Determine the values of the variable \( x \) such that the four expressions \( 2x - 6 \), \( x^2 - 4x + 5 \), \( 4x - 8 \), and \( 3x^2 - 12x + 11 \) differ from each other by the same constant amount, regardless of the order of the expressions. Find all possible integer values of \( x \).


When \( m = -5, -4, -3, -1, 0, 1, 3, 23, 124, 1000 \), the equation \((2m + 1)x + (2 - 3m)y + 1 - 5m = 0\) yields 10 linear equations in \( x \) and \( y \). The common solution for these 10 equations is \( x = \quad \) \( y = \quad \).

Solution: \boxed{x=1, y=-1}


Four friends, One, Two, Five, and Ten, are located on one side of a dark tunnel and have only one flashlight. It takes one minute for person One to walk through the tunnel, two minutes for Two, five minutes for Five, and ten minutes for Ten. The tunnel is narrow, and at most two people can walk at the same time with the flashlight. Whenever two people walk together, they walk at the speed of the slower one. Show that all four friends can go from one side of the tunnel to the other in 17 minutes.

Solution: \boxed{17 \text{ minutes}}


A circle with a radius of 10 cm has rays drawn from point $A$ that are tangent to the circle at points $B$ and $C$ such that triangle $ABC$ is equilateral. Find the area of the triangle.

Solution: \boxed{75 \sqrt{3}}


Teacher Zhang brought some money to buy pens. After arriving at the store, he discovered that the pens were discounted by 25%, and the money he brought was just enough to buy 25 more pens than originally expected. How many pens could he have bought with this money before the discount?


Sixty students went on a trip to the zoo. Upon returning to school, it turned out that 55 of them forgot gloves at the zoo, 52 forgot scarves, and 50 managed to forget hats. Find the smallest number of the most scatterbrained students - those who lost all three items.


It is now 1:00 a.m. What time will it be 2023 minutes from now?

Solution: \boxed{10:43 \text{ a.m.}}


The numbers \(2^{2019}\) and \(5^{2019}\) are written consecutively. How many digits are written in total?


A ballpoint pen costs 10 rubles, a gel pen costs 30 rubles, and a fountain pen costs 60 rubles. What is the maximum number of ballpoint pens that can be bought under the condition that a total of exactly 20 pens must be bought, including pens of all three types, and exactly 500 rubles must be spent?


In the diagram, the two regular octagons have side lengths of 1 and 2. The smaller octagon is completely contained within the larger octagon. What is the area of the region inside the larger octagon and outside the smaller octagon?

Solution: \boxed{6 + 6 \sqrt{2}}


In the figure below, \( AB \) and \( CD \) are parallel chords of a circle with center \( O \) and radius \( r \) cm. It is given that \( AB = 46 \) cm, \( CD = 18 \) cm, and \( \angle AOB = 3 \times \angle COD \). Find the value of \( r \).


Find the point \( M' \) that is symmetric to the point \( M \) with respect to the plane.

Given:
\( M(3, -3, -1) \)

Plane equation:
\( 2x - 4y - 4z - 13 = 0 \)

Solution: \boxed{(2, -1, 1)}


Consider the following number table. The first row contains the numbers from 1 to \( n \). The second row contains the numbers from 1 to \( n \) each multiplied by 2. The rows follow this pattern until the last row, which presents \( n \) times each number from 1 to \( n \).

We will use it to calculate the value of the expression

\[ 1^{3} + 2^{3} + \ldots + 100^{3} \]

In addition to the table, we will use the fact that

\[ 1 + 2 + \ldots + n = \frac{n(n+1)}{2} \]

(a) Determine the sum of all the numbers in row number \( k \). With this, determine an expression for the sum of all the numbers in the table.

(b) Observe sections in the table by separating it into \( L \)-shaped layers. The numbers in a certain layer \( k \) are: \( k, 2k, \ldots, (k-1)k, k^{2}, (k-1)k, \ldots, 2k, k \). Determine the sum of the numbers in this layer as a function of \( k \).

(c) Adding the results of all the layers, we will arrive at the same result as summing all rows. Combining this information, determine the value of the expression:

\[ 1^{3} + 2^{3} + \ldots + 100^{3} \]


Given \( n \in \mathbf{N} \) and \( a \in[0, n] \), under the condition \( \sum_{i=1}^{n} \sin ^{2} x_{i} - a \), find the maximum value of \( \left|\sum_{i=1}^{n} \sin 2 x_{i}\right| \).

Solution: \boxed{2 \sqrt{a (n - a)}}


Let squares of one kind have a side of \(a\) units, another kind have a side of \(b\) units, and the original square have a side of \(c\) units. Then the area of the original square is given by \(c^{2}=n a^{2}+n b^{2}\).

Numbers satisfying this equation can be obtained by multiplying the equality \(5^{2}=4^{2}+3^{2}\) by \(n=k^{2}\). For \(n=9\), we get \(a=4, b=3, c=15\).


\( S \) is the set of all ordered tuples \((a, b, c, d, e, f)\) where \(a, b, c, d, e, f\) are integers and \(a^2 + b^2 + c^2 + d^2 + e^2 = f^2\). Find the largest \( k \) such that \( k \) divides \( a b c d e f \) for all elements in \( S \).


A right triangle and a circle are drawn such that the circle is tangent to the legs of the right triangle. The circle cuts the hypotenuse into three segments of lengths 1, 24, and 3, and the segment of length 24 is a chord of the circle. Compute the area of the triangle.


What is the smallest value that the ratio of the areas of two isosceles right triangles can have, given that three vertices of one of the triangles lie on three different sides of the other triangle?


There is a strip of paper with three types of scale lines that divide the strip into 6 parts, 10 parts, and 12 parts along its length. If the strip is cut along all the scale lines, into how many parts is the strip divided?


Let \(a, b, c\) be positive real numbers satisfying the identity \(a+b+c=1\). Show that:

\[
a \sqrt{b} + b \sqrt{c} + c \sqrt{a} \leq \frac{1}{\sqrt{3}}
\]


Solve for the positive integer(s) \( n \) such that \( \phi\left(n^{2}\right) = 1000 \phi(n) \).


Find the maximum value for: \(\max _{a, b, \cdots \in \mathbf{R}^{+}} \min \left\{\frac{1}{a}, \frac{1}{b^{2}}, \frac{1}{c^{3}}, a+b^{2}+c^{3}\right\}\).


Find all functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) that satisfy:

\[ \forall x, y \in \mathbb{R}, \quad f(f(x+y)) = f(x) + y \]

Solution: \boxed{f(x) = x}


In an arithmetic sequence \(\left\{a_{n}\right\}\), if \(\frac{a_{11}}{a_{10}} < -1\), and its partial sum \(S_{n}\) has a maximum value, then when \(S_{n}\) attains its smallest positive value, \(n = \) \(\qquad\).


Suppose \( P \) is an integer and \( 5 < P < 20 \). If the roots of the equation \( x^{2} - 2(2P - 3)x + 4P^{2} - 14P + 8 = 0 \) are integers, find the value of \( P \).


The simplest fraction \(\frac{a}{b}\) satisfies \(\frac{1}{5}<\frac{a}{b}<\frac{1}{4}\), and \(b\) does not exceed 19. What is the product of the maximum possible value and the minimum possible value of \(a + b\)?


Some years ago, a man spent one hundred English silver coins in Christmas-boxes, giving every person the same amount. This cost him exactly £1, 10s 1d. Can you determine how many people received the present and how he managed the distribution?


A number leaves a remainder of 2 when divided by 5, a remainder of 2 when divided by 6, and a remainder of 3 when divided by 7. Find the smallest natural number that satisfies all three conditions.


A moth starts at vertex \(A\) of a certain cube and is trying to get to vertex \(B\), which is opposite \(A\), in five or fewer "steps," where a step consists of traveling along an edge from one vertex to another. The moth will stop as soon as it reaches \(B\). How many ways can the moth achieve its objective?


The numbers \(a_{1}, a_{2}, \ldots, a_{n}\) are such that the sum of any seven consecutive numbers is negative, and the sum of any eleven consecutive numbers is positive. What is the largest possible \(n\) for which this is true?


In an isosceles trapezoid, the length of the midline is 5, and the diagonals are mutually perpendicular. Find the area of the trapezoid.


Given a set of 4027 points on a plane, called a "Colombian point set," where no three points are collinear. Among these points, 2013 are red, and 2014 are blue. A set of lines drawn on the plane can divide the plane into several regions. A set of lines is considered a "good line set" for a Colombian point set if it satisfies the following two conditions:

1. The lines do not pass through any of the points in the Colombian point set.
2. Each region formed by the lines does not contain both red and blue points simultaneously.

Find the minimum value of $\mathrm{k}$ such that for any Colombian point set, there exists a "good line set" formed by $\mathrm{k}$ lines.


Among the 100 natural numbers from 1 to 100, how many numbers can be represented as \( m \cdot n + m + n \) where \( m \) and \( n \) are natural numbers?


Assume a set of points \( S \) in the plane has the properties: (i) any three points are not collinear; (ii) the distances between any two points are all distinct. For two points \( A \) and \( B \) in \( S \), if there exists \( C \in S \) such that \( |AC| < |AB| < |BC| \), then \( AB \) is called a median edge of \( S \). For three points \( A \), \( B \), and \( C \) in \( S \), if \( AB \), \( AC \), and \( BC \) are all median edges of \( S \), then \( \triangle ABC \) is called a median triangle of \( S \). Find the smallest \( n \) such that any \( n \)-point set \( S \) satisfying properties (i) and (ii) must contain a median triangle.


Given a triangle \(ABC\) where \(AB = AC\) and \(\angle A = 80^\circ\). Inside triangle \(ABC\) is a point \(M\) such that \(\angle MBC = 30^\circ\) and \(\angle MCB = 10^\circ\). Find \(\angle AMC\).


On his birthday, the last guest to arrive was Yana, who gave Andrey a ball, and the second last was Eduard, who gave him a calculator. While using the calculator, Andrey noticed that the product of the total number of gifts he received and the number of gifts he had before Eduard arrived is exactly 16 more than the product of his age and the number of gifts he had before Yana arrived. How many gifts does Andrey have?


A circle with center $O$ has a diameter $AB$. From a point $C$ on the circumference, which is distinct from points $A$ and $B$, a perpendicular is drawn to $AB$ intersecting $AB$ at point $D$. From $O$, a perpendicular is drawn to $BC$, intersecting $BC$ at point $M$. Determine the measure of angle $ABC$ given that $DB = 3 \cdot OM$.


Five boxes contain balls of the same size and weight. In two boxes, there are 6 blue and 4 red balls (these are boxes of type \(H_{1}\)). In two other boxes (type \(H_{2}\)), there are 8 blue and 2 red balls. In one box (type \(H_{3}\)), there are 2 blue and 8 red balls. A box is chosen at random, and a ball is drawn from it. What is the probability that the drawn ball is red?


Two circles, \(\odot O_{1}\) and \(\odot O_{2}\), are externally tangent to each other with radii 7 and 14 respectively. A third circle \(\odot O_{3}\) that is tangent to both \(\odot O_{1}\) and \(\odot O_{2}\) has its center on the line connecting their centers. Find the radius of a fourth circle that is tangent to \(\odot O_{1}\), \(\odot O_{2}\), and \(\odot O_{3}\).


One side of a rectangle is 20% smaller than the other, and its area is \(4 \frac{1}{20}\). Find the longer side of the rectangle.


Two classes are planting trees. Each student in the first class plants 3 trees, and each student in the second class plants 5 trees. A total of 115 trees are planted. What is the maximum possible total number of students in the two classes?


On the island of Misfortune, there are knights who always tell the truth and liars who always lie. One day, a tourist met five residents of the island and asked them: "How many liars are among you?" The first resident answered: "One," the second resident said: "Two," the third resident said: "Three," the fourth resident said: "Four," and the fifth resident said: "Five." How many liars were there actually?


A circle touches the extensions of two sides \( AB \) and \( AD \) of a square \( ABCD \) with a side length of 4 cm. From point \( C \), two tangents are drawn to this circle. Find the radius of the circle if the angle between the tangents is \( 60^{\circ} \).

Solution: \boxed{4 (\sqrt{2} + 1)}


The altitudes of a triangle are 12, 15, and 20. What is the area of the triangle?


Vanya bought balloons, with the number of red ones being 7 times more than the blue ones. On his way home, some balloons burst, and among the burst balloons, there were 3 times fewer red ones than blue ones. What is the minimum number of balloons Vanya could have bought?


Given \(\log _{a} x + 3 \log _{x} a - \log _{x} y = 3 \ (a > 1)\),
(1) Let \(x = a^{t}\), express \(y\) in terms of \(a\) and \(t\);
(2) If \(y\) has a minimum value of 8 when \(0 < t \leqslant 2\), find \(a\) and \(x\).

Solution: \boxed{a = 16, \ x = 64}


Solve the equation for all values of the parameter \( a \):

$$
3 x^{2}+2 a x-a^{2}=\ln \frac{x-a}{2 x}
$$

Solution: \boxed{x = -a}


Two locations $A$ and $B$ are 225 kilometers apart. The price of coal per quintal is 3 korona 75 filler in $A$ and 4 korona 25 filler in $B$, and the transportation cost is 8 filler per ton per kilometer. At which point on the line segment $A B$ does the coal cost the same, whether it is brought from $A$ or $B$? Additionally, show that the coal is most expensive at this point on the entire line segment $A B$.

Solution: \boxed{143.75 \ \text{km}}


In a box, there are 22 kg of cranberries. How, using a single 2-kilogram weight and a two-pan scale, can you measure out 17 kg of cranberries in two weighings?

Solution: \boxed{17\text{ kg}}


From the numbers 1 to 200, one or more numbers were selected to form a group with the following property: if the group contains at least two numbers, then the sum of any two numbers in this group is divisible by 5. What is the maximum number of numbers that can be in the group with this property?


The very hungry caterpillar lives on the number line. For each non-zero integer \(i\), a fruit sits on the point with coordinate \(i\). The caterpillar moves back and forth; whenever he reaches a point with food, he eats the food, increasing his weight by one pound, and turns around. The caterpillar moves at a speed of \(2^{-w}\) units per day, where \(w\) is his weight. If the caterpillar starts off at the origin, weighing zero pounds, and initially moves in the positive \(x\) direction, after how many days will he weigh 10 pounds?


Let \( n \) be the product of the first 10 primes, and let
\[ S = \sum_{xy \mid n} \varphi(x) \cdot y, \]
where \( \varphi(x) \) denotes the number of positive integers less than or equal to \(x\) that are relatively prime to \( x \), and the sum is taken over ordered pairs \((x, y)\) of positive integers for which \( xy \) divides \( n \). Compute \(\frac{S}{n}\).


Decompose the given function into a Fourier series in the specified interval:
1) \(\varphi(x)=\frac{x}{2} ; \quad(0,2 \pi)\)
2) \(\psi(x)=e^{-x} ;(-\pi, \pi)\)
3) \(y=\begin{cases}6 & \text{for } 0<x<2 \\ 3x & \text{for } 2<x<4 \end{cases}\)
4) \(u=|\sin x| ;[-\pi, \pi]\)

Using the obtained decomposition, find the sum \(S\) of the series \(\frac{1}{1 \cdot 3}+\frac{1}{3 \cdot 5}+\frac{1}{5 \cdot 7}+\ldots+\frac{1}{(2k-1)(2k+1)}+\ldots\)


A circle touches the sides \( AB \) and \( AD \) of rectangle \( ABCD \) and passes through vertex \( C \). It intersects side \( DC \) at point \( N \). Find the area of trapezoid \( ABND \), given that \( AB = 9 \) and \( AD = 8 \).


In a regular $n$-gon ($n>5$), the difference between the longest and shortest diagonals is equal to the side length. Find $n$.


The time on a digital clock reads 3:33. What is the shortest length of time, in minutes, until all of the digits are again equal to each other?


In a hockey tournament, there are $n$ teams participating. Each team plays with every other team exactly once, where the winning team is awarded 2 points, a draw gives each team 1 point, and the losing team gets 0 points. What is the maximum point difference that can occur between teams that are ranked in consecutive positions?

Solution: \boxed{n}


Find the curve that passes through the point \((2, 3)\) and has the property that the segment of any of its tangents, intercepted between the coordinate axes, is bisected at the point of tangency.

Solution: \boxed{xy = 6}


Let us call a ticket with a number from 000000 to 999999 excellent if the difference between some two neighboring digits of its number is 5. Find the number of excellent tickets.


Calculate the definite integral

$$
\int_{0}^{2 \pi} \sin ^{4}(3x) \cos ^{4}(3x) \, dx
$$


A secret facility is a rectangle measuring $200 \times 300$ meters. There is one guard at each of the four corners of the facility. An intruder approaches the perimeter of the facility from the outside, and all the guards run towards the intruder by the shortest routes along the outer perimeter (while the intruder remains stationary). Three guards cover a total distance of 850 meters to reach the intruder. How many meters did the fourth guard run to reach the intruder?


In $\triangle ABC$, $\sin A = \frac{3}{5}$, $\cos B = \frac{5}{13}$. Find the value of $\cos C$.


A circle \( \omega \) with radius \( r \) is inscribed in triangle \( ABC \), and the circle is tangent to side \( AB \) at point \( X \). A point \( Y \), which is diametrically opposite to \( X \), is marked on the circle. Line \( CY \) intersects side \( AB \) at point \( Z \). Find the area of triangle \( ABC \), given that \( CA + AZ = 1 \).

Solution: \boxed{r}


Given a positive integer \( n \), let \( x_{1}, x_{2}, \cdots, x_{2n} \) be a permutation of \( 1, 2, \cdots, 2n \) such that
\[
\left| x_{i} - i \right| = 1 \quad (i = 1, 2, \cdots, 2n)
\]
holds for all \( i \). Determine the number of such permutations.


On the hypotenuse \( AB \) of an isosceles right triangle \( ABC \), points \( K \) and \( L \) are marked such that \( AK : KL : LB = 1 : 2 : \sqrt{3} \). Determine \(\angle KCL\).


A natural number \( 1 \leq n \leq 221 \) is called lucky if, when dividing 221 by \( n \), the remainder is wholly divisible by the incomplete quotient (the remainder can be equal to 0). How many lucky numbers are there?


The total mass ratio of two oil barrels, A and B, including their oil, is $4:5$. After pouring 8 kg of oil from barrel B into barrel A, the total mass ratio of barrel A and barrel B, including their oil, becomes $8:7$. Given that the mass of each barrel is 10 kg, find the original mass ratio of the oil in barrels A and B.

Solution: \boxed{3: 4}


Given two geometric progressions, each containing the same number of terms. The first term and the common ratio of the first progression are 20 and \( \frac{3}{4} \), respectively. The first term and the common ratio of the second progression are 4 and \( \frac{2}{3} \), respectively. If we multiply the terms of these progressions with the same indices, the sum of all such products is 158.75. Find the number of terms in these progressions.


In the coordinate plane, the points $A(0, 2)$, $B(1, 7)$, $C(10, 7)$, and $D(7, 1)$ are given. Find the area of the pentagon $ABCDE$, where $E$ is the intersection point of the lines $AC$ and $BD$.


The number of ways to arrange 5 boys and 6 girls in a row such that girls can be adjacent to other girls but boys cannot be adjacent to other boys is \(6! \times k\). Find the value of \(k\).


A rectangle \(ABCD\) with sides \(AB=1\) and \(AD=10\) serves as the base of the pyramid \(SABCD\), where the edge \(SA=4\) is perpendicular to the base. Find a point \(M\) on the edge \(AD\) such that the triangle \(SMC\) has the smallest perimeter. Find the area of this triangle.


A tile, in the shape of a regular polygon, was removed from its place in a panel. It was observed that if this tile were rotated by \( 40^\circ \) or \( 60^\circ \) around its center, it could fit perfectly into the spot that was left vacant on the panel. What is the smallest number of sides that this polygon can have?


From the similarity of the right triangles \(A K C\) and \(B L C\), formed by the segments of the tangent, line \(l\), and the radii \(A K\) and \(B L\) drawn to the points of tangency at time \(T\), we obtain

$$
\frac{a T}{a T - R} = \frac{L + x}{x}
$$

where \(x\) represents the distance \(B C\).

From the given equation, we find \( x = a T \frac{L}{R} - L \). Hence, the speed of the moving point \(C\) at the intersection of the lines is \(a \frac{L}{R}\).

Solution: \boxed{a \frac{L}{R}}


The number of common terms (terms with the same value) between the arithmetic sequences \(2, 5, 8, \ldots, 2015\) and \(4, 9, 14, \ldots, 2014\).


A small garden measuring \(6 \times 7\) meters is divided into 5 square beds. All borders (straight lines dividing the beds) are parallel to the sides of the square, and each side of a bed is an integer number of meters. Find the total length of the resulting borders.

Note: Answer is \(15 \text{ meters}\).

Solution: \boxed{15 \text{ meters}}


Calculate the limit of the numerical sequence:

$$
\lim _{n \rightarrow \infty} \frac{\sqrt{n+3}-\sqrt[3]{8 n^{3}+3}}{\sqrt[4]{n+4}-\sqrt[5]{n^{5}+5}}
$$


In an arithmetic sequence \(\left\{a_{n}\right\}\), if \(\frac{a_{11}}{a_{10}} < -1\), and the sum of its first \(n\) terms \(S_{n}\) has a maximum value. Then, when \(S_{n}\) attains its smallest positive value, \(n =\) ______ .


In a certain company, there are 100 shareholders, and any group of 66 of them collectively owns at least 50% of the company's shares. What is the maximum percentage of all shares that one shareholder can own? (The percentage of the company's shares owned by a shareholder can be a non-integer value.)


Among all simple fractions whose numerator and denominator are two-digit numbers, find the smallest fraction that is greater than $\frac{5}{7}$. Provide its numerator in the answer.


Find the smallest natural number that has the following properties: the remainder when divided by 20 is one less than the remainder when divided by 21, and the remainder when divided by 22 is 2.


Vasily Ivanovich took a loan of 750,000 rubles from a major bank to purchase a car. According to the loan terms, he will repay the initial loan amount plus 37,500 rubles in interest after 5 months. Determine the annual interest rate of the loan (in %), assuming a year consists of 360 days and each month has 30 days.


How many triples (A, B, C) are there of sets with union \(A \cup B \cup C = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}\) and \(A \cap B \cap C = \emptyset\)?


A circle inscribed in a right triangle \(A B C\) \(\left(\angle A B C = 90^{\circ}\right)\) is tangent to sides \(A B\), \(B C\), and \(A C\) at points \(C_{1}\), \(A_{1}\), and \(B_{1}\) respectively. An excircle is tangent to side \(B C\) at point \(A_{2}\). \(A_{0}\) is the center of the circle circumscribed around triangle \(A_{1} A_{2} B_{1}\); similarly, point \(C_{0}\) is defined. Find the angle \(A_{0} B C_{0}\).


We divide two opposite sides of a convex quadrilateral into 100 equal parts, then we connect the corresponding points. The area of the first of the 100 resulting quadrilaterals is 1 unit, and the area of the last, 100th quadrilateral is 2 units. What is the area of the original quadrilateral?


The Antarctican language has an alphabet of just 16 letters. Every word in the language has exactly 3 letters, and no word's first letter equals any word's last letter. Determine the maximum possible number of words in the language.


Given the system of equations

\[
\begin{cases}
|16 + 6x - x^2 - y^2| + |6x| = 16 + 12x - x^2 - y^2 \\
(a + 15)y + 15x - a = 0
\end{cases}
\]

a) Plot on the $(x, y)$ plane the set of points that satisfy the first equation of the system, and find the area of the resulting figure.

b) Find all values of the parameter \(a\) for which the system has exactly one solution.

Solution: \boxed{a = -20, a = -12}


Vitya has five math lessons per week, one on each day from Monday to Friday. Vitya knows that with a probability of \(1 / 2\), the teacher will not check his homework at all during the week, and with a probability of \(1 / 2\), the teacher will check it, but only once on any of the math lessons, with equal chances on any day.

By the end of the math lesson on Thursday, Vitya realized that the teacher has not checked his homework so far this week. What is the probability that the homework will be checked on Friday?


If real numbers \( x \) and \( y \) satisfy the equations \( 2^x + 4x + 12 = 0 \) and \(\log_2((y-1)^3) + 3y + 12 = 0\), what is \( x + y \)?


Two equal cones with a common vertex touch each other and some plane $\alpha$. Let $l$ be the line along which the planes of the bases of the cones intersect. Find the angle between the line $l$ and the plane $\alpha$ if the height of each cone is 2 and the radius of the base is 1.


In how many ways can the faces of a cube be painted in 6 colors such that each color appears exactly once? Two painted cubes are considered identical if they can be rotated to look the same.


In a table tennis tournament, sixth graders and seventh graders participated, with the number of sixth graders being twice the number of seventh graders. The tournament was held in one round. The number of matches won by the seventh graders was $40\%$ greater than the number of matches won by the sixth graders. How many participants were there in the tournament?


There is a large square containing two smaller squares that can move freely within the large square (none of the smaller squares can move outside the large square, and their sides must be parallel to the sides of the large square). If the overlapping area of the two smaller squares ranges from a minimum of 9 to a maximum of 25, and the sum of the side lengths of the three squares (one large square and two smaller squares) is 23, then what is the sum of the areas of the three squares?


Find the number of subsets \( B \) of the set \( \{1, 2, \ldots, 2005\} \) such that the sum of its elements leaves a remainder of 2006 when divided by 2048.


A real-valued function \( f \) satisfies the relation
\[ f(x^2 + x) + 2 f(x^2 - 3x + 2) = 9x^2 - 15x \]
for all real values of \( x \). Find \( f(2011) \).


In a $2 \times 2$ table, each cell is filled with a distinct number. It turns out that the sum of the numbers in the first row is equal to the sum of the numbers in the second row, and the product of the numbers in the first column is equal to the product of the numbers in the second column. What can the sum of all four numbers in the table be? Find all possible values and explain why there are no other possible values.


Paul fills in a \( 7 \times 7 \) grid with the numbers 1 through 49 in a random arrangement. He then erases his work and does the same thing again to obtain two different random arrangements of the numbers in the grid. What is the expected number of pairs of numbers that occur in either the same row as each other or the same column as each other in both of the two arrangements?


In a rectangular parallelepiped $A B C D A_{1} B_{1} C_{1} D_{1}$, the lengths of the edges are known: $A B=14$, $A D=60$, $A A_{1}=40$. A point $E$ is marked at the midpoint of edge $A_{1} B_{1}$, and point $F$ is marked at the midpoint of edge $B_{1} C_{1}$. Find the distance between the lines $A E$ and $B F$.


Specify the smallest number that ends in 37, has a digit sum of 37, and is divisible by 37.


A cube $A B C D-A_{1} B_{1} C_{1} D_{1}$ has an edge length of 1. What is the distance between the lines $A_{1} C_{1}$ and $B D_{1}$?


There is a group of children, each with an integer age. One of them is 10 years old. If the oldest is 13 years old and the total sum of all their ages is 50, and if excluding the 10-year-old child the remaining children's ages form an arithmetic sequence in ascending order, how many children are in the group?


Two triangles - an equilateral triangle with side \( a \) and an isosceles right triangle with legs equal to \( b \) - are positioned in space such that their centers of mass coincide. Find the sum of the squares of the distances from all vertices of one triangle to all vertices of the other.

Solution: \boxed{3a^2 + 4b^2}


In the rectangular coordinate system $xOy$, $F_{1}$ and $F_{2}$ are the left and right foci of the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{4} = 1$ with $a > 0$. The line $l$ passes through the right vertex $A$ of the hyperbola and the point $B(0, 2)$. If the sum of the distances from points $F_{1}$ and $F_{2}$ to the line $l$ is equal to the length of the segment $B F_{2}$, then the coordinates of point $A$ are $\qquad$

Solution: \boxed{(2\sqrt{2}, 0)}


A kindergarten received cards for learning to read: some are labeled "МА", and the rest are labeled "НЯ".

Each child took three cards and started to form words from them. It turned out that 20 children could form the word "МАМА" from their cards, 30 children could form the word "НЯНЯ", and 40 children could form the word "МАНЯ". How many children have all three cards the same?


The product of the four numbers that are the roots of the equations \( x^{2} + 2bx + c = 0 \) and \( x^{2} + 2cx + b = 0 \) is equal to 1. It is known that the numbers \( b \) and \( c \) are positive. Find them.

Solution: \boxed{b = c = 1}


Two circles with radii 2 and 6, centered at points \( O_{1} \) and \( O_{2} \) respectively, are externally tangent at point \( C \). A common external tangent and a common internal tangent to the circles are drawn. These tangents intersect at point \( D \). Find the radius of the inscribed circle in triangle \( O_{1} O_{2} D \).

Solution: \boxed{2(\sqrt{3} - 1)}


Solve the system of equations:
\[
\left\{
\begin{array}{l}
x + 3y + 14 \leq 0, \\
x^4 + 2x^2y^2 + y^4 + 64 - 20x^2 - 20y^2 = 8xy.
\end{array}
\right.
\]

Solution: \boxed{(-2, -4)}


The function \( f: \mathbf{R}^{+} \rightarrow \mathbf{R}^{+} \) satisfies the condition: for any \( x, y, u, v \in \mathbf{R}^{+} \), the following inequality holds:
\[ f\left(\frac{x}{2 u}+\frac{y}{2 v}\right) \leq \frac{1}{2} u f(x) + \frac{1}{2} v f(y). \]
Find all such functions \( f \).

Solution: \boxed{f(x) = \frac{c}{x}}


In $\triangle ABC$, $AC=8$, $BC=10$, and $32 \cos (A - B) = 31$. Find the area of $\triangle ABC$.

Solution: \boxed{15 \sqrt{7}}


In a math competition, there are 5 problems, each with a different natural number score. The smaller the problem number, the lower its score (for example, the score for problem 1 is less than the score for problem 2). Xiao Ming solved all the problems correctly. The total score for the first 2 problems is 10 points, and the total score for the last 2 problems is 18 points. How many points did Xiao Ming score in total?


Pedrinho is playing with three triangular pieces with sides $(5,8,10),(5,10,12)$, and $(5,8,12)$ as shown in the figures below. He can join two pieces by sticking exactly the sides of the same length together. For example, he can join the side 10 of the first piece with the side 10 of the second piece, but he cannot join the side 10 of the first piece with the side 8 of the third piece because they do not have the same length. What is the greatest perimeter Pedrinho can obtain by joining the three pieces?

(Note: The figures provided in the original problem have been omitted for text-based translation purposes.)


From a sequence of natural numbers, all numbers that are squares or cubes of integers have been deleted. Which number is in the 100th position among the remaining numbers?


The team wants to win $\frac{3}{4}$ of all their matches in the season. However, in the first third, they won only $55 \%$ of the matches.

a) What percentage of the remaining matches must the team win to achieve their intended goal?

b) If the team won all the remaining matches, what percentage of their matches would they win in the entire season?


The numbers \(1, 2, \ldots, 20\) are put into a hat. Claire draws two numbers from the hat uniformly at random, \(a < b\), and then puts them back into the hat. Then, William draws two numbers from the hat uniformly at random, \(c < d\). Let \(N\) denote the number of integers \(n\) that satisfy exactly one of \(a \leq n \leq b\) and \(c \leq n \leq d\). Compute the probability \(N\) is even.


In the sequence $\left\{a_{n}\right\}$, $a_{0}=2$, and $a_{n}=(n+2) a_{n-1}$ for $n \geqslant 1$, find $a_{n}$.


Older brother took Misha’s identical unpainted cubes and assembled a large cube from them. Then he painted some (but not all) faces of the large cube completely in red. When the paint dried, Misha disassembled the large cube and found that exactly 343 small cubes had no red faces. How many faces of the large cube did Misha’s brother paint? Justify your answer.


In an ornithological park, there are birds of several species, with a total of 2021 individuals. The birds are seated in a row, and it turns out that between any two birds of the same species, there is an even number of birds. What is the smallest number of bird species that could be?


Vladislav Vladimirovich, having less than 100 rubles, went for a walk. When entering a cafe with \( m \) rubles and \( n \) kopecks, he would spend \( n \) rubles and \( m \) kopecks (where \( m \) and \( n \) are natural numbers). What is the maximum number of cafes Vladislav Vladimirovich could visit?


Find the smallest natural number that starts with the digit five, which becomes four times smaller if this five is removed from the beginning of its decimal representation and appended to the end.

Solution: \boxed{ 512820 }


In a "clearance game," the rules stipulate that in round \( n \), a dice is to be rolled \( n \) times. If the sum of the points of these \( n \) rolls is greater than \( 2^{n} \), the player clears the round.

(1) What is the maximum number of rounds a player can clear in this game?
(2) What is the probability that the player clears the first three rounds consecutively?

(Note: The dice is a fair cube with faces numbered \( 1, 2, 3, 4, 5, 6 \), and the point on the top face after landing indicates the outcome of the roll.)


$n$ points are located at the vertices of a convex $n$-gon. Inside this $n$-gon, $k$ points are marked. It turns out that any three of the $n+k$ points do not lie on the same straight line and form the vertices of an isosceles triangle. What can be the value of $k$?

Solution: \boxed{k = 0 \text{ or } 1}


What is the remainder when $46^{925}$ is divided by $21?$


Vasya wrote twenty threes in a row on a board. By placing "+" signs between some of them, Vasya found that the sum equals 600. How many plus signs did Vasya place?


Given \( x_{1} = 1 \), \( x_{2} = 2 \), and \( x_{n+2} = \frac{x_{n+1}^{2} - 7}{x_{n}} \left( n \in \mathbf{N}^{*} \right) \), find the smallest positive period of the sequence \( \left\{ x_{n} \right\} \).


From the set of three-digit numbers that do not contain the digits 0, 6, 7, 8, or 9, several numbers were written on paper in such a way that no two numbers can be obtained from each other by swapping two adjacent digits. What is the maximum number of such numbers that could have been written?


In a trapezoid, where the diagonals intersect at a right angle, it is known that the midline is 6.5 and one of the diagonals is 12. Find the length of the other diagonal.


A graph has 100 points. Given any four points, there is one joined to the other three. Show that one point must be joined to all 99 other points. What is the smallest number possible of such points (that are joined to all the others)?


Each of the ten cards has a real number written on it. For every non-empty subset of these cards, the sum of all the numbers written on the cards in that subset is calculated. It is known that not all of the obtained sums turned out to be integers. What is the largest possible number of integer sums that could have resulted?


A positive integer \( n \) is called "indivisible" if for every integer \( k \) such that \( 1 < k < n \) and \( \gcd(k,n) = 1 \), \( k \) is a prime number.

How many indivisible numbers greater than 2 are there?


In a plane, there are 5 points, no three of which are collinear. Connecting these points with line segments, there are at least $\qquad$ intersection points other than the 5 points themselves.


Is there an infinite set of points in the plane such that no three points are collinear, and the distance between any two points is rational?

Solution: \boxed{\text{Yes}}


We know the following about three natural numbers:

a) all three are different;

b) their sum is 406;

c) their greatest common divisor is a prime number greater than 2 (let's denote it by \( p \));

d) if each number is divided by \( p \), we get three prime numbers again (let these be \( p_{1}, p_{2}, p_{3} \)). What are these numbers?

Note that 1 is not considered a prime number.

Solution: \boxed{58, 145, 203}


The sum of the positive integers \( a_1, a_2, \cdots, a_{49} \) is 999. Let \( d \) be the greatest common divisor of \( a_1, a_2, \cdots, a_{49} \). What is the maximum value of \( d \)?


Given a triangle with side lengths \( l \), \( m \), and \( n \) that are integers such that \( l > m > n \). It is known that \( \left\{\frac{3^{l}}{10^{4}}\right\} = \left\{\frac{3^{m}}{10^{4}}\right\} = \left\{\frac{3^{n}}{10^{4}}\right\} \), where \( \{x\} = x - [x] \) and \( [x] \) represents the greatest integer less than or equal to \( x \). Find the minimum possible perimeter of such a triangle.


What is the maximum number of bishops that can be placed on an $N \times N$ chessboard so that no two bishops can attack each other?

Solution: \boxed{2N - 2}


Simplify \(\sqrt[3]{5 \sqrt{2} + 7} - \sqrt[3]{5 \sqrt{2} - 7}\) into a rational number.


Calculate

$$
\int_{L}\left(z^{2}+2z \bar{z}\right)d z
$$

where \( L \) is the arc of the circle \( |z|=1 \) with \( \arg z \in[0, \pi] \).


Given a right square prism \(A B C D-A_{1} B_{1} C_{1} D_{1}\) with a base \(A B C D\) that is a unit square, if the dihedral angle \(A_{1}-B D-C_{1}\) is \(\frac{\pi}{3}\), find the length of \(A A_{1}\).


Represent the number 36 as the product of three whole number factors, the sum of which is equal to 4. What is the smallest of these factors?


Given \(1.08 \div 1.2 \div 2.3 = 10.8 \div \square\), find the number represented by \(\square\).


You are given two line segments of length \(2^{n}\) for each integer \(0 \leq n \leq 10\). How many distinct nondegenerate triangles can you form with three of the segments? Two triangles are considered distinct if they are not congruent.


Given a triangle \(ABC\) with an area of 1. The first player chooses a point \(X\) on side \(AB\), the second player chooses a point \(Y\) on side \(BC\), and then the first player chooses a point \(Z\) on side \(AC\). The first player's goal is to maximize the area of triangle \(XYZ\), and the second player's goal is to minimize it. What is the largest area that the first player can guarantee for themselves?


Let \( A = \{0, 1, \cdots, 2016\} \). If there exists a surjective function \( f: \mathbf{N} \rightarrow A \) such that for any \( i \in \mathbf{N} \),
\[
f(i+2017) = f(i),
\]
we call \( f \) a "harmonic function."

Define:
\[
\begin{array}{l}
f^{(1)}(x) = f(x), \\
f^{(k+1)}(x) = f(f^{(k)}(x)) \quad (k \in \mathbf{N}_{+}).
\end{array}
\]

Let \( f \) be a "harmonic mapping" satisfying the conditions: there exists a positive integer \( M \) such that
1. For \( m < M \), if \( i, j \in \mathbf{N} \) and \( i \equiv j + 1 \pmod{2017} \), then \( f^{(m)}(i) - f^{(m)}(j) \not\equiv \pm 1 \pmod{2017} \);
2. If \( i, j \in \mathbf{N} \) and \( i \equiv j + 1 \pmod{2017} \), then \( f^{(M)}(i) - f^{(M)}(j) \equiv \pm 1 \pmod{2017} \).

Find the maximum possible value of \( M \).


Find all values of the parameter \( a \) for which the roots \( x_{1} \) and \( x_{2} \) of the equation

\[ 2x^{2} - 2016(x - 2016 + a) - 1 = a^{2} \]

satisfy the double inequality \( x_{1} < a < x_{2} \).

Solution: \boxed{(2015, 2017)}


Given real numbers \(a_{1}, a_{2}, \cdots, a_{18}\) such that:
\[ a_{1} = 0, \left|a_{k+1} - a_{k}\right| = 1 \text{ for } k = 1, 2, \cdots, 17, \]
what is the probability that \(a_{18} = 13\)?


Money placed in the 1st month will remain on deposit for 6 months and yield a nominal income of $8700\left((1+0.06 / 12)^{\wedge}-1\right)=264.28$ rubles.

Funds placed in the 2nd month will be on deposit for 5 months and yield a nominal income of $8700\left((1+0.06 / 12)^{\wedge} 5-1\right)=219.69$ rubles.

Similarly, for the following months:

$8700\left((1+0.06 / 12)^{\wedge 4-1}\right)=175.31$ rubles.

$8700\left((1+0.06 / 12)^{\wedge} 3-1\right)=131.15$ rubles.

$8700\left((1+0.06 / 12)^{\wedge} 2-1\right)=87.22$ rubles.

$8700((1+0.06 / 12)-1)=43.5$ rubles.

Therefore, the nominal income for 6 months will be:

$264.28+219.69+175.31+131.15+87.22+43.5=921.15$ rubles.


At a certain point on the bank of a wide and turbulent river, 100 meters away from a bridge, a siren is installed that emits sound signals at regular intervals. Another identical siren was taken by Glafira, who got on a bike and positioned herself at the beginning of the bridge on the same bank. Gavrila got into a motorboat, located on the bank halfway between the first siren and the beginning of the bridge. The experimenters start simultaneously, and the speeds of the bicycle and the motorboat relative to the water are both 20 km/h and directed perpendicular to the bank. It turned out that the sound signals from both sirens reach Gavrila simultaneously. Determine the distance from the starting point where Gavrila will be when he is 40 meters away from the bank. Round your answer to the nearest whole number of meters. The riverbank is straight, and the current speed at each point is directed along the bank.


From $m$ boys and $n$ girls $(10 \geqslant m > n \geqslant 4)$, two people are selected as team leaders. Suppose event $A$ represents the selected 2 people being of the same gender, and event $B$ represents the selected 2 people being of different genders. If the probability of event $A$ is equal to the probability of event $B$, what are the possible values of $(m, n)$?

Solution: \boxed{(10, 6)}


There are numbers from 1 to 2013 on the blackboard. Each time, two numbers can be erased and replaced with the sum of their digits. This process continues until there are four numbers left, whose product is 27. What is the sum of these four numbers?


Let \( n \) be a positive integer, and denote by \( f(n) \) the smallest positive integer \( k \) for which \( \left\lfloor \frac{n^2}{k} \right\rfloor = \left\lfloor \frac{n^2}{k+1} \right\rfloor \). Determine the value of \( \left\lfloor \frac{n^2}{f(n)} \right\rfloor + f(n) \).


A square with a side length of one unit has one of its vertices separated from the other three by a line \( e \). The products of the distances of opposite vertex pairs from \( e \) are equal. What is the distance of the center of the square from \( e \)?


Compute the sum of all positive integers \( n < 2048 \) such that \( n \) has an even number of 1's in its binary representation.


Compute the definite integral:

$$
\int_{\pi / 2}^{\pi} 2^{4} \cdot \sin ^{6} x \cos ^{2} x \, dx
$$


Alex wrote all natural divisors of a natural number \( n \) on the board in ascending order. Dima erased several of the first and several of the last numbers of the resulting sequence so that 151 numbers remained. What is the maximum number of these 151 divisors that could be fifth powers of natural numbers?


For a quadratic function \( p(x) = ax^2 + bx + c \), there exists some integer \( n \) such that \( p(n) = p\left(n^2\right) \). Provide an example of a function \( p(x) \) for which the number of such integers \( n \) is the greatest. What is this greatest number of integers \( n \)?


The area of the closed figure enclosed by the parabola \( y^{2} = a x \) (with \( a > 0 \)) and the line \( x = 1 \) is \( \frac{4}{3} \). What is the coefficient of the term containing \( x^{-18} \) in the expansion of the binomial \( \left( x + \frac{a}{x} \right)^{20} \)?


Determine all continuous functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) satisfying the following conditions:
1. \( f(0) = 0 \)
2. \( f(x^2 - y^2) \geq x f(x) - y f(y) \) for all \( x, y \in \mathbb{R} \).

Solution: \boxed{f(x) = kx}


On the Island of Truth and Lies, there are knights who always tell the truth and liars who always lie. One day, 15 islanders lined up in order of height (from tallest to shortest, with the tallest standing first) for a game. Each person had to say one of the following phrases: "There is a liar below me" or "There is a knight above me." Those standing in positions four through eight said the first phrase, and the rest said the second phrase. How many knights were among these 15 people, given that all the residents are of different heights?


There are 2013 cards with the digit 1 and 2013 cards with the digit 2. Vasya creates a 4026-digit number from these cards. In one move, Petya can swap any two cards and pay Vasya 1 ruble. The process ends when Petya forms a number that is divisible by 11. What is the maximum amount of money Vasya can earn if Petya aims to pay as little as possible?


Schoolboy Alexey told his parents that he is already an adult and can manage his finances independently. His mother suggested using a duplicate bank card from her account. For participation in a charitable Christmas program, Alexey wants to buy 40 "Joy" chocolate bars and donate them to an orphanage. However, the bank, where Alexey's parents are clients, has implemented a new system to protect against unauthorized card payments. The protection system analyzes the root mean square (RMS) value of expenses for the last 3 purchases (S) using the formula \(S=\sqrt{\frac{x_{1}^{2}+x_{2}^{2}+x_{3}^{2}}{3}}\), where \(x_{1}, x_{2}\), and \(x_{3}\) are the costs of the previous purchases, and compares the value of \(S\) with the cost of the current purchase. If the cost of the current payment exceeds the value \(S\) by 3 times, the bank blocks the payment and requires additional verification (e.g., a call from mom to the call center). In the last month, payments made on the card were only for cellphone bills in the amount of 300 rubles each. Into how many minimum number of receipts should Alexey split the purchase so that he can buy all 40 "Joy" chocolate bars at a cost of 50 rubles each?


Solve the system of equations:
$$\begin{cases}
x^2 y + x y^2 + 3x + 3y + 24 = 0, \\
x^3 y - x y^3 + 3x^2 - 3y^2 - 48 = 0.
\end{cases}$$

Solution: \boxed{(-3, -1)}


Triangle \( A B C \) is given in the plane. Let \( A D \) be the angle bisector of \( \angle B A C \); let \( B E \) be the altitude from \( B \) to \( A D \), and let \( F \) be the midpoint of \( A B \). Given that \( A B = 28 \), \( B C = 33 \), and \( C A = 37 \), what is the length of \( E F \)?


Find the maximum of $x^{2} y^{2} z$ under the condition that $x, y, z \geq 0$ and $2 x + 3 x y^{2} + 2 z = 36$.


Show that for all \( a, b, c > 0 \) such that \( abc = 1 \), we have:

\[
\left(a^{2} + 1\right)\left(b^{3} + 2\right)\left(c^{6} + 5\right) \geq 36
\]


The game "Geographic Dice" involves each participant rolling a die a number $n$ of times and noting the sequence, with the first roll indicating the distance, in meters, that the participant will walk to the North, the second to the East, the third to the South, the fourth to the West, the fifth to the North, and so on. After this, the distance to the origin is measured. The one who gets closest to the initial position wins.

a) Márcia rolled 214365 (6 rolls). What is her distance from the origin?
b) In 4 rolls, how many possibilities exist for the participant to return to the origin after the last roll?
c) In 5 rolls, how many possibilities exist for the participant to return to the origin after the last roll?


The base of a right prism is a rhombus. The areas of the diagonal sections of this prism are \(P\) and \(Q\). Find the lateral surface area of the prism.

Solution: \boxed{2 \sqrt{P^2 + Q^2}}


Let the set \( S = \{1,2,3, \cdots, 10\} \). Given that subset \( A \) of \( S \) satisfies \( A \cap \{1,2,3\} \neq \varnothing \) and \( A \cup \{4,5,6\} \neq S \), determine the number of such subsets \( A \).

(Note: The original problem includes determining the count of subset \( A \) that meets the given conditions.)


The area of an isosceles obtuse triangle is 8, and the median drawn to one of its equal sides is $\sqrt{37}$. Find the cosine of the angle at the vertex.


Let \( f(x) = x^2 + 12x + 30 \). Find the largest real root of the equation \( f(f(f(x))) = 0 \).

Solution: \boxed{-6 + \sqrt[8]{6}}


There are 11 kilograms of cereal. How can one measure out 1 kilogram of cereal using a balance scale and a single 3-kilogram weight in just two weighings?

Solution: \boxed{1 \text{ kg}}


In triangle $ABC$, it is known that $AB = \sqrt{14}$ and $BC = 2$. A circle is drawn through point $B$, the midpoint $D$ of segment $BC$, and point $E$ on segment $AB$, and it is tangent to side $AC$. Find the ratio in which this circle divides segment $AB$ if $DE$ is the diameter of this circle.


An urn contains 101 balls, exactly 3 of which are red. The balls are drawn one by one without replacement. On which draw is it most likely to pull the second red ball?


On the edge \(A A_{1}\) of cube \(A B C D A_{1} B_{1} C_{1} D_{1}\) with edge length \(3 \sqrt{2}\), point \(K\) is marked. In space, a point \(T\) is marked such that \(T B=7\) and \(T C=\sqrt{67}\). What is the minimum and maximum length of segment \(TK\)? Provide justification for your answer.

Solution: \boxed{1 \text{ and } 13}


After watching a movie, viewers rated the movie sequentially with an integer score from 0 to 10. At any given moment, the movie rating was calculated as the sum of all given scores divided by their number. At a certain moment $T$, the rating turned out to be an integer, and then with each new voting viewer, it decreased by one unit. What is the maximum number of viewers who could have voted after moment $T ?$


We wrote the numbers from 1 to 2009 on a piece of paper. In the second step, we also wrote down twice each of these numbers on the paper, and then we erased the numbers that appeared twice. We repeat this process such that, in the $i$-th step, we write $i$ times each of the numbers from 1 to 2009 on the paper and then erase the numbers that appear twice. How many numbers will be on the paper after the 2009th step?


15 boys and 20 girls sat around a round table. It turned out that the number of pairs of boys sitting next to each other is one and a half times less than the number of pairs of girls sitting next to each other. Find the number of boy-girl pairs sitting next to each other.


What is the expected value of the minimum element of a randomly selected subset of size \( r \) (where \( 1 \leq r \leq n \)) from the set \(\{1, 2, \cdots, n\}\)?


Is it possible to place 8 different numbers from the range 1 to 220 at the vertices of a cube so that the numbers at adjacent vertices have a common divisor greater than 1, while the numbers at non-adjacent vertices do not have one?

Solution: \boxed{\text{No}}


A $5 \times 5$ square is to be cut into rectangles of two types: $1 \times 4$ and $1 \times 3$. How many rectangles can be obtained after the cutting? Justify your answer.


The distance from \(A\) to \(B\) is 999 km. Along the road, there are kilometer markers indicating the distances to \(A\) and \(B\): 0।999, 1।998, \(\ldots, 999।0. How many of these markers have only two different digits?


As shown in the figure, the side length of square ABCD is 10, with AG = CH = 8 and BG = DH = 6. Connect GH, then the length of segment GH is

Solution: \boxed{2 \sqrt{2}}


2011 warehouses are connected by roads such that each warehouse can be reached from any other, possibly by traveling through several roads. Each warehouse contains $x_{1}, \ldots, x_{2011}$ kg of cement, respectively. In one trip, any amount of cement can be transported from one warehouse to another along a connecting road. Ultimately, the plan is to have $y_{1}, \ldots, y_{2011}$ kg of cement at the warehouses, respectively, with the condition that $x_{1} + x_{2} + \ldots + x_{2011} = y_{1} + y_{2} + \ldots + y_{2011}$. What is the minimum number of trips required to execute this plan for any values of $x_{i}$ and $y_{i}$ and any road scheme?


Find the point \( M^{\prime} \) that is symmetric to the point \( M \) with respect to the plane.

\( M(-1, 0, -1) \)

\( 2x + 6y - 2z + 11 = 0 \)

Solution: \boxed{(-2, -3, 0)}


For which integers \( n \geq 2 \) can we arrange the numbers \( 1, 2, \ldots, n \) in a row, such that for all integers \( 1 \leq k \leq n \) the sum of the first \( k \) numbers in the row is divisible by \( k \)?

Solution: \boxed{n = 3}


If \( a > 0 \), \( b > 0 \), and \( a + 2b = 6 \), what is the maximum value of \( \ln a + 2 \ln b \)?

Solution: \boxed{3 \ln 2}


From point \( K \) on side \( AC \) of triangle \( ABC \), perpendiculars \( KL_1 \) and \( KM_1 \) are dropped to sides \( AB \) and \( BC \) respectively. From point \( L_1 \), perpendicular \( L_1 L_2 \) is dropped to \( BC \), and from point \( M_1 \), perpendicular \( M_1 M_2 \) is dropped to \( AB \).

It turns out that triangles \( BL_1M_1 \) and \( BL_2M_2 \) are similar (point \( L_1 \) in the first triangle corresponds to point \( M_2 \) in the second). Additionally, \( BL_2 = 6 \) and \( L_2M_1 = 4 \). Find \( L_1L_2 \).


Solution: \boxed{n = 11}


Consider the function $[x]$ denoting the greatest integer less than or equal to the real number $x$. In the plane, find the area of the shape formed by the points satisfying the equation $[x]^{2} + [y]^{2} = 50$.


In a store, there are 9 headphones, 13 computer mice, and 5 keyboards for sale. Besides these, there are also 4 sets of "keyboard and mouse" and 5 sets of "headphones and mouse." How many ways can you buy three items: headphones, a keyboard, and a mouse? Answer: 646.


In a right triangle, the segments \(a\) and \(b\) are given, which divide the hypotenuse at the point where the incircle of the triangle touches it. Find the area of this triangle.

Solution: \boxed{ab}


Show that for any natural number \( k \), it is possible to find an integer \( n \) such that the numbers \( n+1, \ldots, n+k \) are all composite.

Solution: \boxed{n = (k+1)! + 1}


Mila and Zhenya each thought of a number and wrote down all the natural divisors of their numbers on the board. Mila wrote 10 numbers, Zhenya 9 numbers, and the number 6 appeared twice. How many distinct numbers are on the board in total?


Given that \( n \) is a positive integer, find the smallest positive integer \( k \) satisfying the following condition: In a \( 2n \times 2n \) grid, mark \( k \) cells such that there exists a unique way to partition the \( 2n \times 2n \) grid into \( 1 \times 2 \) and \( 2 \times 1 \) dominoes, with each domino not containing two marked cells.


40 people entered a room with 40 black and white chairs and sat down. They all claimed to be sitting on black chairs. Then they moved around somehow, and exactly 16 claimed to be sitting on a white chair. Each person either lied both times or told the truth both times. How many people who lied were sitting on a black chair the second time?


For which values of the parameter \( a \), the equation

\[
\log _{2}\left(2 x^{2}-x-2 a-4 a^{2}\right)+3 \log _{1 / 8}\left(x^{2}-a x-2 a^{2}\right)=0
\]

has two distinct roots, the sum of whose squares belongs to the interval \( (4, 8) \)?

Solution: \boxed{( \frac{3}{5}, 1 )}


While one lion cub, located 6 minutes away from the watering hole, was heading towards it, the second lion cub, having already quenched its thirst, was heading back along the same road 1.5 times faster than the first. At the same time, a turtle, located 32 minutes away from the watering hole, also headed towards it on the same road. At some point, the first lion cub stepped on the turtle, and 2 minutes and 24 seconds later, the second lion cub did the same. How many minutes after the second incident did the turtle reach the watering hole, assuming all three moved at constant speeds?


Let \(\omega_{1}\) and \(\omega_{2}\) be two non-intersecting circles. Suppose the following three conditions hold:
- The length of a common internal tangent of \(\omega_{1}\) and \(\omega_{2}\) is equal to 19.
- The length of a common external tangent of \(\omega_{1}\) and \(\omega_{2}\) is equal to 37.
- If two points \(X\) and \(Y\) are selected on \(\omega_{1}\) and \(\omega_{2}\), respectively, uniformly at random, then the expected value of \(XY^{2}\) is 2023.
Compute the distance between the centers of \(\omega_{1}\) and \(\omega_{2}\).


The numbers \(1000^{2}, 1001^{2}, 1002^{2}, \ldots\) have their last three digits discarded. How many of the first terms in the resulting sequence form an arithmetic progression?


How many factors of 2 are in the prime factorization of 1984!?


If the union of two sets \( A \) and \( B \) has two elements, let \( f(A) \) denote the number of elements in \( A \) and \( f(B) \) denote the number of elements in \( B \). How many such pairs \((f(A), f(B))\) exist?


If \(a, b\), and \(c\) are random real numbers from 0 to 1, independently and uniformly chosen, what is the average (expected) value of the smallest of \(a, b\), and \(c\)?


Given the functions \( f_{0}(x)=|x|, f_{1}(x)=\left| f_{0}(x) - 1 \right|, f_{2}(x)= \left| f_{1}(x) - 2 \right| \), find the area of the enclosed region between the graph of \( y=f_{2}(x) \) and the x-axis.


Find all three-digit numbers \( \mathrm{N} \) such that the sum of the digits of \( \mathrm{N} \) is 11 times less than the number \( \mathrm{N} \) itself.


We use \( S_{k} \) to represent an arithmetic sequence with the first term \( k \) and common difference \( k^{2} \). For example, \( S_{3} \) is \( 3, 12, 21, \cdots \). If 306 is a term in \( S_{k} \), the sum of all possible \( k \) that satisfy this condition is ____.


Calculate the lengths of the arcs of the curves given by the equations in polar coordinates.

$$
\rho=6 \cos \varphi, 0 \leq \varphi \leq \frac{\pi}{3}
$$


Find pairs of positive integers \((a, b)\) that make \(\frac{a b^{2}}{a+b} (a \neq b)\) a prime number.

Solution: \boxed{(6, 2)}


Let the system of equations be satisfied for positive numbers \( x, y, z \):

\[
\left\{
\begin{array}{l}
x^{2} + x y + y^{2} = 108 \\
y^{2} + y z + z^{2} = 49 \\
z^{2} + x z + x^{2} = 157
\end{array}
\right.
\]

Find the value of the expression \( x y + y z + x z \).


Point \( M \) lies on the edge \( AB \) of cube \( ABCD A_1 B_1 C_1 D_1 \). A rectangle \( MNLK \) is inscribed in square \( ABCD \) such that one of its vertices is point \( M \) and the other three vertices are located on different sides of the square base. Rectangle \( M_1 N_1 L_1 K_1 \) is the orthogonal projection of rectangle \( MNLK \) onto the plane \( A_1 B_1 C_1 D_1 \) of the top face. The plane of quadrilateral \( MK_1 L_1 N \) forms an angle \( \alpha \) with the plane of the cube's base, where \( \cos \alpha=\sqrt{\frac{2}{11}} \). Find the ratio \( AM : MB \).


In a football tournament, six teams played: each team played once with every other team. Teams that scored twelve or more points advance to the next round. For a victory, 3 points are awarded, for a draw - 1 point, for a loss - 0 points. What is the maximum number of teams that can advance to the next round?


Given 1985 sets, each consisting of 45 elements, where the union of any two sets contains exactly 89 elements.

How many elements are in the union of all these 1985 sets?


Four cars $A, B, C,$ and $D$ start simultaneously from the same point on a circular track. $A$ and $B$ travel clockwise, while $C$ and $D$ travel counterclockwise. All cars move at constant (but pairwise different) speeds. Exactly 7 minutes after the start of the race, $A$ meets $C$ for the first time, and at that same moment, $B$ meets $D$ for the first time. After another 46 minutes, $A$ and $B$ meet for the first time. After how much time from the start of the race will $C$ and $D$ meet for the first time?


Inside a large triangle with a perimeter of 120, several segments were drawn to divide it into nine smaller triangles, as shown in the figure. It turned out that the perimeters of all nine small triangles are equal. What could their perimeters be? List all possible options.

The perimeter of a shape is the sum of the lengths of all its sides.


The lengths of the sides of a quadrilateral are \(2006 \, \text{cm}, 2007 \, \text{cm}, 2008 \, \text{cm}\) and \(x \, \text{cm}\). If \(x\) is an integer, find the largest possible value of \(x\).


The positive integer \( N \) has six digits in increasing order. For example, 124689 is such a number. However, unlike 124689, three of the digits of \( N \) are 3, 4, and 5, and \( N \) is a multiple of 6. How many possible six-digit integers \( N \) are there?


In triangle ABC, angle C is a right angle, and CD is the altitude. Find the radius of the circle inscribed in triangle ABC if the radii of the circles inscribed in triangles ACD and BCD are 6 and 8, respectively.


Two squares of a \( 7 \times 7 \) board are painted yellow and the rest are painted green. Two color schemes are equivalent if one can be obtained from the other by rotating the board. How many non-equivalent color schemes can we obtain?


Let \( ABCD \) be a convex quadrilateral and \( Z \) a point inside it. Denote by \( \alpha_1 \) the angle \( \widehat{ZAD} \), \( \alpha_2 \) the angle \( \widehat{ZAB} \), \( \beta_1 \) the angle \( \widehat{ZBA} \), \( \beta_2 \) the angle \( \widehat{ZBC} \), \( \gamma_1 \) the angle \( \widehat{ZCB} \), \( \gamma_2 \) the angle \( \widehat{ZCD} \), \( \delta_1 \) the angle \( \widehat{ZDC} \), and \( \delta_2 \) the angle \( \widehat{ZDA} \). Show the following equality:

$$
\frac{\sin \alpha_1}{\sin \alpha_2} \cdot \frac{\sin \beta_1}{\sin \beta_2} \cdot \frac{\sin \gamma_1}{\sin \gamma_2} \cdot \frac{\sin \delta_1}{\sin \delta_2} = 1
$$


Given the sequence \(\left\{a_{n}\right\}\) satisfying \(a_{1} = p\), \(a_{2} = p + 1\), and \(a_{n+2} - 2a_{n+1} + a_{n} = n - 20\), where \(p\) is a given real number and \(n\) is a positive integer, find the value of \(n\) such that \(a_{n}\) is minimized.


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


Given $a, b, c, d \in \mathbf{N}$ such that $342(abcd + ab + ad + cd + 1) = 379(bcd + b + d)$, determine the value of $M$ where $M = a \cdot 10^{3} + b \cdot 10^{2} + c \cdot 10 + d$.


In the tetrahedron $ABCD$, $\triangle ABC$ is an equilateral triangle, $AD = BD = 2$, $AD \perp BD$, and $AD \perp CD$. Find the distance from point $D$ to the plane $ABC$.


Is it possible to cover a \( 13 \times 13 \) chessboard with forty-two tiles of size \( 4 \times 1 \) so that only the central square of the chessboard remains uncovered? (It is assumed that each tile covers exactly four squares of the chessboard, and the tiles do not overlap.)

Solution: \boxed{\text{No}}


Compute the definite integral:

$$
\int_{0}^{\pi}\left(x^{2}-3 x+2\right) \sin x \, dx
$$

Solution: \boxed{\pi^2 - 3\pi}


Let \( x_1, x_2, \ldots, x_n \) be real numbers, at least one of which is greater than 1988. Select all terms \( x_i \) such that for some \( j \geq i \), the average of the terms from \( x_i \) to \( x_j \) (inclusive) exceeds 1988. Show that the average of the selected terms exceeds 1988.


In a match without draws, the competition ends when one person wins 2 more games than the other, and the person with more wins is the winner. It is known that in odd-numbered games, the probability that player A wins is $\frac{3}{5}$, and in even-numbered games, the probability that player B wins is $\frac{3}{5}$. Find the expected number of games played when the match ends.


What is the remainder when \(2^{2015}\) is divided by 20? Considering it is difficult to perform this division directly using only paper and pencil, we will search for a way to obtain the answer by analyzing the remainders of powers of 2 modulo 20, hoping to find a pattern.

What is the remainder when \(2^{5}\) is divided by 20?

\[2^{5} = 32 = 1 \cdot 20 + 12\]

Knowing this, it's easy to determine the remainder of \(2^{6}\) modulo 20, as

\[2^{6} = 2 \cdot 2^{5} = 2 \cdot (1 \cdot 20 + 12) = 2 \cdot 20 + 24\]

Since 24 is greater than 20 and cannot be a remainder, we must write

\[2^{6} = 3 \cdot 20 + 4\]

We can extend the previous argument concluding that to determine the remainder of \(2^{i+1}\) modulo 20, it is enough to know the remainder of the product of the remainder of \(2^{i}\) modulo 20. This way, we can build the sequence of powers and remainders modulo 20.

\[
\begin{array}{ccccccc}
n & 2^{1} & 2^{2} & 2^{3} & 2^{4} & 2^{5} & 2^{6} \\
\text{Remainder modulo 20} & 2 & 4 & 8 & 16 & 12 & 4
\end{array}
\]

(a) Determine the remainders when \(2^{7}, 2^{10},\) and \( 2^{13} \) are divided by 20.

(b) Knowing that the remainders repeat periodically, determine the period of repetition, that is, the number of distinct remainders that keep repeating.

(c) Return to the initial question. What is the remainder when \(2^{2015}\) is divided by 20?


Find natural numbers \( k \) and \( n \) if it is known that

\[
k n^{2} - k n - n^{2} + n = 94
\]

Solution: \boxed{k = 48, \, n = 2}


A circle is divided into six sectors, each containing a herring. In one move, you are allowed to move any two herrings in adjacent sectors in opposite directions. Is it possible to gather all the herrings into one sector using this operation?

Solution: \boxed{\text{No}}


Given the sequence \(\left\{x_{n}\right\}\) defined by \(x_{n}=\frac{n}{n+2016}\), if \(x_{2016}=x_{m} x_{n}\), find a solution \(\{m, n\}\) where \(m\) and \(n\) are positive integers.

Solution: \boxed{\{4032, 6048\}}


a) Each vertex at the base of a triangle is connected by straight lines to \( n \) points located on the opposite side. Into how many parts do these lines divide the triangle?

b) Each of the three vertices of a triangle is connected by straight lines to \( n \) points located on the opposite side of the triangle. Into how many parts do these lines divide the triangle, assuming that no three of them intersect at the same point?

Solution: \boxed{3n^2 + 3n + 1}


Find the value of the expression \(\frac{1}{ab} + \frac{1}{bc} + \frac{1}{ac}\) if, regarding \(a, b, c\), it is known that these are three distinct real numbers satisfying the conditions \(a^3 - 2020a^2 + 1010 = 0\), \(b^3 - 2020b^2 + 1010 = 0\), and \(c^3 - 2020c^2 + 1010 = 0\).


Compute the integral

$$
I = \int_{1}^{i} \frac{\ln ^{3} z}{z} \, d z
$$

along the arc of the circle $|z| = 1$ ($\ln z$ is the principal value of the logarithm, $\ln 1 = 0$).


Through every three vertices of a cube, located at the ends of each trio of edges meeting at one vertex, a plane is drawn. Find the volume of the solid bounded by these planes if the edge length of the cube is $a$.


Simplify the following expression:

$$
\left[\frac{(a+b)^{2}+2 b^{2}}{a^{3}-b^{3}}-\frac{1}{a-b}+\frac{a+b}{a^{2}+a b+b^{2}}\right] \cdot\left(\frac{1}{b}-\frac{1}{a}\right)
$$


How many pairs of natural numbers \(a\) and \(b\) exist such that \(a \geq b\) and the equation

\[ \frac{1}{a} + \frac{1}{b} = \frac{1}{6} \]

is satisfied?


For positive integer \( n \), let \( f(n) \) denote the unit digit of \( 1+2+3+\cdots+n \). Find the value of \( f(1)+f(2)+\cdots+f(2011) \).


How many six-digit numbers with the penultimate digit being 1 are divisible by 4?


Let \( f(x) \) be a function defined on \( \mathbf{R} \). If \( f(x) + x^{2} \) is an odd function, and \( f(x) + 2^{x} \) is an even function, then the value of \( f(1) \) is ______.


Calculate the limit of the numerical sequence:

$$
\lim _{n \rightarrow \infty} \sqrt[3]{n}\left(\sqrt[3]{n^{2}}-\sqrt[3]{n(n-1)}\right)
$$


As shown in the figure, $C$ is the midpoint of the semicircular arc $\overparen{AB}$. Point $P$ is any point on the arc $\overparen{BC}$. $CD \perp CP$ and intersects $AP$ at point $D$. Connect $BD$. If $AB = 2$, find the minimum value of $BD$.

Solution: \boxed{\sqrt{5} - 1}


The center of a circle tangent to the legs $AC$ and $BC$ of a right triangle $ABC$ lies on the hypotenuse $AB$. Find the radius of the circle if it is six times smaller than the sum of the legs, and the area of the triangle $ABC$ is 27.


Today is January 30th, and we write down 130. The rule for writing the subsequent numbers is as follows: If the last number written is even, divide it by 2 and add 2 to get the next number. If the last number written is odd, multiply it by 2 and subtract 2 to get the next number. Starting with 130, we get: 130, 67, 132, 68, etc. What is the 2016th number in this sequence?


Let \(A B C\) be a triangle and \(M\) be any point in the plane. Show that \(\frac{M A}{B C}+\frac{M B}{C A}+\frac{M C}{A B} \geqslant \sqrt{3}\).


A motorboat moves on calm water with a speed of \( v_{0} = 20 \, \text{km/h} \). Determine the speed of the motorboat 2 minutes after turning off the engine, given that its speed decreased to \( v_{1} = 8 \, \text{km/h} \) in 40 seconds. The water resistance is proportional to the boat's speed.

Solution: \boxed{1.28 \text{ km/h}}


It is known that the minimum number of straight-line cuts required to divide an obtuse triangle into acute triangles is 7. Show how these cuts should be made in practice.


The numbers \( u, v, w \) are the roots of the equation \( x^{3} - 3x - 1 = 0 \). Find \( u^{9} + v^{9} + w^{9} \).


In the right triangle \( \triangle ABC \), \( \angle B = 90^\circ \). Point \( P \) is on the angle bisector of \( \angle A \) within \( \triangle ABC \). Point \( M \) (distinct from \( A \) and \( B \)) is a point on side \( AB \). The lines \( AP \), \( CP \), and \( MP \) intersect sides \( BC \), \( AB \), and \( AC \) at points \( D \), \( E \), and \( N \) respectively. Given that \( \angle MPB = \angle PCN \) and \( \angle NPC = \angle MBP \), find \( \frac{S_{\triangle APC}}{S_{ACDE}} \).


Find the maximum positive integer $r$ that satisfies the following condition: For any five 500-element subsets of the set $\{1,2, \cdots, 1000\}$, there exist two subsets that have at least $r$ common elements.


A number \( \mathrm{X} \) is called "25-supporting" if for any 25 real numbers \( a_{1}, \ldots, a_{25} \) whose sum is an integer, there is at least one for which \( \left|a_{i} - \frac{1}{2}\right| \geq X \).

Provide the largest 25-supporting \( X \), rounded to the nearest hundredth according to standard mathematical rules.


How many solutions in natural numbers \( x, y \) does the system of equations have

\[
\left\{\begin{array}{l}
\gcd(x, y)=20! \\
\text{lcm}(x, y)=30!
\end{array} \quad (\text{where } n! = 1 \cdot 2 \cdot 3 \cdot \ldots \cdot n) ?\right.
\]


On the hyperbola \( xy = 1 \), let \( A_n \) be the point with the abscissa \( \frac{n}{n+1} \) and \( B_n \) the point with the abscissa \( \frac{n+1}{n} \), where \( n \in \mathbf{N}_{+} \). Furthermore, let \( M \) be the point with coordinates \((1,1)\). Let \( P_n (x_n, y_n) \) be the circumcenter of \( \triangle A_n B_n M \). When \( n \rightarrow +\infty \), find the coordinates \((a, b)\) of the limit point of \( P_n \), where \( a = \lim_{n \rightarrow +\infty} x_n \) and \( b = \lim_{n \rightarrow +\infty} y_n \).


Given two segments \( AB \) and \( A'B' \) on the plane, construct a point \( O \) such that the triangles \( AOB \) and \( A'O B' \) are similar (identical letters denote corresponding vertices of similar triangles).

Solution: \boxed{O}


As shown in the figure, a circular ring made up of two concentric circles is evenly divided into 7 parts, creating a total of 8 regions including the small circle in the middle. To color these 8 regions such that no two adjacent regions have the same color, what is the minimum number of colors needed?


Each of the 33 warriors either always lies or always tells the truth. It is known that each warrior has exactly one favorite weapon: a sword, a spear, an axe, or a bow.

One day, Uncle Chernomor asked each warrior four questions:
- Is your favorite weapon a sword?
- Is your favorite weapon a spear?
- Is your favorite weapon an axe?
- Is your favorite weapon a bow?

13 warriors answered "yes" to the first question, 15 warriors answered "yes" to the second question, 20 warriors answered "yes" to the third question, and 27 warriors answered "yes" to the fourth question.

How many of the warriors always tell the truth?


Vasya replaced the same digits in two numbers with the same letters, and different digits with different letters. It turned out that the number ZARAZA is divisible by 4, and ALMAZ is divisible by 28. Find the last two digits of the sum ZARAZA + ALMAZ.


A triangle with interior angles $60^{\circ}, 45^{\circ}$ and $75^{\circ}$ is inscribed in a circle of radius 2. What is the area of the triangle?

Solution: \boxed{3 + \sqrt{3}}


Let \( n \) be a positive integer. Let \( V_{n} \) be the set of all sequences of 0's and 1's of length \( n \). Define \( G_{n} \) to be the graph having vertex set \( V_{n} \), such that two sequences are adjacent in \( G_{n} \) if and only if they differ in either 1 or 2 places. For instance, if \( n=3 \), the sequences \((1,0,0)\), \((1,1,0)\), and \((1,1,1)\) are mutually adjacent, but \((1,0,0)\) is not adjacent to \((0,1,1)\).

Show that, if \( n+1 \) is not a power of 2, then the chromatic number of \( G_{n} \) is at least \( n+2 \).


Given a positive integer \( n \) such that \( n \leq 2016 \) and \(\left\{\frac{n}{2}\right\}+\left\{\frac{n}{4}\right\}+\left\{\frac{n}{6}\right\}+\left\{\frac{n}{12}\right\}=3\), where \(\{x\} = x - \lfloor x \rfloor\) and \(\lfloor x \rfloor\) denotes the greatest integer less than or equal to \( x \), find the number of such integers \( n \).


Given a tetrahedron \(ABCD\) where one edge has a length of 3, and the remaining five edges each have a length of 2, find the radius of its circumscribed sphere.


The polynomial \( G(x) \) with real coefficients takes the value 2022 at exactly five different points \( x_{1} < x_{2} < x_{3} < x_{4} < x_{5} \). It is known that the graph of the function \( y=G(x) \) is symmetric with respect to the line \( x=-8 \).

(a) (2 points) Find \( x_{1} + x_{3} + x_{5} \).

(b) (2 points) What is the minimum degree that \( G(x) \) can have?


The least common multiple of the natural numbers \( a \) and \( b \) is 140, and the greatest common divisor is 5. What is the maximum value of \( a + b \)?


It is known that in a past chess tournament, all players were matched into pairs in each round, and the loser was eliminated (there were no draws). It is known that the winner played 6 games. How many participants in the tournament won at least 2 more games than they lost?


On Children's Day, Xiaoxue and 9 other children are each dressed as a number from 1 to 10 and lined up in a row. The children to the left of Xiaoxue are all dressed as odd numbers, and those to the right of Xiaoxue are all dressed as even numbers. What is the maximum number of children to the left of Xiaoxue?


In triangle \(ABC\), the known angles are \(\angle B = 30^\circ\) and \(\angle A = 90^\circ\). On side \(AC\), there is a point \(K\), and on side \(BC\), there are points \(L\) and \(M\) such that \(KL = KM\) (point \(L\) lies on segment \(BM\)).

Find the length of segment \(LM\), given that \(AK = 4\), \(BL = 31\), and \(MC = 3\).


Four identical squares were arranged to create a pattern as shown in the image. Each side of the squares is $4 \mathrm{~cm}$ long, and they are either parallel or perpendicular to each other, intersecting at either their quarter points or half points. Libor wants to color the pattern and finds out that the cost to color $1 \mathrm{~cm}^{2}$ of each connected region is proportional to the number of squares that the region overlaps.

How many crowns will it cost to color the pattern?

Solution: \boxed{64 \text{ korun}}


Given \( 5 \sin 2 \alpha = \sin 2^\circ \), find the value of \( \frac{\tan (\alpha + 1^\circ)}{\tan (\alpha - 1^\circ)} \).


Find the logarithmic residue of the function

$$
f(z)=\frac{1+z^{2}}{1-\cos 2 \pi z}
$$

with respect to the circle \( |z|=\pi \).


John M. is sitting at \((0,0)\), looking across the aisle at his friends sitting at \((i, j)\) for each \(1 \leq i \leq 10\) and \(0 \leq j \leq 5\). Unfortunately, John can only see a friend if the line connecting them doesn't pass through any other friend. How many friends can John see?


To find the smallest value of \( n \) such that the system of equations
$$
\left\{\begin{array}{l}
\sin x_{1}+\sin x_{2}+\cdots+\sin x_{n}=0, \\
\sin x_{1}+2 \sin x_{2}+\cdots+n \sin x_{n}=100 .
\end{array}\right.
$$
has a solution.


In the Cartesian coordinate system $xOy$, the line $l$ passes through the origin, and $\bar{n}=(3,1)$ is a normal vector of $l$. Given the sequence $\left\{a_{n}\right\}$ satisfies: for any positive integer $n$, the point $\left(a_{n+1}, a_{n}\right)$ lies on $l$. If $a_{2}=6$, then the value of $a_{1} a_{2} a_{3} a_{4} a_{5}$ is $\qquad$


Given the quadratic function
$$
f(x)=a x^{2}+b x+c \ \ (a, b, c > 0)
$$

which has roots, find the maximum value of
$$
\min \left\{\frac{b+c}{a}, \frac{c+a}{b}, \frac{a+b}{c}\right\}.
$$


Given a triangle \(A B C\) with \(A B = A C\) and \(\angle A = 110^{\circ}\). Inside the triangle, a point \(M\) is chosen such that \(\angle M B C = 30^{\circ}\) and \(\angle M C B = 25^{\circ}\). Find \(\angle A M C\).


Find the sum of all prime numbers whose representation in base 14 has the form $101010...101$ (alternating ones and zeros).


Find the number of multiples of 3 which have six digits, none of which is greater than 5.


Two identical cylindrical sheets are cut open along the dotted lines and glued together to form one bigger cylindrical sheet. The smaller sheets each enclose a volume of 100. What volume is enclosed by the larger sheet?


There is a peculiar four-digit number (with the first digit not being 0). It is a perfect square, and the sum of its digits is also a perfect square. Dividing this four-digit number by the sum of its digits results in yet another perfect square. Additionally, the number of divisors of this number equals the sum of its digits, which is a perfect square. If all the digits of this four-digit number are distinct, what is this four-digit number?


Show that in a right-angled triangle, the diameter of the inscribed circle is:

$$
2r = a + b - c
$$

Solution: \boxed{2r = a + b - c}


Calculate the area of the figure bounded by the graphs of the functions:

$$
y=(x-2)^{3}, \quad y=4x-8
$$


As shown in the diagram, a rectangular grassland is divided into three parts. It is known that the second part is 10 meters wider than the first part. The area of the second part is 1000 square meters, and the third part is 4 meters narrower than the first part. The area of the third part is 650 square meters. What is the area of the first part in square meters?


Let's call a non-empty (finite or infinite) set $A$, consisting of real numbers, complete if for any real numbers $a$ and $b$ (not necessarily distinct and not necessarily belonging to $A$), such that $a+b$ belongs to $A$, the number $ab$ also belongs to $A$. Find all complete sets of real numbers.

Solution: \boxed{\mathbb{R}}


In the cube $A B C D-A_{1} B_{1} C_{1} D_{1}$, what is the measure of the dihedral angle $A-B D_{1}-A_{1}$?


The roots of the equation \(2x^{3} + ax^{2} - 13x + b = 0\) are 2 and -3. Find \(a\) and \(b\).

Solution: \boxed{a = 1, \, b = 6}


A billiard table has the shape of a $2 \times 1$ rectangle, with pockets located at the corners and the midpoints of the longer sides. What is the minimum number of balls that need to be placed inside the rectangle so that each pocket is aligned on the same line with some two balls?


Let \( p \geq 3 \) be a prime number and consider the right triangle with the longer leg \( p^{2}-1 \) and the shorter leg \( 2p \). A semicircle is inscribed in the triangle such that its diameter is along the longer leg and it is tangent to the hypotenuse and the shorter leg. Find the values of \( p \) for which the radius of the semicircle is an integer.

Solution: \boxed{p = 3}


Two people, Jia and Yi, start simultaneously from points A and B and walk towards each other. When Jia reaches the midpoint of AB, they are 5 kilometers apart. When Yi reaches the midpoint of AB, they are $\frac{45}{8}$ kilometers apart. Find the distance between points A and B in kilometers.


The city administration of a certain city launched a campaign that allows the exchange of four empty 1-liter bottles for one full 1-liter bottle of milk. How many liters of milk can a person obtain if they have 43 empty 1-liter bottles by making several of these exchanges?
(a) 11
(b) 12
(c) 13
(d) 14
(e) 15


Define \( n! = 1 \times 2 \times \ldots \times n \), for example \( 5! = 1 \times 2 \times 3 \times 4 \times 5 \). If \(\frac{n! \times (n+1)!}{2}\) (where \( \mathbf{n} \) is a positive integer and \( 1 \leq n \leq 100 \)) is a perfect square, what is the sum of all such \( \mathbf{n} \)?


Given that \(a\) and \(b\) are positive integers, find the minimum positive integer value that
\[ M = 3a^2 - ab^2 - 2b - 4 \]
can take.


A scientist stores the design of their time machine in a computer, and the file is protected by a password that is a permutation of the set $\{1, 2, \ldots, 64\}$. They also design a program such that each time eight integers from 1 to 64 are inputted, the computer will indicate the order among these eight numbers in the password (from left to right). Design an operational plan that will determine this password in at most 45 inputs.


Two numbers, \( x \) and \( y \), are randomly selected from the interval \((0,1)\). What is the probability that \(\left\lfloor \log _{2} x \right\rfloor = \left\lfloor \log _{2} y \right\rfloor \)? If necessary, round the answer to two decimal places. Note that \([a]\) denotes the greatest integer less than or equal to \(a\).


Through a fixed point \( M(m, 0) \), draw an arbitrary line that intersects the parabola \( y^2 = 16x \) at points \( P \) and \( Q \). If \( \frac{1}{|PM|^2} + \frac{1}{|QM|^2} \) is a constant, find all possible values of the real number \( m \).


For every integer \( k \geq 3 \), the Ramsey number \( R(k) \) satisfies
$$
R(k) > 2^{k / 2}.
$$

Solution: \boxed{R(k) > 2^{k/2}}


The quadratic equations \(x^{2} + px + q\) and \(x^{2} + ax + b\) each have one root. Among the numbers \(p, q, a, b\) there are 16, 64, and 1024. What can the fourth number be?

If there are multiple possible answers, input the larger one into the system, and specify all of them in the written solution.


Find all numbers of the form \( 22 \ldots 2 \) that can be expressed as the sum of two perfect squares.


The parabola \( y^{2} = 2px \) (with \( p > 0 \)) has a focus \( F \) and a directrix \( l \). Points \( A \) and \( B \) are on the parabola and satisfy \( \angle AFB = \frac{\pi}{3} \). Let the projection of the midpoint \( M \) of segment \( AB \) onto \( l \) be \( N \). Find the maximum value of \( \frac{|MN|}{|AB|} \).


There are four different taps that can fill a swimming pool. If the first and the second taps are turned on simultaneously, the pool fills up in 2 hours. If the second and the third taps are turned on simultaneously, the pool fills up in 3 hours. If the third and the fourth taps are turned on simultaneously, the pool fills up in 4 hours. How long will it take to fill the pool if the fourth and the first taps are turned on simultaneously?


A pedestrian reported to a traffic officer the number of a car whose driver grossly violated traffic rules. This number is expressed as a four-digit number, where the unit digit is the same as the tens digit, and the hundreds digit is the same as the thousands digit. Moreover, this number is a perfect square. What is this number?


The 83 trainees from the Animath workshop each choose an activity for the free afternoon from 5 activities offered. We know that:

$\triangleright$ Shopping was at least as popular as Laser Tag;

$\Delta$ Movie tickets are sold in batches of 6;

$\Delta$ At most 5 students go to the beach;

$\triangleright$ At most one student visited a Romanesque church.

When writing the list of activities in alphabetical order, the number of students corresponding to each activity is written. How many different lists of numbers can be written?


Find the volume of the region \(\Omega\) bounded by the surfaces

\[ x = 17 \sqrt{2 y}, \quad x = 2 \sqrt{2 y}, \quad z = \frac{1}{2} - y, \quad z = 0 \]


In a football tournament, 15 teams participated, each of which played exactly once with all the others. A win awarded 3 points, a draw 1 point, and a loss 0 points.

After the tournament ended, it turned out that some 6 teams each scored at least $N$ points. What is the largest possible integer value for $N$?


Given the function \( f(x) = \cos^2(\theta x) + \cos(\theta x) \sin(\theta x) \) with the smallest positive period \(\frac{\pi}{2}\), what is the maximum value of \(\theta f(x)\)?

Solution: \boxed{1 + \sqrt{2}}


Given an acute $\triangle ABC$ with circumcenter $O$. Line $AO$ intersects $BC$ at point $D$. Let $E$ and $F$ be the circumcenters of $\triangle ABD$ and $\triangle ACD$, respectively. If $AB > AC$ and $EF = BC$, then $\angle C - \angle B = \qquad$


Specify a six-digit number $N$ consisting of distinct digits such that the numbers $2N$, $3N$, $4N$, $5N$, and $6N$ are permutations of its digits.


For the quadrilateral $ABCD$, it is known that $\angle BAC = \angle CAD = 60^{\circ}$, and $AB + AD = AC$. Additionally, it is known that $\angle ACD = 23^{\circ}$. How many degrees is the angle $\angle ABC$?


Let \( S = \{1, 2, 3, 4\} \). The sequence \( a_{1}, a_{2}, \cdots, a_{n} \) of \( n \) terms has the following property: for any non-empty subset \( B \) of \( S \) (denote the number of elements in \( B \) as \( |B| \)), there are exactly \( |B| \) consecutive terms in the sequence that form the set \( B \). Find the minimum value of \( n \).


Given a positive integer \( n \), find the largest real number \( C \) such that if the sum of the reciprocals of a set of integers greater than 1 (where repetitions are allowed) is less than \( C \), then it is always possible to divide this set of numbers into no more than \( n \) subgroups, such that the sum of the reciprocals of the numbers in each subgroup is less than 1.


Find all odd positive natural numbers $n$ such that $n$ divides $3^{n} + 1$.

Solution: \boxed{n = 1}


A natural number \( A \) gives a remainder of 35 when divided by 1981, and also gives a remainder of 35 when divided by 1982. What is the remainder when \( A \) is divided by 14?


In an isosceles triangle \(ABC\), the base \(AC\) is equal to \(x\), and the lateral side is equal to 12. On the ray \(AC\), point \(D\) is marked such that \(AD = 24\). From point \(D\), a perpendicular \(DE\) is dropped to the line \(AB\). Find \(x\) given that \(BE = 6\).


In $\triangle ABC$, $O$ is the circumcenter and $I$ is the incenter. If $\angle BOC = \angle BIC$, find $\angle A$.


How many pairs of integers solve the system \( |x y| + |x - y| = 2 \) if \( -10 \leq x, y \leq 10 \)?


Find all pairs of primes \((p, q)\) for which \(p - q\) and \(pq - q\) are both perfect squares.

Solution: \boxed{(3, 2)}


Calculate to 3 decimal places the following expressions:

1.

$$
\frac{2 \sqrt{3}}{\sqrt{3}-\sqrt{2}}
$$

2.

$$
\frac{(3+\sqrt{3})(1+\sqrt{5})}{(5+\sqrt{5})(1+\sqrt{3})}
$$


Natural numbers \(a\) and \(b\) are such that \(a^{a}\) is divisible by \(b^{b}\), but \(a\) is not divisible by \(b\). Find the smallest possible value of \(a + b\), given that \(b\) is coprime with 210.


In a tetrahedron \(ABCD\), \(AB = CD = 5\), \(AC = BD = \sqrt{34}\), \(AD = BC = \sqrt{41}\). Find the radius of the circumscribed sphere of the tetrahedron \(ABCD\).


Do the following equations have common roots?

\[
\begin{aligned}
& 3x^{4} + 13x^{3} + 20x^{2} + 17x + 7 = 0 \\
& 3x^{4} + x^{3} - 8x^{2} + 11x - 7 = 0
\end{aligned}
\]

Solve both equations.


The height of a regular quadrilateral prism \( A B C D A_{1} B_{1} C_{1} D_{1} \) is twice as small as the side of its base. Find the maximum value of the angle \( \angle A_{1} M C_{1} \), where \( M \) is a point on edge \( AB \).


In a right triangle \( ABC \) with a right angle at \( B \), bisector \( BL \) and median \( CM \) are drawn, intersecting at point \( D \). Line \( AD \) intersects side \( BC \) at point \( E \). Find the area of triangle \( AEL \) given that \( EL = x \).


Let \( x \) be a positive real number. Find the maximum possible value of
\[
\frac{x^2 + 2 - \sqrt{x^4 + 4}}{x}.
\]

Solution: \boxed{2\sqrt{2} - 2}


A store sells chalk in three types of packages: regular, unusual, and excellent. Initially, the quantitative ratio of the types was 2:3:6. After some packages of regular and unusual chalk—totaling no more than 100 packages—were delivered to the store, and $40\%$ of the excellent chalk packages were sold, the ratio changed to 5:7:4. How many total chalk packages were sold in the store?

Solution: \boxed{24 \text{ patches}}


In a three-digit number, the first digit (hundreds place) was increased by 3, the second digit (tens place) by 2, and the third digit (units place) by 1. As a result, the number was increased by 4 times. Provide an example of the original number.


A semicircle is drawn on each side of a square. The square has sides of length \(2 \pi\). What is the area of the resulting shape?

A) \(2 \pi^{2}(\pi+1)\)
B) \(\pi^{2}(\pi+2)\)
C) \(2 \pi^{2}(2 \pi+1)\)
D) \(\pi^{2}(\pi+4)\)
E) \(2 \pi^{2}(\pi+2)\)

Solution: \boxed{2\pi^2(\pi + 2)}


Given a triangle \(ABC\) with side lengths \(AB = BC = 80\) and \(AC = 96\).

Circle \(Q_1\) is inscribed in the triangle \(ABC\). Circle \(Q_2\) is tangent to \(Q_1\) and to the sides \(AB\) and \(BC\). Circle \(Q_3\) is tangent to \(Q_2\) and also to the sides \(AB\) and \(BC\). Find the radius of circle \(Q_3\).


There are 10 black balls, 9 white balls, and 8 red balls in a box. If you close your eyes and draw balls from the box, how many balls do you need to draw to ensure that you have at least 1 red ball and 1 white ball?


A peculiar plant was cultivated in the laboratory. It grows very rapidly, increasing its mass to 3 times the mass of the previous day plus an additional 4 kilograms each day. After 3 days of cultivation, the plant's mass reached 133 kilograms. What was the initial mass of the plant?


Let \( A = \frac{21 \times 62 + 22 \times 63 + 23 \times 64 + 24 \times 65 + 25 \times 66}{21 \times 61 + 22 \times 62 + 23 \times 63 + 24 \times 64 + 25 \times 65} \times 199 \). Find the integer part of \( A \).


Find the sum of all fractions of the form $\frac{1}{xy}$ where $\operatorname{gcd}(x, y) = 1$, $x \leq n$, $y \leq n$, and $x + y > n$.


A circle is inscribed in an isosceles triangle with a base of 12 cm and a height of 8 cm. A tangent parallel to the base is drawn to the circle. Find the length of the segment of this tangent that lies between the sides of the triangle.

Solution: \boxed{3 \text{ cm}}


Find all \( n \in \mathbb{N}^{*} \) such that \( n \) divides \( 2^{n} - 1 \).

Solution: \boxed{n = 1}


Let $[r, s]$ denote the least common multiple of the positive integers $r$ and $s$. Find the number of ordered triplets $(a, b, c)$ such that $[a, b] = 1000$, $[b, c] = 2000$, and $[c, a] = 2000$.


The national football cup was played using a knockout system: the losing team was eliminated from the competition, and in case of a draw, a replay was scheduled. A total of \(m\) games were played, \(n\) of which were replays. How many teams participated in the cup?

Solution: \boxed{m - n + 1}


Given the sequence $\left\{a_{n}\right\}$ with the sum of its first $n$ terms $S_{n}=2 a_{n}-1$ for $n=1,2, \cdots$, and the sequence $\left\{b_{n}\right\}$ defined by $b_{1}=3$ and $b_{k+1}=a_{k}+b_{k}$ for $k=1,2,3, \cdots$, find the sum of the first $n$ terms of the sequence $\left\{b_{n}\right\}$.

Solution: \boxed{2^n + 2n - 1}


Let \( f: \mathbb{N}^{*} \rightarrow \mathbb{N}^{*} \) be a function satisfying for all \( m, n \in \mathbb{N}^{*} \), \( f(m+n) \geq f(m) + f(f(n)) - 1 \). What values can \( f(2019) \) take?


The school organized a picnic with several participants. The school prepared many empty plates. Each attendee counts the empty plates and takes one empty plate to get food (each person can only take one plate, no more). The first attendee counts all the empty plates, the second attendee counts one less plate than the first attendee, and so on. The last attendee finds that there are 4 empty plates left. It is known that the total number of plates prepared by the school plus the number of attendees equals 2015. How many people attended the picnic?


For what value of \( a \) does the equation \( |x-2|=a x-2 \) have an infinite number of solutions?


Let \(\Gamma\) be a circle in the plane and \(S\) be a point on \(\Gamma\). Mario and Luigi drive around the circle \(\Gamma\) with their go-karts. They both start at \(S\) at the same time. They both drive for exactly 6 minutes at constant speed counterclockwise around the track. During these 6 minutes, Luigi makes exactly one lap around \(\Gamma\) while Mario, who is three times as fast, makes three laps.

While Mario and Luigi drive their go-karts, Princess Daisy positions herself such that she is always exactly in the middle of the chord between them. When she reaches a point she has already visited, she marks it with a banana.

How many points in the plane, apart from \(S\), are marked with a banana by the end of the race?


In the example 4-10, as shown in Figure 4-12(a), $v_{1} v_{2} v_{3}$ are three equidistant points on a circle. These points are to be decorated with beads in three different colors: red, blue, and green. How many distinct ways can this be done?


The height of the Eiffel Tower in Paris is 324 meters, and its weight is 8000 tons. It is made entirely of steel. What will be the height of an exact model of the tower weighing 1 kilogram made of the same steel?

Solution: \boxed{1.62 \text{ meters}}


A boy and his father are standing on the seashore. If the boy stands on tiptoe, his eyes will be at a height of 1 meter above the sea surface, and if he sits on his father's shoulders, his eyes will be at a height of 2 meters. How many times farther will he see in the second case? (Find the answer with an accuracy of up to 0.1, considering the radius of the Earth to be 6000 km.)


If a die is rolled, event \( A = \{1, 2, 3\} \) consists of rolling one of the faces 1, 2, or 3. Similarly, event \( B = \{1, 2, 4\} \) consists of rolling one of the faces 1, 2, or 4.

The die is rolled 10 times. It is known that event \( A \) occurred exactly 6 times.

a) Find the probability that under this condition, event \( B \) did not occur at all.

b) Find the expected value of the random variable \( X \), which represents the number of occurrences of event \( B \).


Find the minimum value of the expression

$$
\sqrt{x^{2}-2 \sqrt{3} \cdot|x|+4}+\sqrt{x^{2}+2 \sqrt{3} \cdot|x|+12}
$$

as well as the values of $x$ at which it is achieved.

Solution: \boxed{2 \sqrt{7}}


The cross-section of a regular hexagonal pyramid SABCDEF is formed by a plane passing through the center of the base ABCDEF and parallel to the median CM of the lateral face SCD and the apothem SN of the lateral face SAF. The side of the base of the pyramid is 8, and the distance from the vertex S to the cutting plane is \(3 \sqrt{\frac{13}{7}}\). Find the cosine of the angle between the cutting plane and the base plane. (20 points)


For $i = 1, 2, \cdots, n$, if $\left|x_{i}\right| < 1$ and $\left|x_{1}\right| + \left|x_{2}\right| + \cdots + \left|x_{n}\right| = 2005 + \left|x_{1} + x_{2} + \cdots + x_{n} \right|$, find the minimum value of the positive integer $n$.

Solution: \boxed{n_{\min} = 2006}


Find the smallest natural number which, when multiplied by 2 becomes a square, and when multiplied by 3 becomes a cube of a natural number.


Find the sum of all possible values of \( a \) such that the following equation has a real root in \( x \):
\[ (x-a)^2 + (x^2 - 3x + 2)^2 = 0 \]


Given six thin wooden sticks, the two longer ones are $\sqrt{3}a$ and $\sqrt{2}a$, while the remaining four are of length $a$. If they are used to form a triangular prism, find the cosine of the angle between the lines containing the two longer edges.


Find the functions \( f: \mathbb{Q}_{+}^{*} \rightarrow \mathbb{Q}_{+}^{*} \) such that for all \( x \in \mathbb{Q}_{+}^{*} \), we have \( f(x+1) = f(x) + 1 \) and \( f(1/x) = 1 / f(x) \).

Solution: \boxed{f(x) = x}


13 children sat around a round table and agreed that boys would lie to girls, but tell the truth to each other, while girls would lie to boys but tell the truth to each other. One of the children said to their right neighbor: "Most of us are boys." That child then said to their right neighbor: "Most of us are girls," and the pattern continued with each child alternately stating "Most of us are boys" or "Most of us are girls" until the last child said to the first: "Most of us are boys." How many boys are at the table?


Two cars, referred to as car A and car B, set off from locations $A$ and $B$ respectively, heading towards each other (car B departs from $B$). The speed of car B is $k$ times the speed of car A $(k > 1)$. Upon reaching their destinations $B$ and $A$, both cars immediately return. On the return trip, car A maintains its original speed, while car B’s speed changes to match car A's speed. During their return, the two cars meet at point $C$. If $D$ is the midpoint of the route from $A$ to $B$, and $\frac{C D}{A D}=\frac{1}{2}$, find the value of $k$. (Please provide the solution process)


Along the path, daisies were growing. A cornflower grew between each two daisies, and then a dandelion grew between each cornflower and daisy. It turned out that now there are 101 flowers along the path. How many daisies are growing along the path?


There are 100 light bulbs arranged in a row on a Christmas tree. The bulbs are then toggled according to the following algorithm: initially, all the bulbs are turned on; after one second, every second bulb is turned off; after another second, every third bulb is toggled (turned off if it was on, turned on if it was off); after another second, every fourth bulb is toggled, and so on. After 100 seconds, the process stops. Find the probability that a randomly selected bulb is on after this process (the bulbs do not burn out or break).