If the odd function \( y=f(x) \) defined on \( \mathbf{R} \) is symmetrical about the line \( x=1 \), and when \( 0 < x \leqslant 1 \), \( f(x)=\log_{3}x \), find the sum of all real roots of the equation \( f(x)=-\frac{1}{3}+f(0) \) in the interval \( (0,10) \).


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

\[ f(x+y) + y \leq f(f(f(x))) \]

holds for all \( x, y \in \mathbb{R} \).

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


If for a natural number \( n (\geqslant 2) \), there exist integers \( a_{1}, a_{2}, \cdots, a_{n} \) such that
\[
a_{1}+a_{2}+\cdots+a_{n}=a_{1} \cdot a_{2} \cdot \cdots \cdot a_{n}=1990,
\]
find the minimum value of \( n \).


If the real numbers \( x \) and \( y \) satisfy \( x^{2} + 2 \cos y = 1 \), what is the range of values that \( x - \cos y \) can take?

Solution: \boxed{[-1, \sqrt{3} + 1]}


Inside the cube $ABCDEFG A_{1} B_{1} C_{1} D_{1}$, there is the center $O$ of a sphere with radius 10. The sphere intersects the face $A A_{1} D_{1} D$ along a circle of radius 1, the face $A_{1} B_{1} C_{1} D_{1}$ along a circle of radius 1, and the face $C D D_{1} C_{1}$ along a circle of radius 3. Find the length of the segment $OD_{1}$.


a) What is the maximum number of knights that can be placed on a chessboard of 64 squares so that no two knights threaten each other?

b) Determine the number of different arrangements of the maximum possible number of knights where no two knights threaten each other.


A printing house determines the cost of printing a book by adding the cost of the cover to the cost of each page and rounding the result up to the nearest whole ruble (for example, if the result is 202 rubles and 1 kopek, it is rounded up to 203 rubles). It is known that the cost of a book with 104 pages is 134 rubles, and a book with 192 pages costs 181 rubles. What is the cost of printing the cover, given that it is a whole number of rubles and the cost of a single page is a whole number of kopeks?


A natural number that can be expressed as the sum of two consecutive non-zero natural numbers and can also be expressed as the sum of three consecutive non-zero natural numbers is called a "good number". What is the largest "good number" less than or equal to 2011?


A $72 \mathrm{db}$ $4,5 \mathrm{~V}$ $2 \Omega$ internal resistance batteries are connected together. How should we connect them so that the current through an external resistance of $R=36 \Omega$ is

a) maximum,

b) minimum?

What are the values of these currents? (Only consider configurations consisting of series-connected groups of equal numbers of batteries connected in parallel.)

Solution: \boxed{0.125 \text{ A}}


In a right trapezoid \(ABCD\), the sum of the lengths of the bases \(AD\) and \(BC\) is equal to its height \(AB\). In what ratio does the angle bisector of \(\angle ABC\) divide the lateral side \(CD\)?


In the Cartesian coordinate system $xOy$, the ellipse $C: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ $(a > b > 0)$ has left and right foci at $F_{1}$ and $F_{2}$, respectively. The chords $ST$ and $UV$ of the ellipse are parallel to the $x$-axis and $y$-axis, respectively, and intersect at point $P$. The lengths of segments $PU$, $PS$, $PV$, and $PT$ are $1$, $2$, $3$, and $6$, respectively. The area of the triangle $\triangle P F_{1} F_{2}$ is...


Let \( H \) be the orthocenter of \( \triangle ABC \), and \( 3 \overrightarrow{HA} + 4 \overrightarrow{HB} + 5 \overrightarrow{HC} = \mathbf{0} \). Determine \( \cos \angle AHB \).


A circle touches the longer leg of a right triangle, passes through the vertex of the opposite acute angle, and has its center on the hypotenuse of the triangle. What is the radius of the circle if the lengths of the legs are 5 and 12?


In what numeral system is the number 11111 a perfect square?

Solution: \boxed{B = 3}


A function \( f(x) \) defined on \(\mathbb{R}\) satisfies the following conditions for any real number \( x \):
$$
\begin{array}{l}
f(x+3) \leqslant f(x) + 3, \\
f(x+2) \geqslant f(x) + 2,
\end{array}
$$

and \( f(1) = 2 \). Let \( a_{n} = f(n) \) for \( n \in \mathbb{Z}_{+} \). Find:
$$
f(2015) =
$$


By multiplying four numbers, Nina obtained a result whose units digit is 0. What numbers did she multiply and what was the result, if the factors are consecutive prime numbers?


If the centroid of the inscribed triangle \( ABC \) of the curve \( y^{2}=4 \sqrt{2} x \) is its focus \( F \), then
\[ |FA|^{2} + |FB|^{2} + |FC|^{2} = \]


Given that $P$ is any point on the ellipse $\frac{x^{2}}{16}+\frac{y^{2}}{15}=1$, and $EF$ is any diameter of the circle $\odot N:(x-1)^{2}+y^{2}=4$, find the range of values for $\overrightarrow{P E} \cdot \overrightarrow{P F}$.

Solution: \boxed{[5, 21]}


Let \( f:[0,1] \rightarrow \mathbb{R} \) be continuous and satisfy:

\[
\begin{aligned}
b f(2 x) & = f(x), & & 0 \leq x \leq 1/2 ; \\
f(x) & = b + (1-b) f(2 x - 1), & & 1/2 \leq x \leq 1,
\end{aligned}
\]

where \( b=\frac{1+c}{2+c}, \, c>0 \). Show that \( 0 < f(x) - x < c \) for every \( x \), \( 0 < x < 1 \).

Solution: \boxed{0 < f(x) - x <c}


Given triangles \(ABC\) and \(A'B'C'\) with areas of 1 and 2025, respectively. It is known that rays \(AB\) and \(A'B'\) are parallel and go in opposite directions. The same is true for pairs \(BC\) and \(B'C'\), \(CA\) and \(C'A'\). Let \(A''\), \(B''\), and \(C''\) be the midpoints of segments \(AA'\), \(BB'\), and \(CC'\). Find the area of triangle \(A''B''C''\).


Given that quadrilateral \(ABCD\) is an isosceles trapezoid with \(AB \parallel CD\), \(AB = 6\), and \(CD = 16\). Triangle \(ACE\) is a right-angled triangle with \(\angle AEC = 90^\circ\), and \(CE = BC = AD\). Find the length of \(AE\).


Two spheres with radius \( r \) are externally tangent to each other. Three spheres with radius \( R \) are externally tangent to each other, with each one tangent to the other two. Each of these spheres is also externally tangent to the first two spheres.

Find the relationship between \( R \) and \( r \).

Solution: \boxed{R = 6r}


Consider the cut lines formed in the initial 100 x 100 square. It is easy to see that cutting the folded square once corresponds to multiple parallel lines cut in the initial square, where the distance between these lines is no less than 2 (since at least one fold line must pass between any two cut lines). Thus, there are no more than 50 such lines, and they divide the square into at most 51 rectangles. With two such cuts, a maximum of $51^{2}=2601$ parts can be formed, as each rectangle from the first division can be split into no more than 51 parts in the second cut. An example with 2601 parts can be achieved by folding the square so that the fold lines form a 2 x 2 grid and then cutting the resulting folded 2 x 2 square along its two central lines.


Person A, Person B, and Person C start from point $A$ to point $B$. Person A starts at 8:00, Person B starts at 8:20, and Person C starts at 8:30. They all travel at the same speed. After Person C has been traveling for 10 minutes, Person A is exactly halfway between Person B and point $B$, and at that moment, Person C is 2015 meters away from point $B$. Find the distance between points $A$ and $B$.


Find the remainder when the integer
$$
1 \times 3 \times 5 \times 7 \times \cdots \times 2003 \times 2005
$$
is divided by 1000.


There are apples in a closed bag. Three friends tried to lift the bag and guess how many fruits were inside. The first guessed that there were 20 apples, the second thought there were 22, and the third guessed 25. When they opened the bag, it turned out that one of them was off by 1, another by 3, and the third by 6. How many apples were in the bag? Find all possible answers.


$AB$ is the diameter of a circle, $BC$ is the tangent. The secant $AC$ is divided by the circle at point $D$ into two equal parts. Find the angle $DAB$.


The numbers 0 and 1 are placed in the cells of the table as shown in the figure.

| 0 | 1 | 0 | 0 |
| - | - | - | - |
| 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 |

Permissible actions:

- Add one to all numbers in a row;
- Add one to all numbers in a column;
- Add one to all numbers along a line parallel to any diagonal (in particular, you can add one to the number in any corner cell).

Is it possible, as a result of several such actions, to obtain a table in which all numbers are equal?

Solution: \boxed{\text{No}}


Given that the real part of the expansion of \((1 + \mathrm{i} x)^{4n + 2}\) (where \(x \in \mathbb{R}\)) is a polynomial in \(x\), find the sum of the coefficients of this polynomial.


Kermit the frog enjoys hopping around the infinite square grid in his backyard. It takes him 1 Joule of energy to hop one step north, south, east, or west. He wakes up one morning on the grid with 100 Joules of energy, and hops till he falls asleep with 0 energy. How many different places could he have gone to sleep?


Given the sets \(A=\left\{(x, y) \mid (x-3)^{2}+(y-4)^{2} \leqslant\left(\frac{5}{2}\right)^{2}\right\}\) and \(B=\left\{(x, y) \mid (x-4)^{2}+(y-5)^{2}>\left(\frac{5}{2}\right)^{2}\right\}\), find the number of lattice points in the set \(A \cap B\).


Let \( A = \{a^2, a+1, -3\} \) and \( B = \{a-3, 2a-1, a^2+1\} \), and suppose \( A \cap B = \{-3\} \). Find the value of the real number \( a \).


Given the real numbers \(\alpha\) and \(\beta\) that satisfy the following system of equations:
\[
\begin{cases}
\alpha^{3} - 3\alpha^{2} + 5\alpha - 17 = 0 \\
\beta^{3} - 3\beta^{2} + 5\beta + 11 = 0
\end{cases}
\]
find the value of \(\alpha + \beta\).


Given three circles, each with a radius of 3, and centers at \((14, 92)\), \((17, 76)\), and \((19, 84)\), find the absolute value of the slope of a line passing through the point \((17, 76)\) such that the sum of the areas of the parts of the circles on one side of the line is equal to the sum of the areas of the parts of the circles on the other side of the line.


In the figure, $ABCD$ is a square, $ADE$ is an equilateral triangle and $E$ is a point outside of the square $ABCD$. If $\angle AEB=\alpha^{\circ}$, determine the value of $\alpha$.


In triangle $ABC$ with an acute angle at vertex $A$, an angle bisector $AE$ and an altitude $BH$ are drawn. It is known that $\angle AEB = 45^{\circ}$. Find the angle $EHC$.


Given real numbers \( x \) and \( y \) satisfying the equation \(\frac{x^{2}}{3}+y^{2}=1\), find the range of values of
$$
P=|2x+y-4|+|4-x-2y|.
$$

Solution: \boxed{[2, 14]}


Given a positive integer $d$ which is square-free, and a fundamental solution $(x_{0}, y_{0})$ to the equation $x^{2} - d y^{2} = 1$. Find the positive integer solutions $(x, y)$ to this equation such that all prime factors of $x$ divide $x_{0}$.

Solution: \boxed{(x_0, y_0)}


Let \( M_n \) be the set of \( n \)-digit pure decimals in decimal notation \( 0.\overline{a_1a_2\cdots a_n} \), where \( a_i \) can only take the values 0 or 1 for \( i=1,2,\cdots,n-1 \), and \( a_n = 1 \). Let \( T_n \) be the number of elements in \( M_n \), and \( S_n \) be the sum of all elements in \( M_n \). Find \( \lim_{n \rightarrow \infty} \frac{S_n}{T_n} \).


A line \( l \) passes through the focus \( F \) of the parabola \( y^2 = 4x \) and intersects the parabola at points \( A \) and \( B \). Point \( M \) is given as \( (4,0) \). Extending \( AM \) and \( BM \) intersects the parabola again at points \( C \) and \( D \), respectively. Find the value of \(\frac{S_{\triangle CDM}}{S_{\triangle ABM}}\).


11 gardeners are planting trees. 2 of them are planting trees on Street A, and the remaining 9 are planting trees on Street B. In the evening, they return to their dormitory.

Gardener A says: "Although there are fewer of us, we spent the same amount of time as you did."
Gardener B says: "Although there are more of us, the length of our street is 5 times the length of your street."

If the intervals at which they plant the trees are the same and each gardener plants the same number of trees, planting only on one side of the road and including both ends of the street, how many trees did these 11 gardeners plant in total?


A tourist travels from point \(A\) to point \(B\) in 1 hour and 56 minutes. The route from \(A\) to \(B\) first goes uphill, then on flat terrain, and finally downhill. What is the length of the flat terrain if the tourist's speed downhill is 6 km/h, uphill is 4 km/h, and on flat terrain is 5 km/h, and the total distance between \(A\) and \(B\) is \(x\) km? The distances uphill and on flat terrain are both whole numbers of kilometers.


1) \(\lim _{x \rightarrow 2}\left(\frac{1}{x-2}-\frac{4}{x^{2}-4}\right)\)

2) \(\lim _{x \rightarrow +\infty}\left(x-\sqrt{x^{2}+5x}\right)\)

3) \(\lim \left(\sqrt{\operatorname{tg}^{2} \alpha+\sec \alpha-\operatorname{tg} \alpha}\right)\)

4) \(\lim _{x \rightarrow 0}(2 \operatorname{cosec} 2x - \operatorname{ctg} x)\)


Let \( LOVER \) be a convex pentagon such that \( LOVE \) is a rectangle. Given that \( OV = 20 \) and \( LO = VE = RE = RL = 23 \), compute the radius of the circle passing through \( R, O \), and \( V \).


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

$$
\begin{aligned}
& M(0, 2, 1) \\
& \frac{x-1.5}{2} = \frac{y}{-1} = \frac{z-2}{1}
\end{aligned}
$$

Solution: \boxed{(-1, 0, 1)}


How many solutions does the equation

$$
\left\lfloor\frac{x}{20}\right\rfloor=\left\lfloor\frac{x}{17}\right\rfloor
$$

have over the set of positive integers?

Here, $\lfloor a\rfloor$ denotes the greatest integer less than or equal to \(a\).


In each cell of a \(5 \times 5\) table, a natural number is written with invisible ink. It is known that the sum of all the numbers equals 200, and the sum of three numbers located inside any \(1 \times 3\) rectangle equals 23. What is the value of the central number in the table?


There are 21 towns. Each airline runs direct flights between every pair of towns in a group of five. What is the minimum number of airlines needed to ensure that at least one airline runs direct flights between every pair of towns?


The lateral face of a regular quadrilateral pyramid forms a $45^{\circ}$ angle with the base plane. Find the angle between the adjacent lateral faces.


In triangle \( ABC \), \( AB = 86 \) and \( AC = 97 \). A circle with center at point \( A \) and radius \( AB \) intersects side \( BC \) at points \( B \) and \( X \). Given that both \( BX \) and \( CX \) have integer lengths, what is the length of \( BC \)?


Let \( x = \sin^4\left(\frac{\pi}{8}\right) + \cos^4\left(\frac{\pi}{8}\right) + \sin^4\left(\frac{7\pi}{8}\right) + \cos^4\left(\frac{7\pi}{8}\right) \). Find the value of \( 36x \).


Call a \(2n\)-digit base-10 number special if we can split its digits into two sets of size \(n\) such that the sum of the numbers in the two sets is the same. Let \(p_{n}\) be the probability that a randomly-chosen \(2n\)-digit number is special. (We allow leading zeros in \(2n\)-digit numbers). The sequence \(p_{n}\) converges to a constant \(c\). Find \(c\).


Let $ABC$ be an isosceles triangle with $A$ as the vertex, $D$ the midpoint of $[BC]$, and $\Gamma$ a circle centered at $D$ tangent to $[AB]$ and $[AC]$. Let $M \in [AB]$ and $N \in [AC]$ be such that $[MN]$ is tangent to $\Gamma$.

Show that $BD^2 = BM \times CN$.

Solution: \boxed{BD^2 = BM \times CN}


The four positive integers \(a, b, c, d\) satisfy \(a < b < c < d\). When the sums of the six pairs of these integers are calculated, the six answers are all different and the four smallest sums are 6, 8, 12, and 21. What is the value of \(d\)?


Let \( f \) be the function that maps a positive integer to the sum of its digits when written in base 10. For example, \( f(537)=15 \). Calculate \( f\left(f\left(f\left(4444^{4444}\right)\right)\right) \).


Calculate the limit of the numerical sequence:

$$
\lim _{n \rightarrow \infty} \sqrt{n+2}(\sqrt{n+3}-\sqrt{n-4})$$


In an isosceles triangle with a lateral side of 100 and a base of 60, a circle is inscribed. Find the distance between the points of tangency located on the lateral sides.


A board for playing "Battleship" has a size of $10 \times 10$. What is the maximum number of ships with a size of $1 \times 4$ that can be placed on it?


If the fraction \(\frac{p}{q}\) (where \( p, q \in \mathbf{Z}_{+} \)) is converted into a decimal as
\[
\frac{p}{q} = 0.198 \cdots,
\]
then, when \( q \) takes its minimum value, what is \( p+q \)?


Suppose there are 997 points given in a plane. If every two points are joined by a line segment with its midpoint colored in red, show that there are at least 1991 red points in the plane. Can you find a special case with exactly 1991 red points?


For any positive integer \( n \), let \( D_n \) denote the set of all positive divisors of \( n \). Define \( f_i(n) (i=1,2) \) as the number of elements in the set
\[ F_i(n) = \{ a \in D_n \mid a \equiv i \pmod{4} \} \]
Find the smallest positive integer \( m \) such that
\[ 2f_1(m) - f_2(m) = 2017. \]

Solution: \boxed{2 \times 5^{2016}}


Given that \( z \) is a complex number, \( |z| = 1 \), and \( u = z^{4} - z^{3} - 3z^{2}i - z + 1 \). Find the maximum value of \( |u| \) and the complex number \( z \) when the maximum value is achieved.


A palindrome is a number, word, or text that reads the same backward as forward. How much time in a 24-hour day display palindromes on a clock, showing time from 00:00:00 to 23:59:59?


Suppose that \( ABCDEF \) is a regular hexagon with sides of length 6. Each interior angle of \( ABCDEF \) is equal to \( 120^{\circ} \).

(a) A circular arc with center \( D \) and radius 6 is drawn from \( C \) to \( E \). Determine the area of the shaded sector.

(b) A circular arc with center \( D \) and radius 6 is drawn from \( C \) to \( E \), and a second arc with center \( A \) and radius 6 is drawn from \( B \) to \( F \). These arcs are tangent (touch) at the center of the hexagon. Line segments \( BF \) and \( CE \) are also drawn. Determine the total area of the shaded regions.

(c) Along each edge of the hexagon, a semi-circle with diameter 6 is drawn. Determine the total area of the shaded regions; that is, determine the total area of the regions that lie inside exactly two of the semi-circles.

Solution: \boxed{18\pi - 27\sqrt{3}}


Find all real polynomials \(P\) such that \(P(0) = 0\) and \(P(X^2 + 1) = P(X)^2 + 1\).

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


Given \(a, b, c \in \mathbf{R}^{+}\) (positive real numbers) such that \(a b c + a + c = b\), determine the maximum value of \(p = \frac{2}{a^{2}+1} - \frac{2}{b^{2}+1} + \frac{3}{c^{2}+1}\).


A swimmer swims against the current of a river and encounters an empty boat drifting with the current. Continuing to swim against the current for another $t$ minutes after the encounter, the swimmer then turns back and catches up with the boat $s$ meters from the point where they met. Find the speed of the river current.

Solution: \boxed{\frac{s}{2 t}}


Snow White entered a room with a round table surrounded by 30 chairs. Some of the chairs were occupied by dwarfs. It turned out that Snow White couldn't sit in a way such that no one was sitting next to her. What is the minimum number of dwarfs that could have been at the table? Explain how the dwarfs must have been seated.


An \(n \times m\) maze is an \(n \times m\) grid in which each cell is one of two things: a wall, or a blank. A maze is solvable if there exists a sequence of adjacent blank cells from the top left cell to the bottom right cell going through no walls. (In particular, the top left and bottom right cells must both be blank.) Determine the number of solvable \(2 \times 2\) mazes.


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


Let \(ABC\) be a triangle with \(AB = 13\), \(BC = 14\), and \(CA = 15\). Let \(I_A\), \(I_B\), and \(I_C\) be the \(A\), \(B\), and \(C\) excenters of this triangle, respectively, and let \(O\) be the circumcenter of the triangle. Let \(\gamma_A\), \(\gamma_B\), and \(\gamma_C\) be the corresponding excircles and \(\omega\) be the circumcircle. \(X\) is one of the intersections between \(\gamma_A\) and \(\omega\). Likewise, \(Y\) is an intersection of \(\gamma_B\) and \(\omega\), and \(Z\) is an intersection of \(\gamma_C\) and \(\omega\). Compute
\[
\cos \angle OXI_A + \cos \angle OYI_B + \cos \angle OZI_C.
\]


Let \( S = \{1, 2, \ldots, 2008\} \). For any nonempty subset \( A \subset S \), define \( m(A) \) to be the median of \( A \) (when \( A \) has an even number of elements, \( m(A) \) is the average of the middle two elements). Determine the average of \( m(A) \), when \( A \) is taken over all nonempty subsets of \( S \).


Solve in integers: \(6x^2 + 5xy + y^2 = 6x + 2y + 7\). Specify the answer for which the value \(|x| + |y|\) is the largest. Write the answer in the form \((x; y)\).

Solution: \boxed{(-8 ; 25)}


Let \( P(x) \) be a polynomial of degree 2008 with leading coefficient 1 such that
\[ P(0) = 2007, \; P(1) = 2006, \; P(2) = 2005, \ldots, P(2007) = 0. \]
Determine the value of \( P(2008) \). You may use factorials in your answer.

Solution: \boxed{2008! - 1}


Find the smallest positive number $\alpha$ such that there exists a positive number $\beta$ for which the inequality $\sqrt{1+x} + \sqrt{1-x} \leq 2 - \frac{x^{\alpha}}{\beta}$ holds for $0 \leq x \leq 1$.


Let the ellipse be given by \(\frac{x^{2}}{5}+\frac{y^{2}}{4}=1\). The locus of the intersection points of two mutually perpendicular tangents to the ellipse is \(C\). Tangents \(PA\) and \(PB\) to the curve \(C\) intersect at point \(P\), touching the curve at points \(A\) and \(B\) respectively. Find the minimum value of \(\overrightarrow{PA} \cdot \overrightarrow{PB}\).

Solution: \boxed{18 \sqrt{2} - 27}


Given three numbers \(a, b, c\) such that \(a \times b = 24\), \(a \times c = 36\), and \(b \times c = 54\), find the value of \(a + b + c\).


Does there exist \( n > 1 \) such that the set of positive integers may be partitioned into \( n \) non-empty subsets so that if we take an arbitrary element from every set but one, then their sum belongs to the remaining set?

Solution: \boxed{\text{No}}


Determine the number of roots of the equation:

$$
\left(21 x - 11 + \frac{\sin x}{100}\right) \cdot \sin(6 \arcsin x) \cdot \sqrt{(\pi - 6x)(\pi + x)} = 0
$$


What is the largest $n$ for which the numbers $1,2, \ldots, 14$ can be colored in red and blue so that for any number $k=1,2, \ldots, n$, there are a pair of blue numbers and a pair of red numbers, each pair having a difference equal to $k$?


A circle is inscribed in a triangle, and the points of tangency with the sides of the triangle are connected to each other. A new circle is inscribed in the resulting triangle, and the points of tangency with its sides are the vertices of a third triangle, which has the same angles as the original triangle. Find these angles.


A sequence of two distinct numbers is extended in two ways: one to form a geometric progression and the other to form an arithmetic progression. The third term of the geometric progression coincides with the tenth term of the arithmetic progression. With which term of the arithmetic progression does the fourth term of the geometric progression coincide?


\( n \) is a positive integer that is not greater than 100 and not less than 10, and \( n \) is a multiple of the sum of its digits. How many such \( n \) are there?


We consider 2015 lines in a plane, no two of which are parallel and no three of which meet at a single point. Let $E$ be the set of their intersection points.

We want to assign a color to each point in $E$ such that any two points on the same line, whose connecting segment contains no other points from $E$, are colored differently.

What is the minimum number of colors needed to achieve such a coloring?


On side \(AD\) of rectangle \(ABCD\), there is a point \(E\). On segment \(EC\), there is such a point \(M\) that \(AB=BM\) and \(AE=EM\). Find the length of side \(BC\), given that \(ED=16\) and \(CD=12\).


In each cell of a $5 \times 5$ table, a natural number is written with invisible ink. It is known that the sum of all the numbers is 200, and the sum of the three numbers within any $1 \times 3$ rectangle is 23. What is the value of the central number in the table?


For the infinite sequence of numbers \( x_{1}, x_{2}, x_{3}, \ldots \), the relationship \( x_{n} = x_{n-1} \cdot x_{n-3} \) holds for all natural numbers \( n \geq 4 \). It is known that \( x_{1} = 1 \), \( x_{2} = 1 \), and \( x_{3} = -1 \). Find \( x_{2022} \).


A player throws a fair die (with each number from 1 to 6 having an equal probability of 1/6 and each throw being independent) repeatedly until their total score is greater than or equal to \( n \). Let \( p(n) \) be the probability that their final score is exactly \( n \). Find \( \lim_{n \to \infty} p(n) \).


Find all functions \( f: \mathbb{R}_{+}^{*} \longrightarrow \mathbb{R}_{+}^{*} \) such that for every \( x \in \mathbb{R}_{+}^{*} \), there exists exactly one \( y \in \mathbb{R}_{+}^{*} \) such that:

\[
x f(y) + y f(x) \leq 2
\]

Solution: \boxed{f(x) = \frac{1}{x}}


A plane parallel to the base of a quadrilateral pyramid with a volume of 81 cuts off a smaller pyramid with a volume of 3. Find the volume of the pyramid whose four vertices coincide with the vertices of the section and whose fifth vertex lies in the plane of the base of the larger pyramid.


The diagonals of a trapezoid are mutually perpendicular, and one of them is equal to 17. Find the area of the trapezoid if its height is 15.


Let the three sides of a triangle be integers \( l \), \( m \), and \( n \) with \( l > m > n \). It is known that \( \left\{\frac{3^l}{10^4}\right\} = \left\{\frac{3^m}{10^4}\right\} = \left\{\frac{3^n}{10^4}\right\} \), where \( \{x\} \) denotes the fractional part of \( x \). Determine the minimum value of the perimeter of the triangle.


In a school chess tournament, boys and girls competed, with the number of boys being 5 times the number of girls. According to the tournament rules, each player played against every other player twice. How many players participated in total, given that the boys scored exactly twice as many points as the girls? (A win in a chess game counts as 1 point, a draw counts as 0.5 points, and a loss counts as 0 points.)

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


The square \(ABCD\) is to be decomposed into \(n\) nonoverlapping triangles, all of whose angles are acute. Find the smallest integer \(n\) for which there exists a solution to this problem and construct at least one decomposition for this \(n\). Additionally, answer whether it is possible to require that (at least) one of these triangles has a perimeter less than an arbitrarily given positive number.


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


Let $\Gamma$ be a circle with chord $[BC]$. Let $A$ be the midpoint of the arc $\overparen{BC}$. From point $A$, draw two arbitrary chords $[AD]$ and $[AE]$ which intersect the segment $[BC]$ at $F$ and $G$ respectively. Show that the quadrilateral $DFGE$ is cyclic.

Solution: \boxed{DFGE}


In a regular truncated quadrilateral pyramid with the height of the lateral face \( a \), a sphere can be inscribed. Find the area of its lateral surface.


Find a three-digit number whose square is a six-digit number, such that each subsequent digit from left to right is greater than the previous one.


Given that the function \( f(x) \) is monotonically increasing on \([2,+\infty)\), and for any real number \( x \), \( f(2+x)=f(2-x) \) always holds. If \( f\left(1-2 x^{2}\right)<f\left(1+2 x-x^{2}\right) \), determine the range of values for \( x \).

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


A triangular prism \( ABC A_1 B_1 C_1 \) with the base \( ABC \) and lateral edges \( AA_1, BB_1, CC_1 \) is intersected by a plane passing through points \( E, F, C \), where point \( E \) is the midpoint of edge \( AA_1 \), and point \( F \) lies on edge \( BB_1 \) such that \( BF:FB_1 = 1:2 \). Find the volume of the part of the prism \( ABC A_1 B_1 C_1 \) that is enclosed between the intersecting plane and the base of this prism, given that the volume of the prism is \( V \).

Solution: \boxed{\frac{5}{18} V}


Take a standard set of dominoes and remove all duplicates and blanks. Then consider the remaining 15 dominoes as fractions. The dominoes are arranged in such a way that the sum of all fractions in each row equals 10. However, unlike proper fractions, you are allowed to use as many improper fractions (such as \(\frac{4}{3}, \frac{5}{2}, \frac{6}{1}\)) as you wish, as long as the sum in each row equals 10.


Given complex numbers \( z \) and \( \omega \) satisfying the following two conditions:
1. \( z + \omega + 3 = 0 \);
2. \( |z|, 2, |\omega| \) form an arithmetic sequence.

Is there a maximum value for \( \cos(\arg z - \arg \omega) \)? If so, find it.


Given the equation:

\[ A = \operatorname{ctg}\left(\frac{11 \pi}{4}+\frac{1}{2} \arccos \frac{2 b}{a}\right)+\operatorname{ctg}\left(\frac{11 \pi}{4}-\frac{1}{2} \arccos \frac{2 b}{a}\right) \]

Find the value of \( A \).

Solution: \boxed{-\frac{a}{b}}


One of the diagonals of an inscribed quadrilateral is a diameter of its circumscribed circle.

Verify that the intersections $E$ and $F$ of the lines perpendicular to $A$ and $C$, respectively, to the line $BD$, satisfy:

$$
DE = BF
$$

Solution: \boxed{DE = BF}


Given a finite set \( A \), there exists a function \( f: \mathbf{N}^{*} \rightarrow A \) with the following property: if \( i, j \in \mathbf{N}^{*} \) and \( |i-j| \) is a prime number, then \( f(i) \neq f(j) \). What is the minimum number of elements that the set \( A \) must have?


What is the largest number of natural numbers less than 50 that can be chosen so that any two of them are relatively prime?


As shown in Figure 1, quadrilateral $ACDB$ and parallelogram $AEDF$ satisfy:
$$
\begin{array}{l}
6 \overrightarrow{A E}+3 \overrightarrow{D E}=2 \overrightarrow{A C}, \\
4 \overrightarrow{F B}+5 \overrightarrow{F D}=2 \overrightarrow{A D}.
\end{array}
$$
Find the ratio of the area of quadrilateral $ACDB$ to the area of parallelogram $AEDF$.


Compute the number of positive integers that divide at least two of the integers in the set \(\left\{1^{1}, 2^{2}, 3^{3}, 4^{4}, 5^{5}, 6^{6}, 7^{7}, 8^{8}, 9^{9}, 10^{10}\right\}\).


In the cryptarithm

$$
\mathbf{K}<\mathrm{O}<\mathbf{P}>\mathbf{O}>\mathbf{H}>\mathbf{A}>\mathbf{B}>\mathbf{U}>\mathbf{P}>\mathbf{y}>\mathbf{C}
$$

different letters represent different digits. How many solutions does the cryptarithm have?


As shown in the figure, three equally sized square sheets of paper in red, yellow, and blue are stacked in a large square box. Given that the exposed areas are 25 for the blue sheet, 19 for the red sheet, and 11 for the yellow sheet, what is the area of the large square box?


There is a unique triplet of positive integers \((a, b, c)\) such that \(a \leq b \leq c\) and
$$
\frac{25}{84}=\frac{1}{a}+\frac{1}{a b}+\frac{1}{a b c}.
$$
Determine \(a + b + c\).


Find the number of pairs of integers \((x, y)\) that satisfy the equation \(5x^2 - 6xy + y^2 = 6^{100}\).


Suppose \(A, B\) are the foci of a hyperbola and \(C\) is a point on the hyperbola. Given that the three sides of \(\triangle ABC\) form an arithmetic sequence, and \(\angle ACB = 120^\circ\), determine the eccentricity of the hyperbola.


Péter took one stamp each of values 1, 2, 3, ..., 37 forints from his stamp collection. He wants to group them so that each group has the same total value of stamps. In how many ways can he do this?


Two ferries cross a river with constant speeds, turning at the shores without losing time. They start simultaneously from opposite shores and meet for the first time 700 feet from one shore. They continue to the shores, return, and meet for the second time 400 feet from the opposite shore. Determine the width of the river.

Solution: \boxed{1700 \text{ feet}}


$A B C D E F$ is a regular hexagon, and point \( O \) is its center. How many different isosceles triangles with vertices among these seven points can be formed? Triangles that differ only in the order of the vertices are considered the same (for example, \( AOB \) and \( BOA \)).

The answer is 20.


Masha thought of a natural number and found its remainders when divided by 3, 6, and 9. The sum of these remainders turned out to be 15.

Find the remainder when the thought number is divided by 18.


$35 \cdot 1220$ blocks of square-shaped building blocks are piled up as shown in the figure. First, the first layer consists of 10 blocks arranged in a triangular pattern; then the second layer is made up of 6 blocks arranged in a triangular pattern placed in the center of the first layer; the third layer is made up of 3 blocks arranged in a triangular pattern placed in the center of the second layer; finally, 1 block is placed in the center of the third layer. The 10 blocks of the bottom layer are marked with numbers from 1 to 10 in a certain order. The number marked on each block on the second, third, and fourth layers is equal to the sum of the numbers on the 3 blocks supporting it from the next lower layer.

The minimum possible value of the number on the block at the very top is
(A) 55
(B) 83
(C) 114
(D) 137
(E) 144

(44th American High School Mathematics Examination, 1993)


Find the number of ordered pairs of positive integers \((x, y)\) with \(x, y \leq 2020\) such that \(3x^2 + 10xy + 3y^2\) is the power of some prime.


We call a set of positive integers "tyû-de-jó" if there are no two numbers in the set whose difference is 2. How many "tyû-de-jó" subsets are there of the set $\{1,2,3, \ldots, 10\}$?


Find the maximum value of the expression \(a + b + c + d - ab - bc - cd - da\), given that each of the numbers \(a, b, c\), and \(d\) belongs to the interval \([0, 1]\).


Find all integers \(a, y \geq 1\) such that \(3^{2a-1} + 3^a + 1 = 7^y\).

Solution: \boxed{(a,y) = (1,1)}


The cube $ABCDEFGH$ standing on a table has its three vertical edges divided by points $K, L$, and $M$ in the ratios $1:2$, $1:3$, and $1:4$, respectively, as shown in the figure. The plane $KLM$ cuts the cube into two parts. What is the ratio of the volumes of the two parts?


Five boys and three girls sit around a circular table. If there are no additional requirements, there are $Q(8,8)=7!=5040$ arrangements. How many arrangements are there if the boy $B_{1}$ does not sit next to the girl $G_{1}$? How many arrangements are there if the three girls do not sit next to each other?


Triangle $ABC$ is an isosceles triangle with $BC$ as its base. Let $F$ be the midpoint of $BC$. Point $D$ is an internal point on segment $BF$. The perpendicular to $BC$ at point $D$ intersects side $AB$ at $M$, and a line passing through point $D$ parallel to $AB$ intersects side $AC$ at $P$. Express the ratio of the areas of triangles $AMP$ and $ABC$ as a function of $k = BD : BC$.

Solution: \boxed{k - 2k^2}


The eleven-digit number ' \(A 123456789 B\) ' is divisible by exactly eight of the numbers \(1, 2, 3, 4, 5, 6, 7, 8, 9\). Find the values of \(A\) and \(B\), explaining why they must have these values.

Solution: \boxed{A = 3, B = 6}


On an island of knights and liars, knights always tell the truth, and liars always lie. In a school on this island, both knights and liars study in the same class. One day the teacher asked four children: Anu, Banu, Vanu, and Danu, who among them did the homework. They responded:

- Anu: Banu, Vanu, and Danu did the homework.
- Banu: Anu, Vanu, and Danu did not do the homework.
- Vanu: Do not believe them, teacher! Anu and Banu are liars!
- Danu: No, teacher, Anu, Banu, and Vanu are knights!

How many knights are among these children?


Given triangle \( \triangle ABC \) with circumcenter \( O \) and orthocenter \( H \), and \( O \neq H \). Let \( D \) and \( E \) be the midpoints of sides \( BC \) and \( CA \) respectively. Let \( D' \) and \( E' \) be the reflections of \( D \) and \( E \) with respect to \( H \). If lines \( AD' \) and \( BE' \) intersect at point \( K \), find the value of \( \frac{|KO|}{|KH|} \).


Four squares with integer side lengths are arranged as shown in the diagram, with three vertices of square \( A \) being the centers of squares \( B \), \( C \), and \( D \). If the total area of the red parts is equal to the area of the green part, what is the minimum side length of square \( A \)?


How many sequences of ten binary digits are there in which neither two zeroes nor three ones ever appear in a row?


Two hunters, $A$ and $B$, went duck hunting. Assume that each of them hits a duck as often as they miss it. Hunter $A$ encountered 50 ducks during the hunt, while hunter $B$ encountered 51 ducks. What is the probability that hunter $B$'s catch exceeds hunter $A$'s catch?


Given the quadratic function \( y = ax^2 + bx + c \) with its graph intersecting the \( x \)-axis at points \( A \) and \( B \), and its vertex at point \( C \):

(1) If \( \triangle ABC \) is a right-angled triangle, find the value of \( b^2 - 4ac \).

(2) Consider the quadratic function
\[ y = x^2 - (2m + 2)x + m^2 + 5m + 3 \]
with its graph intersecting the \( x \)-axis at points \( E \) and \( F \), and it intersects the linear function \( y = 3x - 1 \) at two points, with the point having the smaller \( y \)-coordinate denoted as point \( G \).

(i) Express the coordinates of point \( G \) in terms of \( m \).

(ii) If \( \triangle EFG \) is a right-angled triangle, find the value of \( m \).


In which numeral system is the number \( 11111_{d} \) a perfect square?


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

\[
3^{x^{2}-2ax+a^{2}} = ax^{2}-2a^{2}x+a^{3}+a^{2}-4a+4
\]

have exactly one solution?


In the acute triangle \( \triangle ABC \), \( \sin A = \frac{4 \sqrt{5}}{9} \), and \( a, b, c \) represent the sides opposite to angles \( A, B, \) and \( C \), respectively.
1. Find the value of \( \sin 2(B + C) + \sin^{2} \frac{B+C}{2} \).
2. If \( a = 4 \), find the area \( S_{\triangle ABC} \) of the triangle when \( \overrightarrow{AB} \cdot \overrightarrow{AC} \) reaches its maximum value.


Calculate \(\left\lfloor \sqrt{2}+\sqrt[3]{\frac{3}{2}}+\sqrt[4]{\frac{4}{3}}+\ldots+\sqrt[2009]{\frac{2009}{2008}} \right\rfloor\), where \(\left\lfloor x \right\rfloor\) denotes the greatest integer less than or equal to \(x\).


Using each of the digits 1-9 exactly once, form a two-digit perfect square, a three-digit perfect square, and a four-digit perfect square. What is the smallest four-digit perfect square among them?


Given three quadratic polynomials \( f(x)=a x^{2}+b x+c \), \( g(x)=b x^{2}+c x+a \), and \( h(x)=c x^{2}+a x+b \), where \( a, b, c \) are distinct non-zero real numbers. From these, three equations \( f(x)=g(x) \), \( f(x)=h(x) \), and \( g(x)=h(x) \) are formed. Find the product of all roots of these three equations, assuming each equation has two distinct roots.


Let \( a_{n} \) represent the closest positive integer to \( \sqrt{n} \) for \( n \in \mathbf{N}^{*} \). Suppose \( S=\frac{1}{a_{1}}+\frac{1}{a_{2}}+\cdots+\frac{1}{a_{2000}} \). Determine the value of \( [S] \).


Point \( M \) is the midpoint of side \( BC \) of triangle \( ABC \), where \( AB = 17 \), \( AC = 30 \), and \( BC = 19 \). A circle is constructed with diameter \( AB \). A point \( X \) is chosen arbitrarily on this circle. What is the minimum possible value of the length of segment \( MX \)?


Consider a product \(a_{1} a_{2} \ldots a_{100}\) written on a board, where \(a_{1}, \ldots, a_{100}\) are natural numbers. We consider 99 expressions, each of which is obtained by replacing one of the multiplication signs with an addition sign. It is known that exactly 32 of these expressions are even. What is the maximum number of even numbers among \(a_{1}, a_{2}, \ldots, a_{100}\) that could exist?


Find the volume of a parallelepiped where two faces are rhombuses with side length 1 and an acute angle of $60^{\circ}$, and the other faces are squares.


What is the maximum number of colors needed to paint all the cells of a 4 by 4 square so that for each pair of different colors, there exist two cells of these colors located either in the same row or in the same column of the square?


Given natural numbers \( m \) and \( n \) where \( n > m > 1 \), the last three digits of the decimal representation of \( 1978^m \) and \( 1978^n \) are the same. Find \( m \) and \( n \) such that \( m+n \) is minimized.


A horizontal plane can intersect all 6 lateral edges of a prism. If we tilt this plane in such a way that it intersects the upper base near one of the vertices, it will intersect two edges of the upper base while no longer intersecting one of the lateral edges. This increases the number of intersected edges by 1. Similarly, we can increase this number by 1 at the lower base. Thus, we obtain a plane intersecting 8 edges of the prism.

Why can't we achieve more than 8 intersections?


Point \( O \) is the intersection point of the altitudes of an acute triangle \( ABC \). Find \( OC \), given that \( AB = 4 \) and \( \sin \angle C = \frac{5}{13} \).


The products by four and by five of an integer, when considered together, use each digit from 1 to 9 exactly once. What is this number?


In an isosceles triangle \(ABC\), side \(AC = b\), sides \(BA = BC = a\), \(AM\) and \(CN\) are the angle bisectors of angles \(A\) and \(C\). Find \(MN\).

Solution: \boxed{\frac{ab}{a + b}}


a) Two seventh-grade students participated in a chess tournament along with some eighth-grade students. The two seventh-graders scored a total of 8 points, and each eighth-grader scored the same number of points. How many eighth-graders participated in the tournament? Find all possible solutions.

b) Ninth and tenth-grade students participated in a chess tournament. There were 10 times as many tenth-graders as ninth-graders, and they scored 4.5 times more points in total than all the ninth-graders. How many points did the ninth-graders score?


a) Given a piece of wire 120 cm long, is it possible to construct a frame of a cube with an edge length of 10 cm without breaking the wire?

b) What is the minimum number of times the wire needs to be broken in order to construct the required frame?


Find the volume of an oblique triangular prism with a base that is an equilateral triangle with side length $a$, given that the lateral edge of the prism is equal to the side of the base and is inclined at an angle of $60^{\circ}$ to the plane of the base.

Solution: \boxed{\frac{3 a^3}{8}}


Let $P$ and $Q$ be two monic polynomials satisfying:

$$
P(P(X)) = Q(Q(X))
$$

Show that $P = Q$.

Solution: \boxed{P = Q}


Given the sequence

$$
a_{n}=\frac{(n+3)^{2}+3}{n(n+1)(n+2)} \cdot \frac{1}{2^{n+1}}
$$

defined by its general term, form the sequence

$$
b_{n}=\sum_{k=1}^{n} a_{k}
$$

Determine the limit of the sequence $b_{n}$ as $n$ approaches $+\infty$.


Given a sequence of positive integers $\left\{a_{n}\right\}$ such that:
$$
a_{0}=m, \quad a_{n+1}=a_{n}^{5}+487 \quad (n \in \mathbf{N}),
$$
find the value of the positive integer $m$ such that the sequence $\left\{a_{n}\right\}$ contains the maximum number of perfect squares.


For any real number \( x \), the inequality
$$
|x + a| - |x + 1| \leq 2a
$$
holds. What is the minimum value of the real number \( a \)?


There are 20 chairs in a room of two colors: blue and red. Seated on each chair is either a knight or a liar. Knights always tell the truth, and liars always lie. Each of the seated individuals initially declared that they were sitting on a blue chair. After that, they somehow changed seats, and now half of the seated individuals claim to be sitting on blue chairs while the other half claim to be sitting on red chairs. How many knights are now sitting on red chairs?


Masha's school lessons usually end at 13:00, and her mother picks her up by car to go home. One day, lessons ended at 12:00, and Masha decided to walk home. On her way, she met her mother, who as usual was on her way to pick Masha up at 13:00. They then continued their journey home by car and arrived 12 minutes earlier than usual. At what time did Masha meet her mother on the road? (Assume that both Masha and her mother travel at constant speeds and that no time is spent for Masha to get into the car.)


The range of the function
$$
f(x)=\frac{\sin x-1}{\sqrt{3-2 \cos x-2 \sin x}} \quad (0 \leqslant x \leqslant 2 \pi)
$$
is .

Solution: \boxed{[-1, 0]}


Consider two regular triangular pyramids \( P-ABC \) and \( Q-ABC \) with the same base inscribed in the same sphere. If the angle between the lateral face and the base of the pyramid \( P-ABC \) is \( 45^\circ \), find the tangent of the angle between the lateral face and the base of the pyramid \( Q-ABC \).


A math extracurricular activity group at a certain school designed a tree planting plan on graph paper for a desert as follows: The $k$-th tree is planted at point $P_{k}(x_{k}, y_{k})$, where $x_{1}=1$ and $y_{1}=1$. For $k \geq 2$, the coordinates are determined by:

\[
\begin{cases}
x_{k} = x_{k-1} + 1 - 5 \left[\frac{k-1}{5}\right] + 5 \left[\frac{k-2}{5}\right], \\
y_{k} = y_{k-1} + \left[\frac{k-1}{5}\right] - \left[\frac{k-2}{5}\right],
\end{cases}
\]

where $[a]$ denotes the integer part of the real number $a$, for example, $[2.6] = 2$ and $[0.6] = 0$. According to this plan, the coordinates of the 2008th tree are _____

Solution: \boxed{(3, 402)}


Given plane vectors $\vec{a}, \vec{b}, \vec{c}$ that satisfy the following conditions: $|\vec{a}| = |\vec{b}| \neq 0$, $\vec{a} \perp \vec{b}$, $|\vec{c}| = 2 \sqrt{2}$, and $|\vec{c} - \vec{a}| = 1$, determine the maximum possible value of $|\vec{a} + \vec{b} - \vec{c}|$.


Let \(a, b, c\) be positive integers such that \(ab + bc - ca = 0\) and \(a - c = 101\). Find \(b\).


There are 50 lines on a plane, 20 of which are parallel to each other. The maximum number of regions into which these 50 lines can divide the plane is ________.


In the Karabas Barabas theater, a chess tournament took place among the actors. Each participant played exactly one match against each of the other participants. A win awarded one solido, a draw awarded half a solido, and a loss awarded nothing. It turned out that among every three participants, there was one chess player who earned exactly 1.5 solidos in the games against the other two. What is the maximum number of actors that could have participated in such a tournament?


What is the greatest number of natural numbers, not exceeding 2016, that can be selected such that the product of any two selected numbers is a perfect square?


Given the numbers \(\log _{\sqrt{x+34}}(2x+23), \log _{(x+4)^{2}}(x+34), \log _{\sqrt{2x+23}}(-x-4)\). For which values of \(x\) are two of these numbers equal and the third one greater by 1?


In hexagon $ABCDEF$, $AC$ and $CE$ are two diagonals. Points $M$ and $N$ divide $AC$ and $CE$ internally such that $\frac{AM}{AC}=\frac{CN}{CE}=r$. Given that points $B$, $M$, and $N$ are collinear, find $r$.


Petya writes different three-digit natural numbers on the board, each of which is divisible by 3, and the first two digits differ by 2. What is the maximum number of such numbers he can write down if they end in 6 or 7?


What is the smallest number of digits that need to be appended to the right of the number 2014 so that the resulting number is divisible by all natural numbers less than 10?


Among all the proper fractions where both the numerator and denominator are two-digit numbers, find the smallest fraction that is greater than $\frac{4}{5}$. In your answer, provide its numerator.


A line in the plane is called strange if it passes through \((a, 0)\) and \((0, 10-a)\) for some \(a\) in the interval \([0,10]\). A point in the plane is called charming if it lies in the first quadrant and also lies below some strange line. What is the area of the set of all charming points?


Let $O$ be the center of the circumscribed circle around the triangle $ABC$. Find the angle $OAC$ if: a) $\angle B=50^{\circ}$; b) $\angle B=126^{\circ}$.


Given the circumcenter $O$ of $\triangle ABC$, and $2 \overrightarrow{OA} + 3 \overrightarrow{OB} + 4 \overrightarrow{OC} = \mathbf{0}$, find the value of $\cos \angle BAC$.


The natural numbers \(a\) and \(b\) are such that \(5 \, \text{LCM}(a, b) + 2 \, \text{GCD}(a, b) = 120\). Find the largest possible value of \(a\).


2000 people registered on a new website. Each person invited 1000 others to be friends. Two people are considered friends if and only if both have invited each other to be friends. What is the minimum number of friend pairs that could have been formed?


Let \( ABC \) be a triangle with orthocenter \( H \), and let \( G \) be the point such that the quadrilateral \( ABGH \) is a parallelogram. Let \( I \) be the point on the line \( GH \) such that the line \( AC \) intersects the segment \( HI \) at its midpoint. The line \( AC \) intersects the circumcircle of triangle \( GCI \) at point \( J \). Show that \( IJ = AH \).

Solution: \boxed{IJ = AH}


The given quadratic polynomial \( f(x) \) has two distinct roots. Can it be the case that the equation \( f(f(x)) = 0 \) has three distinct roots, while the equation \( f(f(f(x))) = 0 \) has seven distinct roots?

Solution: \boxed{\text{No}}


Two spheres touch the plane of triangle \( A B C \) at points \( A \) and \( B \) and are located on opposite sides of this plane. The sum of the radii of these spheres is 11, and the distance between their centers is \( \sqrt{481} \). The center of a third sphere with radius 9 is located at point \( C \), and this sphere touches each of the first two spheres externally. Find the radius of the circumcircle of triangle \( A B C \).

Solution: \boxed{3 \sqrt{10}}


Real numbers \(a, b, c\) and positive number \(\lambda\) are such that \(f(x) = x^3 + ax^2 + bx + c\) has three real roots \(x_1, x_2, x_3\), and satisfy:
(1) \(x_2 - x_1 = \lambda\);
(2) \(x_3 > \frac{1}{2}(x_1 + x_2)\).
Find the maximum value of \(\frac{2a^3 + 27c - 9ab}{\lambda^3}\).


Given a cyclic quadrilateral \(ABCD\), it is known that \(\angle ADB = 48^{\circ}\) and \(\angle BDC = 56^{\circ}\). Inside the triangle \(ABC\), a point \(X\) is marked such that \(\angle BCX = 24^{\circ}\), and the ray \(AX\) is the angle bisector of \(\angle BAC\). Find the angle \(CBX\).


In triangle \(ABC\), \(AB = 28\), \(BC = 21\), and \(CA = 14\). Points \(D\) and \(E\) are on \(AB\) with \(AD = 7\) and \(\angle ACD = \angle BCE\). Find the length of \(BE\).


Solve the equation for integer values: \(\underbrace{\sqrt{n+\sqrt{n+\ldots \sqrt{n}}}}_{\text{1964 times}} = m\)


The magician taught Kashtanka to bark the number of times he secretly showed her. When Kashtanka correctly answered how much two times two is, he hid a delicious cake in a suitcase with a combination lock and said:

- The eight-digit code for the suitcase is the solution to the puzzle УЧУЙ $=\text{KE} \times \text{KS}$. You need to replace the same letters with the same digits and different letters with different digits so that the equality is true. Bark the correct number of times for each of the eight characters, and you will get the treat.

But then there was a confusion. Kashtanka, out of excitement, barked 1 more time for each letter than she should have. Of course, the suitcase did not open. Suddenly, a child's voice said: "It's not fair! The dog solved the puzzle correctly!" And indeed, if each digit of the solution the magician had in mind is increased by 1, another solution to the puzzle emerges!

Is it possible to determine: a) what exact solution the magician had in mind; b) what the value of the number УЧУЙ was in that solution?

Solution: \boxed{\text{УЧУЙ} = 2021}


The quadrilateral \(P Q R S\) is inscribed in a circle. Diagonals \(P R\) and \(Q S\) are perpendicular and intersect at point \(M\). It is known that \(P S = 13\), \(Q M = 10\), and \(Q R = 26\). Find the area of the quadrilateral \(P Q R S\).


A palace is shaped like a square and divided into $2003 \times 2003$ rooms, similar to the squares on a large chessboard. There is a door between two rooms if and only if they share a wall. The main entrance allows entry into the palace from the outside into the room located at the northwest corner. A person enters the palace, visits some rooms, and then exits the palace through the entrance, returning to the northwest corner for the first time. It turns out this person visited each of the other rooms exactly 100 times, except for the room located at the southeast corner. How many times has the visitor been in the southeast corner room?


From a point \( M \), located inside triangle \( ABC \), perpendiculars are drawn to sides \( BC \), \( AC \), and \( AB \), with lengths \( k \), \( l \), and \( m \) respectively. Find the area of triangle \( ABC \), given that \(\angle CAB = \alpha\) and \(\angle ABC = \beta\). If the answer is not an integer, round it to the nearest whole number.

$$
\alpha = \frac{\pi}{6}, \beta = \frac{\pi}{4}, k = 3, l = 2, m = 4
$$


Given \( |\boldsymbol{a}| = 1 \), \( |\boldsymbol{b}| = |\boldsymbol{c}| = 2 \), \( \boldsymbol{b} \cdot \boldsymbol{c} = 0 \), \( \lambda \in (0, 1) \), find the minimum value of
\[
|a - b + \lambda(b - c)| + \left| \frac{1}{2}c + (1 - \lambda)(b - c) \right|
\]

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


For the quadrilateral $ABCD$, it is known that $AB = BD$, $\angle ABD = \angle DBC$, and $\angle BCD = 90^\circ$. On segment $BC$, point $E$ is marked such that $AD = DE$. Find the length of segment $BD$ if $BE = 7$ and $EC = 5$.


A sphere of radius 4 with center at point $Q$ touches three parallel lines at points $F$, $G$, and $H$. It is known that the area of triangle $QGH$ is $4 \sqrt{2}$, and the area of triangle $FGH$ is greater than 16. Find the angle $GFH$.


Given a triangle $ABC$ with $O$ as the incenter. Find the angle $A$ if the circumradii of triangles $ABC$ and $BOC$ are equal.


An archipelago consisting of an infinite number of islands is spread out along the southern shore of an endless sea in a chain. The islands are connected by an infinite chain of bridges, and each island is also connected to the shore by a bridge. In the event of a strong earthquake, each bridge independently has a probability $p=0.5$ of being destroyed. What is the probability that, after a strong earthquake, it will be possible to travel from the first island to the shore using the remaining bridges?


Given \( n \) squares on a plane, where:

1. All the squares are congruent.
2. If two squares share a common intersection point \( P \), then \( P \) must be a vertex of each of the squares.
3. Any square intersects exactly three other squares.

The collection of these \( n \) squares is called "three-connected."

Determine the number of integers \( n \) within the range \( 2018 \leq n \leq 3018 \) for which there exists a three-connected set of \( n \) squares.


Select four vertices of a cube such that no two of them are on the same edge. A point inside the cube has distances of \(\sqrt{50}\), \(\sqrt{70}\), \(\sqrt{90}\), and \(\sqrt{110}\) from these four vertices. What is the edge length of the cube?

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


Let \( AH_a \) and \( BH_b \) be the altitudes, and \( AL_a \) and \( BL_b \) be the angle bisectors of triangle \( ABC \). It is known that \( H_aH_b \parallel L_aL_b \). Is it true that \( AC = BC \)?

Solution: \boxed{AC = BC}


A coin is tossed 10 times. Find the probability that two heads never appear consecutively.


Let \( z \) be a complex number with a modulus of 1. Then the maximum value of \(\left|\frac{z+\mathrm{i}}{z+2}\right|\) is \(\ \ \ \ \ \ \).


(1) For which positive integers \( n \) does there exist a finite set of \( n \) distinct positive integers \( S_n \) such that the geometric mean of any subset of \( S_n \) is an integer?

(2) Is there an infinite set of distinct positive integers \( S \) such that the geometric mean of any finite subset of \( S \) is an integer?

Solution: \boxed{\text{No}}


To celebrate 2019, Faraz gets four sandwiches shaped in the digits 2, 0, 1, and 9 at lunch. The four digits get reordered (but not flipped or rotated) on his plate, and he notices that they form a 4-digit multiple of 7. What is the greatest possible number that could have been formed?


In the pyramid \(ABCD\), the dihedral angle at edge \(AC\) is \(90^{\circ}\), \(AB = BC = CD\), and \(BD = AC\). Find the dihedral angle at edge \(AD\).


In triangle \(ABC\), the bisector \(BD\) is drawn, and in triangles \(ABD\) and \(CBD\), the bisectors \(DE\) and \(DF\) are drawn, respectively. It turns out that \(EF \parallel AC\). Find the angle \(\angle DEF\).

Solution: \boxed{45 \text{ degrees}}


In the addition below, identical letters represent the same digit and different letters represent different digits. Find the number $ABCDE$.

\[ \begin{array}{cccccc}
    & & & & A & B & C & D & E \\
    + & & & & B & C & D & E \\
    + & & & & & C & D & E \\
    + & & & & & & D & E \\
    + & & & & & & & E \\
    \hline
    & & A & A & A & A & A \\
\end{array} \]


If the complex number \( z \) satisfies \( |z+\mathrm{i}| + |z-2-\mathrm{i}| = 2 \sqrt{2} \), then the minimum value of \( |z| \) is ______.


Find all pairs of integers \( n, m \) such that \( 7^m = 5^n + 24 \).

Solution: \boxed{(2, 2)}


The diagram shows a square with sides of length \(4 \text{ cm}\). Four identical semicircles are drawn with their centers at the midpoints of the square’s sides. Each semicircle touches two other semicircles. What is the shaded area, in \(\text{cm}^2\)?

A) \(8 - \pi\)

B) \(\pi\)

C) \(\pi - 2\)

D) \(\pi - \sqrt{2}\)

E) \(8 - 2\pi\)

Solution: \boxed{8 - 2\pi}


Given sets of real numbers \( A, B \), define \( A \otimes B = \{ x \mid x = ab + a + b, \, a \in A, b \in B \} \). Let \( A = \{0, 2, 4, \cdots, 18\} \) and \( B = \{98, 99, 100\} \). Find the sum of all elements in \( A \otimes B \).


For which integers $n > 2$ is the following statement true? "Any convex $n$-gon has a side such that neither of the two angles at its endpoints is an acute angle."

Solution: \boxed{n \geq 7}


Show that the last digits of the numbers $\left(n^{n}\right)^{n}$ for $n = 1, 2, 3, \ldots$ repeat periodically. Determine the period.


Given real numbers \(x\) and \(y\) satisfying \(x^{2}+(y-2)^{2} \leq 1\), determine the range of values for \(\frac{x+\sqrt{3} y}{\sqrt{x^{2}+y^{2}}}\).

Solution: \boxed{[1, 2]}


Find the functions \( f: \mathbb{N} \rightarrow \mathbb{N} \) such that \( f(2n) = 2f(n) \) and \( f(2n+1) = 2f(n) + 1 \) for all \( n \in \mathbb{N} \).

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


Triangle \(ABC\) has \(AB = 10\), \(BC = 17\), and \(CA = 21\). Point \(P\) lies on the circle with diameter \(AB\). What is the greatest possible area of \( \triangle APC \)?


Fifteen students numbered $1, 2, 3, \ldots, 15$ stand in a circle facing inward. On the first turn, the student with number 1 turns around to face outward. On the second turn, the students numbered 2 and 3 turn around. On the third turn, the students numbered $4, 5, 6$ turn around, and so on. On the 15th turn, all students turn around. After the 12th turn, how many students are still facing outward?


The digits \(1, 2, 3, 4, 5, 6\) are randomly chosen (without replacement) to form the three-digit numbers \(M = \overline{ABC}\) and \(N = \overline{DEF}\). For example, we could have \(M = 413\) and \(N = 256\). Find the expected value of \(M \cdot N\).


In a certain meeting, there are 30 participants. Each person knows at most 5 other people among the remaining participants; for any group of five people, at least two people are not acquainted with each other. Find the largest positive integer $k$ such that among these 30 people, there exists a group of $k$ people where no two people are acquainted with each other.


Each of the four volleyball teams has six players, including a captain and a setter, who are different people. In how many ways can a team of six players be formed from these four teams, where there is at least one player from each team and at least one captain-setter pair from one team?


For given numbers \( n \geq 2 \) and \( a > 0 \), find the maximum value of the sum \( \sum_{i=1}^{n-1} x_i x_{i+1} \) subject to the conditions \( x_i \geq 0 \) for \( i = 1, \ldots, n \) and \( x_1 + \ldots + x_n = a \).


Determine all prime numbers \( p \) for which there exists a unique \( a \) in \( \{1, \ldots, p\} \) such that \( a^{3} - 3a + 1 \) is divisible by \( p \).


What is the smallest possible area of a figure on the xy-plane, located between the lines \( x = -5 \) and \( x = 1 \), bounded below by the x-axis and above by the tangent line to the graph of the function \( y = 7 - 6x - x^2 \) at a point of tangency \( x_0 \), where \( -5 \leq x_0 \leq 1 \)?


A 99-sided polygon has its sides painted in sequence with the colors red, blue, red, blue, $\cdots$, red, blue, yellow. Each side is painted in one color. Then, the following operation is allowed: ensuring that any two adjacent sides have different colors, one can change the color of one side each time. Can the coloring of the 99 sides be changed to red, blue, red, blue, $\cdots$, red, yellow, blue after a number of operations?

Solution: \boxed{\text{No}}


A quadrilateral is inscribed in a circle with radius 1. Two opposite sides are parallel. The difference between their lengths is \( d > 0 \). The distance from the intersection of the diagonals to the center of the circle is \( h \). Find \(\sup \frac{d}{h}\) and describe the cases in which it is attained.


All natural numbers from 1 to 20 are divided into pairs, and the numbers in each pair are summed. What is the maximum number of the resulting ten sums that can be divisible by 11? Justify your answer.


Find the smallest possible sum of digits for a number of the form \(3n^2 + n + 1\) (where \(n\) is a positive integer). Does there exist a number of this form with a sum of digits 1999?


A stack contains 300 cards: 100 white, 100 black, and 100 red. For each white card, the number of black cards below it is counted; for each black card, the number of red cards below it is counted; and for each red card, the number of white cards below it is counted. Find the maximum possible value of the sum of these 300 counts.


Four statements were made about triangle \(ABC\), and we know that out of these statements, two are true and two are false. What could be the perimeter of triangle \(ABC\) if \(BC = 1\)?

I. Triangle \(ABC\) is a right triangle.
II. The angle at \(A\) is \(30^\circ\).
III. \(AB = 2 \times BC\).
IV. \(AC = 2 \times BC\).


In the Magic Land, one of the magical laws of nature states: "A flying carpet will only fly if it has a rectangular shape." Prince Ivan had a flying carpet of dimensions $9 \times 12$. One day, an evil dragon cut off a small piece of this carpet with dimensions $1 \times 8$. Prince Ivan was very upset and wanted to cut off another piece $1 \times 4$ to make a rectangle of $8 \times 12$, but the wise Vasilisa suggested a different approach. She cut the carpet into three parts and used magic threads to sew them into a square flying carpet of dimensions $10 \times 10$. Can you figure out how Vasilisa the Wise transformed the damaged carpet?

Solution: \boxed{10 \times 10}


What is the greatest number of natural numbers not exceeding 2016 that can be selected such that the product of any two selected numbers is a perfect square?


At the class reunion, 45 people attended. It turned out that any two of them, who have the same number of acquaintances among the attendees, do not know each other. What is the largest number of pairs of acquaintances that could have been among the attendees?


For sets \( S \), let \( |S| \) represent the number of elements in \( S \), and let \( n(S) \) represent the number of subsets of \( S \), including the empty set and \( S \) itself. If the sets \( A, B, C \) satisfy \( n(A) + n(B) + n(C) = n(A \cup B \cup C) \) with \( |A| = |B| = 100 \), find the minimum possible value of \( |A \cap B \cap C| \).


In a triangle \(ABC\), angle \(A\) is twice the measure of angle \(B\), angle \(C\) is obtuse, and the side lengths are integers. What is the smallest possible perimeter of this triangle?


For any real numbers \( a \) and \( b \), the inequality \( \max \{|a+b|,|a-b|,|2006-b|\} \geq C \) always holds. Find the maximum value of the constant \( C \). (Note: \( \max \{x, y, z\} \) denotes the largest among \( x, y, \) and \( z \).)


In triangle $ABC$, $AB = 6$ and $BC = 10$ units. At what distance from $B$ does the line connecting the foot of the angle bisector from $B$ and the midpoint of $AB$ intersect line $BC$?


The set of five-digit numbers \( \{ N_1, \ldots, N_k \} \) is such that any five-digit number with all digits in increasing order coincides in at least one digit with at least one of the numbers \( N_1, \ldots, N_k \). Find the smallest possible value of \( k \).


Find the maximum real number \(\lambda\) such that for the real-coefficient polynomial
$$
f(x) = x^3 + ax^2 + bx + c
$$
with all roots being non-negative real numbers, the inequality
$$
f(x) \geqslant \lambda(x - a)^3 \quad \text{for all} \; x \geqslant 0
$$
holds. Also, determine when equality holds in this inequality.


Vanya wrote the number 1 on the board and then added several more numbers. Each time Vanya writes a new number, Mitya calculates the median of the existing set of numbers and writes it down in his notebook. At a certain moment, the following numbers are recorded in Mitya's notebook: $1 ; 2 ; 3 ; 2.5 ; 3 ; 2.5 ; 2 ; 2 ; 2 ; 5$.

a) What number was written on the board fourth?

b) What number was written on the board eighth?


In $\triangle ABC$, find the value of $a^{3} \sin (B-C) + b^{3} \sin (C-A) + c^{3} \sin (A-B)$.


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

\[
\left\{
\begin{array}{l}
x^{2} + x y + y^{2} = 27 \\
y^{2} + y z + z^{2} = 25 \\
z^{2} + x z + x^{2} = 52
\end{array}
\right.
\]

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


A regular triangular prism \( ABC A_1 B_1 C_1 \) with a base \( ABC \) and side edges \( AA_1, BB_1, CC_1 \) is inscribed in a sphere of radius 3. The segment \( CD \) is the diameter of this sphere. Find the volume of the prism, given that \( AD = 2 \sqrt{6} \).


Two adjacent faces of a tetrahedron, which are isosceles right triangles with a hypotenuse of 2, form a dihedral angle of 60 degrees. The tetrahedron rotates around the common edge of these faces. Find the maximum area of the projection of the rotating tetrahedron onto a plane containing the given edge.


The smallest positive odd number that cannot be represented as \(7^{x}-3 \times 2^{y} (x, y \in \mathbb{Z}_{+})\) is $\qquad$ .


Find the integers \( n \geq 1 \) for which \( n \) divides \( 2^{n} - 1 \).

Solution: \boxed{n = 1}


Given that in triangle \( \triangle ABC \), \( a = 2b \), \( \cos B = \frac{2 \sqrt{2}}{3} \), find the value of \( \sin \frac{A-B}{2} + \sin \frac{C}{2} \).


Given a regular tetrahedron $\mathrm{ABCD}$ with a point $\mathrm{P}$ inside it such that $P A=P B=\sqrt{11}$ and $P C=P D=\sqrt{17}$. What is the side length of the tetrahedron $\mathrm{ABCD}$?


The side length \( BC \) of triangle \( ABC \) is 12 cm. A circle with radius 10 cm is circumscribed around the triangle. Find the lengths of sides \( AB \) and \( AC \) of the triangle, given that the radius \( OA \) of the circle bisects the side \( BC \) into two equal segments.


How many pairs of integers $(x, y)$ satisfy the equation \(\sqrt{x-\sqrt{x+23}}=2\sqrt{2}-y\)?


Point \(A\) lies on the line \(y=\frac{12}{5} x-9\), and point \(B\) lies on the parabola \(y=x^{2}\). What is the minimum length of segment \(AB\)?


A mason has bricks with dimensions \( 2 \times 5 \times 8 \) and other bricks with dimensions \( 2 \times 3 \times 7 \). She also has a box with dimensions \( 10 \times 11 \times 14 \). The bricks and the box are all rectangular parallelepipeds. The mason wants to pack bricks into the box, filling its entire volume with no bricks sticking out. Find all possible values of the total number of bricks that she can pack.


A group of adventurers displays their loot. It is known that exactly 9 adventurers have rubies; exactly 8 have emeralds; exactly 2 have sapphires; exactly 11 have diamonds. Additionally, it is known that:

- If an adventurer has diamonds, they either have rubies or sapphires (but not both simultaneously);
- If an adventurer has rubies, they either have emeralds or diamonds (but not both simultaneously).

What is the minimum number of adventurers that could be in this group?


How many ordered pairs of integers \((a, b)\) satisfy all of the following inequalities?
\[
\begin{array}{l}
a^2 + b^2 < 16 \\
a^2 + b^2 < 8a \\
a^2 + b^2 < 8b
\end{array}
\]


In the number $2 * 0 * 1 * 6 * 0 * 2 *$, each of the 6 asterisks needs to be replaced with any of the digits $0, 2, 4, 5, 7, 9$ (digits may repeat), so that the resulting 12-digit number is divisible by 75. How many ways can this be done?


Tolya arranged in a row 101 coins of denominations 1, 2, and 3 kopecks. It turned out that between any two 1-kopeck coins there is at least one coin, between any two 2-kopeck coins there are at least two coins, and between any two 3-kopeck coins there are at least three coins. How many 3-kopeck coins could Tolya have?


Given a plane intersects all 12 edges of a cube at an angle $\alpha$, find $\sin \alpha$.


Let \( S \) be the set of points \((x, y)\) in the plane such that the sequence \( a_n \) defined by \( a_0 = x \), \( a_{n+1} = \frac{a_n^2 + y^2}{2} \) converges. What is the area of \( S \)?

Solution: \boxed{4 + \pi}


When \( N \) takes all the values from 1, 2, 3, \ldots, 2015, how many numbers of the form \( 3^{n} + n^{3} \) are divisible by 7?


What is the smallest number of digits that need to be appended to the right of the number 2014 so that the resulting number is divisible by all natural numbers less than 10?


In the figure, a cake in the shape of a rectangle with sides $3 \times 3$ is shown. A straight line (cut) is illustrated, crossing the maximum number of squares, which is 5. The maximum number of pieces of the cake after the first cut is equal to $2 \cdot 5 + 4 = 14$. Thus, it is enough to use two straight lines to cut all 9 squares.

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


Drop perpendiculars IP and IQ from point I onto sides BC and AB respectively. Points P and Q are points of tangency of these sides with the inscribed circle, and the quadrilateral PIQB is a square. The angles KIL and PIQ are right angles, hence angles PIK and QIL are equal. Consequently, right triangles PIK and QIL are congruent. According to Thales' theorem, the length KP=QL is half of the length BP=BQ, and the length AQ is twice the length BQ=BP.

Therefore, the length of side AB is \( AL + LB = \frac{6}{5}AL = \frac{6}{5}AO = \frac{3}{5}AC \). From the Pythagorean theorem, \( BC = \frac{4}{5}AC \). Hence, \( AB:BC:CA = 3:4:5 \).


On a $5 \times 5$ board, two players alternately mark numbers on empty cells. The first player always marks 1's, the second 0's. One number is marked per turn, until the board is filled. For each of the nine $3 \times 3$ squares, the sum of the nine numbers on its cells is computed. Denote by \( A \) the maximum of these sums. How large can the first player make \( A \), regardless of the responses of the second player?


Let \(a, b, c\) be the side lengths of a right triangle, with \(a \leqslant b < c\). Determine the maximum constant \(k\) such that the inequality \(a^{2}(b+c) + b^{2}(c+a) + c^{2}(a+b) \geqslant k a b c\) holds for all right triangles, and specify when equality occurs.

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


Let $M$ be the midpoint of the base $AC$ of an isosceles triangle $ABC$. Points $E$ and $F$ are marked on sides $AB$ and $BC$ respectively, such that $AE \neq CF$ and $\angle FMC = \angle MEF = \alpha$. Find $\angle AEM$.

Solution: \boxed{\alpha}


The trapezoid \(ABCD\) with base \(AD = 6\) is inscribed in a circle. The tangent to the circle at point \(A\) intersects lines \(BD\) and \(CD\) at points \(M\) and \(N\), respectively. Find \(AN\) if \(AB \perp MD\) and \(AM = 3\).


We shuffle a deck of 52 French playing cards, and then draw one by one from the pile until we find an ace of black color. In what position is it most likely for the first black ace to appear?


Let \( p \) be an odd prime. An integer \( x \) is called a quadratic non-residue if \( p \) does not divide \( x - t^{2} \) for any integer \( t \).

Denote by \( A \) the set of all integers \( a \) such that \( 1 \leq a < p \), and both \( a \) and \( 4 - a \) are quadratic non-residues. Calculate the remainder when the product of the elements of \( A \) is divided by \( p \).


Points \( A, B, C, D \) lie on a circle in that order such that \(\frac{AB}{BC} = \frac{DA}{CD}\). If \(AC = 3\) and \(BD = BC = 4\), find \(AD\).


In $\triangle ABC$, given that $x \sin A + y \sin B + z \sin C = 0$, find the value of $(y + z \cos A)(z + x \cos B)(x + y \cos C) + (y \cos A + z)(z \cos B + x)(x \cos C + y)$.


Let \( x, y, z \) be the lengths of the midline segments connecting the midpoints of opposite edges of a tetrahedron \( P-ABC \). If the sum of the squares of the six edge lengths of the tetrahedron is 300, find \( x^{2} + y^{2} + z^{2} \).


At a physical education lesson, 29 seventh graders attended, some of whom brought one ball each. During the lesson, sometimes one seventh grader would give their ball to another seventh grader who did not have a ball.

At the end of the lesson, $N$ seventh graders said, "I received balls less often than I gave them away!" Find the largest possible value of $N$, given that no one lied.


For any 4 distinct points \(P_{1}, P_{2}, P_{3}, P_{4}\) on a plane, find the minimum value of the ratio
\[
\frac{\sum_{1 \leq i < j \leq 4} P_{i}P_{j}}{\min_{1 \leq i < j \leq 4} P_{i}P_{j}}
\]

Solution: \boxed{5 + \sqrt{3}}


What is the smallest positive integer \( n \) such that \( n^{2} \) and \( (n+1)^{2} \) both contain the digit 7 but \( (n+2)^{2} \) does not?


Given that
$$
S=\left|\sqrt{x^{2}+4 x+5}-\sqrt{x^{2}+2 x+5}\right|,
$$
for real values of \(x\), find the maximum value of \(S^{4}\).


Using the digits $0$ to $9$ exactly once, form several composite numbers. What is the minimum sum of these composite numbers?


A cryptarithm is given: ЛЯЛЯЛЯ + ФУФУФУ = ГГЫГЫЫР. Identical letters represent the same digits, different letters represent different digits. Find the sum of ЛЯ and ФУ.


Five contestants $A, B, C, D, E$ participate in a "Voice" competition, and they stand in a row for a group appearance. Each contestant has a number badge on their chest, the sum of the five numbers is 35. It is known that the sum of the numbers of the contestants standing to the right of $\mathrm{E}$ is 13; the sum of the numbers of the contestants standing to the right of $D$ is 31; the sum of the numbers of the contestants standing to the right of $A$ is 21; and the sum of the numbers of the contestants standing to the right of $C$ is 7. What is the sum of the numbers of the contestants at the far left and the far right?


Find the triangle. The sides and the height of a certain triangle are expressed by four consecutive integers. What is the area of this triangle?


Given the sequence $\left\{a_{n}\right\}$ satisfying $a_{1}=1$ and $a_{n+1}=a_{n}+\frac{1}{2 a_{n}}$, find $\lim _{n \rightarrow \infty}\left(a_{n}-\sqrt{n}\right)$.


Given that 7 divides 111111. If \( b \) is the remainder when \(\underbrace{111111 \ldots 111111}_{a \text{-times }}\) is divided by 7, find the value of \( b \). If \( c \) is the remainder of \( \left\lfloor(b-2)^{4 b^{2}}+(b-1)^{2 b^{2}}+b^{b^{2}}\right\rfloor \) divided by 3, find the value of \( c \). If \( |x+1|+|y-1|+|z|=c \), find the value of \( d=x^{2}+y^{2}+z^{2} \).


Consider an alphabet of 2 letters. A word is any finite combination of letters. We will call a word unpronounceable if it contains more than two of the same letter in a row. How many unpronounceable 7-letter words are there?


Find the number of ordered pairs of integers \((a, b) \in \{1, 2, \ldots, 35\}^{2}\) (not necessarily distinct) such that \(ax + b\) is a quadratic residue modulo \(x^2 + 1\) and 35.


From an 8x8 chessboard, 10 squares were cut out. It is known that among the removed squares, there are both black and white squares. What is the maximum number of two-square rectangles (dominoes) that can still be guaranteed to be cut out from this board?


Given that \( x, y, z \) are positive numbers such that \( x^2 + y^2 + z^2 = 1 \), find the minimum value of the expression:
\[
S = \frac{xy}{z} + \frac{yz}{x} + \frac{zx}{y}.
\]


Square \(CASH\) and regular pentagon \(MONEY\) are both inscribed in a circle. Given that they do not share a vertex, how many intersections do these two polygons have?


Consider a $4 \times 4$ grid of squares, each of which is originally colored red. Every minute, Piet can jump on one of the squares, changing the color of it and any adjacent squares (two squares are adjacent if they share a side) to blue. What is the minimum number of minutes it will take Piet to change the entire grid to blue?


In a quadrilateral $ABCD$ lying in the plane, $AB=\sqrt{3}$, $AD=DC=CB=1$. The areas of triangles $ABD$ and $BCD$ are $S$ and $T$ respectively. What is the maximum value of $S^{2} + T^{2}$?


Let \(a\) and \(b\) be two strictly positive integers with \((a, b) \neq (1,1)\) such that \(ab-1\) divides \(a^2 + b^2\). Show that \(a^2 + b^2 = 5ab - 5\).

Solution: \boxed{a^2 + b^2 = 5ab - 5}


$a$ and $b$ are natural numbers. Show that if $4ab - 1$ divides $(4a^2 - 1)^2$, then $a = b$.

Solution: \boxed{a = b}


A needle lies on a plane. It is allowed to rotate the needle by $45^{\circ}$ around either of its ends. Is it possible, after making several such rotations, to return the needle to its original position with its ends swapped?

Solution: \boxed{\text{No}}


Let \(R\) be the rectangle in the Cartesian plane with vertices at \((0,0), (2,0), (2,1)\), and \((0,1)\). The resulting figure has 7 segments of unit length, connecting neighboring lattice points (those lying on or inside \(R\)). Compute the number of paths from \((0,1)\) (the upper left corner) to \((2,0)\) (the lower right corner) along these 7 segments, where each segment can be used at most once.


Let $a_{1}, a_{2}, a_{3}, \cdots$ be a non-decreasing sequence of positive integers. For $m \geqslant 1$, define
$$
b_{m}=\min \left\{n \mid a_{n} \geqslant m\right\},
$$
which means $b_{m}$ is the smallest $n$ such that $a_{n} \geqslant m$. Given that $a_{19}=85$, find the maximum value of $a_{1} + a_{2} + \cdots + a_{19} + b_{1} + b_{2} + \cdots + b_{85}$.


Given that \( f(x) \) is an odd function defined on \((-1, 1)\), is monotonically decreasing on the interval \([0, 1)\), and satisfies \( f(1-a) + f\left(1-a^2\right) < 0 \), determine the range of the real number \( a \).

Solution: \boxed{(0, 1)}


On a plane, there are a finite number of polygons. If any two of these polygons have a line passing through the origin that intersects both of them, then these polygons are said to be properly placed. Find the smallest natural number \( m \) such that for any set of properly placed polygons, we can draw \( m \) lines passing through the origin such that each of these polygons intersects with at least one of these \( m \) lines.


Among the numbers from 1 to 1000, how many are divisible by 4 and do not contain the digit 4 in their representation?


Fluffball and Shaggy the squirrels ate a basket of berries and a pack of seeds containing between 50 and 65 seeds, starting and finishing at the same time. Initially, Fluffball ate berries while Shaggy ate seeds. Later, they swapped tasks. Shaggy ate berries six times faster than Fluffball, and seeds three times faster. How many seeds did Shaggy eat if Shaggy ate twice as many berries as Fluffball?


Solution: \boxed{x^2 - 1}


Find the probability that a randomly selected five-digit natural number with non-repeating digits, composed of the digits \(1,2,3,4,5,6,7,8\), is divisible by 8 without a remainder.


In a chess tournament, 29 participants are playing. A win in a match is awarded one point, a draw is awarded half a point, and a loss is zero points. If a player has no opponent, they receive 1 point, but according to the rules, a point without a game can be awarded to a player no more than once throughout the tournament. In the next round, participants with the same number of points play against each other (if this is not possible, opponents are chosen randomly). A total of 9 rounds were played. No draws were recorded in any round.

(a) Could two players with 8 points each play in the final round?

(b) Starting from which round, regardless of the results of the previous rounds, could no pair of players who had won all previous rounds meet?


What is the largest possible area of a triangle with sides \(a\), \(b\), and \(c\) that are within the following limits:

$$
0 < a \leq 1 \leq b \leq 2 \leq c \leq 3
$$


In $\triangle ABC$, angle bisectors $BD$ and $CE$ intersect at $I$, with $D$ and $E$ located on $AC$ and $AB$ respectively. A perpendicular from $I$ to $DE$ intersects $DE$ at $P$, and the extension of $PI$ intersects $BC$ at $Q$. If $IQ = 2 IP$, find $\angle A$.


Let \( N \) be the set of natural numbers \(\{1, 2, 3, \ldots \}\). Let \( Z \) be the integers. Define \( d : N \to Z \) by \( d(1) = 0 \), \( d(p) = 1 \) for \( p \) prime, and \( d(mn) = m d(n) + n d(m) \) for any integers \( m, n \). Determine \( d(n) \) in terms of the prime factors of \( n \). Find all \( n \) such that \( d(n) = n \). Define \( d_1(m) = d(m) \) and \( d_{n+1}(m) = d(d_n(m)) \). Find \( \lim_{n \to \infty} d_n(63) \).

Solution: \boxed{\infty}


The opposite sides of a quadrilateral inscribed in a circle intersect at points \( P \) and \( Q \). Find the length of the segment \( |PQ| \), given that the tangents to the circle drawn from \( P \) and \( Q \) are \( a \) and \( b \) respectively.

Solution: \boxed{\sqrt{a^2 + b^2}}


Find all natural values of \( n \) for which
$$
\cos \frac{2 \pi}{9}+\cos \frac{4 \pi}{9}+\cdots+\cos \frac{2 \pi n}{9}=\cos \frac{\pi}{9}, \text { and } \log _{2}^{2} n+45<\log _{2} 8 n^{13}
$$

Record the sum of the obtained values of \( n \) as the answer.


For a given positive integer \( k \), let \( f_{1}(k) \) represent the square of the sum of the digits of \( k \), and define \( f_{n+1}(k) = f_{1}\left(f_{n}(k)\right) \) for \( n \geq 1 \). Find the value of \( f_{2005}\left(2^{2006}\right) \).


Define \( P(n) = n^2 + n + 1 \). For any positive integers \( a \) and \( b \), the set
\[ \{ P(a), P(a+1), P(a+2), \ldots, P(a+b) \} \]
is said to be fragrant if none of its elements is relatively prime to the product of the other elements. Determine the smallest size of a fragrant set.


Using the equality \( \lg 11 = 1.0413 \), find the smallest number \( n > 1 \) for which among the \( n \)-digit numbers, there is not a single one equal to some natural power of the number 11.


Positive numbers \( a, b, c \) satisfy \( a^2 b + b^2 c + c^2 a = 3 \). Find the minimum value of the expression:

\[ A = \frac{\sqrt{a^6 + b^4 c^6}}{b} + \frac{\sqrt{b^6 + c^4 a^6}}{c} + \frac{\sqrt{c^6 + a^4 b^6}}{a}. \]

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


The following figure shows a cube.

Calculate the number of equilateral triangles that can be formed such that their three vertices are vertices of the cube.


What is the minimum number of points that must be marked inside a convex $n$-gon so that each triangle with vertices at the vertices of this $n$-gon contains at least one marked point?


A function \( g \) is ever more than a function \( h \) if, for all real numbers \( x \), we have \( g(x) \geq h(x) \). Consider all quadratic functions \( f(x) \) such that \( f(1)=16 \) and \( f(x) \) is ever more than both \( (x+3)^{2} \) and \( x^{2}+9 \). Across all such quadratic functions \( f \), compute the minimum value of \( f(0) \).


29 boys and 15 girls attended a ball. Some boys danced with some girls (no more than once in each pair). After the ball, each child told their parents how many times they danced. What is the maximum number of distinct numbers the children could have reported?


When dividing the polynomial \( x^{1051} - 1 \) by \( x^4 + x^3 + 2x^2 + x + 1 \), what is the coefficient of \( x^{14} \) in the quotient?


Fold a rectangular piece of paper which is 16 cm long and 12 cm wide as shown in the figure. Find the length of the crease \( GF \) in cm.


Among six different quadratic trinomials, differing by permutation of coefficients, what is the maximum number that can have two distinct roots?


Given a set \( A \) of \( n \) points in the plane, no three collinear, show that we can find a set \( B \) of \( 2n - 5 \) points such that a point of \( B \) lies in the interior of every triangle whose vertices belong to \( A \).


Let \( a, b, c, d, e, f \) be integers selected from the set \( \{1,2, \ldots, 100\} \), uniformly and at random with replacement. Set
\[ M = a + 2b + 4c + 8d + 16e + 32f. \]
What is the expected value of the remainder when \( M \) is divided by 64?


Suppose a sequence of positive real numbers \( x_{0}, x_{1}, \cdots, x_{1995} \) satisfies the following two conditions:
(1) \( x_{0} = x_{1995} \);
(2) \( x_{i-1} + \frac{2}{x_{i-1}} = 2 x_{i} + \frac{1}{x_{i}} \) for \( i = 1, 2, \cdots, 1995 \).

Find the maximum value of \( x_{0} \) among all sequences that satisfy the above conditions.


Two concentric circles have radii of 1 and 2 units, respectively. What is the minimum possible area of a cyclic quadrilateral inscribed in the larger circle that also contains the smaller circle?

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


Compute the limit of the numerical sequence:

$$
\lim _{n \rightarrow \infty}(n \sqrt{n}-\sqrt{n(n+1)(n+2)})
$$

Solution: \boxed{-\infty}


The pentagon $ABCDE$ is inscribed in a circle. Points $M, Q, N,$ and $P$ are the bases of the perpendiculars dropped from vertex $E$ to the sides $AB, BC, CD$ (or their extensions) and diagonal $AD$ respectively. It is known that $|EP|=d$, and the ratio of the area of triangle $MQE$ to the area of triangle $PNE$ is $k$. Find $|EM|$.

Solution: \boxed{d \sqrt{k}}


Let \( q \) be a positive rational number. Two ants start from the same point \( X \) on a plane and each ant moves in one of the four directions (east, south, west, or north) a distance of \( q^n \) meters in the \( n \)-th minute \((n=1,2, \cdots)\). Suppose that after some integer minutes, they meet again, but their paths are not completely identical. Find all possible values of \( q \).


Natural numbers \( a, b, c \) are chosen such that \( a < b < c \). It is also known that the system of equations \( 2x + y = 2033 \) and \( y = |x-a| + |x-b| + |x-c| \) has exactly one solution. Find the minimum possible value of \( c \).


Show that if the numbers \( a_{1}, a_{2}, \ldots \) are not all zeros and satisfy the relation \( a_{n+2} = \left|a_{n+1}\right| - a_{n} \), then from some point on they are periodic and the smallest period is 9.


The edge of cube \( ABCD A_1 B_1 C_1 D_1 \) is 12. Point \( K \) lies on the extension of edge \( BC \) at a distance of 9 from vertex \( C \). Point \( L \) on edge \( AB \) is at a distance of 5 from \( A \). Point \( M \) divides segment \( A_1 C_1 \) in a ratio of 1:3, starting from \( A_1 \). Find the area of the cross-section of the cube by the plane passing through points \( K \), \( L \), and \( M \).


On the board, there are two-digit numbers. Each number is composite, but any two numbers are coprime. What is the maximum number of such numbers that can be written?


A circle is tangent to two adjacent sides \(AB\) and \(AD\) of square \(ABCD\) and cuts off a segment of length 4 cm from vertices \(B\) and \(D\) at the points of tangency. On the other two sides, the circle intersects cutting off segments from the vertices of lengths 2 cm and 1 cm, respectively. Find the radius of the circle.


Find the smallest natural number divisible by 99, all of whose digits are even.


Maria ordered a certain number of televisions at $R$ \$ 1994.00 each. She noticed that in the total amount to be paid, there are no digits 0, 7, 8, or 9. What was the smallest number of televisions she ordered?


From the center \( O \) of the inscribed circle of a right triangle, the half of the hypotenuse that is closer to \( O \) appears at a right angle. What is the ratio of the sides of the triangle?

Solution: \boxed{3 : 4 : 5}


Given a cyclic quadrilateral \(ABCD\). The rays \(AB\) and \(DC\) intersect at point \(K\). It turns out that points \(B\), \(D\), and the midpoints of segments \(AC\) and \(KC\) lie on the same circle. What values can the angle \(ADC\) take?


Suppose \( x \neq y \), and the sequences \( x, a_{1}, a_{2}, a_{3}, y \) and \( b_{1}, x, b_{2}, b_{3}, y, b_{4} \) are both arithmetic sequences. What is \( \frac{b_{4}-b_{3}}{a_{2}-a_{1}} \) equal to?


For each pair of distinct natural numbers \(a\) and \(b\), not exceeding 20, Petya drew the line \( y = ax + b \) on the board. That is, he drew the lines \( y = x + 2, y = x + 3, \ldots, y = x + 20, y = 2x + 1, y = 2x + 3, \ldots, y = 2x + 20, \ldots, y = 3x + 1, y = 3x + 2, y = 3x + 4, \ldots, y = 3x + 20, \ldots, y = 20x + 1, \ldots, y = 20x + 19 \). Vasia drew a circle of radius 1 with center at the origin on the same board. How many of Petya’s lines intersect Vasia’s circle?


Given 2022 lines in the plane, such that no two are parallel and no three are concurrent. We denote \( E \) as the set of their intersection points. We want to assign a color to each point in \( E \) such that any two points on the same line, whose connecting segment does not contain any other point from \( E \), have different colors. What is the minimum number of colors required to achieve this coloring?


Solution: \boxed{12 + 4\sqrt{3}}


How many triangles can be formed by the vertices and the intersection point of the diagonals of a given rectangle (which is not a square), with all these triangles having a common vertex at a given fixed vertex of the rectangle? How many of these triangles are right-angled? How does the problem change if we use any interior point of the rectangle instead of the intersection of the diagonals?


Given a trapezoid \(ABCD\) where \(AD \parallel BC\), \(BC = AC = 5\), and \(AD = 6\). The angle \(ACB\) is twice the measure of angle \(ADB\). Find the area of the trapezoid.


$n$ trains travel in the same direction along a circular track at equal intervals. Stations $A$, $B$, and $C$ are located at the vertices of an equilateral triangle along this track (in the direction of travel). Ira boards at station $A$ and Alex boards at station $B$ to catch the nearest trains. It is known that if they arrive at the stations just as the engineer Roma is passing through the forest, then Ira will board a train before Alex; otherwise, Alex boards before Ira or at the same time. What portion of the track passes through the forest?


Given a triangle \( \triangle ABC \) with internal angles \( A \), \( B \), and \( C \) such that \( \cos A = \sin B = 2 \tan \frac{C}{2} \), determine the value of \( \sin A + \cos A + 2 \tan A \).


Given a rectangular prism with a base $A B C D$ and a top face parallel midpoint line $E F$ where the midpoint of this segment is $G$. The reflection of point $X$ on segment $E G$ across point $G$ is $Y$. For which position of point $X$ will the sum $A X + D X + X Y + Y B + Y C$ be minimized?

Solution: \boxed{x = G}


A pair of natural numbers is called "good" if one of the numbers is divisible by the other. The numbers from 1 to 30 are divided into 15 pairs. What is the maximum number of good pairs that could be formed?


Four points in the order \( A, B, C, D \) lie on a circle with the extension of \( AB \) meeting the extension of \( DC \) at \( E \) and the extension of \( AD \) meeting the extension of \( BC \) at \( F \). Let \( EP \) and \( FQ \) be tangents to this circle with points of tangency \( P \) and \( Q \) respectively. Suppose \( EP = 60 \) and \( FQ = 63 \). Determine the length of \( EF \).


A triangle \( EGF \) is inscribed in a circle with center \( O \), and the angle \( \angle EFG \) is obtuse. There exists a point \( L \) outside the circle such that \( \angle LEF = \angle FEG \) and \( \angle LGF = \angle FGE \). Find the radius of the circumcircle of triangle \( ELG \), given that the area of triangle \( EGO \) is \( 81 \sqrt{3} \) and \( \angle OEG = 60^\circ \).


Find the area of the triangle that is cut off by the line \( y = 2x + 2 \) from the figure defined by the inequality \( |x-2| + |y-3| \leq 3 \).


Calculate the definite integral:

$$
\int_{0}^{2 \pi} \sin^{6} x \cos^{2} x \, dx
$$


Given nine different numbers, how many different values of third-order determinants can be formed using all of them?


In triangle \( ABC \), side \( AB \) is longer than side \( BC \), and angle \( B \) is \( 40^\circ \). Point \( P \) is taken on side \( AB \) such that \( BP = BC \). The angle bisector \( BM \) intersects the circumcircle of triangle \( ABC \) at point \( T \). Find the angle \( MPT \).


As shown in the figure, on a rectangular table with dimensions $9 \mathrm{~cm}$ in length and $7 \mathrm{~cm}$ in width, a small ball is shot from point $A$ at a 45-degree angle. Upon reaching point $E$, it bounces off at a 45-degree angle and continues to roll forward. Throughout its motion, the ball bounces off the table edges at a 45-degree angle each time. Starting from point $A$, after how many bounces does the ball first reach point $C$?


Given \( x, y, z > 0 \) and \( x + y + z = 1 \), find the maximum value of
$$
f(x, y, z) = \sum \frac{x(2y - z)}{1 + x + 3y}
$$


The axial section $SAB$ of a conical frustum is an equilateral triangle with side length 2. $O$ is the center of the base, and $M$ is the midpoint of $SO$. The moving point $P$ is on the base of the conical frustum (including the circumference). If $AM \perp MP$, find the length of the locus formed by point $P$.


In a $28 \times 35$ table, some $k$ cells are colored red, some $r$ cells are colored pink, and the remaining $s$ cells are colored blue. It is known that:

- $k \geqslant r \geqslant s$
- Each boundary cell has at least 2 neighbors of the same color
- Each non-boundary cell has at least 3 neighbors of the same color

What is the smallest possible value of $k - s$?

(A cell is called a boundary cell if it is adjacent to the border of the table. Neighbors are cells that share a common side.)


Let \( X \) be the set of residues modulo 17. We regard two members of \( X \) as adjacent if they differ by 1, so 0 and 16 are adjacent. We say that a permutation of \( X \) is dispersive if it never maps two adjacent values to two adjacent values, and connective if it always maps two adjacent values to two adjacent values. What is the largest \( N \) for which we can find a permutation \( p \) on \( X \) such that \( p \), \( p^2 \), ..., \( p^{N-1} \) are all dispersive and \( p^N \) is connective?


Let $\{x\}$ denote the fractional part of the real number $x$. Given $a=(5 \sqrt{2}+7)^{2017}$, find the value of $a\{a\}$.


A $100 \times 100$ square is divided into $2 \times 2$ squares. Then it is divided into dominos (rectangles $1 \times 2$ and $2 \times 1$). What is the smallest number of dominos that could have been inside the divided squares?


At a height \( BH \) of triangle \( ABC \), a point \( D \) is marked. Line \( AD \) intersects side \( BC \) at point \( E \), and line \( CD \) intersects side \( AB \) at point \( F \). It is known that \( BH \) divides segment \( FE \) in the ratio \( 1:3 \) starting from point \( F \). Find the ratio \( FH:HE \).


In $\triangle ABC$, the side lengths opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. If $c - a$ is equal to the height $h$ from vertex $A$ to side $BC$, then the value of $\sin \frac{C - A}{2} + \cos \frac{C + A}{2}$ is

(A) 1
(B) $\frac{1}{2}$
(C) $\frac{1}{3}$
(D) -1


In the Chinese idioms "虚有其表", "表里如一", "一见如故", and "故弄玄虚", each Chinese character represents one of 11 consecutive non-zero natural numbers. Identical characters represent the same number, and different characters represent different numbers. The order of the numbers is such that "表" > "一" > "故" > "如" > "虚". Additionally, the sum of the numbers represented by the four characters in each idiom is 21. What is the maximum number that "弄" can represent?


The diagonals of a convex quadrilateral $ABCD$, inscribed in a circle, intersect at point $E$. It is known that diagonal $BD$ is the angle bisector of $\angle ABC$ and that $BD = 25$ and $CD = 15$. Find $BE$.


Using three colors $\mathbf{R}$, $\mathbf{G}$, and $\mathbf{B}$ to color a $2 \times 5$ table in a way that two squares sharing a common edge must be colored differently. How many different coloring methods are there?


In the rectangular coordinate plane, find the number of integer points that satisfy the system of inequalities
\[
\left\{
\begin{array}{l}
y \leqslant 3x \\
y \geqslant \frac{x}{3} \\
x + y \leqslant 100
\end{array}
\right.
\]


Calculate the surface integrals of the first kind:
a) \(\iint_{\sigma}|x| dS\), where \(\sigma\) is defined by \(x^2 + y^2 + z^2 = 1\), \(z \geqslant 0\).

b) \(\iint_{\sigma} (x^2 + y^2) dS\), where \(\sigma\) is defined by \(x^2 + y^2 = 2z\), \(z = 1\).

c) \(\iint_{\sigma} (x^2 + y^2 + z^2) dS\), where \(\sigma\) is the part of the cone defined by \(z^2 - x^2 - y^2 = 0\), \(z \geqslant 0\), truncated by the cylinder \(x^2 + y^2 - 2x = 0\).

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


Calculate the area of the common part of two rhombuses, where the lengths of the diagonals of the first one are 4 cm and 6 cm, and the second one is obtained by rotating the first one by 90 degrees around its center.

Solution: \boxed{9.6 \ \text{cm}^2}


Through point \( A \) of a circle with a radius of 10, two mutually perpendicular chords \( AB \) and \( AC \) are drawn. Calculate the radius of a circle that is tangent to the given circle and the constructed chords, given \( AB = 16 \).


Find the largest natural number in which all digits are different, and the sum of any two of its digits is a prime number.


In how many ways can the numbers \(1, 2, \ldots, 2002\) be placed at the vertices of a regular 2002-gon so that no two adjacent numbers differ by more than 2? (Rotations and reflections are considered distinct.)


Natural numbers \(a, b, c\) are chosen such that \(a < b < c\). It is also known that the system of equations \(2x + y = 2037\) and \(y = |x-a| + |x-b| + |x-c|\) has exactly one solution. Find the minimum possible value of \(c\).