Kelvin the Frog is going to roll three fair ten-sided dice with faces labelled $0,1,2, \ldots, 9$. First he rolls two dice, and finds the sum of the two rolls. Then he rolls the third die. What is the probability that the sum of the first two rolls equals the third roll?


In Mexico, ecologists have achieved the passage of a law whereby each car must not be driven at least one day a week (the owner informs the police of the car’s number and "day off" for that car). In a certain family, all adults wish to drive daily (each for their own errands!). How many cars (at minimum) does the family need if there are

a) 5 adults? b) 8 adults?


The city's bus network is organized as follows:

1) From each stop, it is possible to reach any other stop without transfer.
2) For each pair of routes, there is exactly one unique stop where passengers can transfer from one route to the other.
3) Each route has exactly three stops.

How many bus routes are in the city? (It is known that there is more than one route.)


In a box, there are red and black socks. If two socks are randomly taken from the box, the probability that both of them are red is $1/2$.

a) What is the minimum number of socks that can be in the box?

b) What is the minimum number of socks that can be in the box, given that the number of black socks is even?


Solve the inequality \(\sqrt{x-4}+\sqrt{x+1}+\sqrt{2x}-\sqrt{33-x} > 4\). Provide the sum of all its integer solutions in the answer.


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

\[ \left\{\begin{array}{l}
x^{2}+x y+y^{2}=48 \\
y^{2}+y z+z^{2}=25 \\
z^{2}+x z+x^{2}=73
\end{array}\right. \]

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


There is an expression written on the board, $\frac{a}{b} \cdot \frac{c}{d} \cdot \frac{e}{f}$, where $a, b, c, d, e, f$ are natural numbers. If the number $a$ is increased by 1, the value of this expression increases by 3. If the number $c$ in the original expression is increased by 1, its value increases by 4. If the number $e$ in the original expression is increased by 1, its value increases by 5. What is the smallest value that the product $b d f$ can take?


The settlements Arkadino, Borisovo, and Vadimovo are connected pairwise by straight roads. Adjacent to the road between Arkadino and Borisovo is a square field, one side of which coincides entirely with this road. Adjacent to the road between Borisovo and Vadimovo is a rectangular field, one side of which coincides entirely with this road, and the other side is 4 times longer. Adjacent to the road between Arkadino and Vadimovo is a rectangular forest, one side of which coincides entirely with this road, and the other side is 12 km long. The area of the forest is 45 square kilometers larger than the sum of the areas of the fields. Find the total area of the forest and fields in square kilometers.


Two numbers \(x\) and \(y\) are randomly chosen from the interval \([0, 2]\). Find the probability that these numbers satisfy the inequalities \(x^{2} \leq 4y \leq 4x\).


Inside the cube \( ABCD A_1 B_1 C_1 D_1 \) is located the center \( O \) of a sphere with a radius of 10. The sphere intersects the face \( A A_1 D_1 D \) in a circle with a radius of 1, the face \( A_1 B_1 C_1 D_1 \) in a circle with a radius of 1, and the face \( C D D_1 C_1 \) in a circle with a radius of 3. Find the length of the segment \( O D_1 \).


Find the probability that a randomly selected 8-digit number composed of 0s and 1s has the sum of the digits in even positions equal to the sum of the digits in odd positions.


In the sequence $\left\{a_{n}\right\}$, $a_{1}=2$, $a_{2}=-19$, and $a_{n+2}=\left|a_{n+1}\right|-a_{n}$ for all positive integers $n$. Find $a_{2019}$.


A regular triangle \(ABC\) is inscribed in a circle. Another, smaller circle is inscribed in the sector bounded by chord \(BC\), and it touches the larger circle at point \(M\) and the chord \(BC\) at point \(K\). Ray \(MK\) intersects the larger circle a second time at point \(N\). Find the length of \(MN\) if the sum of the distances from point \(M\) to the ends of chord \(BC\) is 6.


As shown in Figure 18-3, in a regular hexagon consisting of six regions, ornamental plants are to be planted with the requirement that the same type of plant is planted within each region and different types of plants are planted in adjacent regions. Given that there are 4 different types of plants available, how many different planting schemes are possible?


A natural number was squared and 600 was subtracted from the result. The same operation was then performed on the resulting number, and so on. What could the original number be, if it is known that after several such operations, the same number was obtained again?


In the convex pentagon \(ABCDE\), the angles \(\angle ABC\) and \(\angle CDE\) are \(90^\circ\), \(BC = CD = AE = 1\), and \(AB + ED = 1\). What is the area of the pentagon?


In a certain game, the "magician" asks a person to randomly think of a three-digit number ($abc$), where $a, b, c$ are the digits of this number. They then ask this person to arrange the digits into 5 different numbers: $(acb)$, $(bac)$, $(bca)$, $(cab)$, and $(cba)$. The magician then asks for the sum of these 5 numbers, denoted as $N$. Once the magician knows $N$, they can determine the original number ($abc$).

Given that $N=3194$, assume the role of the "magician" and determine the original number ($abc$).


A plane intersects each side of the quadrilateral \(ABCD\) at an internal point. Following a traversing direction, we write down the ratio in which each intersection point divides its corresponding side. What will be the product of the four ratios?


In a geometric sequence where the common ratio is greater than 1, what is the maximum number of terms that are integers between 100 and 1000?


The numbers \(a, b, c, d\) belong to the interval \([-11.5, 11.5]\). Find the maximum value of the expression \(a + 2b + c + 2d - ab - bc - cd - da\).


Let \( A B C D E F \) be a regular hexagon of area 1. Let \( M \) be the midpoint of \( D E \). Let \( X \) be the intersection of \( A C \) and \( B M \), let \( Y \) be the intersection of \( B F \) and \( A M \), and let \( Z \) be the intersection of \( A C \) and \( B F \). If \([P]\) denotes the area of polygon \( P \) for any polygon \( P \) in the plane, evaluate \([B X C] + [A Y F] + [A B Z] - [M X Z Y]\).


Let \( f(x) = \frac{x + a}{x^2 + \frac{1}{2}} \), where \( x \) is a real number and the maximum value of \( f(x) \) is \( \frac{1}{2} \) and the minimum value of \( f(x) \) is \( -1 \). If \( t = f(0) \), find the value of \( t \).


Xiaopang arranges the 50 integers from 1 to 50 in ascending order without any spaces in between. Then, he inserts a "+" sign between each pair of adjacent digits, resulting in an addition expression: \(1+2+3+4+5+6+7+8+9+1+0+1+1+\cdots+4+9+5+0\). Please calculate the sum of this addition expression. The result is ________.


Lil Wayne, the rain god, determines the weather. If Lil Wayne makes it rain on any given day, the probability that he makes it rain the next day is 75%. If Lil Wayne doesn't make it rain on one day, the probability that he makes it rain the next day is 25%. He decides not to make it rain today. Find the smallest positive integer \( n \) such that the probability that Lil Wayne makes it rain \( n \) days from today is greater than 49.9%.


The solution set of the inequality \(\log _{a}\left(a-\frac{x^{2}}{2}\right)>\log _{a}(a-x)\) is \(A\), and \(A \cap \mathbf{Z}=\{1\}\). What is the range of values for \(a\)?

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


Let \( M \) be a set of \( n \) points on a plane, satisfying the following conditions:
1. There are 7 points in \( M \) that are the vertices of a convex heptagon.
2. For any 5 points in \( M \), if these 5 points are the vertices of a convex pentagon, then the interior of this convex pentagon contains at least one point from \( M \).

Find the minimum value of \( n \).


A lieutenant is training recruits in marching drills. Upon arriving at the parade ground, he sees that all the recruits are arranged in several rows, with each row having the same number of soldiers, and that the number of soldiers in each row is 5 more than the number of rows. After finishing the drills, the lieutenant decides to arrange the recruits again but cannot remember how many rows there were initially. So, he orders them to form as many rows as his age. It turns out that each row again has the same number of soldiers, but in each row, there are 4 more soldiers than there were in the original arrangement. How old is the lieutenant?


Let the complex number \( z = \cos \theta + \mathrm{i} \sin \theta \) where \( 0^{\circ} \leqslant \theta \leqslant 180^{\circ} \). The complex number \( z \), \( (1+\mathrm{i})z \), and \( 2\bar{z} \) correspond to the points \( P \), \( Q \), and \( R \) on the complex plane, respectively. When \( P \), \( Q \), and \( R \) are not collinear, the fourth vertex of the parallelogram formed with line segments \( PQ \) and \( PR \) as two sides is \( S \). Find the maximum distance from point \( S \) to the origin.


The numbers $1978^{n}$ and $1978^{m}$ have the same last three digits. Find the positive integers $n$ and $m$ such that $m+n$ is minimized, given that $n > m \geq 1$.


Find the smallest \( n > 2016 \) such that \( 1^{n} + 2^{n} + 3^{n} + 4^{n} \) is not divisible by 10.


If \( a \geq b \geq c \), \( a + b + c = 0 \), and \( x_{1} \), \( x_{2} \) are the two real roots of the quadratic equation \( a x^{2} + b x + c = 0 \), then the sum of the maximum and minimum values of \( \left| x_{1}^{2} - x_{2}^{2} \right| \) is \(\quad\) .


Xiaoming puts 127 Go stones into several bags. No matter how many stones (not exceeding 127) a friend wants, Xiaoming can satisfy the request by taking out a few bags. How many bags does Xiaoming need at least?


Every month Ivan pays a fixed amount of his salary for a mortgage, and the remaining portion of his salary is spent on current expenses. In December, Ivan paid 40% of his salary for the mortgage. In January, Ivan's salary increased by 9%. By what percentage did the amount spent on current expenses increase in January compared to December?


For \( x, y \in [1,3] \), find the minimum value of the expression

\[
A = \frac{(3xy + x^{2}) \sqrt{3xy + x - 3y} + (3xy + y^{2}) \sqrt{3xy + y - 3x}}{x^{2} y + y^{2} x}
\]


The sequence \(\{a_{n}\}\) satisfies \(a_{1}=6\), and for any positive integer \(n\), \(a_{n+1}+n+1=2(a_{n}+1)\). What is the units digit of \(a_{1}+a_{2}+\cdots+a_{2022}\)?


Let $\alpha, \beta, \gamma \in \mathbf{R}$ satisfy the equation $\sin \alpha \cdot \cos \beta + |\cos \alpha \cdot \sin \beta| = \sin \alpha \cdot |\cos \alpha| + |\sin \beta| \cdot \cos \beta$. Then, find the minimum value of $(\tan \gamma - \sin \alpha)^2 + (\cot \gamma - \cos \beta)^2$.

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


A tank with a mass of $m_{1}=2$ kg rests on a cart with a mass of $m_{2}=10$ kg, which is accelerated with an acceleration of $a=5 \, \text{m/s}^2$. The coefficient of friction between the tank and the cart is $\mu=0.6$. Determine the frictional force acting on the tank from the cart.

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


A cylinder \( C \) and a cone \( K \) are circumscribed around a sphere; the volumes of the cylinder and cone are denoted by \( V_{1} \) and \( V_{2} \), respectively. What is the minimum possible value of \( v \) if \( V_{1} = 1 \)? What is the angle at the apex of the axial section of the circumscribed cone \( K \) when its volume is \(\boldsymbol{0}\)?


Calculate the limit of the function:

\[ \lim_{x \to 0} \frac{e^{3x} - e^{2x}}{\sin 3x - \tan 2x} \]


Find the minimum value of the expression:

\[
\left(\sqrt{2(1+\cos 2x)} - \sqrt{9-\sqrt{7}} \sin x + 1\right) \cdot \left(3 + 2 \sqrt{13 - \sqrt{7}} \cos y - \cos 2y\right)
\]

If the answer is not an integer, round it to the nearest whole number.


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

(1) The integers consist of digits from the set \(\{1,2,3,4,5,6\}\).

(2) No digit occurs more than once in the same integer.

(3) The digits in each integer are in increasing order.

(4) Any two integers have at least one digit in common (possibly at different positions).

(5) There is no digit which appears in all the integers.


Find the largest number such that each digit, starting from the third one, is equal to the sum of the two preceding digits.


On the side \( AC \) of an equilateral triangle \( ABC \), a point \( D \) is marked. On the segments \( AD \) and \( DC \), equilateral triangles \( ADE \) and \( DCF \) are constructed outward from the original triangle. It is known that the perimeter of triangle \( DEF \) is 19, and the perimeter of pentagon \( ABCEF \) is 43.

(a) Find the length of segment \( AB \).

(b) Find the length of segment \( EF \).


Let \( x_{1}, x_{2}, \cdots, x_{n} \in \mathbf{R}^{+} \) and define \( S_{n} = \sum_{i=1}^{n}\left(x_{i}+\frac{n-1}{n^{2}} \frac{1}{x_{i}}\right)^{2} \).

1. Find the minimum value of \( S_{n} \).
2. Under the condition \( x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}=1 \), find the minimum value of \( S_{n} \).
3. Under the condition \( x_{1}+x_{2}+ \cdots+x_{n}=1 \), find the minimum value of \( S_{n} \) and provide a proof.

Solution: \boxed{n}


Birgit has a list of four numbers. Luciano adds these numbers together, three at a time, and gets the sums 415, 442, 396, and 325. What is the sum of Birgit's four numbers?


Petya wants to make an unusual die, which, as usual, should have the shape of a cube with dots drawn on its faces (with different numbers of dots on different faces). However, the number of dots on each pair of adjacent faces must differ by at least two (it is allowed to have more than six dots on some faces). How many dots in total are necessary to draw?


In a game called "set," all possible four-digit numbers consisting of the digits 1, 2, and 3 (each used exactly once) are considered. A triplet of numbers forms a set if for each digit position, either all three numbers have the same digit, or all three numbers have different digits.

For example, the numbers 1232, 2213, 3221 form a set because in the first position, all three digits (1, 2, and 3) are present; in the second position, only the digit 2 is present; in the third position, all three digits are present; and in the fourth position, all three digits are present. The numbers 1123, 2231, and 3311 do not form a set because in the last position there are two 1s and a 3.

How many total sets exist in the game?

(Note: Rearranging the numbers does not create a new set. For instance, 1232, 2213, and 3221 is considered the same set as 2213, 1232, and 3221.)


Out of 24 matchsticks of the same length, use some of them to form 6 triangles in a plane where each side of the equilateral triangle is one matchstick. Then, use the remaining matchsticks to form squares in the plane where each side of the square is one matchstick. What is the maximum number of such squares that can be formed?


What will the inflation be over two years:
$$
\left((1+0,015)^{\wedge} 2-1\right)^{*} 100 \%=3,0225 \%
$$

What will be the real yield of a bank deposit with an extension for the second year:
$$
(1,07 * 1,07 /(1+0,030225)-1) * 100 \%=11,13 \%
$$


Armen paid \$190 to buy movie tickets for a group of \( t \) people, consisting of some adults and some children. Movie tickets cost \$5 for children and \$9 for adults. How many children's tickets did he buy?


The numbers $96, 28, 6, 20$ were written on the board. One of them was multiplied, another was divided, another was increased, and another was decreased by the same number. As a result, all the numbers became equal to a single number. What is that number?


The line segments \(PQRS\) and \(WXY S\) intersect circle \(C_1\) at points \(P, Q, W\) and \(X\). The line segments intersect circle \(C_2\) at points \(Q, R, X\) and \(Y\). The lengths \(QR\), \(RS\), and \(XY\) are 7, 9, and 18 respectively. The length \(WX\) is six times the length \(YS\). What is the sum of the lengths of \(PS\) and \(WS\)?


Given that \(ABCD - A_1B_1C_1D_1\) is a unit cube, determine the distance between the lines \(AC\) and \(A_1D\).


For a positive integer \( n \), define \( s(n) \) as the smallest positive integer \( t \) such that \( n \) is a factor of \( t! \). Compute the number of positive integers \( n \) for which \( s(n) = 13 \).


A certain quadratic polynomial is known to have the following properties: its leading coefficient is equal to one, it has integer roots, and its graph (parabola) intersects the line \( y = 2017 \) at two points with integer coordinates. Can the ordinate of the vertex of the parabola be uniquely determined based on this information?


As shown in Figure 1, a cross-section of cube \(ABCDEFGH\) passes through vertices \(A\), \(C\), and a point \(K\) on edge \(EF\), and divides the cube into two parts with a volume ratio of 3:1. Find the value of \(\frac{EK}{KF}\).


The classrooms at MIT are each identified with a positive integer (with no leading zeroes). One day, as President Reif walks down the Infinite Corridor, he notices that a digit zero on a room sign has fallen off. Let \( N \) be the original number of the room, and let \( M \) be the room number as shown on the sign.

The smallest interval containing all possible values of \( \frac{M}{N} \) can be expressed as \( \left[\frac{a}{b}, \frac{c}{d}\right) \) where \( a, b, c, d \) are positive integers with \( \operatorname{gcd}(a, b) = \operatorname{gcd}(c, d) = 1 \). Compute \( 1000a + 100b + 10c + d \).


Find the sum of all four-digit numbers that can be formed using the digits \(1, 2, 3, 4, 5\) such that each digit appears at most once.


Investigate for which values of $\alpha>0$, the improper integral $\int_{a}^{b} \frac{d x}{(x-a)^{\alpha}}$ (where $b>a$) converges.

Solution: \boxed{\alpha < 1}


Given a natural number \( x = 8^n - 1 \), where \( n \) is a natural number. It is known that \( x \) has exactly three distinct prime divisors, one of which is 31. Find \( x \).


It is known that the only solution to the equation

$$
\pi / 4 = \operatorname{arcctg} 2 + \operatorname{arcctg} 5 + \operatorname{arcctg} 13 + \operatorname{arcctg} 34 + \operatorname{arcctg} 89 + \operatorname{arcctg}(x / 14)
$$

is a natural number. Find it.


Show that we always get a perfect cube when we place \( n \) digits of 9 before the digit 7, \( n \) digits of 0 between the digits 7 and 2, and \( n \) digits of 9 between the digits 2 and 9 in the cube number 729. \( (n=1,2,\ldots) \)

Solution: \boxed{(10^{n+1} - 1)^3}


In the country of Draconia, there are red, green, and blue dragons. Each dragon has three heads; each head always tells the truth or always lies. Each dragon has at least one head that tells the truth. One day, 530 dragons sat around a round table, and each dragon's heads made the following statements:

- 1st head: "To my left is a green dragon."
- 2nd head: "To my right is a blue dragon."
- 3rd head: "There is no red dragon next to me."

What is the maximum number of red dragons that could have been at the table?


Given two linear functions \( f(x) \) and \( g(x) \) such that the graphs \( y=f(x) \) and \( y=g(x) \) are parallel lines, but not parallel to the coordinate axes. Find the minimum value of the function \( (g(x))^2 - 3f(x) \), if the minimum value of the function \( (f(x))^2 - 3g(x) \) is \( \frac{11}{2} \).


Joe has a triangle with area \(\sqrt{3}\). What's the smallest perimeter it could have?


Let \( S-ABC \) be a triangular prism with the base being an isosceles right triangle \( ABC \) with \( AB \) as the hypotenuse, and \( SA = SB = SC = 2 \) and \( AB = 2 \). If \( S \), \( A \), \( B \), and \( C \) are points on a sphere centered at \( O \), find the distance from point \( O \) to the plane \( ABC \).


a) What is the maximum number of bishops that can be placed on a 1000 by 1000 chessboard such that they do not attack each other?

b) What is the maximum number of knights that can be placed on an 8 by 8 chessboard such that they do not attack each other?


Mitya is 11 years older than Shura. When Mitya was as old as Shura is now, he was twice as old as she was then. How old is Mitya?


The side length of an equilateral triangle $ABC$ is 4. Point $D$ is the midpoint of side $BC$. A line passes through point $B$ and intersects side $AC$ at point $M$. Perpendiculars $DH$ and $AK$ are dropped from points $D$ and $A$ onto the line $BM$. Calculate the length of segment $AM$ if
$$
A K^{4} - D H^{4} = 15
$$


For any positive integer \( x_{0} \), three sequences \(\{x_{n}\}, \{y_{n}\}, \{z_{n}\}\) are defined as follows:
1. \( y_{0} = 4 \) and \( z_{0} = 1 \);
2. if \( x_{n} \) is even for \( n \geq 0 \), then
\[
x_{n+1} = \frac{x_{n}}{2}, \quad y_{n+1} = 2 y_{n}, \quad z_{n+1} = z_{n};
\]
3. if \( x_{n} \) is odd for \( n \geq 0 \), then
\[
x_{n+1} = x_{n} - \frac{y_{n}}{2} - z_{n}, \quad y_{n+1} = y_{n}, \quad z_{n+1} = y_{n} + z_{n}.
\]

The integer \( x_{0} \) is said to be good if \( x_{n} = 0 \) for some \( n \geq 1 \). Find the number of good integers less than or equal to 1994.


The teacher wrote a four-digit number on a piece of paper for Xiaowei to guess. They had four rounds of questions and answers.
Xiaowei: "Is it 8765?"
Teacher: "You guessed two digits correctly, but both are in the wrong positions."
Xiaowei: "Is it 1023?"
Teacher: "You guessed two digits correctly, but both are in the wrong positions."
Xiaowei: "Is it 8642?"
Teacher: "You guessed two digits correctly, and both are in the correct positions."
Xiaowei: "Is it 5430?"
Teacher: "None of the digits are correct."
What is this four-digit number $\qquad$?


Given that \( x = x_{0} \) and \( y = y_{0} \) satisfy the system of equations

\[
\left\{\begin{array}{l}
\frac{x}{3}+\frac{y}{5}=1 \\
\frac{x}{5}+\frac{y}{3}=1
\end{array}\right.
\]

If \( B=\frac{1}{x_{0}}+\frac{1}{y_{0}} \), find the value of \( B \).


Does \( n^2 \) have more divisors that are congruent to \( 1 \mod 4 \) or \( 3 \mod 4 \)?

Solution: \boxed{ 1 \mod 4 }


Calculate the areas of the figures bounded by the curves:

a)
\[
\left\{
\begin{array}{l}
y = x \sqrt{9 - x^{2}} \\
y = 0 \quad (0 \leq x \leq 3)
\end{array}
\right.
\]

b)
\[
\left\{
\begin{array}{l}
y = 2x - x^{2} + 3 \\
y = x^{2} - 4x + 3
\end{array}
\right.
\]


There are 101 natural numbers written in a circle. It is known that among any three consecutive numbers, there is at least one even number. What is the minimum number of even numbers that can be among the written numbers?


Given that \(a, b, c, d\) are positive numbers and
\[ a + 20b = c + 20d = 2, \]
find the minimum value of \(\frac{1}{a} + \frac{1}{bcd}\).

Solution: \boxed{ \frac{441}{2} }


Little children were eating candies. Each ate 7 fewer candies than all the others combined, but still more than one candy. How many candies were eaten in total?


Find all positive integers \( M \) less than 10 such that 5 divides \( 1989^M + M^{1889} \).

Solution: \boxed{1 \text{ and } 4}


For which real \( a \) are there distinct reals \( x \) and \( y \) such that \( x = a - y^2 \) and \( y = a - x^2 \) ?

Solution: \boxed{a > \frac{3}{4}}


If a die is rolled 500 times, what is the most probable number of times that the face showing 1 dot will appear?


Given the sequences $\{a_n\}$ and $\{b_n\}$ such that $a_1 = -1$, $b_1 = 2$, and the recurrence relations $a_{n+1} = -b_n$, $b_{n+1} = 2a_n - 3b_n$ for $n \in \mathbf{N}^*$, find the value of $b_{2015} + b_{2016}$.

Solution: \boxed{-3 \cdot 2^{2015}}


Let \( a \) and \( b \) be positive real numbers such that \(\frac{a}{1+a}+\frac{b}{1+b}=1\). Show that

\[ \frac{a}{1+b^{2}}-\frac{b}{1+a^{2}}=a-b \]

Solution: \boxed{a - b}


The shorter side of a rectangle is equal to 1, and the acute angle between the diagonals is $60^\circ$. Find the radius of the circle circumscribed around the rectangle.


In the figure, segment \(DE\) divides square sheet \(ABCD\) into \(\triangle ADE\) and quadrilateral \(EDCB\). Given that the area ratio \(S_{\triangle ADE} : S_{EDCB} = 5 : 19\), determine the ratio of the perimeter of \(\triangle ADE\) to the perimeter of quadrilateral \(EDCB\).

Solution: \boxed{ \frac{15}{22} }


If the real numbers \( x \) and \( y \) satisfy \( x^{2} + y^{2} - 2x + 4y = 0 \), find the maximum value of \( x - 2y \).


Given ten points on the boundary line of a half-plane, how many ways can the points be paired such that the points paired together can be connected by non-intersecting lines within the half-plane?


In how many ways can all natural numbers from 1 to 200 be painted in red and blue so that no sum of two different numbers of the same color equals a power of two?


Given the sets \( A = \{(x, y) \mid y^2 - x - 1 = 0\} \), \( B = \{(x, y) \mid 4x^2 + 2x - 2y + 5 = 0\} \), and \( C = \{(x, y) \mid y = kx + b\} \), find all non-negative integers \( k \) and \( b \) such that \( (A \cup B) \cap C = \emptyset \).

Solution: \boxed{(k, b) = (1, 2)}


Calculate the limit of the numerical sequence:
$$\lim _{n \rightarrow \infty} \frac{\sqrt{n^{5}-8}-n \sqrt{n\left(n^{2}+5\right)}}{\sqrt{n}}$$


The sum of the first three terms of an increasing arithmetic progression is 15. If 1 is subtracted from each of the first two terms and 1 is added to the third term, the resulting three numbers form a geometric progression. Find the sum of the first ten terms of the arithmetic progression.


The faces of a 12-sided die are numbered 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12 such that the sum of the numbers on opposite faces is 13. The die is meticulously carved so that it is biased: the probability of obtaining a particular face \( F \) is greater than \( \frac{1}{12} \), the probability of obtaining the face opposite \( F \) is less than \( \frac{1}{12} \) while the probability of obtaining any one of the other ten faces is \( \frac{1}{12} \).

When two such dice are rolled, the probability of obtaining a sum of 13 is \( \frac{29}{384} \).

What is the probability of obtaining face \( F \)?


The sides of rectangle $ABCD$ are $AB=3$ and $BC=2$. Point $P$ is on side $AB$ such that line $PD$ touches the circle with diameter $BC$ at point $E$. The line passing through the center of the circle and point $E$ intersects side $AB$ at point $Q$. What is the area of triangle $PQE$?


If the maximum value and the minimum value of the function \( f(x) = \frac{a + \sin x}{2 + \cos x} + b \tan x \) sum up to 4, then find \( a + b \).


n people each have exactly one unique secret. How many phone calls are needed so that each person knows all n secrets? You should assume that in each phone call the caller tells the other person every secret he knows, but learns nothing from the person he calls.


The bases \(AB\) and \(CD\) of the trapezoid \(ABCD\) are 367 and 6 respectively, and its diagonals are mutually perpendicular. Find the dot product of the vectors \(\overrightarrow{AD}\) and \(\overrightarrow{BC}\).


Given a non-empty subset family \( U \) of \( S = \{a_1, a_2, \ldots, a_n\} \) that satisfies the property: if \( A \in U \) and \( A \subseteq B \), then \( B \in U \); and a non-empty subset family \( V \) of \( S \) that satisfies the property: if \( A \in V \) and \( A \supseteq B \), then \( B \in V \). Find the maximum possible value of \( \frac{|U \cap V|}{|U| \cdot |V|} \).


Find all integers \( k \geq 1 \) such that there exists a pair of integers \( (n, m) \) for which \( 9n^6 = 2^k + 5m^2 + 2 \).


Let \( d \) be a randomly chosen divisor of 2016. Find the expected value of
\[
\frac{d^{2}}{d^{2}+2016}
\]


Given the sequence \( a_{n}=\left\lfloor(\sqrt{2}+1)^{n}+\left(\frac{1}{2}\right)^{n}\right\rfloor \) for \( n \geq 0 \), where \(\left\lfloor x \right\rfloor\) denotes the greatest integer less than or equal to \( x \), compute the sum:
\[ \sum_{n=1}^{\infty} \frac{1}{a_{n-1} a_{n+1}} \]


Someone wrote down two numbers $5^{2020}$ and $2^{2020}$ consecutively. How many digits will the resulting number contain?


For any positive integer \( n \), let \( a_{n} \) denote the number of triangles with integer side lengths whose longest side is \( 2n \).

(1) Find an expression for \( a_{n} \) in terms of \( n \);

(2) Given the sequence \( \{b_{n}\} \) satisfies
$$
\sum_{k=1}^{n}(-1)^{n-k} \binom{n}{k} b_{k}=a_{n}\quad (n \in \mathbf{Z}_{+}),
$$
find the number of positive integers \( n \) such that \( b_{n} \leq 2019 a_{n} \).


An ant crawls along the edges of a cube with side length 1 unit. Starting from one of the vertices, in each minute the ant travels from one vertex to an adjacent vertex. After crawling for 7 minutes, the ant is at a distance of \(\sqrt{3}\) units from the starting point. Find the number of possible routes the ant has taken.


Four cars $A$, $B$, $C$, and $D$ start simultaneously from the same point on a circular track. Cars $A$ and $B$ travel clockwise, while cars $C$ and $D$ travel counterclockwise. All cars move at constant but distinct speeds. Exactly 7 minutes after the race starts, $A$ meets $C$ for the first time, and at the same moment, $B$ meets $D$ for the first time. After another 46 minutes, $A$ and $B$ meet for the first time. How long after the race starts will $C$ and $D$ meet for the first time?


\[
\frac{\sin ^{2}\left(135^{\circ}-\alpha\right)-\sin ^{2}\left(210^{\circ}-\alpha\right)-\sin 195^{\circ} \cos \left(165^{\circ}-2 \alpha\right)}{\cos ^{2}\left(225^{\circ}+\alpha\right)-\cos ^{2}\left(210^{\circ}-\alpha\right)+\sin 15^{\circ} \sin \left(75^{\circ}-2 \alpha\right)}=-1
\]


In triangle \(ABC\), the sides are \(AB = 10\), \(AC = 24\), and \(BC = 26\). The medians \(AM\) and \(CN\) are drawn, where points \(M\) and \(N\) are the midpoints of sides \(BC\) and \(AB\), respectively. Point \(I\) lies on side \(AC\), and \(BI\) is the angle bisector. Find the area of triangle \(MNI\).


At 9:00, a pedestrian set off on a journey. An hour later, a cyclist set off from the same starting point. At 10:30, the cyclist caught up with the pedestrian and continued ahead, but after some time, the bicycle broke down. After repairing the bike, the cyclist resumed the journey and caught up with the pedestrian again at 13:00. How many minutes did the repair take? (The pedestrian's speed is constant, and he moved without stopping; the cyclist's speed is also constant except for the repair interval.)

Solution: \boxed{100 \text{ minutes}}


\(3.420 \sin 10^{\circ} \cdot \sin 20^{\circ} \cdot \sin 30^{\circ} \cdot \sin 40^{\circ} \cdot \sin 50^{\circ} \cdot \sin 60^{\circ} \cdot \sin 70^{\circ} \cdot \sin 80^{\circ} = \frac{3}{256} \cdot\)


Determine the number of permutations that can be made using 3 green balls, 2 red balls, 2 white balls, and 3 yellow balls such that no two yellow balls are adjacent.


A regular triangular prism $ABC A_1B_1C_1$ is inscribed in a sphere, with base $ABC$ and lateral edges $AA_1, BB_1, CC_1$. Segment $CD$ is a diameter of this sphere, and point $K$ is the midpoint of edge $AA_1$. Find the volume of the prism if $CK = 2 \sqrt{6}$ and $DK = 4$.


When \( s \) and \( t \) range over all real numbers, what is the minimum value of \( (s+5-3|\cos t|)^{2}+(s-2|\sin t|)^{2} \)?


Several students decided to buy a tape recorder priced between 170 and 195 dollars. However, at the last moment, two students decided not to participate in the purchase, so each of the remaining students had to contribute 1 dollar more. How much did the tape recorder cost?


Calculate the area of the region bounded by the graphs of the functions:
$$
x=\sqrt{e^{y}-1}, x=0, y=\ln 2
$$

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


Given the function
\[ f(x) = \sqrt{2x^{2} + 2x + 41} - \sqrt{2x^{2} + 4x + 4} \quad (x \in \mathbb{R}), \]
determine the maximum value of \( f(x) \).


Lele's family raises some chickens and ducks. Grabbing any 6 of them, there are at least 2 that are not ducks; Grabbing any 9 of them, there is at least 1 that is a duck. What is the maximum number of chickens and ducks in Lele's family?


The product of the digits of any multi-digit number is always less than this number. If we calculate the product of the digits of a given multi-digit number, then the product of the digits of this product, and so on, we will necessarily reach a single-digit number after some number of steps. This number of steps is called the persistence of the number. For example, the number 723 has a persistence of 2 because $7 \cdot 2 \cdot 3 = 42$ (1st step) and $4 \cdot 2 = 8$ (2nd step).

1. Find the largest odd number with distinct digits that has a persistence of 1.
2. Find the largest even number with distinct nonzero digits that has a persistence of 1.
3. Find the smallest natural number that has a persistence of 3.


Given that the equation \(|x| - \frac{4}{x} = \frac{3|x|}{x}\) has \(k\) distinct real root(s), find the value of \(k\).


Luis wrote the sequence of natural numbers starting from 1:

$$
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, \cdots
$$

When did he write the digit 3 for the 25th time?


Show that if \(a, b, c,\) and \(d\) are four positive real numbers such that \( abcd = 1 \), then

\[
a^{2}+b^{2}+c^{2}+d^{2}+ab+ac+ad+bc+bd+cd \geq 10
\]


A die is rolled twice continuously, resulting in numbers $a$ and $b$. What is the probability $p$, in numerical form, that the cubic equation in $x$, given by $x^{3}-(3 a+1) x^{2}+(3 a+2 b) x-2 b=0$, has three distinct real roots?


Place 6 points inside a rectangle with dimensions $4 \times 3$. Show that we can find two points whose distance is less than or equal to $\sqrt{5}$.


Given the sequence $\left\{a_{n}\right\}$ such that:
$$
a_{1}=-2,
$$

and $S_{n}=\frac{3}{2} a_{n}+n$ (where $S_{n}$ is the sum of the first $n$ terms of the sequence $\left\{a_{n}\right\}$). Let $f(x)$ be an odd function defined on $\mathbf{R}$, which satisfies:
$$
f(2-x)=f(x).
$$

Find $f\left(a_{2021}\right)=$ $\qquad$


If the 200th day of some year is a Sunday and the 100th day of the following year is also a Sunday, what day of the week was the 300th day of the previous year? Provide the answer as the number of the day of the week (if Monday, then 1; if Tuesday, then 2, etc.).


From the eight natural numbers 1 to 8, how many ways are there to choose three numbers such that no two numbers are consecutive?


The solutions to the equation \( x^3 - 4 \lfloor x \rfloor = 5 \), where \( x \) is a real number, are denoted by \( x_1, x_2, x_3, \ldots, x_k \) for some positive integer \( k \). Find \( \sum_{i=1}^{k} x_{i}^{3} \).


In right triangle \( ABC \), a point \( D \) is on hypotenuse \( AC \) such that \( BD \perp AC \). Let \(\omega\) be a circle with center \( O \), passing through \( C \) and \( D \) and tangent to line \( AB \) at a point other than \( B \). Point \( X \) is chosen on \( BC \) such that \( AX \perp BO \). If \( AB = 2 \) and \( BC = 5 \), then \( BX \) can be expressed as \(\frac{a}{b}\) for relatively prime positive integers \( a \) and \( b \). Compute \( 100a + b \).


Several points, including points \(A\) and \(B\), are marked on a line. All possible segments with endpoints at the marked points are considered. Vasya counted that point \(A\) is inside 50 of these segments, and point \(B\) is inside 56 of these segments. How many points were marked? (The endpoints of a segment are not considered its internal points.)


If a two-digit number is divided by a certain integer, the quotient is 3 and the remainder is 8. If the digits of the dividend are swapped and the divisor remains the same, the quotient is 2 and the remainder is 5. Find the original value of the dividend.


Given the set \( S = \{1, 2, \cdots, 2005\} \), and a subset \( A \subseteq S \) such that the sum of any two numbers in \( A \) is not divisible by 117, determine the maximum value of \( |A| \).


There are two types of tables in a restaurant: a square table can seat 4 people, and a round table can seat 9 people. The restaurant manager calls a number a "wealth number" if the total number of diners can exactly fill a certain number of tables. How many "wealth numbers" are there among the numbers from 1 to 100?


The function \( f \) maps the set of positive integers onto itself and satisfies the equation

\[
f(f(n)) + f(n) = 2n + 6
\]

What could this function be?

Solution: \boxed{f(n) = n + 2}


Urn A contains 4 white balls and 2 red balls. Urn B contains 3 red balls and 3 black balls. An urn is randomly selected, and then a ball inside of that urn is removed. We then repeat the process of selecting an urn and drawing out a ball, without returning the first ball. What is the probability that the first ball drawn was red, given that the second ball drawn was black?


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

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


Along the school corridor hangs a Christmas garland consisting of red and blue bulbs. Next to each red bulb, there must be a blue bulb. What is the maximum number of red bulbs that can be in this garland if there are a total of 50 bulbs?


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

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


Determine the largest prime factor of the sum \(\sum_{k=1}^{11} k^{5}\).


The journey from Petya's home to school takes him 20 minutes. One day, on his way to school, Petya remembered that he had forgotten a pen at home. If he continues his journey at the same speed, he will arrive at school 3 minutes before the bell rings. However, if he returns home for the pen and then goes to school at the same speed, he will be 7 minutes late for the start of the lesson. What fraction of the way to school had he covered when he remembered about the pen?


Let's define the distance between two numbers as the absolute value of their difference. It is known that the sum of the distances from twelve consecutive natural numbers to a certain number \(a\) is 358, and the sum of the distances from these same twelve numbers to another number \(b\) is 212. Find all possible values of \(a\), given that \(a + b = 114.5\).

Solution: \boxed{ \frac{190}{3}}


Given a triangular pyramid \( S-ABC \) with vertex \( S \). The projection of \( S \) onto the base \( \triangle ABC \) is the orthocenter \( H \) of \( \triangle ABC \). Additionally, \( BC = 2 \), \( SB = SC \), and the dihedral angle between the face \( SBC \) and the base is \( 60^\circ \). Determine the volume of the pyramid.


On the island of Misfortune, there live knights who always tell the truth and liars who always lie. One day, 2023 natives, among whom $N$ are liars, stood in a circle and each said, "Both of my neighbors are liars". How many different values can $N$ take?


There are large, medium, and small cubic pools, with inner edge lengths of 6 meters, 3 meters, and 2 meters, respectively. When two piles of gravel are submerged in the medium and small pools, the water levels in the respective pools rise by 6 centimeters and 4 centimeters. If these two piles of gravel are both submerged in the large pool, by how many centimeters will the water level rise?


Determine the minimum value of \( \sum_{k=1}^{50} x_{k} \), where the summation is done over all possible positive numbers \( x_{1}, \ldots, x_{50} \) satisfying \( \sum_{k=1}^{50} \frac{1}{x_{k}} = 1 \).


Find the largest real number \( k \) such that there exists a sequence of positive real numbers \(\{a_i\}\) for which \(\sum_{n=1}^{\infty} a_n\) converges but \(\sum_{n=1}^{\infty} \frac{\sqrt{a_n}}{n^k}\) does not.


Let \( ABC \) be a right triangle at \( C \). On the line \( (AC) \), place a point \( D \) such that \( CD = BC \), with \( C \) positioned between \( A \) and \( D \). The perpendicular to \( (AB) \) passing through \( D \) intersects \( (BC) \) at \( E \). Show that \( AC = CE \).

Solution: \boxed{AC = CE}


A flock of geese was flying. At each lake, half of the geese and half a goose landed. The rest continued flying. All the geese landed on $n$ lakes.

How many geese were there in the flock in total?

Solution: \boxed{2^n - 1}


To determine the roots of the following system of equations with an error less than $0.01$, how many decimal places must we consider for the value of $\sqrt{2}$?

$$
\begin{aligned}
& \sqrt{2} x + 8.59 y = 9.98 \\
& 1.41 x + 8.59 y = 10
\end{aligned}
$$


Given acute angles \(A\) and \(B\) such that \(\tan (A+B) = 2 \tan A\), what is the maximum value of \(\tan B\)?


The sum of a set of numbers is the sum of all its elements. Let \( S \) be a set consisting of positive integers not exceeding 15, where the sums of any two disjoint subsets of \( S \) are not equal. Among all sets with this property, find the \( S \) with the maximum sum, and determine the sum of this set.


The triangle \(ABC\) is isosceles with \(AB = BC\). The point \(D\) is a point on \(BC\), between \(B\) and \(C\), such that \(AC = AD = BD\). What is the size of angle \(ABC\)?


A plane divides the medians of the faces $ABC$, $ACD$, and $ABD$ of a tetrahedron $ABCD$ emanating from vertex $A$ in the ratios $1:2$, $1:1$, and $1:2$ respectively, considering from point $A$. Find the ratio of the volumes of the parts into which this plane divides the tetrahedron.


The center of a semicircle, inscribed in a right triangle such that its diameter lies on the hypotenuse, divides the hypotenuse into segments of 30 cm and 40 cm. Find the length of the arc of the semicircle that lies between the points where it touches the legs of the triangle.


Given that \( a, b, c \) are all positive numbers, find the maximum value of \( y = \frac{ab + 2bc}{a^2 + b^2 + c^2} \).


On the radius \( AO \) of a circle with center \( O \), a point \( M \) was chosen. On one side of \( AO \) on the circle, points \( B \) and \( C \) were chosen such that \( \angle AMB = \angle OMC = \alpha \). Find the length of \( BC \), given that the radius of the circle is \( 14 \) and \( \sin \alpha = \frac{\sqrt{33}}{7} \).


Let \( x, y \in \mathbf{R} \). Denote the minimum value among \( 2^{-x}, 2^{x-y}, 2^{y-1} \) as \( P \). When \( 0 < x < 1 \) and \( 0 < y < 1 \), what is the maximum value of \( P \)?


In a chess tournament, \( n \) women and \( 2n \) men participated. Everyone played exactly one game with each other. There were no draws, and the number of games won by the women is in the ratio of \( 7: 5 \) to the number of games won by the men. What is \( n \)?


What is the largest prime divisor of a number in the form $\overline{xyxyxy}$ in the decimal system?


In the tetrahedron \( P-ABC \), \( PA=BC=\sqrt{6} \), \( PB=AC=\sqrt{8} \), and \( PC=AB=\sqrt{10} \). Find the radius of the circumscribed sphere of the tetrahedron.


Consider a large cube of dimensions \(4 \times 4 \times 4\) composed of 64 unit cubes. Select 16 of these unit cubes and color them red, ensuring that within every \(1 \times 1 \times 4\) rectangular prism formed by 4 unit cubes, exactly 1 unit cube is colored red. How many different ways can the 16 unit cubes be colored red? Provide a justification for your answer.


In how many ways can all natural numbers from 1 to 200 be painted in red and blue so that no sum of two different numbers of the same color equals a power of two?


Determine the mass of a sphere with radius \( r \), given that the density at each point is proportional to its distance from the center of the sphere.

Solution: \boxed{k \pi r^4}


How many integers at a minimum must be selected from the set $\{1,2, \ldots, 20\}$ to ensure that this selection includes two integers \( a \) and \( b \) such that \( a - b = 2 \)?


Show that if I take 3 integers, I can find 2 of them whose sum is even. Then, if I take 5 integers, I can find 3 of them whose sum is divisible by 3. Then show that if I take 11 integers, I can find 6 of them whose sum is divisible by 6. Find the smallest integer \( n \) such that if I take \( n \) integers, I can always find 18 of them whose sum is divisible by 18.


Does there exist a positive integer $m$ such that the equation $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{a b c} = \frac{m}{a + b + c}$ has infinitely many solutions in positive integers?


Person A and Person B decided to go to a restaurant. Due to high demand, Person A arrived first and took a waiting number, while waiting for Person B. After a while, Person B arrived but did not see Person A, so he also took a waiting number. While waiting, Person B saw Person A, and they compared their waiting numbers. They found that the digits of these two numbers are two-digit numbers in reverse order, and the sum of the digits of both numbers is 8. Additionally, Person B's number is 18 greater than Person A's. What is Person A's number? $\qquad$


Consider a square of side length 1. Draw four lines that each connect a midpoint of a side with a corner not on that side, such that each midpoint and each corner is touched by only one line. Find the area of the region completely bounded by these lines.


Two squares are arranged as shown in the figure. If the overlapping part is subtracted from the smaller square, $52\%$ of its area remains. If the overlapping part is subtracted from the larger square, $73\%$ of its area remains. Find the ratio of the side of the smaller square to the side of the larger square.


A castle has a number of halls and \( n \) doors. Every door leads into another hall or outside. Every hall has at least two doors. A knight enters the castle. In any hall, he can choose any door for exit except the one he just used to enter that hall. Find a strategy allowing the knight to get outside after visiting no more than \( 2n \) halls (a hall is counted each time it is entered).


The base and one side of a triangle are 30 and 14, respectively. Find the area of this triangle if the median drawn to the base is 13.


From a three-digit number \( A \), which does not contain zeroes in its digits, a two-digit number \( B \) is obtained by replacing the first two digits with their sum (for example, the number 243 becomes 63). Find \( A \) if it is known that \( A = 3B \).


The sports event lasted for \( n \) days, during which \( m \) medals were distributed. On the first day, 1 medal was given out, plus \(\frac{1}{7}\) of the remaining \((m-1)\) medals. On the second day, 2 medals were given out, plus \(\frac{1}{7}\) of the remaining medals after that. This pattern continued for subsequent days. On the last day, the \( n \)-th day, the remaining \( n \) medals were given out. How many days did the event last, and how many medals were distributed in total?

Solution: \boxed{n = 6, \, m = 36}


For a natural number \( N \), if at least five out of the nine natural numbers \( 1 \) through \( 9 \) can divide \( N \) evenly, then \( N \) is called a "Five Sequential Number." What is the smallest "Five Sequential Number" greater than 2000?


Given a natural number \( x = 6^n + 1 \), where \( n \) is an odd natural number. It is known that \( x \) has exactly three distinct prime divisors, one of which is 11. Find \( x \).


In the number \(2 * 0 * 1 * 6 * 07 *\), each of the 5 asterisks needs to be replaced by any of the digits \(0, 2, 4, 5, 6, 7\) (digits can repeat) so that the resulting 11-digit number is divisible by 75. In how many ways can this be done?


On the parade ground, 2018 soldiers are lined up in a single row. The commander can order either all soldiers standing in even positions or all soldiers standing in odd positions to leave the formation. After this order, the soldiers remaining in formation close ranks into a single row. In how many ways can the commander issue a series of 8 orders so that there are exactly 7 soldiers left in formation?


Mitya is 11 years older than Shura. When Mitya was as old as Shura is now, he was twice as old as she was then. How old is Mitya?


Let \( f \) be a function such that
\[
f(x) - \frac{1}{2} f\left(\frac{1}{x}\right) = \log x
\]
for all \( x > 0 \), where \( \log \) denotes the logarithm base 10. Find \( f(1000) \).


A 1962-digit number divisible by 9 is randomly taken. Let the sum of its digits be $a$, the sum of the digits of $a$ be $b$, and the sum of the digits of $b$ be $c$. What is the value of $c$?


Let $n$ be a fixed integer such that $n \geq 2$.

(1) Determine the smallest constant $c$ such that the inequality
$$
\sum_{1 \leq i<j \leq n} x_{i} x_{j}\left(x_{i}^{2} + x_{j}^{2}\right) \leq c \cdot \left(\sum_{i=1}^{n} x_{i}\right)^{4}
$$
holds for all non-negative real numbers $x_{1}, x_{2}, \cdots, x_{n}$.

(2) For this constant $c$, determine the necessary and sufficient conditions for equality to hold.


Let \( S(n) \) denote the sum of the digits of a natural number \( n \). For example, \( S(123) = 1 + 2 + 3 = 6 \). If two different positive integers \( m \) and \( n \) satisfy the following conditions:
\[
\begin{cases}
m < 100 \\
n < 100 \\
m > n \\
m + S(n) = n + 2S(m)
\end{cases}
\]
then \( m \) and \( n \) form a pair \( \langle m, n \rangle \).

How many such pairs \( \langle m, n \rangle \) are there?


Eight strangers are preparing to play bridge. How many ways can they be grouped into two bridge games, meaning into unordered pairs of unordered pairs of people?


Given that \( M \) is an arbitrary point on the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 (a>b>0)\), \( F_{1} \) and \( F_{2} \) are the left and right foci of the ellipse, respectively. The lines \( M F_{1} \) and \( M F_{2} \) intersect the ellipse at points \( A \) and \( B \), respectively. Then \(\frac{b^{2}}{a^{2}}\left(\frac{\left|MF_{1}\right|}{\left|F_{1}A\right|}+\frac{\left|MF_{2}\right|}{\left|F_{2}B\right|}+2\right)=\) ______.


All integers are written in sequence starting from one. Determine which digit is in the 206788th position.


What is the largest result we can find when we divide a 2-digit number by the sum of its digits?


Calculate: \( \arctan \frac{1}{3}+\arctan \frac{1}{5}+\arcsin \frac{1}{\sqrt{50}}+\arcsin \frac{1}{\sqrt{65}} = \).


Find the 6-digit number beginning and ending in the digit 2 that is the product of three consecutive even integers.


In the right triangle \( KLM \), a circle is constructed on the leg \( KM \) as the diameter, which intersects the hypotenuse \( KL \) at point \( G \). A tangent to the circle is drawn through point \( G \), intersecting the leg \( ML \) at point \( F \). Find the length of \( FL \), given that \( KG = 5 \) and \( LG = 4 \).


Let \( \left\{ \frac{1}{(n+1) \sqrt{n} + n \sqrt{n+1}} \right\} \) be a sequence and \( S_n \) be the sum of the first \( n \) terms of this sequence. How many rational numbers are there in the first 2016 terms of the sequence \( \left\{ S_n \right\} \)?


A seagull is being fed from a moving boat. A piece of bread is thrown down, the seagull picks it up from the surface of the sea in 3 seconds, and then catches up with the boat in 12 seconds. Upon entering a bay, the boat's speed is halved. How much time will it now take for the seagull to catch up with the boat, after it picks up the piece of bread?

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


In a chess tournament, two 7th-grade students and several 8th-grade students participated. Each student played with every other participant once. The two 7th graders together scored 8 points, and all 8th graders scored the same number of points. How many 8th graders participated in the tournament?


Rachel and Steven play games of chess. If either wins two consecutive games, they are declared the champion. The probability that Rachel will win any given game is 0.6, the probability that Steven will win any given game is 0.3, and the probability that any given game is drawn is 0.1. Find the value of \(1000P\), where \(P\) is the probability that neither is the champion after at most three games.


In front of Vasya, there is a stack of 15 red, 15 blue, and 15 yellow cards. Vasya needs to choose 15 out of all 45 cards to earn the maximum number of points. Points are awarded as follows: for each red card, Vasya earns one point. For each blue card, Vasya earns points equal to twice the number of red cards chosen, and for each yellow card, Vasya earns points equal to three times the number of blue cards chosen. What is the maximum number of points Vasya can earn?


For \( x, y \in [1,3] \), find the minimum value of the expression

\[
A = \frac{(3xy + x^{2}) \sqrt{3xy + x - 3y} + (3xy + y^{2}) \sqrt{3xy + y - 3x}}{x^{2} y + y^{2} x}
\]


What is the smallest sum that nine consecutive natural numbers can have if this sum ends in 2050306?


There are 30 people studying in the Slytherin House at Hogwarts. Some of them are friends (friendship is mutual, i.e., if A is friends with B, then B is also friends with A), but no three people are pairwise friends with each other. For New Year, everyone sent postcards to all their friends. What is the maximum number of postcards that could have been sent?


Let's draw the diagonals AC and BD of the parallelogram. Let them intersect at point O. Then CO, BN, DM are medians of triangle BCD. Let them intersect at point K. By the property of medians, BK:KN = 2:1. Since DM is perpendicular to AC, triangle CKD is a right triangle. KN is the median drawn to the hypotenuse. Therefore, KN = 1/2 CD. Then BN:CD = 3:2. Thus, BN = 1.5 CD = 9.


The instructor of a summer math camp brought several shirts, several pairs of trousers, several pairs of shoes, and two jackets for the entire summer. In each lesson, he wore trousers, a shirt, and shoes, and he wore a jacket only on some lessons. On any two lessons, at least one piece of his clothing or shoes was different. It is known that if he had brought one more shirt, he could have conducted 18 more lessons; if he had brought one more pair of trousers, he could have conducted 63 more lessons; if he had brought one more pair of shoes, he could have conducted 42 more lessons. What is the maximum number of lessons he could conduct under these conditions?


In triangle \(ABC\), bisector \(CD\) is drawn from the right angle at \(ACB\); \(DM\) and \(DN\) are the altitudes of triangles \(ADC\) and \(BDC\) respectively.

Find \(AC\), given that \(AM = 4\) and \(BN = 9\).


The watermelon seller weighed 20 watermelons (weighing 1 kg, 2 kg, 3 kg, ..., 20 kg) one by one while waiting for customers, balancing a watermelon on one pan of the scale with one or two weights on the other pan (possibly identical). At the same time, the seller noted down which weights he used. What is the minimum number of different numbers that could have appeared in his records if the weight of each weight is an integer number of kilograms?


Given a triangle \( \triangle ABC \) with orthocenter \( H \) and circumcenter \( O \). The radius of the circumcircle is 3, and the distance \( OH \) is 1. The side lengths of the triangle are \( |BC| = a \), \( |CA| = b \), and \( |AB| = c \). Find the value of \( a^2 + b^2 + c^2 \).


Josh takes a walk on a rectangular grid of \( n \) rows and 3 columns, starting from the bottom left corner. At each step, he can either move one square to the right or simultaneously move one square to the left and one square up. In how many ways can he reach the center square of the topmost row?


A circle $\omega$ is circumscribed around the triangle $ABC$. A circle $\omega_{1}$ is tangent to the line $AB$ at point $A$ and passes through point $C$, and a circle $\omega_{2}$ is tangent to the line $AC$ at point $A$ and passes through point $B$. A tangent to circle $\omega$ at point $A$ intersects circle $\omega_{1}$ again at point $X$ and intersects circle $\omega_{2}$ again at point $Y$. Find the ratio $\frac{AX}{XY}$.


Let \(a\), \(b\), and \(c\) be positive integers, no two of which have a common divisor greater than 1. Show that \(2abc - ab - bc - ca\) is the largest integer which cannot be expressed in the form \(xbc + yca + zab\), where \(x\), \(y\), \(z\) are non-negative integers.

Solution: \boxed{2abc - ab - bc - ca}


Calculate the double integral

$$
\iint_{D}\left(54 x^{2} y^{2}+150 x^{4} y^{4}\right) d x d y
$$

where the region \(D\) is bounded by the lines \(x=1, y=x^{3}\), and \(y=-\sqrt{x}\).


In an angle containing $60^{\circ}$, five circles are inscribed such that each subsequent circle, starting from the second, touches the previous one. By what factor is the sum of the areas of all five circles greater than the area of the smallest circle?


Calculate the indefinite integral:

$$
\int(3x + 4)e^{3x} \, dx
$$

Solution: \boxed{(x + 1) e^{3x} + C}


Lydia likes a five-digit number if none of its digits are divisible by 3. Find the total sum of the digits of all five-digit numbers that Lydia likes.


Two irreducible fractions have their denominators equal to 600 and 700. Find the minimum value for the denominator of the sum of the fractions.


The angle at vertex $C$ is $120^\circ$. A circle with radius $R$ touches the sides of the angle at points $A$ and $B$. Find the length of $AB$.

Solution: \boxed{R}


To which number can the expression

$$
\frac{(a+b)(b+c)(a+c)}{a b c}
$$

be equal if the numbers \( a \), \( b \), and \( c \) satisfy the condition of problem 84?


Find the number of natural numbers \( k \), not exceeding 333300, such that \( k^{2} - 2k \) is exactly divisible by 303.


Let \( a \) and \( b \) be two positive integers whose least common multiple is 232848. How many such ordered pairs \( (a, b) \) are there?


The base of a triangle is 20; the medians drawn to the lateral sides are 18 and 24. Find the area of the triangle.


Study the sequence defined by \(\mathfrak{u}_{0}=0\) and \(\mathfrak{u}_{n+1}=\sqrt{12+\mathfrak{u}_{n}}\) for all \(n \in \mathbb{N}\).


How many positive-integer pairs \((x, y)\) are solutions to the equation \(\frac{xy}{x+y} = 1000\)?


The integer sequence \(\left\{a_{n}\right\}\) is defined as \(a_{1}=1\), \(a_{2}=2\), and \(a_{n+2}=5a_{n+1}+a_{n}\). Evaluate the expression \(\left[\frac{a_{2}}{a_{1}}\right] \left\{\left[\frac{a_{3}}{a_{2}}\right\} \left\{\frac{a_{4}}{a_{3}}\right\} \cdots \left\{\left[\left\{\frac{a_{2025}}{a_{2024}}\right\}\left[\frac{a_{2024}}{a_{2}}\right]\right.\right.\right.\).


The children went to the forest to pick mushrooms. If Anya gives half of her mushrooms to Vitya, all the children will have the same number of mushrooms. But if Anya instead gives all her mushrooms to Sasha, Sasha will have as many mushrooms as all the others combined. How many children went to pick mushrooms?


Given \( m > n \geqslant 1 \), find the minimum value of \( m+n \) such that \( 1000 \mid 1978^{m} - 1978^{n} \).


Points \( M \) and \( N \) are the midpoints of sides \( BC \) and \( AD \) of quadrilateral \( ABCD \). It is known that \(\angle B = 150^\circ\), \(\angle C = 90^\circ\), and \(AB = CD\). Find the angle between the lines \(MN\) and \(BC\).


We brought a cat named Vilda to our intended cottage to catch mice. On Monday, he caught $\frac{1}{2}$ of all the mice, on Tuesday $\frac{1}{3}$ of the remaining mice, on Wednesday $\frac{1}{4}$ of those left after Tuesday's catch, and on Thursday $\frac{1}{5}$ of the rest. On Friday, the remaining mice decided to move away. How many mice were originally at the cottage, given that on Friday, two more mice moved away than Vilda caught on Tuesday? Ensure that each day the number of mice caught was an integer.


Let three non-identical complex numbers \( z_1, z_2, z_3 \) satisfy the equation \( 4z_1^2 + 5z_2^2 + 5z_3^2 = 4z_1z_2 + 6z_2z_3 + 4z_3z_1 \). Denote the lengths of the sides of the triangle in the complex plane, with vertices at \( z_1, z_2, z_3 \), from smallest to largest as \( a, b, c \). Find the ratio \( a : b : c \).


The sequence \(a_{1}, a_{2}, \ldots\) consists of positive numbers and its limit is 0. What is the limit of the following sequence?

$$
b_{n}=\frac{1}{n}\left(\frac{a_{1}}{1+a_{1}}+\ldots+\frac{a_{n}}{1+a_{n}}\right)
$$


The archipelago consists of $N \geqslant 7$ islands. Any two islands are connected by no more than one bridge. It is known that no more than 5 bridges lead from each island and that among any 7 islands, there are always two that are connected by a bridge. What is the maximum possible value of $N$?


In a table tennis singles tournament, it was initially planned that each pair of players would play one match. However, 3 players each played 2 matches and then withdrew from the tournament. As a result, 50 matches were played in total. How many matches were played among these 3 players?


Vitya and Masha were born in the same year in June. Find the probability that Vitya is at least one day older than Masha.


Does there exist a three-digit number whose cube ends in three sevens?


In the parallelogram \(ABCD\), the diagonals \(AC = 15\) and \(BD = 9\) are known. The radius of the circumcircle of triangle \(ADC\) is 10. Find the radius of the circumcircle of triangle \(ABD\).


Compute the indefinite integral:

$$
\int(1-6 x) e^{2 x} \, dx
$$

Solution: \boxed{(2 - 3x) e^{2x} + C}


A set of composite numbers from the set $\{1,2,3,4, \ldots, 2016\}$ is called good if any two numbers in this set do not have common divisors (other than 1). What is the maximum number of numbers that a good set can have?


A smooth sphere with a radius of 1 cm is dipped in red paint and released between two absolutely smooth concentric spheres with radii of 4 cm and 6 cm respectively (this sphere is outside the smaller sphere but inside the larger one). Upon contact with both spheres, the sphere leaves a red mark. During its movement, the sphere followed a closed path, resulting in a red-bordered area on the smaller sphere with an area of 37 square cm. Find the area of the region bordered by the red contour on the larger sphere. Give your answer in square centimeters, rounded to the nearest hundredth if necessary.


A table can seat 6 people. Two tables joined together can seat 10 people. Three tables joined together can seat 14 people. Following this pattern, if 10 tables are arranged in two rows with 5 tables in each row, how many people can sit?


There are 10 ping pong players participating in a round-robin tournament. The results show that there are no draws, and among any group of 5 players, there is one player who defeats the other 4 and one player who loses to the other 4. How many players have exactly two wins?


Given the hyperbola \(\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1\) with the left vertex \(A\) and the right focus \(F\), let \(P\) be an arbitrary point on the hyperbola in the first quadrant. If it is always true that \(\angle PFA = 2 \angle FAP\), find the eccentricity of the hyperbola.


By permuting the digits of 20130518, how many different eight-digit positive odd numbers can be formed?


How many ways are there to place three rooks on an \(8 \times 8\) chessboard such that the rooks are in different columns and different rows?


On an island, there are knights who always tell the truth and liars who always lie. One day, 15 natives, among whom there were both knights and liars, formed a circle. Each of them said, "Of the two people standing opposite me, one is a knight and the other is a liar." How many knights are there among them?


Expanding the expression \((1+\sqrt{5})^{209}\) using the binomial theorem yields terms of the form \(C_{209}^{k}(\sqrt{5})^{k}\). Find the value of \(k\) for which such a term takes the maximum value.


Find all positive integers \( n \) such that \( 2^n + 3 \) is a perfect square. Also, find all positive integers \( n \) such that \( 2^n + 1 \) is a perfect square.

Solution: \boxed{n = 3}


A boy tells his friend: You can easily remember my phone number the following way. There is an increasing quadruple of natural numbers \(a_{1}, a_{2}, a_{3}, a_{4}\) where both \(a_{2}\) and \(a_{3}\) are the harmonic means of their neighbors, respectively. When you write these numbers next to each other in order, you get a six-digit number. This six-digit number is my phone number. What is it? (The harmonic mean \(a_{2}\) of \(a_{1}\) and \(a_{3}\) is defined as:

$$
\frac{1}{a_{2}}=\frac{1}{2}\left(\frac{1}{a_{1}}+\frac{1}{a_{3}}\right)
$$


In a certain kingdom, the workforce is composed exclusively of the dwarf clan and the elf clan. Historically in this kingdom, dwarves and elves have always worked separately, and no enterprise ever hired both dwarves and elves simultaneously. The aggregate supply of labor resources for dwarves is represented by the function \( w_{\text{dwar}}^S = 1 + \frac{L}{3} \), and the aggregate supply of labor resources for elves is \( w_{\text{elf}}^S = 3 + L \). Conversely, the inverse function of aggregate demand for dwarf labor is \( w_{\text{dwar}}^D = 10 - \frac{2L}{3} \), and the inverse function of aggregate demand for elf labor is \( w_{\text{elf}}^D = 18 - 2L \).

The newly ascended king is very concerned about the difference in wage rates for his subjects and therefore decrees that the wages of elves and dwarves must be the same, ensuring that workers are not discriminated against based on their clan affiliation. The king believes that wage regulation may negatively impact the kingdom's economy and mandates that all his subjects behave completely competitively. By how many times will the wage of the group of workers, whose wage was lower before the king's intervention, increase if the firms in the kingdom are indifferent to hiring either elves or dwarves?


Initially, there are three cards in a box, each with numbers $1, 2, 3$. Each time, two cards are drawn from the box, and their sum is written on another blank card, then one card is put back into the box. After performing this operation 5 times, all cards except the last one have been drawn at least once but not more than twice. What is the maximum number on the card in the box at this point?


Given two lines \( A_{1} x + B_{1} y = 1 \) and \( A_{2} x + B_{2} y = 1 \) intersect at point \( P(-7, 9) \), find the equation of the line passing through points \( P_{1}\left(A_{1}, B_{1}\right) \) and \( P_{2}\left(A_{2}, B_{2}\right) \).

Solution: \boxed{-7x + 9y = 1}


Given non-empty sets
$$
\begin{array}{l}
A = \{x \mid m+1 \leqslant x \leqslant 2m-1\}, \\
B = \{x \mid x^2 - 2x - 15 \leqslant 0\},
\end{array}
$$

and \( A \subseteq B \).

Determine the range of the real number \( m \).

Solution: \boxed{ [2, 3] }


Find the intersection point of the line and the plane.

\(\frac{x-1}{2}=\frac{y+2}{-5}=\frac{z-3}{-2}\)

\(x + 2y - 5z + 16 = 0\)

Solution: \boxed{(3, -7, 1)}


Let \( M = \{1, 2, 3, \ldots, 1995\} \). Subset \( A \) of \( M \) satisfies the condition: If \( x \in A \), then \( 15x \notin A \). What is the maximum number of elements in \( A \)?


\( w, x, y, z \) are real numbers such that
\[
\begin{aligned}
w+x+y+z & =5 \\
2 w+4 x+8 y+16 z & =7 \\
3 w+9 x+27 y+81 z & =11 \\
4 w+16 x+64 y+256 z & =1
\end{aligned}
\]
What is the value of \( 5w + 25x + 125y + 625z \)?


It is known that the only solution to the equation

$$
\pi / 4 = \operatorname{arcctg} 2 + \operatorname{arcctg} 5 + \operatorname{arcctg} 13 + \operatorname{arcctg} 34 + \operatorname{arcctg} 89 + \operatorname{arcctg}(x / 14)
$$

is a natural number. Find it.


A palindromic number is a number that reads the same when the order of its digits is reversed. What is the difference between the largest and smallest five-digit palindromic numbers that are both multiples of 45?


There are five people, \( \mathbf{A} \), \( \mathbf{B} \), \( \mathbf{C} \), \( \mathbf{D} \), and \( \mathbf{E} \), each of whom either always tells the truth or always lies, and they all know each other's behavior. \( \mathrm{A} \) says that \( \mathrm{B} \) is a liar, \( \mathrm{B} \) says that \( \mathrm{C} \) is a liar, \( \mathrm{C} \) says that \( \mathrm{D} \) is a liar, and \( \mathrm{D} \) says that \( \mathbf{E} \) is a liar. How many liars can there be at most among these five people?


$n$ mushroom gatherers went into the forest and collectively brought back 450 mushrooms (each one brought home at least one mushroom). A boy named Petya, upon learning this, declared: "There must be at least two of them who brought back the same number of mushrooms!" What is the smallest $n$ for which Petya is certainly correct? Don't forget to justify your answer.


In triangle \( T_{0} \), a triangle was formed using its midlines and named \( T_{1} \). In triangle \( T_{1} \), a triangle was formed using its midlines and named \( T_{2} \). Continuing in this manner, triangle \( T_{10} \) was obtained. Find the ratio of the sum of the areas of all these eleven triangles to the area of the last triangle. If the answer is a fractional number, it should be written as a decimal using a dot, for example, \(« 0.15 »\).


The side $AB$ of triangle $ABC$ is extended beyond vertex $B$ and a point $A_1$ is chosen on the ray $AB$ such that point $B$ is the midpoint of segment $AA_1$. The side $BC$ is extended beyond vertex $C$ and a point $B_1$ is marked on the extension such that $C$ is the midpoint of $BB_1$. Similarly, the side $CA$ is extended beyond vertex $A$ and a point $C_1$ is marked on the extension such that $A$ is the midpoint of $CC_1$. Find the area of triangle $A_1B_1C_1$ if the area of triangle $ABC$ is 1.


The distance from $A$ to $B$ is 999 km. Along the road, there are kilometer markers that indicate the distance to $A$ and to $B$ as follows:

$0|999, 1|998, \ldots, 999|0$.

How many of these markers have signs that contain only two different digits?


Two individuals undertake a certain task and work for an equal amount of time. $A$ misses 2 days and earns 80 forints in total, while $B$ misses 5 days and earns 63 forints. If $A$ had missed 5 days and $B$ had missed 2 days, then $A$ would earn 2 forints more than $B$. How many days did the work last?


Which of the following numbers is closest to \(2.015 \times 510.2\)?
A) 0.1
B) 1
C) 10
D) 100
E) 1000


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


The sequence \(a_{n} = b[\sqrt{n+c}] + d\) is given, where the successive terms are
\[
1, 3, 3, 3, 5, 5, 5, 5, 5, \cdots
\]
In this sequence, each positive odd number \(m\) appears exactly \(m\) times consecutively. The integers \(b\), \(c\), and \(d\) are to be determined. Find the value of \(b+c+d\).


Let \( ABC \) be a right triangle with \( \angle A \) being the right angle. Show the equality

\[ BC^{2} = AC^{2} + AB^{2} \]

Hint: Introduce the foot of one of the altitudes.

Solution: \boxed{BC^2 = AC^2 + AB^2}


How many natural numbers \( \mathrm{N} \) greater than 900 exist such that among the numbers \( 3\mathrm{N}, \mathrm{N} - 900, \mathrm{N} + 15, 2\mathrm{N} \) exactly two are four-digit numbers?


$C$ is a point on the extension of diameter $A B$, $C D$ is a tangent, and the angle $A D C$ is $110^{\circ}$. Find the angular measure of arc $B D$.


Determine the two-digit numbers that are 3 less than the sum of the cubes of their digits!


Petya cut an 8x8 square along the borders of the cells into parts of equal perimeter. It turned out that not all parts are equal. What is the maximum possible number of parts he could get?


During breaks, schoolchildren played table tennis. Any two schoolchildren played no more than one game against each other. At the end of the week, it turned out that Petya played half, Kolya - a third, and Vasya - one fifth of the total number of games played during the week. What could be the total number of games played during the week if it is known that at least two games did not involve Vasya, Petya, or Kolya?


As shown in the diagram, $E$ is the midpoint of the leg $AB$ of trapezoid $ABCD$. $DF \perp EC$, $DF=10$, and $EC=24$. Find the area of trapezoid $ABCD$.


Solve the system:
\[
\begin{aligned}
|x + y| + |1 - x| &= 6, \\
|x + y + 1| + |1 - y| &= 4.
\end{aligned}
\]

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


Let \( p \) and \( q \) be distinct prime numbers. In how many ways can the fraction \( \frac{1}{pq} \) be decomposed into the sum of the reciprocals of 2 different natural numbers?


Find all positive integers \( n \) such that there exist positive integers \( x_{1}, x_{2}, \cdots, x_{2012} \) satisfying
\[ x_{1} < x_{2} < \cdots < x_{2012} \]
and
\[ \frac{1}{x_{1}} + \frac{2}{x_{2}} + \cdots + \frac{2012}{x_{2012}} = n. \]

Solution: \boxed{1, 2, \ldots, 2012}


Given that there are three points on the coordinate plane: \( O(0,0) \), \( A(12,2) \), and \( B(0,8) \). A reflection of \( \triangle OAB \) along the straight line \( y = 6 \) creates \( \triangle PQR \). If the overlapped area of \( \triangle OAB \) and \( \triangle PQR \) is \( m \) square units, find the value of \( m \).


Given that the quadratic equation $a x^{2}+b x+c=0$ has no real roots. Person A, due to an error in reading the coefficient of the $x^2$ term, incorrectly found the roots to be 2 and 4. Person B, due to an error in reading the sign of a certain coefficient, incorrectly found the roots to be -1 and 4. What is the value of $\frac{2 b+3 c}{a}$?


A number is the product of four prime numbers. What is this number if the sum of the squares of the four prime numbers is 476?


Distribute 16 identical books among 4 classes, with each class receiving at least one book and each class receiving a different number of books. How many different distribution methods are there? (Answer in digits)


Point \( O \) is the center of the circumscribed circle of the acute-angled triangle \( ABC \), and \( H \) is the orthocenter of the triangle. It turned out that line \( OH \) is parallel to side \( BC \). On the plane, a point \( K \) was marked such that \( ABHK \) is a parallelogram. The segments \( OK \) and \( AC \) intersect at point \( L \). In what ratio does the perpendicular dropped from point \( L \) to segment \( AH \) divide \( AH \)?


Two cars simultaneously set off towards each other from cities $A$ and $B$, which are 250 km apart. The cars travel at speeds of 50 km/h and 90 km/h. At what distance from the halfway point between cities $A$ and $B$, denoted as point $C$, will the cars meet? Give the answer in kilometers, rounded to the nearest hundredth if necessary.


Cover the following $2 \times 10$ grid with $1 \times 2$ tiles. How many different ways are there to cover it?


Find all integer values that the expression

$$
\frac{p q + p^{p} + q^{q}}{p + q}
$$

where \(p\) and \(q\) are prime numbers.


If \(n\) is a positive integer, the symbol \(n!\) (read "n factorial") represents the product of the integers from 1 to \(n\). For example, \(4! = (1)(2)(3)(4)\) or \(4! = 24\). Determine
$$
\frac{1}{\log_{2} 100!} + \frac{1}{\log_{3} 100!} + \cdots + \frac{1}{\log_{100} 100!}
$$


Solve the equation $\arcsin \frac{x \sqrt{5}}{3}+\arcsin \frac{x \sqrt{5}}{6}=\arcsin \frac{7 x \sqrt{5}}{18}$.

Solution: \boxed{0, \pm \frac{8}{7}}


Suppose \( x \) satisfies \( x^{3} + x^{2} + x + 1 = 0 \). What are all possible values of \( x^{4} + 2x^{3} + 2x^{2} + 2x + 1 \)?


If the acute angle \(\alpha\) satisfies \(\frac{1}{\sqrt{\tan \frac{\alpha}{2}}}=\sqrt{2 \sqrt{3}} \sqrt{\tan 10^{\circ}}+\sqrt{\tan \frac{\alpha}{2}}\), then the measure of the angle \(\alpha\) in degrees is \(\qquad\)


Verify that 2 is a solution of the equation:

$$
(x+1)^{3}+(x+2)^{3}+(x+3)^{3}=(x+4)^{3}
$$

Does this equation have any other integer solutions?


The three-tiered "pyramid" shown in the image is built from $1 \mathrm{~cm}^{3}$ cubes and has a surface area of $42 \mathrm{~cm}^{2}$. We made a larger "pyramid" based on this model, which has a surface area of $2352 \mathrm{~cm}^{2}$. How many tiers does it have?


Let \( n \in \mathbf{N}^{*} \). Consider the set \( S = \{1, 2, \ldots, 2n\} \) and its \( k \) subsets \( A_{1}, A_{2}, \ldots, A_{k} \) that satisfy the following conditions:

1. For any \( i \neq j \) (with \( i, j \in \{1, 2, \ldots, k\} \)), the intersection \( A_{i} \cap A_{j} \) has exactly an odd number of elements.
2. For any \( i \) (where \( i = 1, 2, \ldots, k \)), \( i \notin A_{i} \).
3. If \( i \in A_{j} \), then \( j \in A_{i} \).

Determine the maximum value of \( k \).

Solution: \boxed{2n - 1}


Find the number of different monic quadratic trinomials (i.e. with the leading coefficient equal to 1) with integer coefficients that have two different roots, which are powers of the number 3 with natural exponents, and their coefficients in absolute value do not exceed $27^{45}$.


For the positive integer \( n \), if the expansion of \( (xy - 5x + 3y - 15)^n \) is combined and simplified, and \( x^i y^j \) (where \( i, j = 0, 1, \ldots, n \)) has at least 2021 terms, what is the minimum value of \( n \)?


Circles \( C_1, C_2, C_3 \) have radius 1 and centers \( O, P, Q \) respectively. \( C_1 \) and \( C_2 \) intersect at \( A \), \( C_2 \) and \( C_3 \) intersect at \( B \), \( C_3 \) and \( C_1 \) intersect at \( C \), in such a way that \( \angle A P B = 60^\circ \), \( \angle B Q C = 36^\circ \), and \( \angle C O A = 72^\circ \). Find angle \( A B C \) (degrees).


Given a sequence \( A = (a_1, a_2, \cdots, a_{10}) \) that satisfies the following four conditions:
1. \( a_1, a_2, \cdots, a_{10} \) is a permutation of \{1, 2, \cdots, 10\};
2. \( a_1 < a_2, a_3 < a_4, a_5 < a_6, a_7 < a_8, a_9 < a_{10} \);
3. \( a_2 > a_3, a_4 > a_5, a_6 > a_7, a_8 > a_9 \);
4. There does not exist \( 1 \leq i < j < k \leq 10 \) such that \( a_i < a_k < a_j \).

Find the number of such sequences \( A \).


For a positive integer \( n \), if there exist positive integers \( a \) and \( b \) such that \( n = a + b + a \times b \), then \( n \) is called a "good number". For example, \( 3 = 1 + 1 + 1 \times 1 \), so 3 is a "good number". Among the 100 positive integers from 1 to 100, there are \(\qquad\) "good numbers".


Alice, Bob, and Charlie each pick a 2-digit number at random. What is the probability that all of their numbers' tens digits are different from each other's tens digits and all of their numbers' ones digits are different from each other's ones digits?


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


Let \(\mathcal{P}\) be a regular 17-gon. We draw in the \(\binom{17}{2}\) diagonals and sides of \(\mathcal{P}\) and paint each side or diagonal one of eight different colors. Suppose that there is no triangle (with vertices among vertices of \(\mathcal{P}\)) whose three edges all have the same color. What is the maximum possible number of triangles, all of whose edges have different colors?


Find all pairs of positive numbers \(a\) and \(b\) for which the numbers \(\sqrt{ab}, \frac{a+b}{2},\) and \(\sqrt{\frac{a^{2}+b^{2}}{2}}\) can form an arithmetic progression.

Solution: \boxed{a = b}


The side \( AB \) of triangle \( ABC \) is equal to 3, \( BC = 2AC \), \( E \) is the point of intersection of the extension of the angle bisector \( CD \) of this triangle with the circumscribed circle around it, \( DE = 1 \). Find \( AC \).


Given that $a \in A$, and $a-1 \notin A$ and $a+1 \notin A$, $a$ is called an isolated element of set $A$. How many four-element subsets of the set $M=\{1,2, \cdots, 9\}$ have no isolated elements?


Let \( p \) and \( q \) be two distinct odd prime numbers. Determine the (multiplicative) order of \( 1 + pq \) modulo \( p^2 q^3 \).


Divide each natural number by the sum of the squares of its digits (for single-digit numbers, divide by the square of the number). Is there a smallest quotient among the obtained quotients, and if so, which one is it?


Petya came up with the reduced quadratic equation \(x^{2} + px + q\), whose roots are \(x_{1}\) and \(x_{2}\). He informed Vasya of three out of the four numbers \(p, q, x_{1}, x_{2}\) without specifying which is which. These numbers turned out to be \(1, 2, -6\). What was the fourth number?


The Pythagorean school believed that numbers are the origin of all things, and they called numbers like $1, 3, 6, 10, \cdots$ triangular numbers. What is the sum of the first 100 triangular numbers?


Solve the equation \((9-3x) \cdot 3^x - (x-2)\left(x^2 - 5x + 6\right) = 0\) in the set of real numbers.

Solution: \boxed{x = 3}


In a triangle, the lengths of the three sides are integers $l, m, n$, 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 $\{x\}=x-[x]$, and $[x]$ denotes the greatest integer not exceeding $x$. Find the minimum perimeter of such a triangle.


Let's conduct \( DL \perp AC \), \(LK \parallel CC_1\) (where \( K \in AC_1 \)), and \( PK \parallel DL \). By marking segment \( BQ = PD_1 \) on the lateral edge \( BB_1 \), we form a parallelogram \( PAQC_1 \), which will be the intersection of minimum area; here, \( AC_1 \) is the larger diagonal, and \( PQ \) is the smaller diagonal, both given in the problem statement.

Given:
\[ AC_1 = 3, \quad PQ = \sqrt{3}, \quad \varphi = 30^\circ \]
\[ \sin \varphi = \frac{1}{2}, \quad \cos \varphi = \frac{\sqrt{3}}{2} \]

Calculations:
\[ DL = PK = \frac{1}{2} \cdot PQ \cdot \sin \varphi = \frac{1}{2} \cdot \sqrt{3} \cdot \frac{1}{2} = \frac{\sqrt{3}}{4} \]
\[ OK = \frac{1}{2} \cdot PQ \cdot \cos \varphi = \frac{1}{2} \cdot \sqrt{3} \cdot \frac{\sqrt{3}}{2} = \frac{3}{4} \]

\[ \frac{CL}{AL} = \frac{C_1 K}{AK} = \frac{C_1 O - OK}{C_1 O + OK} = \frac{3/2 - 3/4}{3/2 + 3/4} = \frac{1}{3} \]

Let \( CL = x \), then \( AL = 3x \), \( AC = 4x \).

\[ DL^2 = CL \cdot AL \]
\[ \frac{3}{16} = 3x^2 \]
\[ x = \frac{1}{4}, \quad AC = 1 \]
\[ CC_1 = \sqrt{AC_1^2 - AC^2} = \sqrt{9 - 1} = 2\sqrt{2} \]

Volume \( V = AC \cdot DL \cdot CC_1 = 1 \cdot \frac{\sqrt{3}}{4} \cdot 2\sqrt{2} = \sqrt{\frac{3}{2}} = \frac{\sqrt{6}}{2} \).


Dad, Masha, and Yasha are walking to school. While Dad takes 3 steps, Masha takes 5 steps. While Masha takes 3 steps, Yasha takes 5 steps. Masha and Yasha counted that together they have taken 400 steps. How many steps did Dad take?


Five individuals are assigned to five specific seats, but they do not know their assigned numbers. When these five individuals sit randomly in these five seats, find:
(1) The probability that exactly 3 out of the 5 individuals sit in their assigned seats.
(2) If the probability that all 5 individuals sit in their assigned seats is not less than $\frac{1}{6}$, what is the maximum number of people who can sit in their assigned seats?


In a math competition, there are 8 fill-in-the-blank questions worth 4 points each if answered correctly, and 6 short-answer questions worth 7 points each if answered correctly. If 400 people participated in the competition, what is the minimum number of people who have the same total score?


Find the smallest real number \( A \) such that for every quadratic polynomial \( f(x) \) satisfying the condition
\[
|f(x)| \leq 1 \quad (0 \leq x \leq 1),
\]
the inequality \( f^{\prime}(0) \leq A \) holds.


Find all real numbers \( p \) such that the cubic equation \( 5x^{3} - 5(p+1)x^{2} + (71p - 1)x + 1 = 66p \) has three roots, all of which are positive integers.


On the segment \( A C \), point \( B \) is marked. Semicircles are constructed on the segments \( A B \), \( B C \), and \( A C \) as diameters in the same half-plane. A circle with center at point \( O \) is tangent to all these semicircles (see figure). Find the radius of this circle, given that \( A B = 4 \) and \( B C = 2 \). If necessary, round your answer to the nearest hundredth.


Given an isosceles triangle \(ABC\) with \(AB = AC\) and \(\angle ABC = 53^\circ\), find the measure of \(\angle BAM\). Point \(K\) is such that \(C\) is the midpoint of segment \(AK\). Point \(M\) is chosen such that:

- \(B\) and \(M\) are on the same side of line \(AC\);
- \(KM = AB\);
- \(\angle MAK\) is the maximum possible.

How many degrees is \(\angle BAM\)?


Find the smallest 10-digit number such that the sum of its digits is greater than that of any smaller number.


Find the smallest positive integer n such that n has exactly 144 positive divisors including 10 consecutive integers.


Determine the number $ABCC$ (written in decimal system) given that

$$
ABCC = (DD - E) \cdot 100 + DD \cdot E
$$

where $A, B, C, D,$ and $E$ are distinct digits.


Let \( n \) be a positive integer greater than 1. If \( 2n \) is divided by 3, the remainder is 2. If \( 3n \) is divided by 4, the remainder is 3. If \( 4n \) is divided by 5, the remainder is 4. If \( 5n \) is divided by 6, the remainder is 5. What is the least possible value of \( n \)?


At a math competition, three problems were presented: \(A\), \(B\), and \(C\). There were 25 students, each of whom solved at least one problem. Among the students who did not solve problem \(A\), twice as many solved problem \(B\) as those who solved problem \(C\). Exactly one more student solved only problem \(A\) than the number of other students who also solved \(A\). Half of the students who solved only one problem did not solve problem \(A\). How many students solved only problem \(B\)?


In the equation \(\overline{ABC} \times \overline{AB} + C \times D = 2017\), the same letters represent the same digits, and different letters represent different digits. If the equation holds true, what two-digit number does \(\overline{\mathrm{AB}}\) represent?


A $7 \times 7$ grid is colored black and white. If the number of columns with fewer black squares than white squares is $m$, and the number of rows with more black squares than white squares is $n$, find the maximum value of $m + n$.


The function \( g \) defined on the set of integers satisfies the following conditions:

1) \( g(1) - 1 > 0 \)
2) \( g(x) g(y) + x + y + xy = g(x+y) + x g(y) + y g(x) \) for any \( x, y \in \mathbb{Z} \)
3) \( 3 g(x+1) = g(x) + 2x + 3 \) for any \( x \in \mathbb{Z} \).

Find \( g(-6) \).


Solution: \boxed{42 \sqrt{3}}


Given the plane point set \( A = \{ (x, y) \mid x = 2 \sin \alpha + 2 \sin \beta, y = 2 \cos \alpha + 2 \cos \beta \} \) and \( B = \{ (x, y) \mid \sin (x + y) \cos (x + y) \geq 0 \} \), find the area of the region represented by \( A \cap B \).


Point \( K \) lies on side \( BC \) of parallelogram \( ABCD \), and point \( M \) lies on its side \( AD \). Segments \( CM \) and \( DK \) intersect at point \( L \), and segments \( AK \) and \( BM \) intersect at point \( N \). Find the maximum value of the ratio of the areas of quadrilateral \( KLMN \) to \( ABCD \).


Find the base of an isosceles triangle if its side is equal to \( a \), and the height dropped onto the base is equal to the segment connecting the midpoint of the base with the midpoint of the side.

Solution: \boxed{x = a \sqrt{3}}


Find the smallest real number \( r \) such that there exists a sequence of positive real numbers \(\{x_n\}\) satisfying \(\sum_{i=1}^{n+1} x_i \leq r \cdot x_n\) for any \( n \in \mathbb{N}^+ \).


Calculate the limit of the numerical sequence:
\[ \lim _{n \rightarrow \infty} \sqrt{n^{3}+8}\left(\sqrt{n^{3}+2}-\sqrt{n^{3}-1}\right) \]


29 boys and 15 girls attended a ball. Some boys danced with some girls (at most once per pair). After the ball, each person told their parents how many times they danced. What is the maximum number of different numbers that the children could have mentioned?


A local high school math club has 12 students in it. Each week, 6 of the students go on a field trip.
(a) Jeffrey, a student in the math club, has been on a trip with each other student in the math club. Determine the minimum number of trips that Jeffrey could have gone on.
(b) If each pair of students have been on at least one field trip together, determine the minimum number of field trips that could have happened.


There are 17 people standing in a circle: each of them is either truthful (always tells the truth) or a liar (always lies). All of them said that both of their neighbors are liars. What is the maximum number of liars that can be in this circle?


Using the digits from 1 to 9 exactly once, form the smallest nine-digit number that is divisible by 11.


At a market in Egypt, a tourist is negotiating with a seller for a souvenir worth 10,000 Egyptian pounds. The tourist first reduces the price by \( x \) percent \((0 < x < 100)\), then the seller increases the price by \( x \) percent, and so on. The value of \( x \) does not change during the bargaining process, and the seller increases the price at least once. The negotiation continues until one of the participants gets a non-integer value for the price of the souvenir. Find the largest possible number of price changes during such a negotiation (including the last non-integer price change).


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

Solution: \boxed{\text{No}}


Suppose that the pair of positive integers \((x, y)\) satisfies \(\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{y}} = \frac{1}{\sqrt{20}}\). How many different possible values are there for \(xy\)?


Given two sets \( A = \{1, 2, 3, \ldots, 15\} \) and \( B = \{0, 1\} \), find the number of mappings \( f: A \rightarrow B \) with 1 being the image of at least two elements of \( A \).


Given \( f_{1}(x)=|1-2 x| \) for \( x \in [0,1] \) and \( f_{n}(x)=f_{1}(f_{n-1}(x)) \), determine the number of solutions to the equation \( f_{2005}(x)=\frac{1}{2} x \).


Calculate the volumes of the solids formed by rotating the regions bounded by the graphs of the functions around the $y$-axis.

$$
y = x^3, \quad y = x^2
$$


There is a basket of apples. The first time, half of the apples plus 2 were taken out. The second time, half of the remaining apples minus 3 were taken out. There are 24 apples left in the basket. How many apples were originally in the basket?


Find the largest root of the equation
$$
3 \sqrt{x-2} + 2 \sqrt{2x+3} + \sqrt{x+1} = 11
$$


Find the sum of all even factors of 1152.


In the nodes of a grid, numbers \( 1, 2, 3, \ldots \) are placed in a spiral order. Then, at the center of each cell, the sum of the numbers in its nodes is written (see Fig. 4). Is it true that the numbers divisible by 52 will appear infinitely many times at the centers of the cells?

Solution: \boxed{Yes}


A cell can divide into 42 or 44 smaller cells. How many divisions are needed to obtain exactly 1993 cells from a single cell?


Compute the number of sequences of numbers \( a_{1}, a_{2}, \ldots, a_{10} \) such that:
I. \( a_{i}=0 \) or 1 for all \( i \),
II. \( a_{i} \cdot a_{i+1}=0 \) for \( i=1,2, \ldots, 9 \),
III. \( a_{i} \cdot a_{i+2}=0 \) for \( i=1,2, \ldots, 8 \).


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


Find all natural numbers whose own divisors can be paired such that the numbers in each pair differ by 545. An own divisor of a natural number is a natural divisor different from one and the number itself.


Find the largest natural number that cannot be represented as the sum of two composite numbers. (Recall that a natural number is called composite if it is divisible by some natural number other than itself and one.)


A technique often used to calculate summations is the Telescoping Sum. It involves "decomposing" the terms of a sum into parts that cancel out. For example,

$$
\begin{aligned}
& \frac{1}{1 \cdot 2}+\frac{1}{2 \cdot 3}+\frac{1}{3 \cdot 4}+\frac{1}{4 \cdot 5}= \\
& \left(\frac{1}{1}-\frac{1}{2}\right)+\left(\frac{1}{2}-\frac{1}{3}\right)+\left(\frac{1}{3}-\frac{1}{4}\right)+\left(\frac{1}{4}-\frac{1}{5}\right)= \\
& \frac{1}{1}-\frac{1}{5}= \\
& \frac{4}{5}
\end{aligned}
$$

Using this technique, we can find a way to sum consecutive odd numbers. Let's see:

a) By listing the odd numbers one by one and starting from 1, verify that the number in position \( m \) is equal to \( m^{2}-(m-1)^{2} \).

b) Calculate the sum of all the odd numbers between 1000 and 2014.


In a regular polygon with 1994 sides, a positive number is written at each vertex. Each number is either the arithmetic mean or the geometric mean of its two adjacent numbers. If one of these numbers is 32, what is the number adjacent to it?


The surface area of a cube is 24. What is the volume of the cube?
(A) 4
(B) \( 3 \sqrt{3} \)
(C) 9
(D) 16
(E) 8


Let \( F_{1} \) and \( F_{2} \) be the foci of the ellipse \(\frac{x^{2}}{9} + \frac{y^{2}}{4} = 1\). \( P \) is a point on the ellipse such that \(\left|P F_{1}\right| : \left|P F_{2}\right| = 2 : 1\). Find the area of triangle \(\triangle P F_{1} F_{2}\).


How many six-digit numbers exist such that the sum of their digits is 51?


Find the smallest natural number \( N \) such that \( N+2 \) is divisible by 2, \( N+3 \) is divisible by 3, \ldots, \( N+10 \) is divisible by 10.


In triangle \( A B C \), the angle \( \angle B \) is equal to \( \frac{\pi}{3} \). A circle with radius 3 cm is drawn through points \( A \) and \( B \), touching the line \( A C \) at point \( A \). Another circle with radius 4 cm is drawn through points \( B \) and \( C \), touching the line \( A C \) at point \( C \). Find the length of side \( A C \).


A four-digit number satisfies the following conditions:
(1) If you simultaneously swap its unit digit with the hundred digit and the ten digit with the thousand digit, the value increases by 5940;
(2) When divided by 9, the remainder is 8.
Find the smallest odd four-digit number that satisfies these conditions.
(Shandong Province Mathematics Competition, 1979)


Find the number of pairs of integers $(a, b)$ such that \(1 \leq a \leq 80\) and \(1 \leq b \leq 30\), and the area \( S \) of the figure defined by the system of inequalities

\[
\begin{cases}
\frac{x}{a} + \frac{y}{b} \geq 1 \\
x \leq a \\
y \leq b
\end{cases}
\]

is such that the number \( 2S \) is a multiple of 5.


A chord of a circle is at a distance \( h \) from the center. In each of the segments subtended by the chord, a square is inscribed such that two adjacent vertices of the square lie on the arc, and the other two lie on the chord or its extension. What is the difference in the side lengths of these squares?


On a rotating round table, there are 8 white teacups and 7 black teacups. Fifteen dwarves wearing hats (8 white hats and 7 black hats) are sitting around the table. Each dwarf picks a teacup of the same color as their hat and places it in front of them. After this, the table is rotated randomly. What is the maximum number of teacups that can be guaranteed to match the color of the dwarf's hat after the table is rotated? (The dwarves are allowed to choose their seating, but they do not know how the table will be rotated.)


What will the inflation be over two years:
$$
\left((1+0,015)^{\wedge} 2-1\right)^{*} 100 \%=3,0225 \%
$$

What will be the real yield of a bank deposit with an extension for the second year:
$$
(1,07 * 1,07 /(1+0,030225)-1) * 100 \%=11,13 \%
$$


Given the sequence \(\left\{x_{n}\right\}\) satisfies:
\[ x_{1}=a, \, x_{2}=b, \, x_{n+2}=3 x_{n+1}+2 x_{n}, \]

if there exists an integer \(k \geq 3\) such that \(x_{k}=2019\), then determine the number of such positive integer ordered pairs \((a, b)\).


Given the hyperbola \( C: \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 \) (where \( a > 0 \) and \( b > 0 \)), its eccentricity is \( \frac{\sqrt{5}}{2} \). The points \( F_{1} \) and \( F_{2} \) are the left and right foci of \( C \), respectively. A line \( l \) passing through \( F_{2} \) intersects \( C \) at points \( A \) and \( B \) (with point \( A \) in the first quadrant), and \(\overrightarrow{A F_{2}} = 3 \overrightarrow{F_{2} B} \). Given that the area of triangle \( \triangle F_{1} A B \) is \( \frac{32}{3} \), find the radius of the incircle of triangle \( \triangle F_{1} A B \).


In triangle \( \triangle ABC \), \( |\overrightarrow{AB}| = 2 \), \( |\overrightarrow{BC}| = 3 \), \( |\overrightarrow{CA}| = 4 \). Find the value of \( \overrightarrow{AB} \cdot \overrightarrow{BC} + \overrightarrow{BC} \cdot \overrightarrow{CA} + \overrightarrow{CA} \cdot \overrightarrow{AB} \).


Let \( f(x) \) be an even function and \( g(x) \) be an odd function, and suppose \( f(x) = -g(x+c) \) where \( c > 0 \). Determine the smallest positive period of the function \( f(x) \).


Vasya has selected 8 squares on a chessboard such that no two squares are in the same row or column. On his turn, Petya places 8 rooks on the board in such a way that none of them attack each other, and then Vasya points out all the rooks that are standing on the selected squares. If the number of rooks pointed out by Vasya on this turn is even (i.e., 0, 2, 4, 6, or 8), Petya wins; otherwise, all pieces are removed from the board and Petya takes the next turn. What is the minimum number of turns in which Petya can guarantee a win?

(I. Bogdanov)


The increasing sequence $1, 3, 4, 9, 10, 12, 13, \cdots$ consists of positive integers that are either powers of 3 or the sum of some distinct powers of 3. Find the 100th term of this sequence.


There are 22 people standing in a circle, and each of them is either a knight (who always tells the truth) or a liar (who always lies). Each person says: "The next 10 people clockwise from me are liars." How many liars are there among these 22 people?


Let \( a \in \mathbf{R} \). The equation \( ||x-a|-a|=2 \) has exactly three distinct solutions. Find the value of \( a \).


Find the largest even three-digit number \( x \) that leaves a remainder of 2 when divided by 5 and satisfies the condition \(\text{GCD}(30, \text{GCD}(x, 15)) = 3\).


On an $8 \times 8$ board, several dominoes (rectangles of $2 \times 1$) can be placed without overlapping. Let $N$ be the number of ways to place 32 dominoes in this manner, and $T$ be the number of ways to place 24 dominoes in this manner. Which is greater: $-N$ or $M$? Configurations that can be derived from one another by rotating or reflecting the board are considered distinct.

Solution: \boxed{T}


Let \([x]\) denote the greatest integer less than or equal to \(x\). Find the value of \(\sum_{k=0}^{\infty}\left[\frac{n+2^{k}}{2^{k+1}}\right]\), where \(n\) is any natural number.

Solution: \boxed{n}


Find the sum of all irreducible fractions with a denominator of 3 that are between the positive integers \( m \) and \( n \) (where \( m < n \)).

Solution: \boxed{n^2 - m^2}