bad cell". For each arrangement, let $S$ be the number of "bad cells". Find the minimum value of $S$.
(2002, Vietnam Mathematical Olympiad)

Answer: (2001 - m)(2002 - n)


13. Given the ellipse $C: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$ passes through the point $(0, \sqrt{3})$, and the eccentricity is $\frac{1}{2}$. The line $l$ passes through the right focus $F$ of the ellipse $C$ and intersects the ellipse at points $A, B$. The projections of points $A, F, B$ on the line $x=4$ are $D, K, E$ respectively.
(1) Find the equation of the ellipse $C$;
(2) Connect $A E$ and $B D$. Explore whether the lines $A E$ and $B D$ intersect at a fixed point when the inclination angle of line $l$ changes. If so, find the coordinates of the fixed point and provide a proof; if not, explain the reason.

Answer: \dfrac{x^2}{4} + \dfrac{y^2}{3} = 1


Let real numbers \( x_{1}, x_{2}, \ldots, x_{n} \) satisfy \( 0 < x_{1} < x_{2} < \cdots < x_{n} < 1 \) and set \( x_{0}=0, x_{n+1}=1 \). Suppose that these numbers satisfy the following system of equations:
\[
\sum_{j=0, j \neq i}^{n+1} \frac{1}{x_{i}-x_{j}}=0 \quad \text{where } i=1,2, \ldots, n.
\]

Prove that \( x_{n+1-i}=1-x_{i} \) for \( i=1,2, \ldots, n \).

Answer: x_{n+1-i} = 1 - x_i \text{ for } i = 1, 2, \ldots, n


12. Place 2 $a$s and 2 $b$s in the 16 small squares shown in the figure, with at most one letter in each small square. If the same letters must not be in the same row or column, the number of different ways to fill the squares is $\qquad$ (answer in digits).

Answer: 3960


Example 7 (2003 Belarusian Mathematical Olympiad) Given a convex pentagon $A B C D E$ satisfying $A B=B C, C D=$ $D E, \angle A B C=150^{\circ}, \angle C D E=30^{\circ}, B D=2$. Find the area of pentagon $A B C D E$.

Answer: 1


12.169. In the square \(ABCD\), an isosceles triangle \(AEF\) is inscribed; point \(E\) lies on side \(BC\), point \(F\) lies on side \(CD\), and \(AE = EF\). The tangent of angle \(AEF\) is 2. Find the tangent of angle \(FEC\).

![](https://cdn.mathpix.com/cropped/2024_05_21_3edb7be591bff54ce350g-0923.jpg?height=466&width=470&top_left_y=78&top_left_x=96)

Fig. 12.39

![](https://cdn.mathpix.com/cropped/2024_05_21_3edb7be591bff54ce350g-0923.jpg?height=458&width=656&top_left_y=82&top_left_x=609)

Fig. 12.40

Answer: 3 - \sqrt{5}


Find the largest positive integer $n$ for which the inequality
\[ \frac{a+b+c}{abc+1}+\sqrt[n]{abc} \leq \frac{5}{2}\]
holds true for all $a, b, c \in [0,1]$. Here we make the convention $\sqrt[1]{abc}=abc$.

Answer: 3


The lengths of the sides of a trapezoid are $5, 15, 15, 20 \mathrm{~cm}$ in some order. We drew the angle bisectors and marked the intersection points of adjacent angle bisector pairs. What quadrilateral is defined by these four points? What fraction of the trapezoid's area is the area of this quadrilateral?

Answer: \dfrac{1}{45}


Example 8 has 1987 sets, each set has 45 elements, and the union of any two sets has 89 elements. How many elements are there in the union of these 1987 sets.

Answer: 87429


We have a $10 \mathrm{Ft}$ coin and several (at least four) $20 \mathrm{Ft}$ coins in our pocket. We take out one coin at a time randomly until the amount we have taken out is not a positive integer multiple of $30 \mathrm{Ft}$. On average, how many draws are needed for this?

Answer: 3


Given a tetrahedron, prove that there exists another tetrahedron \( K L M N \) such that its edges \( K L, L M, M N, \) and \( N K \) are perpendicular to the corresponding faces of the given tetrahedron and their lengths are numerically equal to the areas of those faces. Find the volume of tetrahedron \( K L M N \) if the volume of the given tetrahedron is \( \boldsymbol{V} \).

Answer: \dfrac{3V^2}{4}


[Examples and Counterexamples. Constructions] [ Proof by Contradiction ]

## Auto: Zassavsiy A.A.

In a volleyball tournament, each team met every other team once. Each match consisted of several sets - until one of the teams won three times. If a match ended with a score of $3: 0$ or $3: 1$, the winning team received 3 points, and the losing team received 0. If the set score was

$3: 2$, the winner received 2 points, and the loser received 1 point. By the end of the tournament, it turned out that the team "Cunning" scored the most points, while the team "Simpletons" scored the least. However, "Cunning" won fewer matches than they lost, while "Simpletons" had more wins than losses. What is the smallest number of teams for which this is possible?

Answer: 6


1. Can the first 8 natural numbers be arranged in a circle so that each number is divisible by the difference of its neighbors?

Answer: Yes


# Task 4.

Four brigades were developing an open coal deposit for three years, working with a constant productivity for each brigade. In the second year, due to weather conditions, work was not carried out for four months, and for the rest of the time, the brigades worked in rotation (one at a time). The ratio of the working times of the first, second, third, and fourth brigades and the amount of coal mined respectively are: in the first year 4:1:2:5 and 10 million tons; in the second year $2: 3: 2: 1$ and 7 million tons; in the third year 5:2:1:4 and 14 million tons. How much coal would these four brigades have mined in 4 months if they worked together?

Answer: 12


$7 \cdot 69$ Let set $M$ consist of all points in the plane except for 3 distinct points $A, B, C$. Find the minimum number of convex sets such that their union equals $M$.

The translation is as follows:

$7 \cdot 69$ Let set $M$ consist of all points in the plane except for 3 distinct points $A, B, C$. Find the minimum number of convex sets such that their union equals $M$.

Answer: 3


3. Two equal rectangles $P Q R S$ and $P_{1} Q_{1} R_{1} S_{1}$ are inscribed in triangle $A B C$ (with points $P$ and $P_{1}$ lying on side $A B$, points $Q$ and $Q_{1}$ lying on side $B C$, and points $R, S, R_{1}$ and $S_{1}$ lying on side $A C$). It is known that $P S=12, P_{1} S_{1}=3$. Find the area of triangle $A B C$.

Answer: \dfrac{225}{2}


3. Linglong Tower has 7 floors, and there is a direct elevator in the tower. Qisi and Wangwang go sightseeing from the 1st floor to the 7th floor. Qisi chooses to wait in line for the elevator, while Wangwang chooses to climb the stairs. When Wangwang reaches the 5th floor, Qisi finally gets on the elevator. In the end, both arrive at the 7th floor at the same time. The upward speed of the elevator is $\qquad$ times the speed of Wangwang climbing the stairs.

Answer: 3


In a running race, three people started: $X, Y$, and $Z$. $X$ performed the best, $Y$ came in second, while $Z$ lagged a bit behind. During the race, $Z$'s position changed six times, $X$'s position changed five times, and in the end, $Y$ reached the finish line before $X$.

What was the final result of the race?

Answer: Y


6. Given $t$ is a real number. Find all functions $f: \mathbf{R} \rightarrow \mathbf{R}$ such that
$$
f(x+t+f(y))=f(f(x))+f(t)+y .
$$
(2014, Croatian Mathematical Olympiad)

Answer: f(x) = x


Example 3. Solve the equation $y^{\text {IV }}-16 y=0$.

Answer: y(x) = C_1 e^{2x} + C_2 e^{-2x} + C_3 \cos(2x) + C_4 \sin(2x)


Let \( A B C \) be a triangle and \( M \) an interior point. Prove that \( \min \{M A, M B, M C\} + M A + M B + M C < A B + A C + B C \).

Answer: \min \{MA, MB, MC\} + MA + MB + MC < AB + AC + BC


11.3. The teacher is going to give the children a problem of the following type. He will inform them that he has thought of a polynomial $P(x)$ of degree 2017 with integer coefficients, the leading coefficient of which is 1. Then he will tell them $k$ integers $n_{1}, n_{2}, \ldots, n_{k}$, and separately inform them of the value of the expression $P\left(n_{1}\right) \cdot P\left(n_{2}\right) \cdot \ldots \cdot P\left(n_{k}\right)$. Based on this data, the children are supposed to find the polynomial that the teacher could have thought of. For what smallest $k$ can the teacher formulate such a problem so that the polynomial found by the children necessarily coincides with the one thought of?

(G. Zhukov)

Answer: 2017


4. Photographs are archived in the order of their numbering in identical albums, with exactly 4 photographs per page. In this case, the 81st photograph in sequence landed on the 5th page of one of the albums, and the 171st photograph landed on the 3rd page of another. How many photographs can each album hold?

#

Answer: 32


Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ that satisfy
$$ (f(a)-f(b))(f(b)-f(c))(f(c)-f(a))=f\left(a b^{2}+b c^{2}+c a^{2}\right)-f\left(a^{2} b+b^{2} c+c^{2} a\right) $$
for all real numbers $a, b, c$. Answer: $f(x)=\alpha x+\beta$ or $f(x)=\alpha x^{3}+\beta$ where $\alpha \in\{-1,0,1\}$ and $\beta \in \mathbb{R}$.

Answer: f(x) = \alpha x + \beta


4. The numbers $x$ and $y$ are such that the equalities $\sin y + \cos x = \sin 3x$ and $\sin 2y - \sin 2x = \cos 4x - \cos 2x$ hold. What is the smallest value that the sum $\cos y + \sin x$ can take?

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


Let $S$ be the set of $2\times2$-matrices over $\mathbb{F}_{p}$ with trace $1$ and determinant $0$. Determine $|S|$.

Answer: p(p+1)


4. Given a parallelogram $A B C D$. A line passing through point $D$ intersects segment $A C$ at point $G$, segment $B C$ at point $F$, and ray $A B$ at point $E$ such that triangles $BEF$ and $CGF$ have the same area. Determine the ratio $|A G|:|G C|$.

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


Rectangles \( P \) and \( Q \) have the same area, but the diagonal of \( P \) is longer. Two copies of \( P \) can cover \( Q \). Prove that two copies of \( Q \) can cover \( P \).

Answer: \text{Two copies of } Q \text{ can cover } P


3. Let $n<100$. Then the largest integer $n$ such that the expansion of $(a+b)^{n}$ has three consecutive terms with coefficients in arithmetic progression is $\qquad$ . . .

Answer: 98


Answer: (F_{n+2})^2


3. In the expansion of $(\sqrt{3}+i)^{10}$, the sum of all odd terms is $\qquad$ .

Answer: 512


## 47.

One day, Alice met the Lion and the Unicorn resting under a tree. They made the following statements.

Lion. Yesterday was one of the days when I lie.

Unicorn. Yesterday was one of the days when I also lie.

From these two statements, Alice (a very smart girl) was able to deduce what day of the week it was yesterday. What day was it?

Answer: Thursday


7$\cdot$70 Alice and Bob came to a hardware store, where colored tassels are sold to be tied on keys to distinguish different keys. Below is a segment of their conversation:
Alice: Are you going to buy some colored tassels to tie on your keys?
Bob: I would like to do that, but the tassels come in only 7 different colors, and I have 8 keys.
Alice: That’s no problem, because even if two keys are tied with red tassels, you can still distinguish them by noticing whether they are adjacent to a key with a green tassel or a key with a blue tassel.
Bob: Of course, you know that all my keys are on the same keyring, and the keyring can be flipped and rotated, so when talking about “adjacent” or “three keys in front,” one must be careful.
Alice: Even so, you don’t need 8 different colors of tassels.
Try to determine: To distinguish $n$ keys on the same keyring, what is the minimum number of different colors of tassels needed?

Answer: 3


15. (6 points) The poetry lecture lasted for 2 hours $m$ minutes, and at the end, the positions of the hour and minute hands on the clock were exactly swapped compared to when it started. If $[x]$ represents the integer part of the decimal number $x$, then $[m]=$ $\qquad$ .

Answer: 46


1. In a certain country, the alphabet consists of three letters: "M", "G", and "U". A word is any finite sequence of these letters, in which two consonants cannot stand next to each other and two vowels cannot stand next to each other. How many 200-letter words are there in this country that contain each of the three letters at least once?

Answer: 2^{101} - 4


4. The base of a right quadrilateral prism is a rhombus with side length 2 and an angle of $30^{\circ}$, and the height is 1. A section is made through the base edge at an angle of $60^{\circ}$ to the base. The area of this section is $\qquad$ .

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


$$
\begin{array}{l}
\text { 3. In quadrilateral } A B C D, \angle A B D=70^{\circ}, \\
\angle C A D=20^{\circ}, \angle B A C=48^{\circ}, \angle C B D=40^{\circ} \text {. Then } \\
\angle A C D=
\end{array}
$$

Answer: 22


1. As shown in Figure 2, quadrilateral $ABCD$ is a rectangle, and $AB = 2BC$. Points $M$ and $N$ are the midpoints of sides $BC$ and $CD$, respectively. $AM$ intersects $BN$ at point $E$. If the area of the shaded region is $a$, then the area of rectangle $ABCD$ is $\qquad$

Answer: 5a


10. Find the number of triples $(a, b, c)$ of real numbers such that the equation $|a x+b y+c z|+|b x+c y+a z|+|c x+a y+b z|=|x|+|y|+|z|$ holds for all real numbers $x, y, z$.
(1 mark)
Find the number of triples $(a, b, c)$ of real numbers such that the equation $|a x+b y+c z|+|b x+c y+a z|+|c x+a y+b z|=|x|+|y|+|z|$ holds for all real numbers $x, y, z$.

Answer: 6


There are $4 n\left(n \in \mathbf{Z}_{+}\right)$ circles with equal radii on a plane, where any two circles do not intersect with the other three circles. Let the number of intersection points of these circles be $f(n)$. Find the minimum value of $f(n)$.

Answer: 4n


Let $m$ be a positive integer and let $A$, respectively $B$, be two alphabets with $m$, respectively $2m$ letters. Let also $n$ be an even integer which is at least $2m$. Let $a_n$ be the number of words  of length $n$, formed with letters from $A$, in which appear all the letters from $A$, each an even number of times. Let $b_n$ be the number of words of length $n$, formed with letters from $B$, in which appear all the letters from $B$, each an odd number of times. Compute $\frac{b_n}{a_n}$.

Answer: 2^{n - m}


6. Cut a square with a side of 4 into rectangles, the sum of the perimeters of which is 25.

Answer: 25


12. The diagram shows a 16 metre by 16 metre wall. Three grey squares are painted on the wall as shown.

The two smaller grey squares are equal in size and each makes an angle of $45^{\circ}$ with the edge of the wall. The grey squares cover a total area of $B$ metres squared.
What is the value of $B$ ?

Answer: 128


20. Patrick asks SpongeBob: "What is your lucky number?" SpongeBob says: "There is a sequence of numbers: $0,1,3,8$, $21,55,144,377,987, \cdots \cdots$ Starting from the third number, each number multiplied by 3 is exactly equal to the sum of the numbers immediately before and after it. My lucky number is the remainder when the 2020th number in this sequence is divided by 6." So, SpongeBob's lucky number is $\qquad$ . (If it is divisible, then the remainder is 0)

Answer: 2


10. A positive integer $n$ is called "flippant" if $n$ does not end in 0 (when written in decimal notation) and, moreover, $n$ and the number obtained by reversing the digits of $n$ are both divisible by 7 . How many flippant integers are there between 10 and 1000 ?

Answer: 17


5. July 1, 2021 is the 100th anniversary of the founding of our Party, recorded as 20210701, where the sum of the digits in the odd positions is $1+7+1+0=9$ and the sum of the digits in the even positions is $0+0+2+2=4$, with a ratio of $9: 4$; among the 8-digit numbers formed by $0 \sim 9$ (no repetition), how many satisfy the condition that the sum of the digits in the odd positions to the sum of the digits in the even positions is $9: 4$?

Answer: 2304


Four people sit at the four sides of a table and the 32-card Hungarian deck is dealt out equally among them. If one of the selected players does not get an ace, what is the probability that at least one other player has exactly two (no more, no less) aces?

Answer: \dfrac{8}{11}


3.100. The base of the pyramid is a rectangle, where the angle between the diagonals is $\alpha$. A sphere of radius $R$ is circumscribed around this pyramid. Find the volume of the pyramid if all its lateral edges form an angle $\beta$ with the base.

Answer: \dfrac{16}{3} R^3 \sin^4 \beta \cos^2 \beta \sin \alpha


4. A convex quadrilateral $A B C D$ is given with a point $E$ inside side $A B$ such that $|\angle A D E|=|\angle D E C|=|\angle E C B|$. The areas of triangles $A E D$ and $C E B$ are $18 \mathrm{~cm}^{2}$ and $8 \mathrm{~cm}^{2}$, respectively. Determine the area of triangle $E C D$.

The regional round of category C takes place

on Tuesday, April 8, 2014

so that it starts in the morning and the participants have 4 hours of pure time to solve the problems. Allowed aids are writing and drawing supplies and school MF tables. Calculators, laptops, and any other electronic aids are not allowed. Each problem can earn the participant 6 points; the score boundaries $\mathrm{k}$ for determining successful solvers and successful participants will be set centrally after evaluating the statistical results of the scores from all regions. These details will be communicated to the students before the competition begins.

Answer: 12


## 163. The Cat and the Mouse.

- One of these barrels has a mouse in it,- said the dog.
- Which one? - asked the cat.
- Well, the five hundredth.
- What do you mean by that? There are only five barrels here.
- The barrel I mean will be the five hundredth if you start counting forward and backward like this.

And the dog explained how exactly one should count:

| 1 | 2 | 3 | 4 | 5 |
| ---: | ---: | ---: | ---: | ---: |
| 9 | 8 | 7 | 6 |  |
|  | 10 | 11 | 12 | 13 |

For example, the seventh barrel will coincide with the one that has the number 3, and the twelfth barrel with the one that has the number 4.

- This will take a lot of time,- said the cat and began to count patiently. Several times she got confused and started all over again.
- Damn it!-exclaimed the dog.-Hurry up, or it will be too late!
- Be damned yourself! You made me lose my count again, now I have to start all over.

Meanwhile, the mouse, having heard the entire conversation, gnawed a hole and slipped away just as the cat jumped into the right barrel.

- I knew it would happen, - said the dog. - Your education I would not dare to call too brilliant. A little familiarity with arithmetic would not hurt any cat, just as it does not hurt any dog. What am I saying! Even some snakes are so diligent in this science that they have to wear glasses!

Which of the barrels was the five hundredth? Could you find the answer without counting to 500?

![](https://cdn.mathpix.com/cropped/2024_05_21_56f37ca27ac3d928838dg-049.jpg?height=354&width=628&top_left_y=1228&top_left_x=720)

Answer: 4


Given positive real numbers \(a_{1}, a_{2}, \ldots, a_{n}\) with \(n > 1\), let \(g_{n}\) denote their geometric mean. Define \(A_{k} = \frac{\sum_{i=1}^{k} a_{i}}{k}\) for \(1 \leq k \leq n\), and let \(G_{n}\) be the geometric mean of \(A_{1}, A_{2}, \ldots, A_{n}\). Prove that \(n \cdot \sqrt[n]{\frac{G_{n}}{A_{n}}} + \frac{g_{n}}{G_{n}} \leq n + 1\), and determine the conditions under which equality holds.

Answer: n \cdot \sqrt[n]{\frac{G_{n}}{A_{n}}} + \frac{g_{n}}{G_{n}} \leq n + 1


## Task A-1.2.

Let $S$ be the center of the circle $k$ with radius of length 1. Vertices $A$ and $B$ of the square $ABCD$ lie on the circle $k$, and the side $\overline{CD}$ passes through the point $S$. Determine the length of the side of the square $ABCD$.

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


13. Let $x[n]$ denote $x^{x^{\prime}}$, where there are $n$ terms of $x$. What is the minimum value of $n$ such that $9[9]<3[n]$ ?
(For example, $3[2]=3^{3}=27 ; 2[3]=2^{2^{2}}=16$.)

Answer: 10


7. Three parallel lines touch a sphere with center at point O at points K, L, and M. It is known that the radius of the sphere is $5 \sqrt{2}$, the area of triangle OKL is 7, and the area of triangle KLM is greater than 50. Find the angle KML.

Answer. $\angle \mathrm{KML}=\arccos \frac{1}{5 \sqrt{2}}$.

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