Define a "position" as a point $(x, y)$ on the Cartesian plane, where $x$ and $y$ are positive integers not exceeding 20. Initially, all 400 positions are empty. Two players, A and B, take turns placing stones, starting with A. Each time it is A's turn, he places a new red stone on an empty position such that any two red stones are not at a distance of $\sqrt{5}$. Each time it is B's turn, he places a new blue stone on any empty position (the distance between the position of a blue stone and other stones can be any value). The game continues until one player can no longer place a stone. Determine the largest integer $K$ such that no matter how B places the blue stones, A can always guarantee placing at least $K$ red stones.


Let \( k(a) \) denote the number of points \((x, y)\) in the coordinate system such that \(1 \leq x \leq a\) and \(1 \leq y \leq a\) are relatively prime integers. Determine the following sum:

$$
\sum_{i=1}^{100} k\left(\frac{100}{i}\right)
$$


Given that the difference between two 2-digit numbers is 58 and the last two digits of the squares of these two numbers are the same, find the smaller number.


What is the smallest square of an integer that ends with the longest sequence of the same digits?

For example, if the longest sequence of the same digits were five, then a suitable number would be 24677777 (of course, if it were the smallest square, but it is not). Zero is not considered an acceptable digit.

Solution: \boxed{38^2 = 1444}


The center of ellipse \( \Gamma \) is at the origin \( O \). The foci of the ellipse lie on the x-axis, and the eccentricity \( e = \sqrt{\frac{2}{3}} \). A line \( l \) intersects the ellipse \( \Gamma \) at two points \( A \) and \( B \) such that \( \overrightarrow{C A} = 2 \overrightarrow{B C} \), where \( C \) is a fixed point \((-1, 0)\). When the area of the triangle \( \triangle O A B \) is maximized, find the equation of the ellipse \( \Gamma \).

Solution: \boxed{x^2 + 3y^2 = 5}


Given the function \( f(x)=\left(x^{2}-4\right)^{-\frac{1}{2}}(x>2), f^{-1}(x) \) is the inverse function of \( f(x) \).

(1) If the sequence \(\{a_{n}\}\) is defined by \(a_{1}=1\) and \(\frac{1}{a_{n+1}}=f^{-1}\left(a_{n}\right) \left(n \in \mathbf{N}_{+}\right)\), find \(a_{n}\).

(2) Let \(S_{n}=a_{1}^{2}+a_{2}^{2}+\cdots+a_{n}^{2}\) and \(b_{n}=S_{2n+1}-S_{n}\). Does there exist a maximum positive integer \(p\) such that \(b_{n}<\frac{1}{p}\) holds for all \(n \in \mathbf{N}_{+}\)? Provide an explanation or proof.

Solution: \boxed{p = 3}


How many times does 24 divide into 100 factorial (100!)?


In a trapezoid, the lengths of the diagonals are 6 and 8, and the length of the midline is 5. Find the height of the trapezoid.


Leo the fox has a 5 by 5 checkerboard grid with alternating red and black squares. He fills in the grid with the numbers \(1, 2, 3, \ldots, 25\) such that any two consecutive numbers are in adjacent squares (sharing a side) and each number is used exactly once. He then computes the sum of the numbers in the 13 squares that are the same color as the center square. Compute the maximum possible sum Leo can obtain.


In the arithmetic sequence \(\left(a_{n}\right)\) where \(a_{1}=1\) and \(d=4\),

Calculate

\[
A=\frac{1}{\sqrt{a_{1}}+\sqrt{a_{2}}}+\frac{1}{\sqrt{a_{2}}+\sqrt{a_{3}}}+\ldots+\frac{1}{\sqrt{a_{1579}}+\sqrt{a_{1580}}}
\]

Report the smallest integer greater than \(A\).


On the island of Unfortune, there live knights who always tell the truth and liars who always lie. One day, 2022 natives gathered at a round table, and each of them made the statement:

"Next to me sit a knight and a liar!"

It is known that three knights made a mistake (i.e., unintentionally lied). What is the maximum number of knights that could have been at the table?


Find the number of real solutions of the equation
\[ x^{2}+\frac{1}{x^{2}}=2006+\frac{1}{2006}. \]


Petrov and Vasechkin were solving the same arithmetic problem. A certain number had to be divided by 2, multiplied by 7, and subtracted by 1001. Petrov performed all the operations correctly, but Vasechkin mixed up everything: he divided by 8, squared it, and also subtracted 1001. It is known that Petrov ended up with a prime number. What number did Vasechkin get?


The master and his apprentice were tasked with producing a batch of identical parts. After the master worked for 7 hours and the apprentice for 4 hours, it was found that they had completed $5 / 9$ of the total work. After working together for an additional 4 hours, they determined that $1 / 18$ of the work remained. How long would it take for the apprentice to complete the entire work if working alone?

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


A poplar tree, a willow tree, a locust tree, a birch tree, and a phoenix tree are planted in a row, with a distance of 1 meter between any two adjacent trees. The distance between the poplar tree and the willow tree is equal to the distance between the poplar tree and the locust tree. The distance between the birch tree and the poplar tree is equal to the distance between the birch tree and the locust tree. What is the distance between the phoenix tree and the birch tree? $\qquad$ meters.


Vasya throws three dice (each die has numbers from 1 to 6 on its faces) and sums up the rolled numbers. Furthermore, if all three rolled numbers are different, he can roll all three dice again and add the rolled numbers to the already accumulated sum. This process continues until at least two of the three rolled numbers are the same. What is the expected value of Vasya's total result?


On weekdays, a scatterbrained scientist takes the Moscow metro's circular line from the "Taganskaya" station to the "Kievskaya" station, and back in the evening. He boards the first arriving train. Trains run at approximately equal intervals in both directions. The northern route (via "Belorusskaya") takes 17 minutes between "Kievskaya" and "Taganskaya" or vice-versa, and the southern route (via "Paveletskaya") takes 11 minutes.

The scientist has calculated that:

- A counterclockwise train arrives at "Kievskaya" on average 1 minute and 15 seconds after a clockwise train. The same is true for "Taganskaya";
- The average journey from home to work takes 1 minute less than the return journey.

Find the expected interval between trains going in the same direction.

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


For positive integer \(n\), let \(D(n)\) be the eventual value obtained when the digits of \(n\) (in base 10) are added up recursively until a one-digit number is obtained. For example \(D(4)=4\), \(D(2012)=D(5)=5\) and \(D(1997)=D(26)=D(8)=8\). If \(x\) denotes the 2012th Fibonacci number (i.e. the 2012th term of the sequence \(1, 1, 2, 3, 5, 8, 13, \ldots)\), find \(D(x)\).


2015 people are sitting around a round table, each of whom is either a knight or a liar. Knights always tell the truth, liars always lie. They were each given a card, with a unique number written on it. After looking at the cards of their neighbors, each person at the table said: "My number is greater than that of each of my two neighbors." After this, $k$ of the people said: "My number is less than that of each of my two neighbors." What is the maximum possible value of $k?


Solve the following equation:

$$
1+\log x=\log (1+x)
$$


We place a coin in one of three identical boxes and then put these three boxes into two drawers such that each drawer contains at least one box. What is the probability that if someone randomly selects a drawer and then takes a box out from that drawer, they will find the coin in that box,

a) if the box containing the coin is alone in one drawer?

b) if the box containing the coin is in a drawer with one other box?

c) if the 3 boxes are randomly placed into the two drawers without knowing which box contains the coin?


The base side length of the regular triangular prism \( ABC-A_1B_1C_1 \) is \( a \), and the side edge length is \( \sqrt{2}a \). Find the angle between \( AC_1 \) and the lateral face \( ABB_1A_1 \).


Given the ellipse \(\frac{x^{2}}{6}+\frac{y^{2}}{3}=1\) and a point \(P\left(1, \frac{1}{2}\right)\) inside the ellipse, draw a line through \(P\) that does not pass through the origin and intersects the ellipse at points \(A\) and \(B\). Find the maximum area of triangle \(OAB\), where \(O\) is the origin.


Let the ellipse with foci \( F_1(-1,0) \) and \( F_2(1,0) \) have an eccentricity \( e \). A parabola with vertex at \( F_1 \) and focus at \( F_2 \) intersects the ellipse at a point \( P \). If \( \frac{|P F_1|}{|P F_2|} = e \), what is the value of \( e \)?


Calculate the integral

$$
\int_{L} \frac{\sin z}{z\left(z-\frac{\pi}{2}\right)} d z
$$

where \( L \) is the rectangle bounded by the lines \( x=2 \), \( x=-1 \), \( y=2 \), and \( y=-1 \).


A square is drawn on a plane with its sides parallel to the horizontal and vertical axes. Inside the square, several line segments parallel to its sides are drawn such that no two segments lie on the same line or intersect at an interior point of both segments. These segments divide the square into rectangles in such a manner that any vertical line that intersects the square and does not contain any of the dividing segments intersects exactly $k$ of these rectangles, and any horizontal line that intersects the square and does not contain any of the dividing segments intersects exactly $\ell$ of these rectangles. What could be the total number of rectangles formed by this division?

Solution: \boxed{k \ell}


In the Cartesian coordinate plane \(xOy\), points \(A\) and \(B\) are on the parabola \(y^2 = 4x\), satisfying \(\overrightarrow{OA} \cdot \overrightarrow{OB} = -4\). \(F\) is the focus of the parabola. Find \(S_{\triangle OFA} \cdot S_{\triangle OFB} = \) ________.


Integers \(a, b, c, d,\) and \(e\) satisfy the following three properties:
(i) \(2 \leq a<b<c<d<e<100\)
(ii) \(\operatorname{gcd}(a, e)=1\)
(iii) \(a, b, c, d, e\) form a geometric sequence.
What is the value of \(c\)?


A sphere with a radius of \(\sqrt{3}\) has a cylindrical hole drilled through it; the axis of the cylinder passes through the center of the sphere, and the diameter of the base of the cylinder is equal to the radius of the sphere. Find the volume of the remaining part of the sphere.

Solution: \boxed{\frac{9 \pi}{2}}


Polina custom makes jewelry for a jewelry store. Each piece of jewelry consists of a chain, a stone, and a pendant. The chains can be silver, gold, or iron. Polina has stones - cubic zirconia, emerald, quartz - and pendants in the shape of a star, sun, and moon. Polina is happy only when three pieces of jewelry are laid out in a row from left to right on the showcase according to the following rules:

- There must be a piece of jewelry with a sun pendant on an iron chain.
- Next to the jewelry with the sun pendant there must be gold and silver jewelry.
- The three pieces of jewelry in the row must have different stones, pendants, and chains.

How many ways are there to make Polina happy?


Find the number of pairs of integers \((x, y)\) that satisfy the equation \(x^{2} + 7xy + 6y^{2} = 15^{50}\).


What is the smallest positive integer $n$ such that

$$
\sqrt{n}-\sqrt{n-1}<0.01 ?
$$


A point \( M \) is chosen on the diameter \( AB \). Points \( C \) and \( D \), lying on the circumference on one side of \( AB \), are chosen such that \(\angle AMC=\angle BMD=30^{\circ}\). Find the diameter of the circle given that \( CD=12 \).


Let \( S \) be a set of sequences of length 15 formed by using the letters \( a \) and \( b \) such that every pair of sequences in \( S \) differ in at least 3 places. What is the maximum number of sequences in \( S \)?


Find all the ways to assign an integer to each vertex of a 100-gon such that among any three consecutive numbers written down, one of the numbers is the sum of the other two.


The edge length of cube \( ABCD-A_1B_1C_1D_1 \) is 1. Find the distance between the line \( A_1C_1 \) and the line \( BD_1 \).


Find all prime numbers \( p \) such that both \( p + 28 \) and \( p + 56 \) are also prime.


There is a bag of grapes with more than one hundred. If each student in the choir group is given 3 grapes, there will be 10 grapes left; if each student in the art group is given 5 grapes, there will be 12 grapes left; if each student in the math group is given 7 grapes, there will be 18 grapes left. How many grapes are there in the bag at most?


Let \( S \) denote the set of all triples \( (i, j, k) \) of positive integers where \( i + j + k = 17 \). Compute
$$
\sum_{(i, j, k) \in S} i j k
$$


Point \( C \) divides the chord \( AB \) of a circle with radius 6 into segments \( AC = 4 \) and \( CB = 5 \). Find the minimum distance from point \( C \) to the points on the circle.


Select some numbers from \(1, 2, 3, \cdots, 9, 10\) such that each number from \(1, 2, 3, \cdots, 19, 20\) is equal to the sum of one or two of the selected numbers (they can be the same). Find the minimum number of selected numbers needed.


At most, how many interior angles greater than $180^\circ$ can a 2006-sided polygon have?


Find the integer part of the number \(a + \frac{9}{b}\), where \(a\) and \(b\) are the integer and fractional parts, respectively, of the number \(\sqrt{76 - 42 \sqrt{3}}\).


The length of the escalator is 200 steps. When Petya walks down the escalator, he counts 50 steps. How many steps will he count if he runs twice as fast?

Solution: \boxed{80 \text{ steps}}


In an isosceles triangle \( \triangle ABC \), the base \( AB \) is the diameter of the circle, which intersects the legs \( AC \) and \( CB \) at points \( D \) and \( E \) respectively. Find the perimeter of the triangle \( ABC \), given \( AD = 2 \) and \( AE = \frac{8}{3} \).


There are fewer than 30 students in a class. The probability that a randomly selected girl is an honor student is $\frac{3}{13}$, and the probability that a randomly selected boy is an honor student is $\frac{4}{11}$. How many honor students are there in the class?


Find the smallest natural number \( n \) for which \( n^{2} + 20n + 19 \) is divisible by 2019.


There is a checkers championship in Anchuria that consists of several rounds. The days and cities where the rounds are held are determined by a draw. According to the rules of the championship, no two rounds can take place in the same city, and no two rounds can take place on the same day. A lottery is held among the fans: the main prize is awarded to the person who correctly guesses, before the championship begins, in which cities and on which days all the rounds will take place. If no one guesses correctly, the main prize will be retained by the organizing committee. There are eight cities in Anchuria, and the championship is allocated a total of eight days. How many rounds should the championship consist of to maximize the probability that the main prize will be retained by the organizing committee?

Solution: \boxed{6 \text{ rounds}}


The two diagonals of a quadrilateral are perpendicular. The lengths of the diagonals are 14 and 30. What is the area of the quadrilateral?


Integer \( n \) such that the polynomial \( f(x) = 3x^3 - nx - n - 2 \) can be factored into a product of two non-constant polynomials with integer coefficients. Find the sum of all possible values of \( n \).


If real numbers \(x\) and \(y\) satisfy \(x^2 + y^2 = 20\), what is the maximum value of \(xy + 8x + y\)?


100 people participated in a quick calculation test consisting of 10 questions. The number of people who answered each question correctly is given in the table below:

\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}
\hline
Problem Number & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\
\hline
Number of Correct Answers & 93 & 90 & 86 & 91 & 80 & 83 & 72 & 75 & 78 & 59 \\
\hline
\end{tabular}

Criteria: To pass, one must answer at least 6 questions correctly. Based on the table, calculate the minimum number of people who passed.


Find the number of real values of \( a \) such that for each \( a \), the cubic equation \( x^{3} = ax + a + 1 \) has an even root \( x \) with \( |x| < 1000 \).


Several people were seated around a round table such that the distances between neighboring people were equal. One of them was given a card with the number 1, and the rest were given cards with numbers 2, 3, and so on, in a clockwise direction.

The person with the card numbered 31 noticed that the distance from him to the person with the card numbered 7 is the same as the distance to the person with the card numbered 14. How many people are seated at the table in total?


Given the universal set \( U = \{1, 2, 3, 4, 5\} \) and the set \( I = \{X \mid X \subseteq U\} \), two different elements \( A \) and \( B \) are randomly selected from set \(I\). What is the probability that the intersection \( A \cap B \) has exactly 3 elements?


Consider a $4 \times 4$ square grid with 25 points. Determine the number of distinct lines passing through at least 3 of these points.


Given a polynomial \( P(x) \) with integer coefficients. It is known that \( P(1) = 2013 \), \( P(2013) = 1 \), and \( P(k) = k \), where \( k \) is some integer. Find \( k \).


Given the curve \( x y^{2} = 1 (y > 0) \), the slope of the tangent line at a point \((x_0, y_0)\) on the curve is \( k = -\frac{1}{2 \sqrt{x_0^{3}}} \). A line parallel to the \( y \)-axis through the point \( P_1 (1, 0) \) intersects the curve at \( Q_1 \). The tangent line to the curve at \( Q_1 \) intersects the \( x \)-axis at \( P_2 \). A line parallel to the \( y \)-axis through \( P_2 \) intersects the curve at \( Q_2 \). This process is repeated, generating a sequence of points \( P_1, P_2, \cdots \) and \( Q_1, Q_2, \cdots \). Let \( l_n = \left| P_n Q_n \right| \). Find the value of \( l_{2003} \).


How many integer solutions does the equation \((2x + y)^{2} = 2017 + x^{2}\) have for \(x\) and \(y\)?


Find the positive integer solution of the equation \( x^{2y} + (x+1)^{2y} = (x+2)^{2y} \).

Solution: \boxed{(3, 1)}


There are some pieces in a box, less than 50 in total. Xiaoming and Xiaoliang take turns taking pieces from the box. If Xiaoming takes 2 pieces, Xiaoliang takes 2 pieces, Xiaoming takes 2 pieces, and Xiaoliang takes 2 pieces, in this manner, Xiaoming will have 2 more pieces than Xiaoliang in the end. If Xiaoming takes 3 pieces, Xiaoliang takes 3 pieces, Xiaoming takes 3 pieces, and Xiaoliang takes 3 pieces, in this manner, both will end up taking the same number of pieces. How many pieces are there at most in the box?


Given the complex number \( z \) satisfying
$$
\left|\frac{z^{2}+1}{z+\mathrm{i}}\right|+\left|\frac{z^{2}+4 \mathrm{i}-3}{z-\mathrm{i}+2}\right|=4,
$$
find the minimum value of \( |z - 1| \).


Several consecutive natural numbers are written on the board. Exactly 52% of them are even. How many even numbers are written on the board?


In a tetrahedron \( ABCD \), \( AB = AC = AD = 5 \), \( BC = 3 \), \( CD = 4 \), \( DB = 5 \). Find the volume of this tetrahedron.


Given that \( b\left[\frac{1}{1 \times 3}+\frac{1}{3 \times 5}+\cdots+\frac{1}{1999 \times 2001}\right]=2 \times\left[\frac{1^{2}}{1 \times 3}+\frac{2^{2}}{3 \times 5}+\cdots+\frac{1000^{2}}{1999 \times 2001}\right] \), find the value of \( b \).


Find the smallest value \( x \) such that, given any point inside an equilateral triangle of side 1, we can always choose two points on the sides of the triangle, collinear with the given point and a distance \( x \) apart.


What is the maximum number of sides of a convex polygon that can be divided into right triangles with acute angles measuring 30 and 60 degrees?


Draw two concentric circles, one with a radius measuring $1 \text{ cm}$ and the other with a radius measuring $3 \text{ cm}$. In the region outside the circle of radius $1 \text{ cm}$ and inside the circle of radius $3 \text{ cm}$, draw circles that are simultaneously tangent to both circles, as shown in the given figure.

(a) What should be the radius of these circles?

(b) What is the maximum number of these circles that can be drawn without overlapping?


Consider the collection of all 5-digit numbers whose sum of the digits is 43. One of these numbers is chosen at random. What is the probability that it is a multiple of 11?


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


A quadrilateral has three sides with lengths \(a=4 \sqrt{3}\), \(b=9\), and \(c=\sqrt{3}\). The angle between sides \(a\) and \(b\) is \(30^{\circ}\), and the angle between sides \(b\) and \(c\) is \(90^{\circ}\). What is the angle between the diagonals of the quadrilateral?


Given that the modulus of the complex number \( z \) is 1, find the minimum value of \( |z-4|^{2} + |z+3i|^{2} \).


A rectangular painting is placed in a rectangular frame made of strips of the same width. The area of the painting is equal to the area of the frame. What is the ratio of the length to the width of the painting in the frame if, without the frame, the ratio of the length of the painting to its width is $2:3$?


Find all polynomials \( P \in \mathbb{R}[X] \) such that \( P(X^2 + 1) = P(X)^2 + 1 \) with \( P(0) = 0 \).

Bonus (difficult): Remove the condition \( P(0) = 0 \).

Solution: \boxed{P(X) = X}


There are 11 children sitting in a circle playing a game. They are numbered clockwise from 1 to 11. The game starts with child number 1, and each child has to say a two-digit number. The number they say cannot have a digit sum of 6 or 9, and no child can repeat a number that has already been said. The game continues until someone cannot say a new number, and the person who cannot say a new number loses the game. Who will be the last person in the game?


The quadrilateral \(ABCD\) is circumscribed around a circle with a radius of \(1\). Find the greatest possible value of \(\left| \frac{1}{AC^2} + \frac{1}{BD^2} \right|\).


If both \( 7n+1 \) and \( 8n+1 \) can be represented as the sum of three distinct positive integers in a geometric progression, what is the smallest value of the positive integer \( n \)?


We colored every side of unit squares using one of the colors red, yellow, blue, and green in such a way that each square has all four sides in different colors. The squares colored in this way can be assembled along their sides with the same color.

Can a $99 \times 99$ square and a $100 \times 100$ square be assembled from such colored squares such that the outer sides of the assembled square are single-colored and each side has a different color?

Solution: \boxed{\text{Yes}}


Determine the order of operations in the expression

$$
1891-(1600: a+8040: a) \times c
$$

and calculate its value when \( a = 40 \) and \( c = 4 \). Show how the expression can be modified without changing its numerical value.


In how many ways can the numbers \(1, 2, 3, 4, 5, 6\) be arranged in a row so that for any three consecutive numbers \(a, b, c\), the expression \(ac - b^2\) is divisible by 7? Answer: 12.


Edge \( AB \) of tetrahedron \( ABCD \) is the diagonal of the base of a quadrilateral pyramid. Edge \( CD \) is parallel to the other diagonal of this base, and its ends lie on the lateral edges of the pyramid. Find the smallest possible volume of the pyramid if the volume of the tetrahedron is \( V \).


a) Find all the divisors of the number 572 based on its prime factorization.

b) How many divisors does the number \(572 a^{3} b c\) have if:

   I. \(a, b\), and \(c\) are prime numbers greater than 20 and different from each other?

   II. \(a = 31\), \(b = 32\), and \(c = 33\)?


On the circle, points $A, B, C, D, E, F, G$ are arranged clockwise as shown in the diagram. It is known that $AE$ is the diameter of the circle. It is also known that $\angle ABF = 81^{\circ}$ and $\angle EDG = 76^{\circ}$. How many degrees is the angle $FCG$?


Given the sequence $\left\{a_{n}\right\}$ which satisfies $a_{1}=0$ and $a_{n+1}=a_{n}+4 \sqrt{a_{n}+1}+4$ for $n \geq 1$, find $a_{n}$.

Solution: \boxed{a_n = 4n^2 - 4n}


Given \(\sin x \sin 2x \sin 3x + \cos x \cos 2x \cos 3x = 1\), find the value of \(x\).

Solution: \boxed{x = k\pi}


There are two alloys of copper and zinc. In the first alloy, there is twice as much copper as zinc, and in the second alloy, there is five times less copper than zinc. In what ratio should these alloys be combined to obtain a new alloy in which zinc is twice as much as copper?

Solution: \boxed{1 : 2}


From the vertex of the obtuse angle \( A \) of triangle \( ABC \), the height \( AD \) is dropped. A circle with center \( D \) and radius \( DA \) is drawn, which intersects the sides \( AB \) and \( AC \) again at points \( M \) and \( N \) respectively. Find \( AC \), given that \( AB = c \), \( AM = m \), and \( AN = n \).

Solution: \boxed{\frac{mc}{n}}


Calculate the area of the figures bounded by the curves given in polar coordinates.

$$
r=2 \sin 4 \phi
$$


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.


The sequence of consecutive positive integers starting from 1 was written on the board up to a certain number. Then, one number was erased. The arithmetic mean of the remaining numbers is $\frac{602}{17}$. What was the erased number?


Karel and Vojta discovered that the kitchen clock at the cottage runs 1.5 minutes fast every hour, and the bedroom clock runs 0.5 minutes slow every hour. At exactly noon, they synchronized both clocks to the correct time. Both the kitchen and bedroom clocks have a standard twelve-hour dial. Determine when the clocks will next show the correct time without further adjustments:

1. The kitchen clock will again show the correct time.
2. The bedroom clock will again show the correct time.
3. Both clocks will again show the same (possibly incorrect) time.

(M. Volfová)

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


a) Determine the sum of the multiplicative contents of all subsets of \(\{1,2,3,4\}\).

b) Determine the sum of the multiplicative contents of all subsets of
\[
\left\{1, \frac{1}{2}, \frac{1}{3}, \ldots, \frac{1}{2016}\right\}
\]


Two cars, A and B, start from points A and B respectively and travel towards each other at the same time. They meet at point C after 6 hours. If car A maintains its speed and car B increases its speed by 5 km/h, they will meet 12 km away from point C. If car B maintains its speed and car A increases its speed by 5 km/h, they will meet 16 km away from point C. What was the original speed of car A?


\((\cos 8 \alpha \cdot \tan 4 \alpha - \sin 8 \alpha)(\cos 8 \alpha \cdot \cot 4 \alpha + \sin 8 \alpha)\).


The largest angle of a right trapezoid is $120^{\circ}$, and the longer leg is 12. Find the difference between the bases of the trapezoid.


A novice economist-cryptographer received a cryptogram from the ruler, which contained the next secret decree about introducing a per unit tax in a certain market. The cryptogram specified the amount of tax revenue that needed to be collected and emphasized that it would be impossible to collect a higher amount of tax revenue in this market. Unfortunately, the economist-cryptographer deciphered the cryptogram incorrectly — the digits in the tax revenue amount were determined in the wrong order. Based on these erroneous data, a decision was made to introduce a per unit tax on the consumer at a rate of 30 monetary units per unit of goods. The market supply is given by $Qs=6P-312$, and the market demand is linear. Additionally, it is known that when the price changes by one unit, the change in demand is 1.5 times less than the change in supply. After the tax was introduced, the consumer price rose to 118 monetary units.

1) Restore the market demand function.
2) Determine the amount of tax revenue collected at the chosen rate.
3) Determine the rate of the per unit tax that would achieve the ruler's decree.
4) What is the amount of tax revenue the ruler indicated to collect?

Solution: \boxed{T_{\max} = 8640}


The Mad Hatter's clock gains 15 minutes per hour, while the March Hare's clock loses 10 minutes per hour. One day, they set their clocks according to the Dormouse's clock, which is stopped and always shows 12:00, and agreed to meet at 5 o'clock in the evening for their traditional five-o'clock tea. How long will the Mad Hatter wait for the March Hare if both arrive exactly at 17:00 according to their own clocks?

Solution: \boxed{2 \text{ hours}}


A chessboard has 13 rows and 17 columns. Each small square contains a number, starting from the top-left corner. The first row is filled sequentially with $1, 2, \cdots, 17$; the second row with $18, 19, \cdots, 34$, and so on, until the last row. Now, the numbers are rewritten starting from the top-left corner. The first column is filled sequentially from top to bottom with $1, 2, \cdots, 13$; the second column with $14, 15, \cdots, 26$, and so on, until the last column.
Some of the small squares will have the same number in both the original and the rewritten grid. Find the sum of all such numbers.


On the board is written the number 98. Every minute the number is erased and replaced with the product of its digits increased by 15. What number will be on the board in an hour?


Given 100 numbers. Each number is increased by 2. The sum of the squares of the numbers remains unchanged. Each resulting number is then increased by 2 again. How has the sum of the squares changed now?


Find the value of the expression $\frac{p}{q}+\frac{q}{p}$, where $p$ and $q$ are the largest and smallest roots of the equation $x^{3}+6x^{2}+6x=-1$ respectively.


There are $n$ people, and it is known that any two of them can make at most one phone call to each other. For any $n-2$ people among them, the total number of phone calls is equal and is $3^k$ (where $k$ is a positive integer). Find all possible values of $n$.


Given the vectors \(\boldsymbol{a} = (x, 1)\), \(\boldsymbol{b} = (2, y)\), and \(\boldsymbol{c} = (1,1)\), and knowing that \(\boldsymbol{a} - \boldsymbol{b}\) is collinear with \(\boldsymbol{c}\). Find the minimum value of \( |\boldsymbol{a}| + 2|\boldsymbol{b}| \).


In the complex plane, the complex number \( z_{1} \) moves along the line segment connecting \( 1+2i \) and \( 1-2i \), and the complex number \( z_{2} \) moves along the circumference of a circle centered at the origin with radius 1. What is the area of the shape formed by the trajectory of \( z_{1} + z_{2} \)?


A square is divided into 2016 triangles, with no vertex of any triangle lying on the sides or inside any other triangle. The sides of the square are sides of some of the triangles in the division. How many total points, which are the vertices of the triangles, are located inside the square?


Find the ratio \(\frac{b^{2}}{a c}\) given that one of the roots of the equation \(a x^{2} + b x + c = 0\) is 4 times the other root \((a \neq 0, c \neq 0)\).


Suppose I have a closed cardboard box in the shape of a cube. By cutting 7 out of its 12 edges with a razor blade (it must be exactly 7 edges), I can unfold the box onto a plane, and the unfolding can take various forms. For instance, if I cut along the edges shown by the bold lines in the diagram and along the invisible edge indicated by a dashed line, I will get unfolding $A$. By cutting the box differently, one can obtain unfolding $B$ or $C$. It is easy to see that unfolding $D$ is simply a flipped version of unfolding $C$, so we consider these two unfoldings to be identical.

How many distinct unfoldings can be obtained in this way?


Primes like $2, 3, 5, 7$ are natural numbers greater than 1 that can only be divided by 1 and themselves. We split 2015 into the sum of 100 prime numbers, requiring that the largest of these prime numbers be as small as possible. What is this largest prime number?


Divide each three-digit (natural) number by the sum of its digits in your mind. What is the smallest resulting quotient?


a) In the cube \( ABCD A_{1} B_{1} C_{1} D_{1} \), a common perpendicular \( MN \) is drawn to the lines \( A_{1} B \) and \( B_{1} C \) (point \( M \) lies on the line \( A_{1} B \) ). Find the ratio \( A_{1} M : M B \).

b) Given a cube \( ABCD A_{1} B_{1} C_{1} D_{1} \). Points \( M \) and \( N \) are taken on the segments \( A A_{1} \) and \( B C_{1} \) respectively, such that the lines \( MN \) and \( B_{1} D \) intersect. Find the difference between the ratios \( B C_{1} : B N \) and \( A M : A A_{1} \).


a) Each vertex of a cube has a number, either 1 or 0. On each face of the cube, the sum of the four numbers at its vertices is written. Can it be that all the numbers written on the faces are different?

b) The same question, but with the numbers 1 or -1 written at the vertices.

Solution: \boxed{No}


Eliane wants to choose her schedule for swimming. She wants to attend two classes per week, one in the morning and one in the afternoon, not on the same day nor on consecutive days. In the morning, there are swimming classes from Monday to Saturday at 9 AM, 10 AM, and 11 AM, and in the afternoon, from Monday to Friday at 5 PM and 6 PM. How many different ways can Eliane choose her schedule?


Simplify the following expression:

$$
\frac{\cos \left(2 x+\frac{\pi}{2}\right) \sin \left(\frac{3 \pi}{2}-3 x\right)-\cos (2 x-5 \pi) \cos \left(3 x+\frac{3 \pi}{2}\right)}{\sin \left(\frac{5 \pi}{2}-x\right) \cos 4 x+\sin x \cos \left(\frac{5 \pi}{2}+4 x\right)}
$$


A quality controller checks products for standard compliance. It is known that the probability of a product meeting the standard is 0.95. Find the probability that:
a) Both of the two checked products will meet the standard, assuming the events of the products meeting the standard are independent.
b) Out of the two checked products, only one will meet the standard.


Candies are in the shape of $1 \times 1 \times 1$ cubes. The teacher arranged them into a $3 \times 4 \times 5$ rectangular prism and asked the children to pick candies. In the first minute, Petya took one of the corner candies. Each subsequent minute, the children took all candies that had a neighboring face with already missing candies (for example, 3 candies were taken in the second minute). How many minutes did it take for the children to take all the candies?

Solution: \boxed{10 \text{ minutes}}


Find all pairs of integers \((x, y)\) that are solutions to the equation

$$
7xy - 13x + 15y - 37 = 0.
$$

Indicate the sum of all found values of \(x\).


As shown in the figure to the right, three circles intersect to form seven regions. Assign the integers $0 \sim 6$ to these seven regions such that the sum of the four numbers within each circle is equal. What is the maximum possible sum? $\qquad$


There are $n$ people, and it is known that any two of them communicate at most once. The number of communications among any $n-2$ of them is equal and is $3^k$ (where $k$ is a positive integer). Find all possible values of $n$.


Neo has an infinite supply of red pills and blue pills. When he takes a red pill, his weight will double, and when he takes a blue pill, he will lose one pound. If Neo originally weighs one pound, what is the minimum number of pills he must take to make his weight 2015 pounds?


Given \( f(x)=a \sin x+\sqrt[2019]{x}+1 \) where \( a \in \mathbf{R} \), and \( f\left(\lg \log _{2} 10\right)=3 \), find \( f(\lg \lg 2) \).


In how many ways can 25 forints be paid using 1, 2, 5, 10, and 20 forint coins?


Anna and Berta are playing a game where they take turns removing marbles from the table. Anna makes the first move. If at the beginning of a turn there are \( n \geq 1 \) marbles on the table, then the player whose turn it is takes \( k \) marbles, where \( k \geq 1 \) is either an even number with \( k \leq \frac{n}{2} \) or an odd number with \( \frac{n}{2} \leq k \leq n \). A player wins the game if she takes the last marble from the table.

Determine the smallest number \( N \geq 100000 \) such that Berta can force a win if there are initially exactly \( N \) marbles on the table.


Through point \( A \) located on a circle, a diameter \( AB \) and a chord \( AC \) are drawn, where \( AC = 8 \) and \( \angle BAC = 30^\circ \).

Find the length of the chord \( CM \), which is perpendicular to \( AB \).


Three sides \(OAB, OAC\) and \(OBC\) of a tetrahedron \(OABC\) are right-angled triangles, i.e. \(\angle AOB = \angle AOC = \angle BOC = 90^\circ\). Given that \(OA = 7\), \(OB = 2\), and \(OC = 6\), find the value of
\[
(\text{Area of }\triangle OAB)^2 + (\text{Area of }\triangle OAC)^2 + (\text{Area of }\triangle OBC)^2 + (\text{Area of }\triangle ABC)^2.
\]


Calculate the area of the figure bounded by the curves given by the equations:

$$
\begin{aligned}
& \left\{\begin{array}{l}
x=8(t-\sin t) \\
y=8(1-\cos t)
\end{array}\right. \\
& y=12(0<x<16 \pi, y \geq 12)
\end{aligned}
$$


Positive real numbers \( a, b, c \) are in a geometric progression \((q \neq 1)\), and \( \log _{a} b, \log _{b} c, \log _{c} a \) are in an arithmetic progression. Find the common difference \( d \).


Point \( M \) is located on the lateral side \( AB \) of trapezoid \( ABCD \), such that \( AM : BM = 2 : 1 \). A line passing through point \( M \) parallel to the bases \( AD \) and \( BC \) intersects the lateral side \( CD \) at point \( N \). Find \( MN \), if \( AD = 18 \) and \( BC = 6 \).


Let \( n \) be an integer with \( n \geqslant 2 \). On a slope of a mountain, \( n^{2} \) checkpoints are marked, numbered from 1 to \( n^{2} \) from the bottom to the top. Each of two cable car companies, \( A \) and \( B \), operates \( k \) cable cars numbered from 1 to \( k \); each cable car provides a transfer from some checkpoint to a higher one. For each company, and for any \( i \) and \( j \) with \( 1 \leqslant i < j \leqslant k \), the starting point of car \( j \) is higher than the starting point of car \( i \); similarly, the finishing point of car \( j \) is higher than the finishing point of car \( i \). Say that two checkpoints are linked by some company if one can start from the lower checkpoint and reach the higher one by using one or more cars of that company (no movement on foot is allowed).

Determine the smallest \( k \) for which one can guarantee that there are two checkpoints that are linked by each of the two companies.

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


Among the natural numbers from 1 to 1000 (inclusive), how many are divisible by 2 or 3 or 5, but not by 6?


Let \( f: \mathbb{N} \rightarrow \mathbb{N} \) be a surjective function and \( g: \mathbb{N} \rightarrow \mathbb{N} \) be an injective function such that for all \( n \in \mathbb{N} \), \( f(n) \geq g(n) \). Show that \( f = g \).

Solution: \boxed{f = g}


A circle is inscribed in a right triangle. The point of tangency divides the hypotenuse into two segments of lengths $6 \mathrm{~cm}$ and $7 \mathrm{~cm}$. Calculate the area of the triangle.

Solution: \boxed{42 \text{ cm}^2}


Can the difference of the squares of two prime numbers be equal to 4048?

Solution: \boxed{No}


A chord \( AB \) of fixed length slides its ends along a circle of radius \( R \). A point \( C \) on this chord, which is at distances \( a \) and \( b \) from the ends \( A \) and \( B \) of the chord respectively, traces another circle as the chord makes a full rotation. Compute the area of the annulus enclosed between the given circle and the circle traced by point \( C \).

Solution: \boxed{\pi ab}


Calculate the areas of figures bounded by lines given in polar coordinates.

$$
r = 1 + \sqrt{2} \sin \phi
$$


If \( 2^{n+2} \cdot 3^n + 5n - a \) can be divided by 25, what is the smallest positive value of \( a \)?


Find the value of the expression \(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\), given that \(a, b, c\) are three distinct real numbers satisfying the conditions:

\[ a^{3} - 2022a + 1011 = 0, \]
\[ b^{3} - 2022b + 1011 = 0, \]
\[ c^{3} - 2022c + 1011 = 0. \]


Let the sequence $\left\{a_{n}\right\}$ satisfy $a_{1}=a, a_{2}=b, 2a_{n+2}=a_{n+1}+a_{n}$. If $\lim _{n \rightarrow \infty}\left(a_{1}+a_{2}+\cdots+a_{n}\right)=4$, find the values of $a$ and $b$.

Solution: \boxed{a = 6, b = -3}


Given \( y z \neq 0 \) and the set \( \{2x, 3z, xy\} \) can also be represented as \( \{y, 2x^2, 3xz\} \), find \( x \).


In a regular truncated quadrilateral pyramid, the height is 2, and the sides of the bases are 3 and 5. Find the diagonal of the truncated pyramid.


Let \( F \) be the left focus of the ellipse \( E: \frac{x^{2}}{3} + y^{2} = 1 \). A line \( l \) with positive slope passes through \( F \) and intersects the ellipse \( E \) at points \( A \) and \( B \). From points \( A \) and \( B \), lines \( AM \) and \( BN \) are drawn respectively, where \( AM \perp l \) and \( BN \perp l \), intersecting the x-axis at points \( M \) and \( N \). Find the minimum value of \( |MN| \).


The number of moles of potassium hydroxide and nitric acid in the solutions:

$$
\begin{gathered}
v(\mathrm{KOH})=\omega \rho V / M=0,062 \cdot 1,055 \text { g/ml } \cdot 22.7 \text { ml / } 56 \text { g/ mol }=0.0265 \text { mol. } \\
v\left(\mathrm{HNO}_{3}\right)=\text { C }_{\mathrm{M}} \cdot \mathrm{V}=2.00 \text { mol/l } \cdot 0.0463 \text { l }=0.0926 \text { mol. }
\end{gathered}
$$

Since $v(\mathrm{KOH})<v(\mathrm{HNO} 3)$, and the coefficients for these substances in the reaction equation are equal to one, potassium hydroxide is in deficit, and its quantity will determine the thermal effect of the reaction, which will be:

$$
\text { Q = 55.6 kJ/mol } \cdot \text { 0.0265 mol = 1.47 kJ. }
$$

Solution: \boxed{1.47 \, \text{kJ}}


Ann and Anne are in bumper cars starting 50 meters apart. Each one approaches the other at a constant ground speed of $10 \mathrm{~km} / \mathrm{hr}$. A fly starts at Ann, flies to Anne, then back to Ann, and so on, back and forth until it gets crushed when the two bumper cars collide. When going from Ann to Anne, the fly flies at $20 \mathrm{~km} / \mathrm{hr}$; when going in the opposite direction, the fly flies at $30 \mathrm{~km} / \mathrm{hr}$ (thanks to a breeze). How many meters does the fly fly?

Solution: \boxed{55 \text{ meters}}


Let the sequences \(\left\{a_{n}\right\}\) and \(\left\{b_{n}\right\}\) be defined as follows:
\[ a_{1} = 3, \quad b_{1} = 1 \]

and for any \( n \in \mathbb{Z}_{+} \), we have
\[
\begin{cases}
a_{n+1} = a_{n} + b_{n} + \sqrt{a_{n}^{2} - a_{n} b_{n} + b_{n}^{2}}, \\
b_{n+1} = a_{n} + b_{n} - \sqrt{a_{n}^{2} - a_{n} b_{n} + b_{n}^{2}}.
\end{cases}
\]

(1) Find the general terms of the sequences \(\left\{a_{n}\right\}\) and \(\left\{b_{n}\right\}\).

(2) Let \([x]\) denote the greatest integer less than or equal to the real number \(x\). Define \(S_{n} = \sum_{i=1}^{n} \left[a_{i}\right]\) and \(T_{n} = \sum_{i=1}^{n} \left[b_{i}\right]\). Find the smallest \(n \in \mathbb{Z}_{+}\) such that
\[
\sum_{k=1}^{n} \left( S_{k} + T_{k} \right) > 2017.
\]


The numbers \( x \), \( y \), and \( z \) satisfy the equations

\[
xy + yz + zx = xyz, \quad x + y + z = 1
\]

What values can the sum \( x^3 + y^3 + z^3 \) take?


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?


Find the number of pairs of positive integers $(x, y)$ which satisfy the equation $2x + 3y = 2007$.


What is the minimum number of weights necessary to be able to measure any number of grams from 1 to 100 on a balance scale, given that weights can be placed on both sides of the scale?


On side \( BC \) of the rhombus \( ABCD \), a point \( M \) is chosen. Lines drawn through \( M \) perpendicular to the diagonals \( BD \) and \( AC \) intersect line \( AD \) at points \( P \) and \( Q \) respectively. It turned out that lines \( PB, QC \), and \( AM \) intersect at one point. What can the ratio \( BM : MC \) be?


Calculate the limit of the numerical sequence:

$$
\lim _{n \rightarrow \infty} \frac{\sqrt{3 n-1}-\sqrt[3]{125 n^{3}+n}}{\sqrt[3]{n}-n}
$$


Given the sequence \(\left\{a_{n}\right\}\) satisfies: \(a_{n+1} \leq \frac{a_{n+2} + a_{n}}{2}\), with \(a_{1}=1\) and \(a_{404}=2016\), find the maximum value of \(a_{6}\).


On a line, there are blue and red points, with at least 5 red points. It is known that on any interval with endpoints at red points that contains a red point inside, there are at least 4 blue points. Additionally, on any interval with endpoints at blue points that contains 3 blue points inside, there are at least 2 red points. What is the maximum number of blue points that can be on an interval with endpoints at red points, not containing other red points inside?


As shown in the figure, the abacus has three sections, each with 10 beads. Divide the beads in each section into an upper and lower part to form two three-digit numbers. The upper part must form a three-digit number with distinct digits, and it must be a multiple of the three-digit number formed by the lower part. What is the three-digit number in the upper part? $\qquad$


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


Let \( S = \{1, 2, 3, 4\} \) and let the sequence \( a_{1}, a_{2}, \cdots, a_{n} \) have the following property: For any non-empty subset \( B \) of \( S \) (with the number of elements in set \( B \) denoted as \( |B| \)), there exist \( |B| \) consecutive terms in the sequence that exactly form the set \( B \). Find the smallest possible value of \( n \).


Two circles of radius \( r \) touch each other. Additionally, each of them is externally tangent to a third circle of radius \( R \) at points \( A \) and \( B \) respectively.

Find the radius \( r \), given that \( AB = 12 \) and \( R = 8 \).


ABCD is a rectangle. AEB is isosceles with E on the opposite side of AB to C and D and lies on the circle through A, B, C, D. This circle has radius 1. For what values of |AD| do the rectangle and triangle have the same area?


As shown in Figure 2, in triangle \( \triangle ABC \), angle \( \angle B \) is an acute angle. Circles \( \odot O \) and \( \odot I \) are the circumcircle and excircle corresponding to angle \( \angle B \), respectively. Let \( R \) and \( r \) be the radii of \( \odot O \) and \( \odot I \), respectively. Then:
$$
OI^{2} > R^{2} - 2 R r
$$

Solution: \boxed{OI^2 > R^2 - 2Rr}


In the set of positive integers less than 10,000, how many integers \( x \) are there such that \( 2^x - x^2 \) is divisible by 7?


The six-digit number begins with the digit 2. If this digit is moved from the first place to the last, keeping the order of the other five digits, the resulting number will be three times the original number. Find the original number.


There is a five-digit odd positive integer \( x \). By changing all 2s in \( x \) to 5s and all 5s in \( x \) to 2s, while keeping all other digits unchanged, we obtain a new five-digit number \( y \). If \( x \) and \( y \) satisfy the equation \( y = 2(x+1) \), then \( x \) is ________.


A perfect power is an integer \( n \) that can be represented as \( a^{k} \) for some positive integers \( a \geq 1 \) and \( k \geq 2 \). Find the sum of all prime numbers \(0 < p < 50\) such that \( p \) is 1 less than a perfect power.


How many triangles exist that have a perimeter of 15 units and sides measuring integer numbers?


There are two positive integers, \(A\) and \(B\). The sum of the digits of \(A\) is \(19\), the sum of the digits of \(B\) is \(20\), and their addition results in carrying over twice. What is the sum of the digits of \((\mathbf{A} + B)\)?


Let \( l \) and \( m \) be two skew lines. On line \( l \), there are three points \( A \), \( B \), and \( C \) such that \( AB = BC \). From points \( A \), \( B \), and \( C \), perpendiculars \( AD \), \( BE \), and \( CF \) are drawn to line \( m \) with feet \( D \), \( E \), and \( F \) respectively. Given that \( AD = \sqrt{15} \), \( BE = \frac{7}{2} \), and \( CF = \sqrt{10} \), find the distance between lines \( l \) and \( m \).


Anya, Vanya, Dania, Manya, Sanya, and Tanya were picking apples. It turned out that each of them picked a whole number of percentages of the total number of apples collected, and all these numbers are different and greater than zero. What is the minimum number of apples that could have been collected?


There are two types of containers: 27 kg and 65 kg. How many containers of the first and second types were there in total, if the load in the containers of the first type exceeds the load of the container of the second type by 34 kg, and the number of 65 kg containers does not exceed 44 units?


In the quadrilateral \(ABCD\), \(AB = 1\), \(BC = 2\), \(CD = \sqrt{3}\), \(\angle ABC = 120^\circ\), and \(\angle BCD = 90^\circ\). What is the exact length of side \(AD\)?


The exterior angles of a triangle are proportional to the numbers 5: 7: 8. Find the angle between the altitudes of this triangle drawn from the vertices of its smaller angles.


The first \( n \) terms of the expansion of \((2 - 1)^{-n}\) are \(2^{-n} \left( 1 + \frac{n}{1!} \left(\frac{1}{2}\right) + \frac{n(n + 1)}{2!} \left(\frac{1}{2}\right)^2 + \ldots + \frac{n(n + 1) \cdots (2n - 2)}{(n - 1)!} \left(\frac{1}{2}\right)^{n-1} \right)\). Show that they sum to \(\frac{1}{2}\).


Given the point \( M_{1}(7, -8) \) and the normal vector of the line \( \vec{n} = (-2, 3) \), find the equation of the line.

Solution: \boxed{2x - 3y - 38 = 0}


During intervals, students played table tennis. Any two students played against each other no more than one game. At the end of the week, it turned out that Petya played half, Kolya played a third, and Vasya played a fifth of all the games played during the week. How many games could have been played during the week if it is known that Vasya did not play with either Petya or Kolya?


Solution: \boxed{x = -\frac{1}{2}}


Find all values of \( x \) for each of which one of the three given numbers \( \log _{x^{2}}\left(x^{2}-10 x+21\right) \), \( \log _{x^{2}} \frac{x^{2}}{x-7} \), and \( \log _{x^{2}} \frac{x^{2}}{x-3} \) is equal to the sum of the other two.

Solution: \boxed{x = 8}


For each positive integer \( n \), define the point \( P_{n} \) to have coordinates \(\left((n-1)^{2}, n(n-1)\right)\) and the point \( Q_{n} \) to have coordinates \(\left((n-1)^{2}, 0\right)\). For how many integers \( n \) with \( 2 \leq n \leq 99 \) is the area of trapezoid \( Q_{n} P_{n} P_{n+1} Q_{n+1} \) a perfect square?


In $\triangle ABC$, if $\overrightarrow{AB} \cdot \overrightarrow{AC} = 7$ and $\left|\overrightarrow{AB} - \overrightarrow{AC}\right| = 6$, then the maximum area of $\triangle ABC$ is


While waiting for customers, a watermelon seller sequentially weighed 20 watermelons (with masses of 1 kg, 2 kg, 3 kg, ..., up to 20 kg), balancing a watermelon on one side of the scale with one or two weights on the other side (possibly identical weights). The seller recorded on paper the mass of the weights he used. What is the minimum number of different numbers that could appear in his records, given that the mass of each weight is an integer in kilograms?


Given two regular triangular pyramids \( P-ABC \) and \( Q-ABC \) with the same base \( ABC \) such that both are inscribed in the same sphere. If the angle between the side face and the base of the regular triangular pyramid \( P-ABC \) is \( 45^{\circ} \), find the tangent of the angle between the side face and the base of the regular triangular pyramid \( Q-ABC \).


One evening, 21 people communicate by phone $n$ times. It is known that among them, there are $m$ ($m$ is an odd number) people $a_{1}, a_{2}, \cdots, a_{m}$ such that $a_{i}$ communicates with $a_{i+1}$ $\left(i=1,2, \cdots, m; a_{m+1}=a_{1}\right)$. If none of these 21 people had a three-way conversation, find the maximum value of $n$.


Let \( S \) be a set of \( n \) points in the plane such that the greatest distance between two points of \( S \) is 1. Show that at most \( n \) pairs of points of \( S \) are a distance 1 apart.

Solution: \boxed{n}


Paris is a city with 2,300,000 inhabitants. There are at most 500,000 hairs on the head of a Parisian. Show that there are at least 5 Parisians who have the same number of hairs.


Color 8 small squares on a $4 \times 4$ chessboard black, such that each row and each column has exactly 2 black squares. How many different ways are there to color the chessboard?


Ten points are given in the plane, and no three points are collinear. Four distinct segments connecting pairs of these points are chosen at random, all with the same probability. What is the probability that three of the chosen segments will form a triangle?


Through vertex $C$ of square $ABCD$, a line passes, intersecting diagonal $BD$ at point $K$ and the perpendicular bisector of side $AB$ at point $M$ ($M$ lies between $C$ and $K$). Find the angle $\angle DCK$ if $\angle AKB = \angle AMB$.


Zhenya had 9 cards with numbers from 1 to 9. He lost the card with the number 7. Is it possible to arrange the remaining 8 cards in a row so that any two adjacent cards form a number divisible by 7?

Solution: \boxed{\text{No}}


The area of triangle \( ABC \) is \( S \). Find the area of the triangle whose sides are equal to the medians of triangle \( ABC \).

Solution: \boxed{\frac{3}{4} S}


Point $B$ is the midpoint of segment $AC$. The square $ABDE$ and the equilateral triangle $BCF$ are located on the same side of the line $AC$. Find (in degrees) the measure of the acute angle between lines $CD$ and $AF$.


There are two docks, A and B, on a river. Dock A is upstream and dock B is downstream. Two people, person 1 and person 2, start rowing from A and B respectively at the same time, rowing towards each other, and meet after 4 hours. If person 1 and person 2 start rowing from A and B respectively at the same time, rowing in the same direction, person 2 catches up with person 1 after 16 hours. Given that person 1's rowing speed in still water is 6 km per hour, determine the speed of person 2's rowing in still water in km per hour.


Given \( k, m, n \in \mathbf{N}^{\cdot}, 1 \leqslant k \leqslant m \leqslant n \), find the value of \( \sum_{i=0}^{n}(-1)^{n+i} \frac{1}{n+k+i} \cdot \frac{(m+n+i)!}{i!(m-i)!(m+i)!} \).


A divisor of a natural number is called proper if it is different from 1 and the number itself. A number is called interesting if it has two proper divisors, one of which is a prime number, and the other is a perfect square, and their sum is also a perfect square. How many interesting numbers are there that do not exceed 1000?


A circle passing through the vertices $B, C,$ and $D$ of parallelogram $ABCD$ touches line $AD$ and intersects line $AB$ at points $B$ and $E$. Find the length of segment $AE$, if $AD = 4$ and $CE = 5$.


A team of athletes, one-third of which are snowboarders, descended from the mountain. Some of them took a cable car that holds no more than 10 people, while the rest descended on their own. The number of people who descended on their own was more than 45% but less than 50% of the total number. Determine the number of snowboarders (if the total number of snowboarders is ambiguous based on the problem's conditions, provide the sum of all possible values).


Find the sum of the first 10 elements that appear both in the arithmetic progression $\{4, 7, 10, 13, \ldots\}$ and the geometric progression $\{10, 20, 40, 80, \ldots\}$. (10 points)


Point \( M \) divides the side \( BC \) of parallelogram \( ABCD \) in the ratio \( BM: MC = 1: 3 \). Line \( AM \) intersects diagonal \( BD \) at point \( K \). Find the area of quadrilateral \( CMKD \) if the area of parallelogram \( ABCD \) is 1.


Given real numbers \( x, y, z \) satisfying
\[
\frac{y}{x-y}=\frac{x}{y+z}, \quad z^{2}=x(y+z)-y(x-y)
\]
find the value of
\[
\frac{y^{2}+z^{2}-x^{2}}{2 y z}.
\]


How many ordered pairs \((b, g)\) of positive integers with \(4 \leq b \leq g \leq 2007\) are there such that when \(b\) black balls and \(g\) gold balls are randomly arranged in a row, the probability that the balls on each end have the same colour is \(\frac{1}{2}\)?


Given the function \( f(x) = \frac{2+x}{1+x} \), let \( f(1) + f(2) + \cdots + f(1000) = m \) and \( f\left(\frac{1}{2}\right) + f\left(\frac{1}{3}\right) + \cdots + f\left(\frac{1}{1000}\right) = n \). What is the value of \( m + n \)?


In triangle \( ABC \), \( AC = 2BC \), \(\angle C = 90^\circ\), and \( D \) is the foot of the altitude from \( C \) onto \( AB \). A circle with diameter \( AD \) intersects the segment \( AC \) at \( E \). Find the ratio \( AE: EC \).


Let \(a_{1}, a_{2}, a_{3}, \ldots \) be the sequence of all positive integers that are relatively prime to 75, where \(a_{1}<a_{2}<a_{3}<\cdots\). (The first five terms of the sequence are: \(a_{1}=1, a_{2}=2, a_{3}=4, a_{4}=7, a_{5}=8\).) Find the value of \(a_{2008}\).


The base of a pyramid is a square with side length \( a = \sqrt{21} \). The height of the pyramid passes through the midpoint of one of the edges of the base and is equal to \( \frac{a \sqrt{3}}{2} \). Find the radius of the sphere circumscribed around the pyramid.


A circular coin \(A\) is rolled, without sliding, along the circumference of another stationary circular coin \(B\) with radius twice the radius of coin \(A\). Let \(x\) be the number of degrees that the coin \(A\) makes around its center until it first returns to its initial position. Find the value of \(x\).


In a plane, two vectors \(\overrightarrow{O A}\) and \(\overrightarrow{O B}\) satisfy \(|\overrightarrow{O A}|=a\) and \(|\overrightarrow{O B}|=b\), with \(a^2 + b^2 = 4\), and \(\overrightarrow{O A} \cdot \overrightarrow{O B} = 0\). If the vector \(\overrightarrow{O C}=\lambda \overrightarrow{O A}+\mu \overrightarrow{O B}\) \((\lambda, \mu \in \mathbb{R})\) satisfies
\[
\left(\lambda-\frac{1}{2}\right)^{2} a^{2}+\left(\mu-\frac{1}{2}\right)^{2} b^{2}=1,
\]
then the maximum value of \(|\overrightarrow{O C}|\) is __________.


Plot the set of points on the plane \((x, y)\) that satisfy the equation \( |3x| + |4y| + |48 - 3x - 4y| = 48 \), and find the area of the resulting figure.


\[
\left(\cos 70^{\circ}+\cos 50^{\circ}\right)\left(\cos 310^{\circ}+\cos 290^{\circ}\right)+\left(\cos 40^{\circ}+\cos 160^{\circ}\right)\left(\cos 320^{\circ}-\cos 380^{\circ}\right)=1
\]


There are 21 nonzero numbers. For each pair of these numbers, their sum and product are calculated. It turns out that half of all the sums are positive and half are negative. What is the maximum possible number of positive products?


Given positive integers \( x, y, z \) and real numbers \( a, b, c, d \), which satisfy \( x \leq y \leq z \), \( x^a = y^b = z^c = 70^d \), and \( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \frac{1}{d} \), determine the relationship between \( x + y \) and \( z \). Fill in the blank with ">", "<" or "=".

Solution: \boxed{=}


It is known that the number of birch trees in a certain mixed forest plot ranges from $13\%$ to $14\%$ of the total number of trees. Find the minimum possible total number of trees in this plot.


In a $10 \times 10$ grid square, the centers of all unit squares (a total of 100 points) are marked. What is the minimum number of lines, not parallel to the sides of the square, needed to pass through all the marked points?


Find all functions \( f: \mathbb{N} \rightarrow \mathbb{N} \) such that for all natural numbers \( n \),

\[
f(n) + f(f(n)) + f(f(f(n))) = 3n
\]

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


Given the function \( f:\{1,2, \cdots, 10\} \rightarrow\{1,2,3,4,5\} \), and for each \( k=1,2, \cdots, 9 \), it is true that \( |f(k+1)-f(k)| \geq 3 \). Find the number of functions \( f \) that satisfy these conditions.


Find the last 3 digits of \(1 \times 3 \times 5 \times 7 \times \cdots \times 2005\).


What is the maximum number of natural numbers that can be written in a row such that the sum of any three consecutive numbers is even, and the sum of any four consecutive numbers is odd?


Calculate the definite integral:

$$
\int_{-\frac{\pi}{2}}^{0} \frac{\sin x \, dx}{(1+\cos x-\sin x)^{2}}
$$

Solution: \boxed{\frac{1}{2} - \ln 2}


The edge length of cube $ABCD A_1 B_1 C_1 D_1$ is 1. On the extension of edge $AD$ beyond point $D$, point $M$ is chosen such that $|AM| = 2\sqrt{2/5}$. Point $E$ is the midpoint of edge $A_1 B_1$, and point $F$ is the midpoint of edge $D D_1$. What is the maximum value of the ratio $|MP|/|PQ|$, where point $P$ lies on segment $AE$, and point $Q$ lies on segment $CF$?


Let $ABC$ be a triangle such that $\angle BAC = 60^\circ$. The perpendicular bisector of $[AC]$ intersects $(AB)$ at $P$, and the perpendicular bisector of $[AB]$ intersects $(AC) at $Q$. Show that $PQ = BC$.

Solution: \boxed{PQ = BC}


The function \( f(x) \) is an odd function defined on \( \mathbf{R} \) and \( f(2)=0 \). For any \( x \in \mathbf{R} \), the equation \( f(x+4)=f(x)+f(4) \) holds. Find \( f(2002) \).


Thirty girls - 13 in red dresses and 17 in blue dresses - were dancing in a circle around a Christmas tree. Afterwards, each of them was asked if her neighbor to the right was in a blue dress. It turned out that only those girls who stood between girls in dresses of the same color answered correctly. How many girls could have answered affirmatively?


In an isosceles triangle \(ABC\), the perpendicular bisector of the lateral side \(BC\) intersects the base \(AB\) at point \(D\) such that \(AC = AD\). Find the angle \(ABC\).

Provide the answer in degrees without the degree symbol.


Given a complex number \( z \) such that \( |z| = 1 \), find the minimum value of \( \left|z^{2} - 2z + 3\right| \).


An archipelago consists of \( N \geq 7 \) islands. Any two islands are connected by no more than one bridge. It is known that no more than 5 bridges lead from each island, and among any 7 islands, there are necessarily two that are connected by a bridge. What is the maximum value that \( N \) can take?


Find all pairs of positive integers \( m, n \geq 3 \) for which there exist infinitely many positive integers \( a \) such that
$$
\frac{a^{m}+a-1}{a^{n}+a^{2}-1}
$$
is itself an integer.

Solution: \boxed{(5, 3)}


Solution: \boxed{2 \sqrt{38}}


Let \(a_i\) and \(b_i\) be real numbers such that \(a_1 b_2 \neq a_2 b_1\). What is the maximum number of possible 4-tuples \((\text{sign}(x_1), \text{sign}(x_2), \text{sign}(x_3), \text{sign}(x_4))\) for which all \(x_i\) are non-zero and \(x_i\) is a simultaneous solution of
\[
a_1 x_1 + a_2 x_2 + a_3 x_3 + a_4 x_4 = 0
\]
and
\[
b_1 x_1 + b_2 x_2 + b_3 x_3 + b_4 x_4 = 0.
\]
Find necessary and sufficient conditions on \(a_i\) and \(b_i\) for this maximum to be achieved.


For which values of the parameter \( a \) does the equation \( x^{3}-11x^{2}+ax-8=0 \) have three distinct real roots that form a geometric progression?


Given the sequence \(\left\{a_{n}\right\}\) defined by: \(a_{1} = 7\) and \(\frac{a_{n+1}}{a_{n}} = a_{n} + 2\) for \(n = 1, 2, 3, \ldots\). Find the smallest positive integer \(n\) such that \(a_{n} > 4^{2018}\).


Every morning, Absent-minded Scientist goes to work, and in the evening, returns home. The Scientist has two umbrellas and takes one with him in only one of these two cases: either it is raining outside, or there is no umbrella where he is going. After some time, the Scientist calculated that he takes an umbrella with him in 20% of the cases. Find the probability of rain.


Consider the paths from \((0,0)\) to \((6,3)\) that only take steps of unit length up and right. Compute the sum of the areas bounded by the path, the \(x\)-axis, and the line \(x=6\) over all such paths.


With a number, it is permissible to perform one of two actions: square it or add one to it. Given the numbers 19 and 98, is it possible to obtain equal numbers from them using the same number of actions?

Solution: \boxed{\text{No}}


In each cell of a square table of size \((2^n - 1) \times (2^n - 1)\), one of the numbers 1 or -1 is placed. The arrangement of numbers is called successful if each number is equal to the product of all its neighboring numbers (neighbors are considered to be the numbers in the cells with a common side). Find the number of successful arrangements.


Is it possible that 1000 participants of an Olympiad solve this problem correctly, with the number of boys being 43 more than the number of girls?

Solution: \boxed{\text{No}}


In Figure 1, \(ABCD\) is a parallelogram with \(BA = 3 \text{ cm}\), \(BC = 4 \text{ cm}\), and \(BD = \sqrt{37} \text{ cm}\). If \(AC = h \text{ cm}\), find the value of \(h\).


Among all the simple fractions with a numerator and denominator that are two-digit numbers, find the smallest fraction greater than $\frac{3}{4}$. Provide its numerator in the answer.


For a four-digit positive integer \(ABCD\), where \(A, B, C,\) and \(D\) are digits with \(A \neq 0\) and \(D \neq 0\), we define the Reverse Digit sum (RD sum) of \(ABCD\) to be the sum of \(ABCD\) and \(DCBA\). For example, the RD sum of 1205 is \(1205 + 5021 = 6226\), while the integer 2300 does not have an RD sum.

(a) Determine the RD sum of 4281.

(b) There are positive integers \(m\) and \(n\) with the property that the RD sum of the integer \(ABCD\) is always equal to \(m \times (A + D) + n \times (B + C)\). State the value of \(m\) and the value of \(n\).

(c) Determine the number of four-digit integers whose RD sum is 3883.

(d) Determine the number of four-digit integers that are equal to the RD sum of a four-digit integer.


Vitya Perestukin always incorrectly calculates percentages during surveys: he divides the number of respondents who answered a certain way by the number of all remaining respondents. For instance, in the survey "What is your name?" conducted among 7 Annas, 9 Olgas, 8 Julias, Vitya calculated 50% Julias.

Vitya conducted a survey in his school: what type of triangle has sides of \(3, 4, 5\)? According to his calculations, 5% answered "acute", 5% "obtuse", and 5% "such a triangle does not exist", 50% "right", and the remaining \(a \%\) answered "depends on the geometry". What is \(a\)?

Solution: \boxed{a = 110}


In the following two equations, identical Chinese characters represent identical digits, and different Chinese characters represent different digits:
$$
\text{数字花园} + \text{探秘} = 2015, \overline{\text{探秘}} + 1 + 2 + 3 + \cdots + 10 = \overline{\text{花园}}
$$

What is the four-digit number represented by 数字花园?


Any finite sequence composed of 0s and 1s is called a word. Repeating a word \( A \) three times to form \( A A A \) is called the tripling of \( A \). For example, if \( A = 101 \), then the tripling of \( A \) is 101101101. Two operations can be performed on words:
1. Add the tripling of any word at any position (including the beginning or the end) of the word.
2. Remove the tripling of any word from any position of the word.

For example, from the word 0001, it is possible to obtain 0111001 or 1, among others.

Question: Is it possible, through a series of the above operations, to transform the word 10 into the word 01?

Solution: \boxed{\text{No}}


Find the smallest positive integer \( n \), or show that no such \( n \) exists, with the following property: there are infinitely many distinct \( n \)-tuples of positive rational numbers \( \left(a_{1}, a_{2}, \ldots, a_{n}\right) \) such that both
\[ a_{1}+a_{2}+\cdots+a_{n} \quad \text{and} \quad \frac{1}{a_{1}}+\frac{1}{a_{2}}+\cdots+\frac{1}{a_{n}} \]
are integers.


Compute the sum of all real numbers \( x \) such that
\[
2x^6 - 3x^5 + 3x^4 + x^3 - 3x^2 + 3x - 1 = 0
\]


Given 5 points, any 4 of which do not lie in the same plane.

a) How many tetrahedrons can be formed whose vertices are 4 of these points?
b) How many planes can be formed passing through triples of these points?
c) How many lines are determined by pairs of these points?


Let \(a, b, c\) be three integers such that \(a + b + c\) is divisible by 13. Show that also

\[ a^{2007} + b^{2007} + c^{2007} + 2 \cdot 2007 \cdot abc \]

is divisible by 13.


Given that the area of triangle $\triangle ABC$ is 1, points $B_{1}$, $B_{2}$, $C_{1}$, and $C_{2}$ are trisection points of sides $AB$ and $AC$ respectively. Connect $B_{1}C$, $B_{2}C$, $BC_{1}$, and $BC_{2}$. Find the area of the quadrilateral they form.


If the positive integers \( m \) and \( n \) satisfy \( \frac{(m+n)!}{n!}=5040 \), then the value of \( m!n \) is ________.


Find the remainder of the polynomial \(P(x)=x^{6 n}+x^{5 n}+x^{4 n}+x^{3 n}+x^{2 n}+x^{n}+1\) when divided by \(Q(x)=x^{6}+x^{5}+x^{4}+x^{3}+x^{2}+x+1\), given that \(n\) is a multiple of 7.


The number \( x \) is such that \( \log _{2}\left(\log _{4} x\right) + \log _{4}\left(\log _{8} x\right) + \log _{8}\left(\log _{2} x\right) = 1 \). Find the value of the expression \( \log _{4}\left(\log _{2} x\right) + \log _{8}\left(\log _{4} x\right) + \log _{2}\left(\log _{8} x\right) \). If necessary, round your answer to the nearest 0.01.


Given \( n \) new students, out of any 3 of them, 2 students know each other, and out of any 4 of them, 2 students do not know each other. Find the maximum value of \( n \).


Given the point sets \( A = \left\{(x, y) \left\lvert\,(x-3)^{2}+(y-4)^{2} \leqslant\left(\frac{5}{2}\right)^{2}\right.\right\} \) and \( B = \left\{(x, y) \left\lvert\,(x-4)^{2}+(y-5)^{2}>\left(\frac{5}{2}\right)^{2}\right.\right\} \), how many integer points (i.e., points where both the x-coordinate and y-coordinate are integers) are in the set \( A \cap B \)?


How many consecutive "0"s are there at the end of the product \(5 \times 10 \times 15 \times 20 \times \cdots \times 2010 \times 2015\)?


Find the smallest positive integer \( n \) such that \( 1^2 + 2^2 + 3^2 + 4^2 + \cdots + n^2 \) is divisible by 100.


A, B, C, D are circles such that A and B touch externally at P, B and C touch externally at Q, C and D touch externally at R, and D and A touch externally at S. A does not intersect C, and B does not intersect D. Show that PQRS is cyclic. If A and C have radius 2, B and D have radius 3, and the distance between the centers of A and C is 6, find the area of PQRS.


ABCD is a quadrilateral. The feet of the perpendiculars from D to AB and BC are P and Q, respectively, and the feet of the perpendiculars from B to AD and CD are R and S, respectively. Show that if ∠PSR = ∠SPQ, then PR = QS.

Solution: \boxed{PR = QS}


Given a triangular pyramid $S-ABC$ with the base being an isosceles right triangle with $AB$ as the hypotenuse, and $SA = SB = SC = 2$, $AB = 2$, let points $S$, $A$, $B$, and $C$ all lie on the surface of a sphere centered at $O$. What is the distance from point $O$ to the plane $ABC$?


There are seven cards on the table. In one move, you are allowed to flip any five cards. What is the minimum number of moves required to flip all the cards?


A fraction $\frac{1}{6}$ had some fraction added to it, and the result was a proper fraction with a denominator less than 6. What is the largest fraction that could have been added?


Simplify the expression:

\[
\frac{\sin 2 \alpha + \cos 2 \alpha - \cos 6 \alpha - \sin 6 \alpha}{\sin 4 \alpha + 2 \sin^2 2 \alpha - 1}
\]

Solution: \boxed{2 \sin 2\alpha}


Let \( k \) be a natural number. Determine the smallest natural number \( n \) that satisfies the following property: In any set of \( n \) integers, there must be two numbers whose sum or difference is divisible by \( 2k + 1 \).


Given positive real numbers \(a\) and \(b\) satisfying \(a(a+b) = 27\), find the maximum value of \(a^{2}b\).


Given that \(x\), \(y\), and \(z\) satisfy the following system of equations:

\[
\begin{cases}
\log_{2}\left(x y z - 3 + \log_{5} x \right) = 5, \\
\log_{3}\left(x y z - 3 + \log_{5} y \right) = 4, \\
\log_{4}\left(x y z - 3 + \log_{5} z \right) = 4,
\end{cases}
\]

find the value of \(\log_{5} (x y z)\).


The highway from Lishan Town to the provincial capital is 189 kilometers long, passing through the county town. It is 54 kilometers from Lishan Town to the county town. At 8:30 AM, a bus left from Lishan Town to the county town and arrived at 9:15 AM. After a 15-minute stop, it continued to the provincial capital and arrived at 11:00 AM. Another bus left from the provincial capital to Lishan Town on the same morning at 8:50 AM, traveling at 60 kilometers per hour. What is the time when the two buses meet?


The parabolas given by the equations \( y = -x^2 + 6x - 10 \) and \( 2y = x^2 + 6x + 13 \) are centrally similar. Find the center of similarity.

Solution: \boxed{(1, 0)}


The sum of the first $n$ terms of an arithmetic series is $A$, and the sum of the first $2n$ terms is $B$. Express the sum of the first $3n$ terms using $A$ and $B$.

Solution: \boxed{C = 3(B - A)}


Given the functions \( f_0(x) = |x| \), \( f_1(x) = \left| f_0(x) - 1 \right| \), and \( f_2(x) = \left| f_1(x) - 2 \right| \), determine the area of the closed region enclosed by the graph of the function \( y = f_2(x) \) and the x-axis.


Find the greatest common divisor of the pair of numbers 123456789 and 987654321 (in other words, the greatest common measure of segments of such lengths).


10.25 people are lined up in a row, each of whom either tells the truth or lies.
The person at the front of the line says: "Everyone behind me is lying."
Everyone else says: "The person in front of me (the one directly next to the speaker) is lying."
Among these 25 people, there are ___ people who are lying.


Two couriers start from two locations $A$ and $B$; the first heading towards $B$, and the second towards $A$. When they meet, the first courier has traveled 12 miles more than the second. If they continue their journeys at their original speeds, the first courier will reach their destination 9 days after the meeting, and the second courier will reach their destination 16 days after the meeting. How far apart are the two locations?


Grisha has 5000 rubles. Chocolate bunnies are sold in a store at a price of 45 rubles each. To carry the bunnies home, Grisha will have to buy several bags at 30 rubles each. One bag can hold no more than 30 chocolate bunnies. Grisha bought the maximum possible number of bunnies and enough bags to carry all the bunnies. How much money does Grisha have left?

Solution: \boxed{20 \text{ rubles}}


Inside a unit cube, eight equal spheres are placed. Each sphere is inscribed in one of the trihedral angles of the cube and touches the three spheres corresponding to the adjacent vertices of the cube. Find the radii of the spheres.


ABCD is a quadrilateral with ∠A = 90°, AD = a, BC = b, AB = h, and area (a+b)h/2. What can we say about ∠B?


From unit cubes, a large parallelepiped with side lengths greater than 3 was assembled. Two cubes are called neighboring if they share a face. Therefore, one cube can have up to 6 neighbors. It is known that the number of cubes with exactly 6 neighbors is 429. Find the number of cubes that have exactly 4 neighbors.


In the plane quadrilateral $\mathrm{ABCD}$, given $\mathrm{AB}=1, \mathrm{BC}=4, \mathrm{CD}=2, \mathrm{DA}=3$, find the value of $\overrightarrow{\mathrm{AC}} \cdot \overrightarrow{\mathrm{BD}}$.


Given that \( A \) and \( B \) are two points on the surface of a sphere with a radius of 5, and \( AB = 8 \). Planes \( O_1AB \) and \( O_2AB \) are perpendicular to each other and pass through \( AB \). The intersections of these planes with the sphere create cross-sections \(\odot O_1\) and \(\odot O_2\). Let the areas of \(\odot O_1\) and \(\odot O_2\) be denoted as \( S_1 \) and \( S_2 \) respectively. Solve for \( S_1 + S_2 \).

Solution: \boxed{41 \pi}


Let \( P \) be a regular 2006-sided polygon. A diagonal of \( P \) is called a "good edge" if both parts of \( P \)'s boundary divided by this diagonal contain an odd number of edges. Each side of \( P \) is also defined as a "good edge."

Given that 2003 non-intersecting diagonals inside \( P \) divide it into several triangles, determine the maximum number of isosceles triangles with two "good edges" in this division.


Draw a pentagon and connect all of its vertices with each other as shown in the figure. How many triangles are there in the resulting figure?


Given a regular rectangular pyramid \(\Gamma\) with a height of 3 and an angle of \(\frac{\pi}{3}\) between its lateral faces and its base, first inscribe a sphere \(O_{1}\) inside \(\Gamma\). Then, sequentially inscribe spheres \(O_{2}, O_{3}, O_{4}, \ldots\) such that each subsequent sphere is tangent to the preceding sphere and to the four lateral faces of \(\Gamma\). Find the sum of the volumes of all the inscribed spheres.

Solution: \boxed{\frac{18 \pi}{13}}


The square of a three-digit number ends with three identical digits different from zero. Write the smallest such three-digit number.


We are approaching a 120-meter high skyscraper on a horizontal road. After traveling 300 meters, we see the building at an angle of elevation that is $45^\circ$ greater than at the start of our journey. How close have we approached the skyscraper?


Given an arithmetic sequence \(a_1, a_2, \cdots, a_{1000}\), the sum of the first 100 terms is 100, and the sum of the last 100 terms is 1000. Find \(a_1\).


Find the smallest prime number \( p \) that cannot be expressed as \( \left|3^{a} - 2^{b}\right| \), where \( a \) and \( b \) are non-negative integers.


Define \([a]\) as the largest natural number no greater than \(a\); for example, \([0.6] = 0\) and \([1.25] = 1\). Given that \([5a - 0.9] = 3a + 0.7\), find the value of \(a\).

Solution: \boxed{a = 1.1}


A sequence \(\left\{a_{n}\right\}\) satisfies \(a_{n+1}=\frac{1}{2-a_{n}}\). If for any positive integer \(n\), \(a_{n+1}>a_{n}\) always holds, what is the range of values for \(a_{1}\)?

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


\( p(z) \) is a polynomial of degree \( n \) with complex coefficients. Its roots (in the complex plane) can be covered by a disk of radius \( r \). Show that for any complex \( k \), the roots of \( n p(z) - k p'(z) \) can be covered by a disk of radius \( r + |k| \).

Solution: \boxed{r + |k|}


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 $T ?$


In an arithmetic sequence \(\{a_{n}\}\), for any positive integer \(n\), it holds that \(a_{n} + 2a_{n+1} + 3a_{n+2} = 6n + 22\). Find the value of \(a_{2017}\).


A cuckoo clock chimes the number of times corresponding to the current hour (e.g., at 19:00, it chimes 7 times). One morning, Maxim approached the clock when it was 9:05 and started moving the minute hand forward until the clock read 7 hours later. How many times did the cuckoo chime during this period?


In the number \(2 * 0 * 1 * 6 * 0 *\), each of the 5 asterisks needs to be replaced by any of the digits \(0,1,2,3,4,5,6,7,8\) (digits can repeat) such that the resulting 10-digit number is divisible by 18. How many ways can this be done?


Given an ellipse \( C_{1}: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 (a > b > 0) \) with foci \( F_{1} \) and \( F_{2} \) on the left and right, respectively, and right vertex \( A \). Let \( P \) be any point on the ellipse \( C_{1} \), and the range for the maximum value of \( \overrightarrow{P F_{1}} \cdot \overrightarrow{P F_{2}} \) is \([c^{2}, 3 c^{2}]\), where \( c = \sqrt{a^{2} - b^{2}} \).

1. Find the range of the eccentricity \( \boldsymbol{e} \) of the ellipse \( C_{1} \):

2. Given a hyperbola \( C_{2} \) with the foci of ellipse \( C_{1} \) as its vertices, and \( B \) being any point on \( C_{2} \) in the first quadrant, when the eccentricity \( e \) takes the minimum value, is there a constant \( \lambda (\lambda > 0) \) such that \( \angle B A F_{1} = \lambda \angle B F_{1} A \) always holds? If it exists, find the value of \( \lambda \); if it does not exist, explain why.

Solution: \boxed{\lambda = 2}


Find all integer values of the parameter \(a\) for which the system
\[
\begin{cases}
x - 2y = y^2 + 2, \\
ax - 2y = y^2 + x^2 + 0.25a^2
\end{cases}
\]
has at least one solution. In the answer, indicate the sum of the found values of the parameter \(a\).


Let \( S \) be the set of all nondegenerate triangles formed from the vertices of a regular octagon with side length 1. Find the ratio of the largest area of any triangle in \( S \) to the smallest area of any triangle in \( S \).

Solution: \boxed{3 + 2\sqrt{2}}


Find the value of $\cos \frac{2 \pi}{7} + \cos \frac{4 \pi}{7} + \cos \frac{6 \pi}{7}$.


Find the maximum for all triangles $ABC$ of

$$
\sin(A) + \sin(B) + \sin(C)
$$


Determine the geometric mean of the following two expressions:

$$
\frac{2\left(a^{2}-a b\right)}{35 b}, \quad \frac{10 a}{7\left(a b-b^{2}\right)}
$$


Four consecutive natural numbers, in ascending order, are multiples of 11, 7, 5, and 3 respectively. Find the minimum possible sum of these four natural numbers.


Two different natural numbers end with 7 zeros and have exactly 72 divisors. Find their sum.


Let \( A B C \) be a triangle with \( AB = 8 \), \( AC = 12 \), and \( BC = 5 \). Let \( M \) be the second intersection of the internal angle bisector of \( \angle B A C \) with the circumcircle of \( A B C \). Let \( \omega \) be the circle centered at \( M \) tangent to \( AB \) and \( AC \). The tangents to \( \omega \) from \( B \) and \( C \), other than \( AB \) and \( AC \) respectively, intersect at a point \( D \). Compute \( AD \).


Given three positive integers \(a\), \(b\), \(c\), we can derive 8 numbers using one addition and one multiplication, and using each number just once: \(a+b+c\), \(a+bc\), \(b+ac\), \(c+ab\), \((a+b)c\), \((b+c)a\), \((c+a)b\), \(abc\). Show that if \(a\), \(b\), \(c\) are distinct positive integers such that \( \frac{n}{2} < a, b, c \leq n \), then the 8 derived numbers are all different. Additionally, show that if \(p\) is a prime and \( n \geq p^2 \), then there are exactly \( d(p-1) \) ways of choosing two distinct numbers \(b, c\) from \(\{p+1, p+2, ... , n\}\) such that the 8 numbers derived from \(p\), \(b\), \(c\) are not all distinct, where \(d(p-1)\) is the number of positive divisors of \(p-1\).


What is the probability that in a group of 30 people, there are two individuals who share the same birthday?


Let positive numbers \( x \), \( y \), and \( z \) satisfy the system of equations:

\[
\left\{\begin{array}{l}
x^{2}+xy+y^{2}=147 \\
y^{2}+yz+z^{2}=9 \\
z^{2}+xz+x^{2}=156
\end{array}\right.
\]

Find the value of the expression \( xy + yz + xz \).


Each interior angle of an \( n \)-sided regular polygon is \( 160^{\circ} \). Find \( n \).

The \( n \)-th day of May in a year is Friday. The \( k \)-th day of May in the same year is Tuesday, where \( 20 < k < 26 \). Find \( k \).


Parabola \(\Pi_{1}\) with upward-facing branches passes through points with coordinates \((10,0)\) and \((13,0)\). Parabola \(\Pi_{2}\) with upward-facing branches also passes through the point with coordinates \((13,0)\). It is also known that the vertex of parabola \(\Pi_{1}\) bisects the segment connecting the origin and the vertex of parabola \(\Pi_{2}\). At what x-coordinate does parabola \(\Pi_{2}\) intersect the \(x\)-axis again?


For each positive integer \( n \) let \( S_{n} \) denote the set \( \{1,2,3, \ldots, n\} \). Compute the number of triples of subsets \( A, B, C \) of \( S_{2006} \) (not necessarily nonempty or proper) such that \( A \) is a subset of \( B \) and \( S_{2006}-A \) is a subset of \( C \).


In $\triangle ABC$, $AB = 33 \text{ cm}$, $AC = 21 \text{ cm}$ and $BC = x \text{ cm}$, where $x$ is an integer. $D$ is a point on segment $AB$ and $E$ is a point on segment $AC$ such that $AD = DE = EC = y \text{ cm}$, where $y$ is also an integer. Find $x$.


There are 40 ones written on the board. Each minute, Karlson erases two random numbers and writes their sum on the board, then eats an amount of candy equal to the product of the two erased numbers. What is the maximum number of candies he could have eaten in 40 minutes?


Quadrilateral \(ABCD\) with mutually perpendicular diagonals \(AC\) and \(BD\) is inscribed in a circle. Find its radius, given that \(AB = 4\) and \(CD = 2\).


Let \( a \oplus b = a b + 10 \). If \( C = (1 \oplus 2) \oplus 3 \), find the value of \( C \).


In how many ways can 10 athletes be divided into two teams of 5 people, given that two athletes wish to play on different teams?


An apple, pear, orange, and banana were placed in four boxes (one fruit in each box). Labels were made on the boxes:

1. An orange is here.
2. A pear is here.
3. If a banana is in the first box, then either an apple or a pear is here.
4. An apple is here.

It is known that none of the labels correspond to reality.

Determine which fruit is in which box. In the answer, record the box numbers sequentially, without spaces, where the apple, pear, orange, and banana are located, respectively (the answer should be a 4-digit number).


Let \(\left(u_{n}\right)\) be the sequence defined by \( u_{0}=5 \) and \( u_{n+1}=u_{n}+\frac{1}{u_{n}} \) for \( n \in \mathbb{N} \). Show that \( u_{1000}>45 \).

Solution: \boxed{u_{1000} > 45}


Determine all functions from \(\mathbb{R}\) to \(\mathbb{R}\) satisfying:
\[ \forall(x, y) \in \mathbb{R}^{2}, \; f(f(x) + 9y) = f(y) + 9x + 24y \]

Solution: \boxed{f(x) = 3x}


Calculate the length of the arc of the curve given by the polar equation

$$
\varrho = 6 \sin \varphi, \quad 0 \leq \varphi \leq \pi / 3
$$


What is the smallest positive integer divisible by 28 that ends with the digits 28 in decimal representation, and whose sum of digits is 28?


Calculate the area of the parallelogram formed by the vectors \(a\) and \(b\).

Given:

\[ a = 6p - q \]
\[ b = p + q \]
\[ |p| = 3 \]
\[ |q| = 4 \]
\[ (\widehat{p, q}) = \frac{\pi}{4} \]


How many natural numbers $N \leq 1000000$ exist such that $N$ is divisible by $\lfloor \sqrt{N} \rfloor$?


The numbers \(2^{0}, 2^{1}, \cdots, 2^{15}, 2^{16}=65536\) are written on a blackboard. You repeatedly take two numbers on the blackboard, subtract one from the other, erase them both, and write the result of the subtraction on the blackboard. What is the largest possible number that can remain on the blackboard when there is only one number left?


Set \( A \) is a subset consisting of 40 elements chosen from \(\{1, 2, 3, \ldots, 50\}\). Let \( S \) be the sum of all elements in set \( A \). Find the number of possible values for \( S \).


Is it possible to cut the figure shown in the picture into four equal parts along the grid lines so that these parts can be rearranged to form a square?

Solution: \boxed{Yes}


In triangle $\triangle P A T$, $\angle P=36^{\circ}$, $\angle A=56^{\circ}$, and $PA=10$. Points $U$ and $G$ are located on sides $TP$ and $TA$ respectively, such that $PU=1$ and $AG=1$. If $M$ and $N$ are the midpoints of $PA$ and $UG$ respectively, what is the acute angle between $MN$ and $PA$?

(A) $76^{\circ}$
(B) $77^{\circ}$
(C) $78^{\circ}$
(D) $79^{\circ}$
(E) $80^{\circ}$


A book contains 30 stories. Each story has a different number of pages under 31. The first story starts on page 1 and each story starts on a new page. What is the largest possible number of stories that can begin on odd page numbers?


The sides of a rectangle were reduced: the length by 10%, the width by 20%. In this case, the perimeter of the rectangle decreased by 12%. By what percentage will the perimeter of the rectangle decrease if its length is reduced by 20% and its width by 10%?


In the tetrahedron \(ABCD\), given that \(AB = 1\), \(CD = \sqrt{3}\), the distance between the lines \(AB\) and \(CD\) is 2, and the angle between them is \(\frac{\pi}{3}\), find the volume of the tetrahedron \(ABCD\).


There are six children in a family. Five of them are older than the youngest by 2, 6, 8, 12, and 14 years respectively. How old is the youngest if the ages of all children are prime numbers?


Among \( n \) knights, every pair of them is either friends or enemies. Each knight has exactly three enemies, and the enemies of their friends are also their enemies. For which \( n \) is this possible?

Solution: \boxed{n=4 \text{ or } n=6}


Seven fishermen stand in a circle. The fishermen have a professional habit of exaggerating numbers. Each fisherman has an exaggeration factor (unique integer) representing how many times the actual number is exaggerated. For example, if a fisherman with an exaggeration factor of 3 catches two fish, he will say he caught six fish. When asked, "How many fish did your left neighbor catch?", the responses (not necessarily in the order the fishermen are sitting) were $12, 12, 20, 24, 32, 42,$ and $56$. When asked, "How many fish did your right neighbor catch?", six of the fishermen responded $12, 14, 18, 32, 48,$ and $70$. What did the seventh fisherman respond?


Given $m$ points on a plane, where no three points are collinear, and their convex hull is an $n$-gon. Connecting the points appropriately can form a mesh region composed of triangles. Let $f(m, n)$ represent the number of non-overlapping triangles in this region. Find $f(2016, 30)$.


Find the sum of all real solutions to \( x^{2} + \cos x = 2019 \).


Let \( x_{1}, x_{2}, x_{3}, x_{4}, x_{5} \) be nonnegative real numbers whose sum is 300. Let \( M \) be the maximum of the four numbers \( x_{1} + x_{2}, x_{2} + x_{3}, x_{3} + x_{4}, \) and \( x_{4} + x_{5} \). Find the least possible value of \( M \).


Joey wrote a system of equations on a blackboard, where each of the equations was of the form \( a + b = c \) or \( a \cdot b = c \) for some variables or integers \( a, b, c \). Then Sean came to the board and erased all of the plus signs and multiplication signs, so that the board reads:

\[
\begin{array}{ll}
x & z = 15 \\
x & y = 12 \\
x & x = 36
\end{array}
\]

If \( x, y, z \) are integer solutions to the original system, find the sum of all possible values of \( 100x + 10y + z \).


The sequence $\left\{a_{n}\right\}$ is defined as follows:
$a_{1}=1$, and $a_{n+1}=a_{n}+\frac{1}{a_{n}}$ for $n \geq 1$.
Find the integer part of $a_{100}$.


In a quadrilateral pyramid \( S A B C D \):

- The lateral faces \( S A B, S B C, S C D, S D A \) have areas 9, 9, 27, and 27 respectively.
- The dihedral angles at the edges \( A B, B C, C D, D A \) are equal.
- The quadrilateral \( A B C D \) is inscribed in a circle, and its area is 36.

Find the volume of the pyramid \( S A B C D \).


In a convex quadrilateral \(ABCD\), \(\angle ABC = 90^\circ\), \(\angle BAC = \angle CAD\), \(AC = AD\), and \(DH\) is the altitude of triangle \(ACD\). In what ratio does the line \(BH\) divide the segment \(CD\)?


A truck and a car are moving in the same direction on adjacent lanes at speeds of 65 km/h and 85 km/h respectively. How far apart will they be 3 minutes after they are even with each other?

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


Given the circle \( O: x^{2}+y^{2}=4 \) and the curve \( C: y=3|x-t| \), and points \( A(m, n) \) and \( B(s, p) \) \((m, n, s, p \in \mathbb{N}^*) \) on the curve \( C \), such that the ratio of the distance from any point on the circle \( O \) to point \( A \) and to point \( B \) is a constant \( k (k>1) \), find the value of \( t \).


Karlson filled a conical glass with lemonade and drank half of it by height (measured from the surface of the liquid to the top of the cone), and the other half was finished by Kid. By what factor did Karlson drink more lemonade than Kid?


Show that:

(a) The number of unordered selections of size \( n \) with "selection with replacement" from a set \( A \) with \( |A| = N \) equals \( C_{N+n-1}^{n} \).

(b) The number of ordered vectors \(\left(n_{1}, \ldots, n_{N}\right)\) with non-negative integers \( n_{i} \) (\( i = 1, \ldots, N \)), satisfying the condition \( n_{1} + \ldots + n_{N} = n \), equals \( C_{N+n-1}^{n} \), where \( n \geqslant 1 \) and \( N \geqslant 1 \).

(c) The number of ways in which \( n \) indistinguishable items can be placed into \( N \) distinguishable bins (with no restrictions on the number of items in each bin) equals \( C_{N+n-1}^{n} \), where \( n \geqslant 1 \) and \( N \geqslant 1 \).


The most common dice is a six-sided die, which is a cube. The 6 faces have 1 to 6 points on them, with the sum of points on opposite faces equal to 7. From a single point in space, you can see multiple faces of the die. The minimum visible sum of points is 1, and the maximum is 15 (15 = 4 + 5 + 6). Determine which sums in the range from 1 to 15 are not possible to see.


Set \( S \) satisfies the following conditions:
1. The elements of \( S \) are positive integers not exceeding 100.
2. For any \( a, b \in S \) where \( a \neq b \), there exists \( c \in S \) different from \( a \) and \( b \) such that \(\gcd(a + b, c) = 1\).
3. For any \( a, b \in S \) where \( a \neq b \), there exists \( c \in S \) different from \( a \) and \( b \) such that \(\gcd(a + b, c) > 1\).

Determine the maximum value of \( |S| \).


Players are dividing chips. The first player takes $m$ chips and one-sixth of the remaining chips; the second player takes $2m$ chips and one-sixth of the new remaining chips; the third player takes $3m$ chips and one-sixth of the new remaining chips, and so on. It turns out that the chips were divided equally in this manner. How many players were there?


Let $n$ be a natural number with the following property: If 50 different numbers are randomly chosen from the numbers $1, 2, \ldots, n$, there will necessarily be two numbers among them whose difference is 7. Find the maximum value of such $n$.


Let $x, y, z$ be positive numbers satisfying the following system of equations:

$$
\left\{\begin{array}{l}
x^{2} + xy + y^{2} = 12 \\
y^{2} + yz + z^{2} = 9 \\
z^{2} + xz + x^{2} = 21
\end{array}\right.
$$

Find the value of the expression $xy + yz + xz$.


You are in a completely dark room with a drawer containing 10 red, 20 blue, 30 green, and 40 khaki socks. What is the smallest number of socks you must randomly pull out in order to be sure of having at least one of each color?


Find the smallest natural number that cannot be represented in the form \(\frac{2^{a} - 2^{b}}{2^{c} - 2^{d}}\), where \(a, b, c, d\) are natural numbers.


To the natural number \( N \), the largest divisor of \( N \) that is less than \( N \) was added, and the result is a power of ten. Find all such \( N \).


Find the largest three-digit number such that the number minus the sum of its digits is a perfect square.


A cuckoo clock produces a number of "cuckoo" sounds equal to the hour it indicates (for example, at 19:00, it sounds "cuckoo" 7 times). One morning, Maxim approaches the clock at 9:05 and starts turning the minute hand forward until the clock shows 7 hours later. How many "cuckoo" sounds are made during this time?


A six-digit number starts with 1. If we move this digit 1 from the first position to the last position on the right, we obtain a new six-digit number, which is three times the original number. What is this number?


In each of the 16 unit squares of a $4 \times 4$ grid, a + sign is written except in the second square of the first row. You can perform the following three operations:

- Change the sign of each square in a row.
- Change the sign of each square in a column.
- Change the sign of each square in a diagonal (not just the two main diagonals).

Is it possible to obtain a configuration in which every square contains a + sign using a finite number of these operations?

Solution: \boxed{\text{No}}


The non-zero numbers \( a, b, \) and \( c \) are such that the doubled roots of the quadratic polynomial \( x^{2}+a x+b \) are the roots of the polynomial \( x^{2}+b x+c \). What can the ratio \( a / c \) equal?


During the draw before the math marathon, the team captains were asked to name the smallest possible sum of the digits in the decimal representation of the number \( n+1 \), given that the sum of the digits of the number \( n \) is 2017. What answer did the captain of the winning team give?


Matvey decided to start eating healthily and each day ate one less bun and one more pear than the previous day. In total, during the period of healthy eating, he ate 264 buns and 187 pears. How many days did Matvey follow his healthy diet?


A carpenter wants to cut a wooden cube with a side length of 3 inches into 27 smaller cubes with a side length of 1 inch each. The question is: "What is the minimum number of cuts required if the pieces can be rearranged arbitrarily during the cutting process, given that the carpenter can make 6 cuts to split the cube into smaller cubes while keeping the pieces together so they don't fall apart?"

Given that without rearranging the pieces, the minimum number of cuts would be 6 because each face needs to be cut to separate the inner cube, what is the minimum number of cuts required if the inner cube is missing and pieces can be rearranged arbitrarily during the process?


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

\[ \left\{\begin{array}{l}
x^{2}+xy+y^{2}=75 \\
y^{2}+yz+z^{2}=4 \\
z^{2}+xz+x^{2}=79
\end{array}\right. \]

Find the value of the expression \( xy + yz + xz \).


In the equations of the line \(\frac{x}{2}=\frac{y}{-3}=\frac{z}{n}\), determine the parameter \(n\) so that this line intersects with the line \(\frac{x+1}{3}=\frac{y+5}{2}=\frac{z}{1}\), and find the point of intersection.

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


Given a tetrahedron \(ABCD\), a sphere with a radius of 1 is drawn through vertex \(D\). This sphere is tangent to the circumsphere of the tetrahedron at point \(D\) and also tangent to the face \(ABC\). Given that \(AD=2\sqrt{3}\), \(\angle BAD = \angle CAD = 45^\circ\), and \(\angle BAC = 60^\circ\), find the radius \(r\) of the circumsphere of the tetrahedron.


A magical checkered sheet of paper of size $2000 \times 70$, initially has all cells grey. A painter stands on a certain cell and paints it red. Every second, the painter makes two steps: one cell to the left and one cell down, and paints the cell he lands on red. If the painter is in the leftmost column and needs to step left, he teleports to the rightmost cell of the same row; if the painter is in the bottom row and needs to step down, he teleports to the top cell of the same column. After several moves, the painter returns to the cell where he started. How many cells have been painted red at this point?