A positive integer is called a "random number" if and only if:
(1) All digits are non-zero.
(2) The number is divisible by 11.
(3) The number is divisible by 12, and if the digits of the number are permuted in any way, the resulting number is still divisible by 12.
How many such ten-digit random numbers are there?


In how many ways can we place 8 digits equal to 1 and 8 digits equal to 0 on a 4x4 board such that the sums of the numbers written in each row and column are the same?


The base of a right parallelepiped is a rhombus. A plane passing through one of the sides of the lower base and the opposite side of the upper base forms an angle of $45^\circ$ with the base plane. The resulting cross-section has an area of $Q$. Determine the lateral surface area of the parallelepiped.


A construction team is building a railway tunnel. When $\frac{1}{3}$ of the task is completed, the team starts using new equipment, which increases the construction speed by $20\%$ but shortens the daily work time to $\frac{4}{5}$ of the original time for equipment maintenance. As a result, the project is completed in a total of 185 days. Based on these conditions, determine how many days it would take to complete the project without using the new equipment.


Find the number of numbers \( N \) from the set \(\{1, 2, \ldots, 2018\}\) for which positive solutions \( x \) exist for the equation

$$
x^{[x]} = N
$$

(where \([x]\) is the integer part of the real number \( x \), i.e., the largest integer not exceeding \( x \)).


A three-digit number is called a "concave number" if the digit in the tens place is smaller than both the digit in the hundreds place and the digit in the units place. For example, 504 and 746 are concave numbers. How many three-digit concave numbers are there if all the digits are distinct?


Solution: \boxed{\text{No}}


Seven old women are going to Rome. Each has seven mules, each mule carries seven bags, each bag contains seven loaves of bread, each loaf contains seven knives, each knife is in seven sheaths. How many total items are there?


Natural numbers \( a, b, c \) are chosen such that \( a < b < c \). It is also known that the system of equations \( 2x + y = 2021 \) and \( y = |x - a| + |x - b| + |x - c| \) has exactly one solution. Find the minimum possible value of \( c \).


Ten elves are sitting around a circular table, each with a basket of nuts. Each elf is asked, "How many nuts do your two neighbors have together?" and the answers, going around the circle, are 110, 120, 130, 140, 150, 160, 170, 180, 190, and 200. How many nuts does the elf who answered 160 have?


The polynomials \( p_n(x) \) are defined by \( p_0(x) = 0 \), \( p_1(x) = x \), \( p_{n+2}(x) = x p_{n+1}(x) + (1 - x) p_n(x) \). Find the real roots of each \( p_n(x) \).


As shown in the figure, the distance between two adjacent points in both the horizontal and vertical directions is $m$. If the area of quadrilateral $ABCD$ is 23, what is the area of pentagon $EFGHI$?


Let $[x]$ denote the greatest integer less than or equal to the real number $x$. Given:
$$
a_{k}=\sum_{i=k^{2}}^{(k+1)^{2}-1} \frac{1}{i} \quad (k=1,2, \cdots)
$$
Find:
$$
\sum_{k=1}^{n}\left(\left[\frac{1}{a_{k}}\right]+\left[\frac{1}{a_{k}}+\frac{1}{2}\right]\right) .
$$


Let \(\mathcal{A}\) denote the set of all polynomials in three variables \(x, y, z\) with integer coefficients. Let \(\mathcal{B}\) denote the subset of \(\mathcal{A}\) formed by all polynomials which can be expressed as
\[ (x + y + z) P(x, y, z) + (xy + yz + zx) Q(x, y, z) + xyz R(x, y, z) \]
with \(P, Q, R \in \mathcal{A}\). Find the smallest non-negative integer \(n\) such that \(x^{i} y^{j} z^{k} \in \mathcal{B}\) for all nonnegative integers \(i, j, k\) satisfying \(i + j + k \geqslant n\).


Point \( A \) lies on the line \( y = \frac{5}{12} x - 7 \), and point \( B \) lies on the parabola \( y = x^2 \). What is the minimum length of segment \( AB \)?


The numbers from 1 to 9 are divided into three groups of three numbers, and then the numbers in each group are multiplied. $A$ is the largest of the three products. What is the smallest possible value of $A$?


A plane \( P \) slices through a cube of side length 2 with a cross section in the shape of a regular hexagon. The inscribed sphere of the cube intersects \( P \) in a circle. What is the area of the region inside the regular hexagon but outside the circle?

Solution: \boxed{3 \sqrt{3} - \pi}


In an isosceles triangle \(ABC\) on the base \(AC\), a point \(M\) is taken such that \(AM = a\) and \(MC = b\). Circles are inscribed in triangles \(ABM\) and \(CBM\). Find the distance between the points of tangency of these circles with the segment \(BM\).


In the triangle \( ABC \), \(\angle B = 90^\circ\), \(\angle C = 20^\circ\), \( D \) and \( E \) are points on \( BC \) such that \(\angle ADC =140^\circ\) and \(\angle AEC =150^\circ\). Suppose \( AD=10 \). Find \( BD \cdot CE \).


Let \( S = \{1, 2, 3, \ldots, 20\} \) be the set of all positive integers from 1 to 20. Suppose that \( N \) is the smallest positive integer such that exactly eighteen numbers from \( S \) are factors of \( N \), and the only two numbers from \( S \) that are not factors of \( N \) are consecutive integers. Find the sum of the digits of \( N \).


Vasily invented a new operation on the set of positive numbers: \( a \star b = a^{\ln b} \). Find the logarithm of the number \(\frac{(a b) \star(a b)}{(a \star a)(b \star b)}\) to the base \( a \star b \).


If \( x, y \), and \( z \) are real numbers such that \( 2 x^{2} + y^{2} + z^{2} = 2 x - 4 y + 2 xz - 5 \), find the maximum possible value of \( x - y + z \).


Given the complex numbers \( Z_1 \) and \( Z_2 \) such that \( \left|Z_1\right| = 2 \) and \( \left|Z_2\right| = 3 \). If the angle between their corresponding vectors is \( 60^\circ \), find the value of \( \left| \frac{Z_1 + Z_2}{Z_1 - Z_2} \right| \).


At a worldwide meteorological conference, each participant sequentially announced the average monthly temperature in their hometown. All other participants recorded the product of the temperatures in the announcer's and their own cities at that moment. A total of 48 positive and 42 negative numbers were recorded. What is the least number of times a positive temperature could have been announced?


How many natural numbers are divisors of the number 1,000,000 and do not end in 0?


Let \(ABCD\) be a convex quadrilateral with all sides and diagonals having integer lengths. Given that \(\angle ABC = \angle ADC = 90^\circ\), \(AB = BD\), and \(CD = 41\), find the length of \(BC\).


Calculate \(\cos (\alpha + \beta)\) if \(\cos \alpha - \cos \beta = -\frac{3}{5}\) and \(\sin \alpha + \sin \beta = \frac{7}{4}\).


Among the integers from 1 to 1000, how many are divisible by 2 but not divisible by 3 or 5?


How many solutions in integers \( x \) and \( y \) does the inequality \( |y - x| + |3x - 2y| \leq a \) have? a) when \( a = 2 \); b) when \( a = 20 \)?


Calculate the sum \(\sum_{n=0}^{502}\left\lfloor\frac{305 n}{503}\right\rfloor\).


Show that in any triangle,

$$
\frac{\operatorname{ctg} \alpha+\operatorname{ctg} \beta}{\operatorname{tg} \alpha+\operatorname{tg} \beta}+\frac{\operatorname{ctg} \beta+\operatorname{ctg} \gamma}{\operatorname{tg} \beta+\operatorname{tg} \gamma}+\frac{\operatorname{ctg} \gamma+\operatorname{ctg} \alpha}{\operatorname{tg} \gamma+\operatorname{tg} \alpha}=1
$$


Let \( \triangle ABC \) have an inscribed circle \( \Gamma \). This inscribed circle is tangent to sides \( AC, BC, \) and \( AB \) at points \( Y, X, \) and \( Z \), respectively. Let \( \gamma_A \) be the A-excircle tangent to sides \( AC, BC, \) and \( AB \) at points \( K_C, T, \) and \( K_B \), respectively. Show that \( BK_A = CX \).

Solution: \boxed{BK_A = CX}


Rectangle \(ABCD\) is congruent to rectangle \(ABEF\), and \(D-AB-E\) forms a dihedral angle. \(M\) is the midpoint of \(AB\). \(FM\) makes an angle \(\theta\) with \(BD\), where \(\sin \theta = \frac{\sqrt{78}}{9}\). Find the value of \(\frac{AB}{BC}\).


Select 5 different numbers from $0,1,2,3,4,5,6,7,8,9$ to form a five-digit number such that this five-digit number is divisible by $3$, $5$, $7$, and $13$. What is the largest such five-digit number?


\(\cos^2 \varphi + \cos^2(\alpha - \varphi) - 2 \cos \alpha \cos \varphi \cos (\alpha - \varphi) = \sin^2 \alpha\)

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


In a castle, there are 16 identical square rooms arranged in a $4 \times 4$ square. Sixteen people—some knights and some liars—each occupy one room. Liars always lie, and knights always tell the truth. Each of these 16 people said, "At least one of my neighboring rooms houses a liar." Rooms are considered neighbors if they share a common wall. What is the maximum number of knights that could be among these 16 people?


Let \( d \) be the greatest common divisor of a natural number \( b \) and 175. If \( 176 \times (b - 11 \times d + 1) = 5 \times d + 1 \), then \( b = \) ________.


If the sum of interior angles of \( m \) regular \( n \)-sided polygons is divisible by 27, what is the minimum value of \( m + n \)?


Given positive real numbers \( x \) and \( y \) satisfying \( x^{2} + y^{2} + \frac{1}{x} + \frac{1}{y} = \frac{27}{4} \), find the minimum value of \( P = \frac{15}{x} - \frac{3}{4y} \).


Choose six out of the ten digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 to fill in the blanks below, so that the equation is true. Each blank is filled with a single digit, and no two digits are the same.

$\square + \square \square = \square \square \square$.

What is the largest possible three-digit number in the equation?


A polygon $\mathcal{P}$ is drawn on the 2D coordinate plane. Each side of $\mathcal{P}$ is either parallel to the $x$ axis or the $y$ axis (the vertices of $\mathcal{P}$ do not have to be lattice points). Given that the interior of $\mathcal{P}$ includes the interior of the circle \(x^{2}+y^{2}=2022\), find the minimum possible perimeter of $\mathcal{P}$.

Solution: \boxed{8 \sqrt{2022}}


According to legend, the poet Li Bai from the Tang Dynasty went to buy wine, singing as he walked down the street. Each time he passed a store, the amount of his wine doubled, and each time he saw a flower, he drank 2 cups. After encountering four stores and flowers in total, he still had 2 cups of wine left at the end. How many cups of wine did he originally have?


In a container, there is a mixture of equal masses of nitrogen $N_{2}$ and helium He under pressure $p$. The absolute temperature of the gas is doubled, and all nitrogen molecules dissociate into atoms. Find the pressure of the gas mixture at this temperature. The molar masses of the gases are $\mu_{\text{He}} = 4$ g/mol and $\mu_{N_{2}} = 28$ g/mol. Assume the gases are ideal.

Solution: \boxed{ \frac{9}{4} p }


There is a caravan with 100 camels, consisting of both one-humped and two-humped camels, with at least one of each kind. If you take any 62 camels, they will have at least half of the total number of humps in the caravan. Let \( N \) be the number of two-humped camels. How many possible values can \( N \) take within the range from 1 to 99?


The longer base of a trapezoid has a length of 24 cm. Find the length of the shorter base, given that the distance between the midpoints of the diagonals of the trapezoid is 4 cm.


Determine the maximal size of a set of positive integers with the following properties:

1. The integers consist of digits from the set {1,2,3,4,5,6}.
2. No digit occurs more than once in the same integer.
3. The digits in each integer are in increasing order.
4. Any two integers have at least one digit in common (possibly at different positions).
5. There is no digit which appears in all the integers.


Xiao Hong asked Da Bai: "Please help me calculate the result of $999 \quad 9 \times 999 \quad 9$ and determine how many zeros appear in it."
2019 nines times 2019 nines
Da Bai quickly wrote a program to compute it. Xiao Hong laughed and said: "You don't need to compute the exact result to know how many zeros there are. I'll tell you it's....."
After calculating, Da Bai found Xiao Hong's answer was indeed correct. Xiao Hong's answer is $\qquad$.


Given a triangle \( \triangle ABC \) with \(\angle B = 90^\circ\). The incircle touches sides \(BC\), \(CA\), and \(AB\) at points \(D\), \(E\), and \(F\) respectively. Line \(AD\) intersects the incircle at another point \(P\), and \(PF \perp PC\). Find the ratio of the side lengths of \(\triangle ABC\).


Determine all primes \( p \) such that
\[ 5^p + 4 \cdot p^4 \]
is a perfect square, i.e., the square of an integer.


On the sides \(AB, BC, CD,\) and \(DA\) of parallelogram \(ABCD\), points \(M, N, K,\) and \(L\) are taken, respectively, such that \(AM : MB = CK : KD = 1 : 2\) and \(BN : NC = DL : LA = 1 : 3\). Find the area of the quadrilateral whose vertices are the intersections of the segments \(AN, BK, CL,\) and \(DM\), given that the area of parallelogram \(ABCD\) is 1.


A function is given by
$$
f(x)=\ln (a x+b)+x^{2} \quad (a \neq 0).
$$
(1) If the tangent line to the curve $y=f(x)$ at the point $(1, f(1))$ is $y=x$, find the values of $a$ and $b$.

(2) If $f(x) \leqslant x^{2}+x$ always holds, find the maximum value of $ab$.


A roulette can land on any number from 0 to 2007 with equal probability. The roulette is spun repeatedly. Let $P_{k}$ be the probability that at some point the sum of the numbers that have appeared in all spins equals $k$. Which number is greater: $P_{2007}$ or $P_{2008}$?


Arrange the natural numbers from 1 to 7 in a row such that each number is either greater than all the numbers before it or less than all the numbers before it. For example, the arrangement 4356271 satisfies this condition: starting from the second position, 3 is less than the first number 4; 5 is greater than the first two numbers 4 and 3; 6 is greater than the first three numbers 4, 3, and 5; 2 is less than the first four numbers 4, 3, 5, and 6; 7 is greater than the first five numbers 4, 3, 5, 6, and 2; 1 is less than the first six numbers 4, 3, 5, 6, 2, and 7.

Determine the number of such arrangements where the number 7 does not occupy the fourth position.


A natural number \( n \) is such that the number \( 100n^2 \) has exactly 55 different natural divisors. How many natural divisors does the number \( 10n \) have?


The numbers \(a, b, c, d\) belong to the interval \([-7.5, 7.5]\). Find the maximum value of the expression \(a + 2b + c + 2d - ab - bc - cd - da\).


A society of \(n\) members is selecting one representative from among themselves.

a) In how many ways can an open vote occur if each member votes for one person (possibly themselves)?

b) Solve the same problem if the vote is secret, i.e., only the number of votes each candidate receives is considered, and it does not matter who voted for whom.


How many ordered pairs of integers \((m, n)\) where \(0 < m < n < 2008\) satisfy the equation \(2008^{2} + m^{2} = 2007^{2} + n^{2}?\)


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}}


Find all prime numbers \( p \) not exceeding 1000 such that \( 2p + 1 \) is a perfect power (i.e., there exist natural numbers \( m \) and \( n \geq 2 \) such that \( 2p + 1 = m^n \)).


Through the vertex \(C\) of the base of a regular triangular pyramid \(SABC\), a plane is drawn perpendicular to the lateral edge \(SA\). This plane forms an angle with the base plane, the cosine of which is \( \frac{2}{3} \). Find the cosine of the angle between two lateral faces.


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

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


Ria has three counters marked 1, 5, and 11. She wants to place them side-by-side to make a four-digit number. How many different four-digit numbers can she make?


Given \( f(x) \) is a function defined on \(\mathbf{R}\), for any \( x, y \in \mathbf{R} \), it always holds that
\[ f(x-f(y)) = f(f(y)) + x f(y) + f(x) - 1 .\]

Find \( f(x) \) and calculate the value of \( f(\sqrt{2014}) \).


A library spends 4500 yuan to buy 300 books of five types: "Zhuangzi," "Kongzi," "Mengzi," "Laozi," and "Sunzi." Their prices per book are 10 yuan, 20 yuan, 15 yuan, 30 yuan, and 12 yuan, respectively. The number of "Zhuangzi" books is the same as "Kongzi" books. The number of "Sunzi" books is 15 more than four times the number of "Laozi" books. How many "Sunzi" books are there in this batch?


Find the minimum value of the function \( f(x, y) = 6\left(x^{2} + y^{2}\right)(x + y) - 4\left(x^{2} + xy + y^{2}\right) - 3(x + y) + 5 \) in the region \( D = \{(x, y) \mid x > 0, y > 0\} \).


In a box, 10 smaller boxes are placed. Some of the boxes are empty, and some contain another 10 smaller boxes each. Out of all the boxes, exactly 6 contain smaller boxes. How many empty boxes are there?


In the triangular prism \( P-ABC \),
\[
\begin{array}{l}
\angle APB = \angle BPC = \angle CPA = 90^{\circ}, \\
PA = 4, \, PB = PC = 3.
\end{array}
\]

Find the minimum sum of the squares of the distances from any point on the base \( ABC \) to the three lateral faces.


Calculate the result of the expression:

\[ 2013 \times \frac{5.7 \times 4.2 + \frac{21}{5} \times 4.3}{\frac{14}{73} \times 15 + \frac{5}{73} \times 177 + 656} \]


Starting with the numbers \(x_{1}=2\) and \(y_{1}=1\), we define two sequences based on the following rules:

\[ x_{k}=2 x_{k-1}+3 y_{k-1}, \quad y_{k}=x_{k-1}+2 y_{k-1} \]

where \(k=2,3,4, \ldots\). Show that for the terms of the two sequences with the same index, the difference \(x_{k}^{2}-3 y_{k}^{2}\) is constant.


Let $ABC$ be a triangle, $M$ the midpoint of segment $[BC]$, and $E$ and $F$ two points on the rays $[AB)$ and $[AC)$ such that $2 \angle AEC = \angle AMC$ and $2 \angle AFB = \angle AMB$. Let $X$ and $Y$ be the intersections of the circumcircle of triangle $AEF$ with the line $(BC)$. Show that $XB = YC$.

Solution: \boxed{X B = Y C}


Find a formula for the sum of the squares of the numbers in the $n$th row of Pascal's triangle (i.e., the numbers $\binom{n}{0}$, $\binom{n}{1}$, ..., $\binom{n}{n}$).


Six people, A, B, C, D, E, and F are each assigned a unique number.

A says: "The six of our numbers exactly form an arithmetic sequence."
B says: "The smallest number in this arithmetic sequence is 2."
C says: "The sum of our six numbers is 42."
D says: "The sum of A, C, and E's numbers is twice the sum of B, D, and F's numbers."
E says: "The sum of B and F's numbers is twice A's number."

What is D's number? ( )

Solution: \boxed{D = 2}


Given the function \( f: \mathbf{R} \rightarrow \mathbf{R} \), for any real numbers \( x, y, z \), the inequality \(\frac{1}{3} f(x y) + \frac{1}{3} f(x z) - f(x) f(y z) \geq \frac{1}{9} \) always holds. Find the value of \(\sum_{i=1}^{100} [i f(i)]\), where \([x]\) represents the greatest integer less than or equal to \( x \).


The teacher wrote the number 1818 on the board. Vasya noticed that if you insert a multiplication sign between the hundreds and tens places, the value of the resulting expression is a perfect square (18 × 18 = 324 = 18²). What is the next four-digit number after 1818 that has the same property?


From "The Greek Anthology": Four springs emerge from the ground. The first fills a pool in one day, the second in 2 days, the third in 3 days, and the fourth in 4 days. In how much time will all four springs together fill the pool?


A triangular pyramid \( S-ABC \) has a base in the shape of an equilateral triangle with side lengths of 4. It is known that \( AS = BS = \sqrt{19} \) and \( CS = 3 \). Find the surface area of the circumscribed sphere of the triangular pyramid \( S-ABC \).


Last year, 10% of the net income from our school's ball was allocated to clubs for purchases, and the remaining part covered the rental cost of the sports field. This year, we cannot sell more tickets, and the rental cost remains the same, so increasing the share for the clubs can only be achieved by raising the ticket price. By what percentage should the ticket price be increased to make the clubs' share 20%?


Solution: \boxed{x = 1}


For what smallest positive value of \(a\) is the inequality \(\frac{\sqrt[3]{\sin ^{2} x} - \sqrt[3]{\cos ^{2} x}}{\sqrt[3]{\tan ^{2} x} - \sqrt[3]{\cot ^{2} x}} < \frac{a}{2}\) satisfied for all permissible \(x \in \left(\frac{3 \pi}{2}, 2 \pi\right)\)? Round the answer to two decimal places if necessary.


Let the product of the digits of a positive integer \( n \) be denoted by \( a(n) \). Find the positive integer solution to the equation \( n^2 - 17n + 56 = a(n) \).


If the price of a product increased from $R \$ 5.00$ to $R \$ 5.55$, what was the percentage increase?


Determine all pairs \((p, q)\) of positive integers such that \(p\) and \(q\) are prime, and \(p^{q-1} + q^{p-1}\) is the square of an integer.

Solution: \boxed{(2, 2)}


A deck of 54 cards is divided into several piles by a magician. An audience member writes a natural number on each card equal to the number of cards in that pile. The magician then shuffles the cards in a special way and redistributes them into several piles. The audience member again writes a natural number on each card equal to the number of cards in the new pile. This process continues. What is the minimum number of times this process needs to be performed so that the (unordered) array of numbers written on the cards becomes unique for each card?


The country Omega grows and consumes only vegetables and fruits. It is known that in 2014, 1200 tons of vegetables and 750 tons of fruits were grown in Omega. In 2015, 900 tons of vegetables and 900 tons of fruits were grown. During the year, the price of one ton of vegetables increased from 90,000 to 100,000 rubles, and the price of one ton of fruits decreased from 75,000 to 70,000 rubles. By what percentage (%) did the real GDP of this country change in 2015, if the base year in Omega is 2014? Round your answer to two decimal places. If the real GDP of the country decreased, put a minus sign in the answer, and if it increased, put a plus sign.


Several points were marked on a line. After that, a point was marked between each pair of neighboring points. This "densification" was repeated two more times (a total of 3 times). As a result, there were 113 points marked on the line. How many points were originally marked?


In the sequence $5, 8, 15, 18, 25, 28, \cdots, 2008, 2015$, how many numbers have a digit sum that is an even number? (For example, the digit sum of 138 is $1+3+8=12$)


In city $\mathrm{N}$, there are exactly three monuments. One day, a group of 42 tourists arrived in this city. Each tourist took no more than one photograph of each of the three monuments. It turned out that any two tourists together had photographs of all three monuments. What is the minimum number of photographs that all the tourists together could have taken?


The perimeter of triangle \(ABC\) is 1. A circle \(\omega\) touches side \(BC\), the extension of side \(AB\) at point \(P\), and the extension of side \(AC\) at point \(Q\). A line passing through the midpoints of \(AB\) and \(AC\) intersects the circumcircle of triangle \(APQ\) at points \(X\) and \(Y\). Find the length of segment \(XY\).


Determine the integers \( n \in \mathbb{N} \) such that \( 2^{n} + 1 \) is a square or a cube.

Solution: \boxed{n = 3}


Through the vertex \( A \) of rectangle \( ABCD \), a line \( \ell \) is drawn, as shown in the figure. From points \( B \) and \( D \), perpendiculars \( BX \) and \( DY \) are dropped to the line \( \ell \). Find the length of segment \( XY \), if it is known that \( BX = 4 \), \( DY = 10 \), and \( BC = 2 AB \).


Find the number of points on the plane \( xOy \) with natural coordinates \( (x, y) \) that lie on the parabola \( y=-\frac{x^{2}}{4}+9x+19 \).


There are three types of plants in the garden: sunflowers, lilies, and peonies.

1. There is only one day in a week when all three types of flowers bloom simultaneously.
2. No type of flower can bloom for three consecutive days.
3. In one week, the number of days when any two types of flowers do not bloom together will not exceed one day.
4. Sunflowers do not bloom on Tuesday, Thursday, and Sunday.
5. Lilies do not bloom on Thursday and Saturday.
6. Peonies do not bloom on Sunday.

On what day of the week do all three types of flowers bloom simultaneously? (Represent the days from Monday to Sunday as numbers 1 to 7.)


Let's introduce the notations: $Q$ - center of the first circle, $F, E, B$ - points of tangency of the circles and lines, $G$ - point of tangency of the circles, $AD$ - common tangent (see the figure). Draw the line $OC$ parallel to the line $FA$. Let $FA = a$, $OG = r$, and $AG = a \ (AG = FA$ as tangents drawn from point $A$ to the same circle). Since $QC = 4 - r, OQ = 4 + r, OA = 8 - r, OC = a$, from the right triangles $QOC$ and $AOG$, we obtain the system:

$$
\left\{\begin{array}{c}
(4 - r)^2 + a^2 = (4 + r)^2 \\
r^2 + a^2 = (8 - r)^2
\end{array}\right.
$$

By subtracting the first equation from the second, we get:

$$
4 \cdot(4 - 2r) = 12 \cdot(4 - 2r)
$$

From which it follows that $r = 2$.


If $\sin \alpha + \sin \beta = 1$ and $\cos \alpha + \cos \beta = 0$, then what is $\cos 2 \alpha + \cos 2 \beta$?


Farmers Ivanov, Petrov, Sidorov, Vasilev and Ermolaev own rectangular plots of land, the areas of which are indicated in the diagram (see the figure). Find the area of the common pasture.

| Ivanov
 24 ha | Forest | Ermolaev
 30 ha |
| :--- | :--- | :--- |
| Petrov
 28 ha | Common
 pasture | Lake |
| Empty | Sidorov
 10 ha | Vasilev
 20 ha |


Calculate the limit of the function:

$$\lim_{x \rightarrow 2\pi}\left(\cos x\right)^{\frac{\cot{2x}}{\sin{3x}}}$$


Simplify the expression:
$$
\frac{1-\cos 2 \alpha + \sin 2 \alpha}{1+\cos 2 \alpha + \sin 2 \alpha}
$$

Solution: \boxed{\tan \alpha}


Let $A B C D$ be a square and $E$ be the point on segment $[B D]$ such that $E B = A B$. Define point $F$ as the intersection of lines $(C E)$ and $(A D)$. Find the value of the angle $\widehat{F E A}$.


If \((2x + 4)^{2n} = \sum_{i=0}^{2n} a_i x^i\) (where \(n \in \mathbf{Z}_+\)), what is the remainder when \(\sum_{i=1}^{n} a_{2i}\) is divided by 3?


In how many ways can we place two rooks of different colors on a chessboard such that they do not attack each other?


Expanding the expression \((1+\sqrt{5})^{206}\) using the binomial theorem, we get terms of the form \(C_{206}^{k}(\sqrt{5})^{k}\). Find the value of \(k\) at which such a term takes on the maximum value.


Give an example of a number $x$ for which the equation $\sin 2017 x - \operatorname{tg} 2016 x = \cos 2015 x$ holds. Justify your answer.


The ratio of the sides of a rectangle is \(3:4\), and its diagonal is 9. Find the shorter side of the rectangle.


In trapezoid \(ABCD\) with bases \(AB\) and \(CD\), it holds that \(|AD| = |CD|\), \(|AB| = 2|CD|\), \(|BC| = 24 \text{ cm}\), and \(|AC| = 10 \text{ cm}\).

Calculate the area of trapezoid \(ABCD\).

Solution: \boxed{180 \text{ cm}^2}


A covered rectangular football field with a length of 90 m and a width of 60 m is being designed to be illuminated by four floodlights, each hanging from some point on the ceiling. Each floodlight illuminates a circle, with a radius equal to the height at which the floodlight is hanging. Determine the minimally possible height of the ceiling, such that the following conditions are met: every point on the football field is illuminated by at least one floodlight, and the height of the ceiling must be a multiple of 0.1 m (for example, 19.2 m, 26 m, 31.9 m, etc.).

Solution: \boxed{27.1 \text{ m}}


Let \( f:[0,1) \rightarrow \mathbb{R} \) be a function that satisfies the following condition: if
\[
x=\sum_{n=1}^{\infty} \frac{a_{n}}{10^{n}}=. a_{1} a_{2} a_{3} \ldots
\]
is the decimal expansion of \( x \) and there does not exist a positive integer \( k \) such that \( a_{n}=9 \) for all \( n \geq k \), then
\[
f(x)=\sum_{n=1}^{\infty} \frac{a_{n}}{10^{2 n}} .
\]
Determine \( f^{\prime}\left(\frac{1}{3}\right) \).


In triangle \( \triangle ABC \), \(\angle C = 90^\circ\). The angle bisectors of \(\angle A\) and \(\angle B\) intersect at point \(P\). \(PE \perp AB\) at point \(E\). Given that \(BC = 2\) and \(AC = 3\), find the value of \(AE \cdot EB\).


In the plane Cartesian coordinate system, the ellipse $\Gamma$: $\frac{x^{2}}{4}+\frac{y^{2}}{3}=1$, point $P$ is inside the ellipse $\Gamma$ and moves on the line $y=x$. Points $K$ and $L$ are on $\Gamma$, such that the directions of $\overrightarrow{P K}$ and $\overrightarrow{P L}$ are along the positive direction of the $x$-axis and $y$-axis respectively. Point $Q$ is such that $P K Q L$ forms a rectangle. Does there exist two points $A$ and $B$ in the plane such that when the rectangle $P K Q L$ changes, $|Q A|-|Q B|$ is a non-zero constant? If such points exist, find all possible lengths of line segment $A B$; if not, explain why.


Let $[a]$ represent the greatest integer less than or equal to $a$. Determine the largest positive integer solution to the equation $\left[\frac{x}{7}\right]=\left[\frac{x}{8}\right]+1$.


Given a triplet of real numbers $(a, b, c)$, we can perform the following action: choose $x$ and $y$ from the triplet and replace them with $(x-y) / \sqrt{2}$ and $(x+y) / \sqrt{2}$. Can we transform the initial triplet $(2, \sqrt{2}, 1 / \sqrt{2})$ into the final triplet $(1, \sqrt{2}, 1+\sqrt{2})$?

Solution: \boxed{\text{No}}


What is the largest number of white and black chips that can be placed on a chessboard so that on each horizontal and each vertical, the number of white chips is exactly twice the number of black chips?

Solution: \boxed{48 \text{ chips}}


Let \( ABCD \) be a parallelogram. Let \( E \) be the midpoint of \( AB \) and \( F \) be the midpoint of \( CD \). Points \( P \) and \( Q \) are on segments \( EF \) and \( CF \), respectively, such that \( A, P \), and \( Q \) are collinear. Given that \( EP = 5 \), \( PF = 3 \), and \( QF = 12 \), find \( CQ \).


For any $n \geq 3$ points $A_{1}$, $A_{2}, \ldots, A_{n}$ in the plane, none of which are collinear, let $\alpha$ be the smallest angle $\angle A_{i} A_{j} A_{k}$ formed by any triple of distinct points $A_{i}, A_{j}, A_{k}$. For each value of $n$, find the maximum value of $\alpha$. Determine the configurations of the points for which this value is attained.


A collection of \( n \) squares on the plane is called tri-connected if the following criteria are satisfied:
(i) All the squares are congruent.
(ii) If two squares have a point \( P \) in common, then \( P \) is a vertex of each of the squares.
(iii) Each square touches exactly three other squares.

How many positive integers \( n \) are there with \( 2018 \leq n \leq 3018 \), such that there exists a collection of \( n \) squares that is tri-connected?


Find the smallest three-digit number ABC that is divisible by the two-digit numbers AB and BC (the digit A cannot be 0, but the digit B can be; different letters do not necessarily represent different digits).


Assume we have a box of matches, each match being 1 unit in length. We are also given a cardboard square with side length \( n \) units. Divide the square into \( n^2 \) smaller square units with straight lines. Arrange the matches on the cardboard subject to the following conditions:
1. Each match must cover the side of one of the small squares.
2. Exactly two sides of each small square (including edge squares) must be covered by matches.
3. Matches cannot be placed on the edge of the cardboard.

For which values of \( n \) does this problem have a solution?

Solution: \boxed{n \text{ is even}}


A regular hexagonal truncated pyramid is inscribed in a sphere of radius \( R \). The plane of the lower base passes through the center of the sphere, and the lateral edge forms an angle of \( 60^{\circ} \) with the plane of the base. Determine the volume of the pyramid.

Solution: \boxed{\frac{21 R^3}{16}}


Boys were collecting apples. Each boy collected either 10 apples or 10% of the total number of apples collected, and there were both types of boys. What is the minimum number of boys that could have been?


The function \( f \) is defined on the natural numbers \( 1, 2, 3, \ldots \) by \( f(1)=1 \) and
\[
f(n)=\left\{\begin{array}{ll}
f\left(\frac{n}{10}\right) & \text{if } 10 \mid n \\
f(n-1)+1 & \text{otherwise}
\end{array}\right.
\]

(a) Calculate \( f(2019) \).
(b) Determine the maximum value of \( f(n) \) for \( n \leq 2019 \).
(c) A new function \( g \) is defined by \( g(1)=1 \) and
\[
g(n)=\left\{\begin{array}{ll}
g\left(\frac{n}{3}\right) & \text{if } 3 \mid n \\
g(n-1)+1 & \text{otherwise}
\end{array}\right.
\]

Determine the maximum value of \( g(n) \) for \( n \leq 100 \).


How many pairs of integers \( x, y \), between 1 and 1000, exist such that \( x^{2} + y^{2} \) is divisible by 7?


Given a natural number \( x = 9^n - 1 \), where \( n \) is a natural number. It is known that \( x \) has exactly three distinct prime divisors, one of which is 13. Find \( x \).


Given that \( a \) and \( b \) are integers, and \( a + b \) is a root of the equation
\[
x^{2} + ax + b = 0
\]
find the maximum possible value of \( b \).


Given a regular triangular pyramid \( P-ABC \) with a volume of \( 9 \sqrt{3} \), the dihedral angle between the face \( PAB \) and the base \( ABC \) is \( 60^\circ \). Point \( D \) is on the line segment \( AB \) such that \( AD = \frac{1}{6} AB \), point \( E \) is on the line segment \( AC \) such that \( AE = \frac{1}{6} AC \), and point \( F \) is the midpoint of \( PC \). The plane \( DEF \) intersects the line segment \( PB \) at point \( G \). Find the area of the quadrilateral \( DEFG \).


The probability of randomly selecting 5 different numbers from the set {1, 2, ..., 20} such that at least two of them are consecutive.


What is the value of the following infinite product?

$$
z=9 \cdot \sqrt[3]{9} \cdot \sqrt[9]{9} \ldots \sqrt[3^{n}]{9} \ldots
$$


A square can be divided into four congruent figures. If each of the congruent figures has an area of 1, what is the area of the square?


In triangle \( \triangle ABC \), \( AB = AC \), \( AD \) and \( BE \) are the angle bisectors of \( \angle A \) and \( \angle B \) respectively, and \( BE = 2 AD \). What is the measure of \( \angle BAC \)?


Given \( z \in \mathbf{C} \). If the equation with respect to \( x \):
$$
4 x^{2}-8 z x+4 i+3=0
$$
has real roots, then the minimum value of \( |z| \) is _______.


In February of a non-leap year, Kirill and Vova decided to eat ice cream according to the following rules:
1. If the day of the month was even and the day of the week was Wednesday or Thursday, they would each eat seven servings of ice cream.
2. If the day of the week was Monday or Tuesday and the day of the month was odd, they would each eat three servings of ice cream.
3. If the day of the week was Friday, the number of servings each of them ate would be equal to the day of the month.

On all other days and under other conditions, eating ice cream was prohibited. What is the maximum number of servings of ice cream that Vova could eat in February under these conditions?


On the island of knights and liars, each inhabitant was asked about each of the others: whether the person is a liar or a knight. A total of 42 answers of "knight" and 48 answers of "liar" were received. What is the maximum number of knights that could have been on the island? Justify your answer. (It is known that knights always tell the truth, and liars always lie.)


There are 20 rooms, some with lights on and some with lights off. The occupants of these rooms prefer to match the majority of the rooms. Starting from room one, if the majority of the remaining 19 rooms have their lights on, the occupant will turn the light on; otherwise, they will turn the light off. Initially, there are 10 rooms with lights on and 10 rooms with lights off, and the light in the first room is on. After everyone in these 20 rooms has had a turn, how many rooms will have their lights off?

A. 0
B. 10
C. 11
D. 20


For every natural number $n$, let $\mathbf{S}(n)$ denote the sum of the digits of $n$. Calculate $\mathbf{S}^{5}(2018^{2018^{2018}})$.


Let \( m \) and \( n \) be two positive integers that satisfy
$$
\frac{m}{n}=\frac{1}{10 \times 12}+\frac{1}{12 \times 14}+\frac{1}{14 \times 16}+\cdots+\frac{1}{2012 \times 2014}
$$
Find the smallest possible value of \( m+n \).


Each of the 10 people is either a knight, who always tells the truth, or a liar, who always lies. Each of them picked an integer. Then the first person said, "My number is greater than 1," the second said, "My number is greater than 2," and so on, with the tenth person saying, "My number is greater than 10." After that, all ten, speaking in some order, said one of the following statements: "My number is less than 1," "My number is less than 2," ..., "My number is less than 10" (each person said exactly one of these ten phrases). What is the maximum number of knights that could be among these 10 people?


Gavrila got on the train with a fully charged smartphone, and by the end of the trip, his smartphone was fully discharged. He spent half of the total time playing Tetris and the other half watching cartoons. It is known that the smartphone completely discharges in 3 hours of video playback or 5 hours of playing Tetris. What distance did Gavrila travel if the train moved at an average speed of 80 km/h for half the way and at an average speed of 60 km/h for the other half? Give the answer in kilometers, rounded to the nearest whole number if necessary.


Points \( M, N, \) and \( K \) are located on the lateral edges \( A A_{1}, B B_{1}, \) and \( C C_{1} \) of the triangular prism \( A B C A_{1} B_{1} C_{1} \) such that \( A M : A A_{1} = 1 : 2, B N : B B_{1} = 1 : 3, \) and \( C K : C C_{1} = 1 : 4 \). Point \( P \) belongs to the prism. Find the maximum possible volume of the pyramid \( M N K P \) if the volume of the prism is 16.


Rearrange the four digits of 2016 to form a four-digit perfect square; then this four-digit perfect square is $\qquad$ .


Through a point in the interior of a triangle \(ABC\), three lines are drawn, one parallel to each side. These lines divide the sides of the triangle into three regions each. Let \(a, b\), and \(c\) be the lengths of the sides opposite \(\angle A, \angle B\), and \(\angle C\), respectively, and let \(a', b'\), and \(c'\) be the lengths of the middle regions of the sides opposite \(\angle A, \angle B\), and \(\angle C\), respectively. Find the numerical value of \(a' / a + b' / b + c' / c\).


Let \(A \cup B \cup C = \{1, 2, 3, 4, 5, 6\}\), and \(A \cap B = \{1, 2\}\). Additionally, \(\{1, 2, 3, 4\} \subseteq B \cup C\). Determine the number of distinct groups \((A, B, C)\) that meet these conditions. (Note: The order of \(A\), \(B\), and \(C\) matters, so different orderings are considered different groups.)


At each vertex of a tetrahedron, there is an ant. At a given moment, each ant randomly chooses an edge and moves along it to the neighboring vertex. What is the probability that two ants will meet either midway on the edge or at the end of their journey?


What is the smallest natural number ending in 4 such that placing the last digit at the front of the number gives a number that is four times the original number?


Find all functions \( f: \mathbb{R} \longrightarrow \mathbb{R} \) such that \( f \) is monotone and there exists an \( n \in \mathbb{N} \) such that for all \( x \):

\[ f^{n}(x) = -x \]

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


Given a triangle \( \triangle ABC \) with \( \angle ABC = 80^\circ \), \( \angle ACB = 70^\circ \), and \( BC = 2 \). A perpendicular line is drawn from \( A \) to \( BC \), and another perpendicular line is drawn from \( B \) to \( AC \). The two perpendicular lines intersect at \( H \). Find the length of \( AH \).

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


Calculate \(\sec \frac{2 \pi}{9} + \sec \frac{4 \pi}{9} + \sec \frac{6 \pi}{9} + \sec \frac{8 \pi}{9}\).


There are 306 different cards with numbers \(3, 19, 3^{2}, 19^{2}, \ldots, 3^{153}, 19^{153}\) (each card has exactly one number, and each number appears exactly once). How many ways can you choose 2 cards such that the product of the numbers on the selected cards is a perfect square?


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


An inscribed circle is drawn inside isosceles trapezoid \(ABCD\) with \(AB = CD\). Let \(M\) be the point where the circle touches side \(CD\), \(K\) be the intersection point of the circle with segment \(AM\), and \(L\) be the intersection point of the circle with segment \(BM\). Calculate the value of \(\frac{AM}{AK} + \frac{BM}{BL}\).


Calculate the limit of the numerical sequence:

\[
\lim _{n \rightarrow \infty} \frac{(n+1)^{3}-(n-1)^{3}}{(n+1)^{2}+(n-1)^{2}}
\]


We will call a date diverse if its representation in the format DD/MM/YY (day-month-year) includes all digits from 0 to 5. How many diverse dates are there in the year 2013?


Find \( g(2021) \) if for any real numbers \( x, y \) the following equation holds:

\[ g(x-y) = g(x) + g(y) - 2022(x + y) \]


Plot the graph of the function \( \mathrm{y} = \sqrt{4 \sin ^{4} x - 2 \cos 2x + 3} + \sqrt{4 \cos ^{4} x + 2 \cos 2x + 3} \).


Find the curvature of the curve:
1) \( x=t^{2}, y=2 t^{3} \) at the point where \( t=1 \);
2) \( y=\cos 2 x \) at the point where \( x=\frac{\pi}{2} \).


For how many natural numbers \( n \) not exceeding 600 are the triples of numbers

\[ \left\lfloor \frac{n}{2} \right\rfloor, \left\lfloor \frac{n}{3} \right\rfloor, \left\lfloor \frac{n}{5} \right\rfloor \quad\text{and}\quad \left\lfloor \frac{n+1}{2} \right\rfloor, \left\lfloor \frac{n+1}{3} \right\rfloor, \left\lfloor \frac{n+1}{5} \right\rfloor \]

distinct? As always, \(\lfloor x \rfloor\) denotes the greatest integer less than or equal to \( x \).


If a zero is appended to the right of a given number and the result is subtracted from 143, the resulting value is three times the given number. What is the given number?


Find the smallest positive integer \( n \) such that in any 2-coloring of \( K_n \), there exist two monochromatic triangles of the same color that do not share any edge.
(1991 China National Training Team Problem)


In a mathematics competition consisting of three problems, A, B, and C, among the 39 participants, each person solved at least one problem. Among those who solved problem A, there are 5 more people who only solved A than those who solved A and any other problems. Among those who did not solve problem A, the number of people who solved problem B is twice the number of people who solved problem C. Additionally, the number of people who only solved problem A is equal to the combined number of people who only solved problem B and those who only solved problem C. What is the maximum number of people who solved problem A?


a) Consider a prime $p$ that divides $10^{n} + 1$ for some positive integer $n$. For example, $p = 7$ divides $10^{3} + 1$. By analyzing the main period of the decimal representation of $\frac{1}{p}$, verify that the number of times the digit $i$ appears is equal to the number of times the digit $9 - i$ appears for each $i \in \{0, 1, 2, \ldots, 9\}$.

b) Consider a prime number $p$ that does not divide 10 and suppose that the period of the decimal representation of $\frac{1}{p}$ is $2k$. Is it always possible to break the period into two numbers that sum up to $10^k - 1$? For example, the period of $\frac{1}{7}$ has a length of $6 = 2k$, as it is equal to 142857. See that $142 + 857 = 999 = 10^3 - 1 = 10^k - 1$.

c) Given

$$
x = \frac{1}{1998} + \frac{1}{19998} + \frac{1}{199998} + \ldots
$$

when we write $2x$ as a decimal number, what is the $59^{\text{th}}$ digit after the decimal point?


Given \(0 \leq x \leq \pi\), and
\[
3 \sin \frac{x}{2} = \sqrt{1 + \sin x} - \sqrt{1 - \sin x}
\]

Find \(\tan x\).


A square is divided into rectangles whose sides are parallel to the sides of the square. For each of these rectangles, the ratio of its shorter side to its longer side is calculated. Show that the sum of these ratios is at least 1.


On a board, the digit 1 is written $n$ times. An operation consists of choosing two numbers $a$ and $b$ written on the board, erasing them, and writing $\frac{a+b}{4}$ in their place. Show that the number written on the board after $n-1$ steps is greater than or equal to $\frac{1}{n}$.


In rectangle \(ABCD\), points \(E\) and \(F\) lie on sides \(AB\) and \(CD\) respectively such that both \(AF\) and \(CE\) are perpendicular to diagonal \(BD\). Given that \(BF\) and \(DE\) separate \(ABCD\) into three polygons with equal area, and that \(EF = 1\), find the length of \(BD\).


A cup is filled with a salt solution at a concentration of 15%. There are three iron balls in large, medium, and small sizes, with their volume ratio being 10:5:3. First, the small ball is placed into the cup of salt solution, causing 10% of the salt solution to overflow. After removing the small ball, the medium ball is placed into the cup and then removed. Next, the large ball is placed into the cup and then removed. Finally, pure water is added to the cup to fill it to the top. What is the final concentration of the salt solution in the cup?

Solution: \boxed{10 \%}


Find the total number of positive integers \( n \) not more than 2013 such that \( n^4 + 5n^2 + 9 \) is divisible by 5.


Find the exact value of \(\tan^{-1}\left(\frac{1}{2}\right) + \tan^{-1}\left(\frac{1}{5}\right) + \tan^{-1}\left(\frac{1}{8}\right)\).


Denote the circles around $A$ and $B$ in problem 1448 by $k_{1}$ and $k_{2}$, respectively, with radii $r_{1}$ and $r_{2}$. Let $k$ be a circle that is tangent to $k_{1}$ and $k_{2}$ from the inside, with radius $r_{3}$. What does the ratio $r_{3} / r_{1}$ approach when, fixing $A$ and $B$, the point $C$ approaches $A$, i.e., as $r_{1}$ approaches 0?


Let's calculate the mass of sodium carbonate:

$$
m\left(\mathrm{Na}_{2} \mathrm{CO}_{3}\right)=n \cdot M=0.125 \cdot 106=13.25 \text{ g}
$$

Next, let's determine the mass fraction of sodium carbonate in the initial solution:

$$
\omega\left(\mathrm{Na}_{2} \mathrm{CO}_{3}\right)=\frac{m\left(\mathrm{Na}_{2} \mathrm{CO}_{3}\right) \cdot 100\%}{m_{\mathrm{p}-\mathrm{pa}}\left(\mathrm{Na}_{2} \mathrm{CO}_{3}\right)}=\frac{13.25 \cdot 100\%}{132.5}=10\%
$$


Using 6 squares with a side length of 1 can form a shape with a side length of 1. To form a regular hexagon with a side length of 5, how many equilateral triangles with a side length of 1 are needed?


A regular tessellation is formed by identical regular \( m \)-polygons for some fixed \( m \). Find the sum of all possible values of \( m \).


Given the quadratic polynomial \( f(x) = a x^{2} - a x + 1 \), it is known that \( |f(x)| \leq 1 \) for \( 0 \leq x \leq 1 \). Find the maximum possible value of \( a \).


If a natural number \( n \) can be uniquely represented as the sum \( a_1 + a_2 + \cdots + a_k \) and the product \( a_1 a_2 \cdots a_k \) of \( k \) natural numbers (\( k \geq 2 \)), then this natural number is called a "good" number. For example, 10 is a "good" number because \( 10 = 5 + 2 + 1 + 1 + 1 = 5 \times 2 \times 1 \times 1 \times 1 \), and this representation is unique. Use the notion of prime numbers to determine which natural numbers are "good" numbers.

Solution: \boxed{n}


Solve the equation \(3^{x} + 4^{y} = 5^{z}\) for positive integers \(x, y,\) and \(z\).

Solution: \boxed{x = 2, y = 2, z = 2}


Jirka drew a square grid with 25 squares. He then wanted to color each square so that no two squares of the same color share a common vertex.

What is the minimum number of colors Jirka had to use?


Given two parallel lines at a distance of 15 units from each other, there is a point \( M \) that is 3 units away from one of them. A circle is drawn through the point \( M \) and tangent to both lines. Find the distance between the projections of the center and the point \( M \) onto one of the given lines.


Given that the interior angles \(A, B, C\) of triangle \(\triangle ABC\) are opposite to the sides \(a, b, c\) respectively, and that \(A - C = \frac{\pi}{2}\), and \(a, b, c\) form an arithmetic sequence, find the value of \(\cos B\).


Evaluate the limit as \( x \) approaches 1 of \( \frac{\sin 3 \pi x}{\sin 2 \pi x} \).


Given that \( p \) and \( q \) are real numbers with \( pq = b \) and \( p^{2}q + q^{2}p + p + q = 70 \). If \( c = p^{2} + q^{2} \), find the value of \( c \).


What is the common ratio of the geometric sequence \( a + \log_{2} 3, a + \log_{4} 3, a + \log_{8} 3 \)?


At the first site, higher-class equipment was used, while at the second site, first-class equipment was used, with higher-class being less than first-class. Initially, 30% of the equipment from the first site was transferred to the second site. Then, 10% of the equipment at the second site was transferred to the first site, with half of the transferred equipment being first-class. After this, the amount of higher-class equipment at the first site exceeded that at the second site by 6 units, and the total amount of equipment at the second site increased by more than 2% compared to the initial amount. Find the total amount of first-class equipment.


Find the number of subsets $\{a, b, c\}$ of $\{1,2,3,4, \ldots, 20\}$ such that $a<b-1<c-3$.


Two circles touch each other internally at point \( C \). A tangent is drawn at point \( P \) on the smaller circle, which intersects the larger circle at points \( A \) and \( B \). Show that \( CP \) intersects the arc \( AB \) of the larger circle at its midpoint \( Q \).

Solution: \boxed{Q}


Given the vectors $\boldsymbol{a}=(0,1)$, $\boldsymbol{b}=\left(-\frac{\sqrt{3}}{2},-\frac{1}{2}\right)$, and $\boldsymbol{c}=\left(\frac{\sqrt{3}}{2},-\frac{1}{2}\right)$. Real numbers $x, y, z$ satisfy $x \boldsymbol{a}+y \boldsymbol{b}+z \boldsymbol{c}=(1,2)$. Find the minimum value of $x^{2}+y^{2}+z^{2}$.


In a circle there are 101 numbers written. It is known that among any five consecutive numbers, there are at least two positive numbers. What is the minimum number of positive numbers that can be among these 101 written numbers?


The diagonal $BD$ of parallelogram $ABCD$ forms angles of $45^{\circ}$ with side $BC$ and with the altitude from vertex $D$ to side $AB$.

Find the angle $ACD$.


Let \( n > 0 \) be an integer. Each face of a regular tetrahedron is painted in one of \( n \) colors (the faces are not necessarily painted different colors.) Suppose there are \( n^3 \) possible colorings, where rotations, but not reflections, of the same coloring are considered the same. Find all possible values of \( n \).

Solution: \boxed{1 \text{ and } 11}


Find the positive real solution \((x, y, z)\) of the system of equations:
$$
\left\{\begin{array}{l}
2 x^{3}=2 y\left(x^{2}+1\right)-\left(z^{2}+1\right), \\
2 y^{4}=3 z\left(y^{2}+1\right)-2\left(x^{2}+1\right), \\
2 z^{5}=4 x\left(z^{2}+1\right)-3\left(y^{2}+1\right)
\end{array}\right.
$$

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


A flea named Kuzya can make a jump exactly 15 mm in any direction on a plane. Its task is to get from point $A$ to point $B$ on the plane, with the distance between them being 2020 cm. What is the minimum number of jumps it must make to accomplish this?


Anya calls a date beautiful if all 6 digits of its record are different. For example, 19.04.23 is a beautiful date, while 19.02.23 and 01.06.23 are not. How many beautiful dates are there in the year 2023?


Let the function \( f(x) = \frac{1}{2} + \log_{2} \frac{x}{1-x} \). Define \( S_n = \sum_{i=1}^{n-1} f\left(\frac{i}{n}\right) \), where \( n \in \mathbf{N}^{*} \) and \( n \geqslant 2 \). Determine \( S_n \).


All positive integer solutions $(a, b)$ of the equation $a^{3}-a^{2} b+a^{2}+2 a+2 b+1=0$ are $\qquad$ .

Solution: \boxed{(5, 7)}


What is the minimum force required to press a cube with a volume of $10 \, \text{cm}^{3}$, floating in water, so that it is completely submerged? The density of the material of the cube is $400 \, \text{kg/m}^{3}$, and the density of water is $1000 \, \text{kg/m}^{3}$. Give the answer in SI units. Assume the acceleration due to gravity is $10 \, \text{m/s}^{2}$.

Solution: \boxed{0.06 \, \text{N}}


The sum of the non-negative real numbers \( s_{1}, s_{2}, \ldots, s_{2004} \) is 2. Additionally, we know that

\[ s_{1} s_{2} + s_{2} s_{3} + \cdots + s_{2003} s_{2004} + s_{2004} s_{1} = 1 \]

Determine the minimum and maximum possible values of the expression

\[ S = s_{1}^2 + s_{2}^2 + \cdots + s_{2004}^2 \]

under these conditions.


Let the odd function \( f(x) \) have a domain of \( [-2,2] \), and be decreasing in the interval \( [-2,0] \), satisfying
$$
f(1-m)+f\left(1-m^{2}\right)<0.
$$
Determine the range of values for the real number \( m \).

Solution: \boxed{[-1, 1)}


Given a triangle \(ABC\). Point \(P\) is the center of the inscribed circle. Find the angle \(B\), given that \(R_{ABC} = R_{APC}\), where \(R_{ABC}\) and \(R_{APC}\) are the circumradii of triangles \(ABC\) and \(APC\) respectively.


In the interval $[0, \pi]$, how many solutions does the trigonometric equation $\cos 7x = \cos 5x$ have?


Roma thought of a natural number for which the sum of its digits is divisible by 8. He then added 2 to the number and again obtained a number whose sum of digits is divisible by 8. Find the smallest number that Roma could have thought of.


As shown in the figure, two ants, A and B, move on the following circumferences. \( AC \) is the diameter of the larger circle. Point \( B \) is on \( AC \), with \( AB \) and \( BC \) being the diameters of two smaller circles respectively. Ant A moves clockwise on the larger circle, while Ant B moves in a "8" pattern along the two smaller circles in the direction indicated by the arrow (A → B → C → B → A). Both ants start from point \( A \) at the same time and keep moving continuously. The speed ratio of Ant A to Ant B is \( v_{\text{A}} : v_{\text{B}} = 3 : 2 \). After \( T_1 \) minutes, the two ants meet. Then, Ant A increases its speed by \(\frac{1}{3}\) while Ant B's speed remains unchanged, and they continue to move along their original paths. After \( T_2 \) minutes, the two ants meet again. Given that \( T_1 + T_2 = 100^3 - 99^3 + 98^3 - 97^3 + \dots + 2^3 - 1^3 \), determine how many minutes it takes for Ant A to complete one round of the larger circle at its original speed (express the answer as an improper fraction).


Even natural numbers \( a \) and \( b \) are such that the greatest common divisor (GCD) of \( a \) and \( b \) plus the least common multiple (LCM) of \( a \) and \( b \) equals \( 2^{23} \). How many distinct values can the LCM of \( a \) and \( b \) take?


After expanding and combining like terms in the expression \((x+y+z)^{2028} + (x-y-z)^{2028}\), how many monomials of the form \(x^{a} y^{b} z^{c}\) have a non-zero coefficient?


How many four-digit numbers satisfy the following two conditions:
(1) The sum of any two adjacent digits is not greater than 2;
(2) The sum of any three adjacent digits is not less than 3.


Let \( ABCDEF \) be a regular hexagon. A frog starts at vertex \( A \) and can jump to one of the two adjacent vertices randomly on each jump. If the frog reaches point \( D \) within 5 jumps, it stops jumping. If the frog cannot reach \( D \) within 5 jumps, it stops jumping after completing 5 jumps. Determine the total number of different jumping sequences possible from the start until it stops.


The Minions need to make jam within the specified time. Kevin can finish the job 4 days earlier if he works alone, while Dave would finish 6 days late if he works alone. If Kevin and Dave work together for 4 days and then Dave completes the remaining work alone, the job is completed exactly on time. How many days would it take for Kevin and Dave to complete the job if they work together?


If digits in different positions are considered different, then in a base-$d$ numbering system, $nd$ digits allow writing $d^n$ numbers (from 0 to $d^n-1$). Which numbering system is the most economical in this respect, i.e., allows writing the most significant number of numbers using a given number of digits? (When comparing numbering systems with bases $d_1$ and $d_2$, we only consider sets of $m$ digits, where $m$ is divisible by both $d_1$ and $d_2$.)


Let \( n \) be a natural number with the following property: from the numbers \( 1, 2, 3, \cdots, n \), any selection of 51 different numbers must include two numbers whose sum is equal to 101. What is the maximum value of \( n \) with this property?


In parallelogram \(ABCD\), \(AB = 1\), \(BC = 4\), and \(\angle ABC = 60^\circ\). Suppose that \(AC\) is extended from \(A\) to a point \(E\) beyond \(C\) so that triangle \(ADE\) has the same area as the parallelogram. Find the length of \(DE\).


Given a fixed point \( A(2,0) \) on a plane and a moving point \( P\left(\sin \left(2 t-60^{\circ}\right), \cos \left(2 t-60^{\circ}\right)\right) \), determine the area of the figure swept by line segment \( AP \) as \( t \) changes from \( 15^{\circ} \) to \( 45^{\circ} \).


Find the value of the expression:

\[ \frac{\sin (2 \alpha + 2 \pi) + 2 \sin (4 \alpha - \pi) + \sin (6 \alpha + 4 \pi)}{\cos (6 \pi - 2 \alpha) + 2 \cos (4 \alpha - \pi) + \cos (6 \alpha - 4 \pi)}. \]

Solution: \boxed{\tan 4\alpha}


In a trapezoid with bases 3 and 4, find the length of the segment parallel to the bases that divides the area of the trapezoid in the ratio $5:2$, counting from the shorter base.


Compute the number of sequences of integers \((a_{1}, \ldots, a_{200})\) such that the following conditions hold:
- \(0 \leq a_{1}<a_{2}<\cdots<a_{200} \leq 202\).
- There exists a positive integer \(N\) with the following property: for every index \(i \in \{1, \ldots, 200\}\) there exists an index \(j \in \{1, \ldots, 200\}\) such that \(a_{i}+a_{j}-N\) is divisible by 203.


The area of a quadrilateral is \( S \). Find the area of a parallelogram whose sides are equal and parallel to the diagonals of the quadrilateral.


Buses. At the stop near Absent-Minded Scientist's house, buses of two routes stop: No. 152 and No. 251. Both go to the metro station. The interval between buses No. 152 is exactly 5 minutes, and the interval between buses No. 251 is exactly 7 minutes. The intervals are strictly observed, but these two routes are not coordinated with each other and their schedules do not depend on each other. At a completely random moment, the Absent-Minded Scientist arrives at the stop and boards the first bus that arrives to get to the metro. What is the probability that the Scientist will board bus No. 251?


Among the numbers greater than 2013, find the smallest even number \( N \) for which the fraction \(\frac{15N-7}{22N-5}\) is reducible.


Given the sets:
$$
\begin{array}{l}
U=\{1,2, \cdots, 8\}, \\
A=\{1,2, \cdots, 5\}, \\
B=\{4,5, \cdots, 8\} .
\end{array}
$$

Find the number of subsets of $U$ that are not subsets of $A$ and also not subsets of $B$.


A circle with its center on the line \( y = b \) intersects the parabola \( y = \frac{3}{4} x^{2} \) at least at three points; one of these points is the origin, and two of the remaining points lie on the line \( y = \frac{3}{4} x + b \). Find all values of \( b \) for which the described configuration is possible.


Find the area of the figure defined on the coordinate plane by the inequality \( 2(2-x) \geq \left| y - x^2 \right| + \left| y + x^2 \right| \).


The binomial expansion of $\left(\sqrt{x}+\frac{1}{2 \sqrt[4]{x}}\right)^{n}$ is arranged in descending powers of $x$. If the coefficients of the first three terms form an arithmetic sequence, then there are ___ terms in the expansion in which the exponent of $x$ is an integer.


Gosha entered a natural number into the calculator. Then, he performed the following operation three times: first, he extracted the square root, and then took the integer part of the resulting number. As a result, he obtained the number 1. What is the largest number that Gosha could have initially entered?

Note: The integer part of a number is the largest integer that does not exceed the given number.


Given that \(\cos \alpha + \cos \beta + \cos \gamma = \sqrt{\frac{1}{5}}\) and \(\sin \alpha + \sin \beta + \sin \gamma = \sqrt{\frac{4}{5}}\), find \(\cos (\alpha - \beta) + \cos (\beta - \gamma) + \cos (\gamma - \alpha)\).


Given \( f(x)=\frac{a^{x}}{a^{x}+\sqrt{a}} \) (where \( a \) is a positive constant), find the value of \( f\left(\frac{1}{1001}\right)+f\left(\frac{2}{1001}\right)+\cdots+ f\left(\frac{1000}{1001}\right) \).


Given the point \( P(x, y) \) satisfies
\[ y = \frac{2x + 2018}{x - 1} \quad (x > y, \; x, y \in \mathbf{Z}_+) , \]
find the number of such points \( P(x, y) \).


Let's accept without proof that for every natural number $n$, there exists a polynomial $P_{n}$ such that the identity $\sin(nx) = P_{n}(\cos x) \cdot \sin x$ holds. Show that $P_{n}(1)=n$.

Solution: \boxed{P_n(1) = n}


Calculate the limit of the function:

$$\lim _{x \rightarrow 1} \frac{\sqrt{x^{2}-x+1}-1}{\operatorname{tg} \pi x}$$


Inside the square $ABCD$, a point $M$ is taken such that $\angle MAB = 60^{\circ}$ and $\angle MCD = 15^{\circ}$. Find $\angle MBC$.


Natural numbers from 1 to 1982 are arranged in some order. A computer examines pairs of adjacent numbers from left to right (the first and second, the second and third, etc. up to the last pair) and swaps the numbers in the examined pair if the larger number is on the left. Then it examines all pairs, moving from the last pair to the first, swapping numbers in the pairs by the same rule. After this examination, the operator received information that the number in the hundredth position did not move both times. Find this number.


On the sides of triangle \(ABC\), points were marked: 10 on side \(AB\), 11 on side \(BC\), and 12 on side \(AC\). None of the vertices of the triangle were marked. How many triangles with vertices at the marked points exist?


Given \( a, b, c \) are all greater than 1, satisfying
$$
\left\{\begin{array}{l}
\lg a+\log_{b} c=3, \\
\lg b+\log_{a} c=4 .
\end{array}\right.
$$

Find the maximum value of \( \lg a \cdot \lg c \).


Evaluate: \( 3.416 \cos \frac{\pi}{33} \cdot \cos \frac{2 \pi}{33} \cdot \cos \frac{4 \pi}{33} \cdot \cos \frac{8 \pi}{33} \cdot \cos \frac{16 \pi}{33} \) and verify that it equals \( \frac{1}{32} \).


Find \(\log _{n}\left(\frac{1}{2}\right) \log _{n-1}\left(\frac{1}{3}\right) \cdots \log _{2}\left(\frac{1}{n}\right)\) in terms of \(n\).


In triangle \( ABC \), the angle bisectors \( BP \) and \( CT \) intersect at point \( O \). It is known that points \( A, P, O \), and \( T \) lie on the same circle. Find angle \( A \).


Find the volume of a regular triangular prism if the side of its base is \(a\) and the lateral surface area is equal to the sum of the areas of the bases.


A motorcyclist departed from point $A$ with an initial speed of 90 km/h, uniformly increasing the speed (i.e., the speed increases by the same amount over equal time intervals). After three hours, the motorcyclist arrived at point $B$, passing through $C$ along the way. Then, the motorcyclist turned around and, still uniformly increasing speed, set off back. Two hours later, he passed by point $C$ again at a speed of 110 km/h and continued on to $A$. Find the distance between points $A$ and $C$.

Solution: \boxed{76 \ \text{km}}


Calculate the degree of ionization using the formula:

$$
\alpha=\sqrt{ } K_{\mathrm{HCN}} \mathrm{C}
$$

Given values:

$$
\alpha_{\text {ion }}=\sqrt{ }\left(7,2 \cdot 10^{-10}\right) / 0,1=\sqrt{ } 7,2 \cdot 10^{-9}=8,5 \cdot 10^{-5}, \text{ or } 8,5 \cdot 10^{-5} \cdot 10^{2}=0,0085\%
$$

Alternatively, if the concentration of ions is known, you can calculate $\alpha$ as:

$$
\mathrm{C} \cdot \alpha=[\mathrm{H}^{+}]=[\mathrm{CN}^{-}], [\mathrm{H}^{+}]=[\mathrm{CN}^{-}]=8,5 \cdot 10^{-6} \text{ mol/L}
$$

Then:

$$
\alpha_{\text{ion }}=8,5 \cdot 10^{-6}, 0,1=8,5 \cdot 10^{-5} \text{ or } 8,5 \cdot 10^{-5} \cdot 10^{2}=0,0085\%
$$


Three students proposed different methods to tackle the problem: "For the inequality $x^{2}+25+\left|x^{3}-5 x^{2}\right| \geqslant a x$ to hold for all $x$ in the interval $[1,12]$, find the range of the real number $a$."

Each student suggested a different approach:
- Student A: "We only need the minimum value on the left-hand side of the inequality to be no smaller than the maximum value on the right-hand side."
- Student B: "Transform the inequality into a function involving variable $x$ on the left side and a constant on the right side, then find the extremum of the function."
- Student C: "Consider both sides of the inequality as functions of $x$ and plot their graphs."

Based on their proposed methods, determine the correct conclusion for the range of $a$.

Solution: \boxed{(-\infty, 10]}


Find the sum $\sin x+\sin y+\sin z$, given that $\sin x=\operatorname{tg} y$, $\sin y=\operatorname{tg} z$, and $\sin z=\operatorname{tg} x$.


Find all possible pairs of prime numbers \((p, q)\) such that the equation

\[ x^{4} + p x^{3} - q = 0 \]

has an integer root.

Solution: \boxed{(2, 3)}


8. On an island of knights, who always tell the truth, and liars, who always lie, a school was opened. All $2N$ students of different heights lined up in pairs (forming two columns). The first two people said: "I am taller than two: my partner and the person behind me." The last two said: "I am also taller than two: my partner and the person in front of me." Finally, all the others said: "And I am taller than three: my partner, the person in front of me, and the person behind me."

a) What is the maximum number of knights in the school?

b) Can only liars study in the school?

Solution: \boxed{\text{No}}


Calculate the limit of the numerical sequence:

\[
\lim_{{n \to \infty}} \frac{2n - \sin n}{\sqrt{n} - \sqrt[3]{n^3 - 7}}
\]


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

$$
r=\sin 3 \phi
$$


Given the real numbers \( a \geq b \geq c \geq d \) with \( a + b + c + d = 9 \) and \( a^2 + b^2 + c^2 + d^2 = 21 \), find the minimum possible value of \( \text{ab} - \text{cd} \).


If \( a \) and \( b \) are positive integers that can each be written as a sum of two squares, then \( ab \) is also a sum of two squares. Find the smallest positive integer \( c \) such that \( c = ab \), where \( a = x^3 + y^3 \) and \( b = x^3 + y^3 \) each have solutions in integers \( (x, y) \), but \( c = x^3 + y^3 \) does not.


Let $n$ be a given natural number. Find all continuous functions $f(x)$ such that
$$
(0) f(x) + (1) f\left(x^{2}\right) + (2) f\left(x^{2^{2}}\right) + \cdots + (n-1) f\left(x^{2^{n-1}}\right) + (n) f\left(x^{2^{n}}\right) = 0.
$$

Solution: \boxed{f(x) = 0}


Find all composite positive integers \( n \) such that all the divisors of \( n \) can be written in the form \( a^{r} + 1 \), where \( a \) and \( r \) are integers with \( a \geq 0 \) and \( r \geq 2 \).


A group of cows and horses are randomly divided into two equal rows. Each animal in one row is directly opposite an animal in the other row. If 75 of the animals are horses and the number of cows opposite cows is 10 more than the number of horses opposite horses, determine the total number of animals in the group.


On an island, there are only 50 knights, who always tell the truth, and 15 civilians, who can either tell the truth or lie. A forgetful professor, who came to the island to give a lecture, forgot what color hat he is wearing. What is the minimum number of local inhabitants that the professor needs to ask about the color of his hat in order to be certain about its color?


In 1900, a reader asked the following question in 1930: He knew a person who died at an age that was $\frac{1}{29}$ of the year of his birth. How old was this person in 1900?


Find the number of ordered integer pairs \((a, b)\) such that the equation \(x^{2} + a x + b = 167 y\) has integer solutions \((x, y)\), where \(1 \leq a, b \leq 2004\).


Let \( n \) be a given positive integer. Determine the minimum of the polynomial \( f(x) = x^{2n} + 2x^{2n-1} + 3x^{2n-2} + \cdots + (2n+1-k)x^k + \cdots + 2n x + (2n+1) \) defined for real numbers.


For what smallest natural number \( a \) are there exactly 50 perfect squares in the numerical interval \( (a, 3a) \)?


An urn contains nine balls, numbered from 1 to 9. José and Maria each simultaneously draw one ball from the urn. They form a two-digit number with the number on José's ball as the tens digit and the number on Maria's ball as the units digit. What is the probability that this number is even?


The sets \( A \) and \( B \) are bounded subsets of the real line. The set \( A \) is the union of \( n \) disjoint and open intervals, and the set \( B \) is the union of \( m \) disjoint and open intervals. What is the maximum number of disjoint intervals that the set \( A \cap B \) can consist of?

Solution: \boxed{n + m - 1}


In a tournament with 10 teams, each team faces each other exactly once. Additionally, there are no ties, and each team has a 50% chance of winning any match. What is the probability that, after counting the scores of the $\frac{10 \cdot 9}{2} = 45$ games, no two players have the same number of wins?


For an integer \( n \geq 0 \), let \( f(n) \) be the smallest possible value of \( |x+y| \), where \( x \) and \( y \) are integers such that \( 3x - 2y = n \). Evaluate \( f(0) + f(1) + f(2) + \cdots + f(2013) \).


A rectangular strip of length 16 was cut into two strips of lengths 9 and 7. These two strips were placed on a table as shown in the picture.

It is known that the area of the part of the table covered only by the left strip is 27, and the area of the part of the table covered only by the right strip is 18. Find the area of the part of the table covered by both strips.


$ABCD$ is a square of side length 1. $P$ is a point on $AC$, and the circumcircle of $\triangle BPC$ intersects $CD$ at $Q$. If the area of $\triangle CPQ$ is $\frac{6}{25}$, find $CQ$.


A cylinder of base radius 1 is cut into two equal parts along a plane passing through the center of the cylinder and tangent to the two base circles. Suppose that each piece's surface area is \( m \) times its volume. Find the greatest lower bound for all possible values of \( m \) as the height of the cylinder varies.


One tea bag can brew either two or three cups of tea. Mila and Tanya equally divided a box of tea bags. Mila brewed 57 cups of tea, and Tanya brewed 83 cups. How many tea bags could have been in the box?


The result of the expression \( 210 \times 6 - 52 \times 5 \) is \(\qquad\)


Let the random variables $\xi$ and $\eta$ denote the lifetimes of the blue and red light bulbs, respectively. The lifetime of the flashlight is equal to the minimum of these two values. Clearly, $\min (\xi, \eta) \leq \xi$. Let's move to the expected values: $\operatorname{E} \min (\xi, \eta) \leq \mathrm{E} \xi=2$. Therefore, the expected lifetime of the flashlight is no more than 2 years.


Consider the sequence \(a_1, a_2, a_3, \cdots, a_n, \cdots \) which satisfies \(a_1 = a_2 = 1\), \(a_3 = 2\), and for any natural number \(n\), \(a_n \cdot a_{n+1} \cdot a_{n+2} \neq 1\). Additionally, it holds that \( a_n \cdot a_{n+1} \cdot a_{n+2} \cdot a_{n+3} = a_n + a_{n+1} + a_{n+2} + a_{n+3} \). What is the value of \(a_1 + a_2 + \cdots + a_{100}\)?


Find the smallest natural number \( N \) that is divisible by \( p \), ends with \( p \), and has a digit sum equal to \( p \), given that \( p \) is a prime number and \( 2p+1 \) is a cube of a natural number.


Katya wrote a four-digit number on the board which was divisible by each of its digits without remainder (the number did not contain any zeros). Then she erased the first and last digits, leaving the number 89 on the board. What could have been the original number on the board? (Find all variants and show that there are no others.)


There are 4 bookstores that subscribed to a total of 400 "Mathematical and Physical World" magazines. Each bookstore subscribed to at least 98 and at most 101 magazines. How many different ways can they subscribe to these magazines?


Find the largest four-digit number that is divisible by 7 and consists of four different digits.


Divide the set \( M = \{1, 2, \ldots, 12\} \) of the first 12 positive integers into four subsets each containing three elements, such that in each subset, one number is the sum of the other two. Find the number of different ways to do this.


Divide 23 cards, each labeled with numbers 1 to 23, into three piles. It is known that the average numbers of the three piles are 13, 4, and 17, respectively. What is the minimum number of cards in the pile with an average number of 13?


The clock shows $00:00$, with both the hour and minute hands coinciding. Considering this coincidence as number 0, determine after what time interval (in minutes) they will coincide for the 21st time. If the answer is not an integer, round the result to the nearest hundredth.


A firecracker was thrown vertically upward with a speed of \(20 \text{ m/s}\). One second after the flight began, it exploded into two fragments of equal mass. The first fragment flew horizontally immediately after the explosion with a speed of 48 m/s. Find the magnitude of the speed of the second fragment (in m/s) immediately after the explosion. The acceleration due to gravity is \(10 \text{ m/s}^2\).


Find all functions \( f: \mathbf{R} \rightarrow \mathbf{R} \) such that for any real numbers \( x \) and \( y \), the following holds:
\[ f(f(x) + y) = 2x + f(f(y) - x). \]

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


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


In a fixed $3 \times 3$ grid, 4 identical chess pieces are to be placed into the cells. No more than one piece can be placed in any single cell. If each row and each column must contain at least one piece, how many different ways can this be done?


There are three batches of parts, each containing 30 parts. The number of standard parts in the first, second, and third batches is 20, 15, and 10, respectively. A part is randomly selected from a randomly chosen batch and it turns out to be standard. Then, a second part is randomly selected from the same batch, and it also turns out to be standard. Find the probability that the parts were drawn from the third batch.


Given that the positive integer \( m \) satisfies that \( m^{4} + 16m + 8 \) can be expressed as the product of two or more consecutive integers, determine the maximum value of \( m \).


To enter Ali Baba's cave, it is necessary to reset 28 counters, each of which is set to a natural number from 1 to 2017. In one move, treasure seekers are allowed to decrease the values of some counters by the same number, which can be changed from move to move. Indicate the minimum number of moves in which the treasure seekers will definitely reset the counters (regardless of the initial values) and enter the cave.


On the sides \( AB, BC \), and \( AC \) of triangle \( ABC \), points \( C_1, A_1 \) and \( B_1 \) are taken respectively, such that \( AC_1 : C_1B = BA_1 : A_1C = CB_1 : B_1A = 2 : 1 \).

Find the area of the triangle whose vertices are the pairwise intersections of segments \( AA_1, BB_1 \), and \( CC_1 \), given that the area of triangle \( ABC \) is 1.


In the figure, \(ABCD\) is a square with a side length of \(1 \text{ cm}\). \(E\) is the midpoint of the diagonal \(BD\), and \(F\) is the midpoint of the segment \(BE\). What is the area of the triangle \(\triangle BCF\)?

Solution: \boxed{0.125 \text{ cm}^2}


Find the minimum value of the expression $\cos (x+y)$, given that $\cos x + $\cos y = \frac{1}{3}$.


A real number \(x\) is chosen uniformly at random from the interval \([0, 1000]\). Find the probability that
\[
\left\lfloor\frac{\left\lfloor\frac{x}{2.5}\right\rfloor}{2.5}\right\rfloor = \left\lfloor\frac{x}{6.25}\right\rfloor.
\]


There are ten cards each of the digits "3", "4", and "5". Choose any 8 cards such that their sum is 27. At most, how many of these cards can be "3"?


In the dream market, a Sphinx offered a traveler seven illusions, two naps, and one nightmare for four dreams. To another traveler, the same Sphinx offered four illusions, four naps, and two nightmares for seven dreams. The Sphinx measures equally for all travelers.

How many illusions did one dream cost?


Is it possible to place 12 ships of size $1 \times 4$ in a $10 \times 10$ square (for a game like "battleship") such that the ships do not touch each other (even at the corners)?

Solution: \boxed{\text{No}}


Find the mass of a plate \(D\) with surface density \(\mu = \frac{x}{y^5}\), bounded by the curves

\[
\frac{x^2}{16} + y^2 = 1, \quad \frac{x^2}{16} + y^2 = 3, \quad y = \frac{x}{4}, \quad x = 0 \quad \left(y \geq \frac{x}{4}, x \geq 0\right)
\]


Let \( A \) be the sum of the digits of the number \( 16^{16} \), and \( B \) be the sum of the digits of the number \( A \). Find the sum of the digits of the number \( B \) without calculating \( 16^{16} \).


Given the sequence $\left\{A_{n}\right\}$ defined by
$$
A_{1}=1, \quad A_{2}=3, \quad A_{n}=4 A_{n-1}-A_{n-2},
$$
find the value of $\sum_{n=1}^{+\infty} \operatorname{arccot} 2 A_{n}^{2}$.

Solution: \boxed{ \frac{\pi}{6} }


Let \( S \) be a subset of \(\{1,2,3, \ldots, 199,200\}\). We say that \( S \) is pretty if, for every pair of elements \( a \) and \( b \) in \( S \), the number \( a - b \) is not a prime number. What is the maximum number of elements in a pretty subset of \(\{1,2,3, \ldots, 199,200\}\)?


We successively write all the numbers from 1 to one million. Then, we replace each number by the sum of its digits. We repeat this process until only single-digit numbers remain. Which digit appears the most frequently?


Kolya drew 10 line segments and marked all their intersection points in red. After counting the red points, he noticed the following property: on each segment, there are exactly three red points.
a) Provide an example of the arrangement of 10 line segments with this property.
b) What can be the maximum number of red points for 10 segments with this property?


Let \( a, b, c, x, y, \) and \( z \) be complex numbers such that
\[ a = \frac{b+c}{x-2}, \quad b = \frac{c+a}{y-2}, \quad c = \frac{a+b}{z-2}. \]

If \( x y + y z + z x = 67 \) and \( x + y + z = 2010 \), find the value of \( x y z \).


Let the function \( y = f(x) \) satisfy: for all \( x \in \mathbb{R} \), \( y = f(x) \geqslant 0 \), and \( f(x+1) = \sqrt{9 - f(x)^2} \). When \( x \in [0,1) \),
$$
f(x) = \begin{cases}
2^x, & 0 \leqslant x < \frac{1}{2}, \\
\log_{10} (x + 31), & \frac{1}{2} \leqslant x < 1
\end{cases}
$$

Find \( f(\sqrt{1000}) \).


We define a sequence \( u_n \) as follows: \( u_1 \) and \( u_2 \) are integers between 1 and 10000 (inclusive), and \( u_{k+1} \) is the smallest absolute value of the pairwise differences of the previous terms. Show that \( u_{21} = 0 \).

Solution: \boxed{u_{21} = 0}


As shown in the figure, \(BC = 3 BE\) and \(AC = 4 CD\). The area of triangle \(ABC\) is how many times the area of triangle \(ADE\)?


It is said that whoever pulls out the sword from the stone can rule England. King Arthur wanted to obtain the sword from the stone, so he sought advice from the prophet Merlin. To test King Arthur's sincerity, Merlin gave him a problem composed entirely of prime numbers:

\[ 5 \times A + 7 \times B = 7 \times 7 \]

Merlin said: "If you can tell me how much greater \( A \) is than \( B \), I will help you." After thinking about it, King Arthur gave Merlin the correct answer. This correct answer is ____.


Two natural numbers \( a \) and \( b \), their least common multiple is 60. Then, the number of possible values for the difference between these two natural numbers is _____.


As shown in Figure 1, in the regular triangular prism \(ABC-A_1B_1C_1\), \(D\) and \(E\) are points on the lateral edges \(BB_1\) and \(CC_1\), respectively. Given that \(EC = BC = 2BD\), find the measure of the dihedral angle between the section \(ADE\) and the base \(ABC\).


Find the number of 7-digit integers formed by some or all of the five digits, namely, 0, 1, 2, 3, and 4, such that these integers contain none of the three blocks 22, 33, and 44.


Let \( a, b, c \) be positive real numbers such that \( a + b + c = 2013 \). Find the maximum value of \( \sqrt{3a + 12} + \sqrt{3b + 12} + \sqrt{3c + 12} \).


In a convex quadrilateral \(ABCD\), side \(AB\) is equal to diagonal \(BD\), \(\angle A=65^\circ\), \(\angle B=80^\circ\), and \(\angle C=75^\circ\). What is \(\angle CAD\) (in degrees)?


A natural number \( A \) has exactly 100 different divisors (including 1 and \( A \) ). Find the product of these divisors.


Given that the real numbers \( x, y \) and \( z \) satisfy the condition \( x + y + z = 3 \), find the maximum value of \( f(x, y, z) = \sqrt{2x + 13} + \sqrt[3]{3y + 5} + \sqrt[4]{8z + 12} \).


Determine the smallest integer $n$ such that there exist $n$ real numbers $x_{1}, \ldots, x_{n}$ all belonging to the interval $(-1, 1)$ and for which

$$
x_{1}+\cdots+x_{n}=0 \quad \text{and} \quad x_{1}^{2}+\cdots+x_{n}^{2}=2020
$$


A natural number is written on the board. If its last digit (in the units place) is erased, the remaining non-zero number is divisible by 20. If the first digit is erased, the remaining number is divisible by 21. What is the smallest number that could be on the board if its second digit is not 0?


At the olympiad, gymnasium students, lyceum students, and regular school students gathered. Some of them stood in a circle. Gymnasium students always lie to regular school students, lyceum students lie to gymnasium students, and regular school students lie to lyceum students. In all other cases, the students tell the truth. Each person told their neighbor to the right, "I am a gymnasium student." How many regular school students were in this circle?


Count the number of permutations \( a_{1} a_{2} \ldots a_{7} \) of 1234567 with the longest decreasing subsequence of length at most two (i.e. there does not exist \( i < j < k \) such that \( a_{i} > a_{j} > a_{k} \)).


Find all values of \( n \) for which the sum

\[
1! + 2! + 3! + \cdots + n!
\]

is a perfect square, where \( n! = 1 \cdot 2 \cdot 3 \cdots n \).

Solution: \boxed{1 \text{ and } 3}


A group of $n$ friends takes $r$ distinct photos (no two photos have exactly the same people) each containing at least one person. Find the maximum $r$ such that each person appears at most once.

Solution: \boxed{r = n}


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

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

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

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


At a certain meeting, a total of \(12k\) people attended. Each person greeted exactly \(3k+6\) other people. For any two individuals, the number of people who greeted both of them is the same. How many people attended the meeting?


Petya wrote down a sequence of ten natural numbers as follows: he wrote the first two numbers randomly, and each subsequent number, starting from the third, was equal to the sum of the two preceding numbers. Find the fourth number if the seventh number is 42 and the ninth number is 110.


Given the sequence \(a_{n}=\sqrt{4+\frac{1}{n^{2}}}+\sqrt{4+\frac{2}{n^{2}}}+\cdots+\sqrt{4+\frac{n}{n^{2}}}-2 n\) where \(n\) is a positive integer, find the value of \(\lim _{n \rightarrow \infty} a_{n}\).


Let \( f: \mathbb{N}^{*} \rightarrow \mathbb{N}^{*} \) be a function that satisfies the following conditions:

1. \( f(1)=1 \)
2. \( f(2n)=f(n) \)
3. \( f(2n+1)=f(n)+1 \)

What is the greatest value of \( f(n) \) for \( 1 \leqslant n \leqslant 2018 \) ?


What is the maximum integer number of liters of water that can be heated to boiling temperature using the heat obtained from burning solid fuel, if during the first 5 minutes of burning the fuel produces 480 kJ, and during each subsequent five-minute period $25\%$ less than during the previous one. The initial temperature of the water is $20^{\circ} \mathrm{C}$, the boiling temperature is $100^{\circ} \mathrm{C}$, and the specific heat capacity of water is 4.2 kJ.


In a $6 \times 6$ toy board, each cell represents a light button. When someone presses a button, it lights up if it is off and turns off if it is on. Additionally, all buttons that share a side with the pressed button also change their state: from on to off or from off to on. Starting with all buttons off and pressing each button on the board exactly once, one at a time and in any order, how many buttons will be lit at the end?


The distance between points $A$ and $B$ is 500 kilometers. Two cyclists, Alpha and Beta, start from point $A$ to point $B$ at the same time. Alpha cycles 30 kilometers per day, and Beta cycles 50 kilometers per day but rests every other day. At the end of the $\qquad$ day, the remaining distance Beta has to travel to point $B$ is twice the remaining distance Alpha has to travel.


The roots of the quadratic equation \( ax^2 + bx + c = 0 \), which has integer coefficients, are distinct and both lie between 0 and 1. Show that if \( a \) is positive, it is at least 5.


In a class of 50 students, each student has either a set square or a ruler. 28 students have rulers. Among those who have set squares, 14 are boys. Given that there are 31 girls in the class, how many girls have rulers?


A rectangle with a perimeter of 100 cm is divided into 88 identical smaller rectangles by seven vertical and ten horizontal cuts. What is the perimeter of each? Additionally, the total length of all cuts is 434 cm.


Let \( S_{n} \) be the sum of the elements of all 3-element subsets of the set \( A=\left\{1, \frac{1}{2}, \frac{1}{4}, \cdots, \frac{1}{2^{n}1}\right\} \). Evaluate \(\lim _{n \rightarrow \infty} \frac{S_{n}}{n^{2}}\).


Given a positive integer \( n \) less than 2006, such that
\[
\left\lfloor \frac{n}{3} \right\rfloor + \left\lfloor \frac{n}{6} \right\rfloor = \frac{n}{2}.
\]
Determine how many such \( n \) there are.


Let \( F(x) = |f(x) \cdot g(x)| \), where \( f(x) = a x^{2} + b x + c \) and \( g(x) = c x^{2} + b x + a \), with \( x \in [-1, 1] \). For any \( a, b, c \) such that \( |f(x)| \leq 1 \), find the maximum value of \( F(x) \) as \( a, b, \) and \( c \) vary.


It is known that the numbers EGGPLANT and TOAD are divisible by 3. What is the remainder when the number CLAN is divided by 3? (Letters denote digits, identical letters represent identical digits, different letters represent different digits).


What is the maximum number of handshakes that could have occurred in a group of 20 people, given that no matter which three people are chosen from the group, there will always be at least two who have not shaken hands?


How many right-angled triangles can Delia make by joining three vertices of a regular polygon with 18 sides?


If \( x \) and \( y \) are integers that satisfy the equation \( y^{2} + 3x^{2}y^{2} = 30x^{2} + 517 \), find the value of \( 3x^{2}y^{2} \).


In triangle \( \triangle ABC \), the side lengths are \( AC = 6 \), \( BC = 9 \), and \( \angle C = 120^\circ \). Find the length of the angle bisector \( CD \) of \( \angle C \).


Find all positive integers $k$ such that for any positive numbers $a, b, c$ that satisfy the inequality $k(a b+b c+c a)>5\left(a^{2}+b^{2}+c^{2}\right)$, there always exists a triangle with side lengths $a, b, c$.


12 Smurfs are sitting around a circular table, and each Smurf dislikes the 2 Smurfs adjacent to him but does not dislike the other 9 Smurfs. Papa Smurf wants to form a team of 5 Smurfs to rescue Smurfette, who was captured by Gargamel. The team must be made up of Smurfs who do not dislike one another. How many ways are there to form such a team?


In city $\mathrm{N}$, there are exactly three monuments. One day, a group of 42 tourists arrived in this city. Each tourist took no more than one photograph of each of the three monuments. It turned out that any two tourists together had photographs of all three monuments. What is the minimum number of photographs that all the tourists together could have taken?


A teacher wrote down three positive integers on the whiteboard: 1125, 2925, and \( N \), and asked her class to compute the least common multiple of the three numbers. One student misread 1125 as 1725 and computed the least common multiple of 1725, 2925, and \( N \) instead. The answer he obtained was the same as the correct answer. Find the least possible value of \( N \).


In an airspace, there are clouds. It turned out that the space can be divided into parts by ten planes so that each part contains no more than one cloud. Through how many clouds could an airplane fly at most while following a straight course?


Find the smallest natural number that is greater than the sum of its digits by 1755.


Find the sum of the digits of the product $\left(1+\frac{1}{2}\right)\left(1+\frac{1}{3}\right)\left(1+\frac{1}{4}\right)\cdots\left(1+\frac{1}{2006}\right)\left(1+\frac{1}{2007}\right)$.


The teacher wrote the number 1818 on the board. Vasya noticed that if you insert a multiplication sign between the hundreds and tens places, the value of the resulting expression is a perfect square (18 × 18 = 324 = 18²). What is the next four-digit number after 1818 that has the same property?


Find the radius of the circumscribed circle around an isosceles trapezoid with bases of lengths 2 cm and 14 cm, and a lateral side of 10 cm.


In a right triangle \(PQR\), the leg \(PQ\) is equal to 12. The length of the angle bisector \(QS\) is \(6\sqrt{5}\). Find the hypotenuse \(QR\).


Given five points inside an equilateral triangle of area 1, show that one can find three equilateral triangles with total area at most 0.64 and sides parallel to the original triangle, so that each of the five points is inside one or more of the new triangles.


Given \( a_{1}, a_{2}, \cdots, a_{n} \) are real numbers such that \( a_{1} + a_{2} + \cdots + a_{n} = k \). Find the minimum value of \( a_{1}^{2} + a_{2}^{2} + \cdots + a_{n}^{2} \).


Two bikers, Bill and Sal, simultaneously set off from one end of a straight road. Neither biker moves at a constant rate, but each continues biking until he reaches one end of the road, at which he instantaneously turns around. When they meet at the opposite end from where they started, Bill has traveled the length of the road eleven times and Sal seven times. Find the number of times the bikers passed each other moving in opposite directions.


Martin, Tomáš and Jirka guessed the result of the following expression:

$$
2.4 - 1.5 \cdot 3.6 \div 2 =
$$

Their average guess differed from the correct result by 0.4. Tomáš guessed $-1.2$ and Jirka guessed 1.7. Find the number that Martin guessed, knowing that the worst guess differed from the correct result by 2.


The number \(a\) is a root of the quadratic equation \(x^{2} - x - 50 = 0\). Find the value of \(a^{3} - 51a\).


If \( a \) is the smallest prime number which can divide the sum \( 3^{11} + 5^{13} \), find the value of \( a \).


Find the sum of the squares of the real roots of the equation \(2x^{4} - 3x^{3} + 7x^{2} - 9x + 3 = 0\).


The probability of event $A$ occurring in each trial is 1/2. Using Chebyshev's inequality, estimate the probability that the number $X$ of occurrences of event $A$ lies between 40 and 60, if 100 independent trials are conducted.


The integers \( x, y, z \), and \( p \) satisfy the equation \( x^2 + xz - xy - yz = -p \). Given that \( p \) is prime, what is the value of \( |y+z| \) in terms of \( p \) ? Your answer should refer to the variable \( p \) so that it works for every prime number \( p \).


A tangent and a secant drawn from the same point to a circle are mutually perpendicular. The length of the tangent is 12, and the internal segment of the secant is 10. Find the radius of the circle.


Find the smallest positive integer \( a \) such that the equation \( ax^2 - bx + c = 0 \) has two distinct roots in the interval \( 0 < x < 1 \) for some integers \( b \) and \( c \).


Determine the smallest natural number \( n \) such that \( n! \) (n factorial) ends with exactly 1987 zeros.


Given a positive integer $n \geq 3$, if an $n$-vertex simple graph $G(V, E)$ is connected such that removing all the edges of any cycle in $G$ makes $G$ disconnected, find the maximum possible value of $|E(G)|$.

Solution: \boxed{2n - 3}


A scatterbrained scientist in his laboratory has developed a unicellular organism, which has a probability of 0.6 of dividing into two identical organisms and a probability of 0.4 of dying without leaving any offspring. Find the probability that after some time the scatterbrained scientist will have no such organisms left.


In a given circle with radius \( R \), draw a regular pentagon \( ABCDE \). The perpendicular from vertex \( A \) to the opposite side \( DC \) intersects the circumference of the circle at \( A' \). Show that

$$
\overline{A^{\prime} B} - \overline{A^{\prime} C} = R.
$$

Solution: \boxed{A'B - A'C = R}


Find the set of pairs of real numbers \((x, y)\) that satisfy the conditions:

$$
\left\{
\begin{array}{l}
3^{-x} y^{4}-2 y^{2}+3^{x} \leq 0 \\
27^{x}+y^{4}-3^{x}-1=0
\end{array}
\right.
$$

Compute the values of the expression \(x_{k}^{3}+y_{k}^{3}\) for each solution \((x_{k}, y_{k})\) of the system and find the minimum among them. In the answer, specify the found minimum value, if necessary rounding it to two decimal places. If the original system has no solutions, write the digit 0 in the answer field.


In an isosceles trapezoid, the midline is equal to \( a \), and the diagonals are perpendicular to each other. Find the area of the trapezoid.


Given that \( p(x) \) is a monic polynomial of degree 2003 with integer coefficients, show that the polynomial \( p(x)^2 - 25 \) cannot have more than 2003 distinct integer roots.


Find the maximum constant $k$ such that $\frac{k a b c}{a+b+c} \leqslant (a+b)^{2} + (a+b+4c)^{2}$ holds for all positive real numbers $a, b, c$.

Solution: \boxed{k_{\max} = 100}


A seven-digit number \(\overline{m0A0B9C}\) is a multiple of 33. We denote the number of such seven-digit numbers as \(a_{m}\). For example, \(a_{5}\) represents the count of seven-digit numbers of the form \(\overline{50A0B9C}\) that are multiples of 33. Find the value of \(a_{2}-a_{3}\).


For how many remaining years this century (after 2025 and up to and including 2099) will the highest common factor of the first two digits and the last two digits be equal to one?


Let \(a_{1}, a_{2}, \ldots, a_{n}\) be a sequence of distinct positive integers such that \(a_{1} + a_{2} + \cdots + a_{n} = 2021\) and \(a_{1} a_{2} \cdots a_{n}\) is maximized. If \(M = a_{1} a_{2} \cdots a_{n}\), compute the largest positive integer \(k\) such that \(2^{k} \mid M\).


On an island, there are 10 people. Some of them are truth-tellers and others are liars. Each person has thought of an integer. The first person says, "My number is greater than 1." The second person says, "My number is greater than 2." ... The tenth person says, "My number is greater than 10."

Afterwards, these ten people stand in a line in some order and each says one statement in sequence: "My number is less than 1," "My number is less than 2," ... "My number is less than 10" (each person makes exactly one statement).

Question: What is the maximum number of truth-tellers that could be among these people?


Calculate the mass of silver nitrate:
\[ m\left(\mathrm{AgNO}_{3}\right) = n \cdot M = 0.12 \cdot 170 = 20.4 \text{ g} \]

Determine the mass fraction of potassium sulfate in the initial solution:
\[ \omega\left(\mathrm{AgNO}_{3}\right) = \frac{m\left(\mathrm{AgNO}_{3}\right) \cdot 100\%}{m_{\mathrm{p}-\mathrm{pa}}\left(\mathrm{AgNO}_{3}\right)} = \frac{20.4 \cdot 100\%}{255} = 8\% \]


Two distinct natural numbers end in 5 zeros and have exactly 42 divisors. Find their sum.


Calculate the angle between the bisectors of the coordinate angles $x O y$ and $y O z$.


A general gathers his troops. When he arranges them in groups of 2, one soldier is left over. When he arranges them in groups of 3, two soldiers are left over. When he arranges them in groups of 5, three soldiers are left over. If the general arranges his soldiers in groups of 30, how many soldiers will be left over?


The real polynomial \( p(x) \) is such that for any real polynomial \( q(x) \), we have \( p(q(x)) = q(p(x)) \). Find \( p(x) \).

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


The set of all positive integers can be divided into two disjoint subsets of positive integers \(\{f(1), f(2), \cdots, f(n), \cdots\}, \{g(1), g(2), \cdots, g(n), \cdots\}\), where \(f(1)<f(2)<\cdots<f(n)<\cdots\) and \(g(1)< g(2)<\cdots<g(n)<\cdots\) and it is given that \(g(n)=f[f(n)]+1\) for \(n \geq 1\). Find \(f(240)\).


In the country of Anchuria, a unified state exam takes place. The probability of guessing the correct answer to each question on the exam is 0.25. In 2011, to receive a certificate, one needed to answer correctly 3 questions out of 20. In 2012, the School Management of Anchuria decided that 3 questions were too few. Now, one needs to correctly answer 6 questions out of 40. The question is, if one knows nothing and simply guesses the answers, in which year is the probability of receiving an Anchurian certificate higher - in 2011 or in 2012?


In a certain country, there are 47 cities. Each city has a bus station from which buses travel to other cities in the country and possibly abroad. A traveler studied the schedule and determined the number of internal bus routes originating from each city. It turned out that if we do not consider the city of Ozerny, then for each of the remaining 46 cities, the number of internal routes originating from it differs from the number of routes originating from other cities. Find out how many cities in the country have direct bus connections with the city of Ozerny.

The number of internal bus routes for a given city is the number of cities in the country that can be reached from that city by a direct bus without transfers. Routes are symmetric: if you can travel by bus from city $A$ to city $B$, you can also travel by bus from city $B$ to city $A$.


Find the number of three-digit numbers which are multiples of 3 and are formed by the digits 0, 1, 2, 3, 4, 5, 6, and 7 without repetition.