Let \(ABC\) be a triangle with \(CA = CB = 5\) and \(AB = 8\). A circle \(\omega\) is drawn such that the interior of triangle \(ABC\) is completely contained in the interior of \(\omega\). Find the smallest possible area of \(\omega\).


In triangle \( \triangle ABC \), \( M \) is the midpoint of side \( BC \), and \( N \) is the midpoint of line segment \( BM \). Given that \( \angle A = \frac{\pi}{3} \) and the area of \( \triangle ABC \) is \( \sqrt{3} \), find the minimum value of \( \overrightarrow{AM} \cdot \overrightarrow{AN} \).

Solution: \boxed{\sqrt{3} + 1}


A ball with a radius of 1 is placed inside a cube with an edge length of 4. The ball can move freely inside the cube, and the cube can also be flipped in any direction. What is the area of the inner surface of the cube that the ball cannot touch?


Given the real number sets \( A_n = \{ x \mid n < x^n < n+1, n \in \mathbf{N} \} \), if the intersection \( A_1 \cap A_2 \cap \cdots \cap A_n \neq \emptyset \), determine the maximum value of \( n \).


In 1968, a high school graduate returned from a written university exam and shared at home that he could not solve the following problem:

Several identical books and identical albums were purchased. 10 rubles and 56 kopecks were paid for the books, and 56 kopecks for the albums. The number of books bought was 6 more than the number of albums. How many books were bought if the price of a book is more than a ruble higher than the price of an album?

How did his younger brother, a fifth-grader, solve the problem and conclude that 8 books were bought?

If you find solving this problem challenging, try to understand and remember the conditions clearly. Seek new approaches to solving the problem. Try to "guess" the answer or approximate it. How many solutions does the problem have? Write down the solution and verify your answer.


Find the lateral surface area of a regular triangular pyramid if the plane angle at its apex is $90^{\circ}$ and the area of the base is $S$.

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


Is it possible? Can three people cover a distance of 60 km in 3 hours if they have a two-seater motorcycle at their disposal? The speed of the motorcycle is 50 km/h, and the speed of a pedestrian is 5 km/h.

Solution: \boxed{Yes}


Let \( D \) be a point inside the acute triangle \( \triangle ABC \), such that \( \angle ADB = \angle ACB + 90^\circ \) and \( AC \cdot BD = AD \cdot BC \). Calculate the ratio \( \frac{AB \cdot CD}{AC \cdot BD} \).


In the triangle \(ABC\), angle \(C\) is obtuse, \(D\) is the intersection point of the line \(DB\) perpendicular to \(AB\) and the line \(DC\) perpendicular to \(AC\). The altitude of triangle \(ADC\) drawn from vertex \(C\) intersects side \(AB\) at point \(M\). Given that \(AM = a\) and \(MB = b\), find \(AC\).

Solution: \boxed{\sqrt{a(a+b)}}


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


Given that the sequence \(\left\{a_{n}\right\}\) is a geometric sequence with all positive terms, and \(a_{50}\) and \(a_{51}\) are two different solutions to the equation \(100 \lg^{2} x = \lg (100 x)\). Find the value of \(a_{1} a_{2} \cdots a_{100}\).


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

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

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

How many of the warriors always tell the truth?


The plane is divided by parallel lines that are a distance of $2a$ apart. A needle of length $2l$ $(l<a)$ is randomly thrown onto the plane. Find the probability that the needle will intersect any of the lines.

Solution: \boxed{\frac{2l}{\pi a}}


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


A rook has traveled across an \( n \times n \) board, visiting each cell exactly once, where each of its moves was exactly one cell. The cells are numbered from 1 to \( n^2 \) in the order of the rook's path. Let \( M \) be the maximum difference between the numbers of adjacent (by side) cells. What is the smallest possible value of \( M \)?

Solution: \boxed{2n - 1}


On the radius $A O$ of a circle with center $O$, a point $M$ is chosen. On one side of $A O$, points $B$ and $C$ are chosen on the circle such that $\angle A M B = \angle O M C = \alpha$. Find the length of $B C$ if the radius of the circle is 12 and $\cos \alpha = \frac{5}{6}$.


Put ping pong balls in 10 boxes. The number of balls in each box must not be less than 11, must not be 17, must not be a multiple of 6, and must be different from each other. What is the minimum number of ping pong balls needed?


Given a plane $\alpha$ and a triangle $ABC$ whose plane is not parallel to the plane $\alpha$, find a point $S$ in space such that the lines $SA, SB, SC$ intersecting the plane $\alpha$ produce points $A', B', C'$ such that triangle $A'B'C'$ is congruent to another given triangle $MNP$.

Solution: \boxed{S}


Determine the smallest possible positive integer \( n \) with the following property: For all positive integers \( x, y, \) and \( z \) with \( x \mid y^{3} \), \( y \mid z^{3} \), and \( z \mid x^{3} \), it also holds that \( x y z \mid (x+y+z)^{n} \).

Solution: \boxed{n = 13}


Given the real number \( x \) satisfies \( 20 \sin x = 22 \cos x \), find the largest integer not greater than the real number \( \left( \frac{1}{\sin x \cos x} - 1 \right)^7 \).


The domain of the function \( f(x) \) is \(\mathbf{R}\) and it satisfies the following conditions: for \( x \in [0, 1) \), \( f(x) = 2^x - x \), and for any real number \( x \), \( f(x) + f(x+1) = 1 \). Let \( a = \log_2 3 \). Find the value of the expression \( f(a) + f(2a) + f(3a) \).


Let \( k \) and \( n \) be integers such that \( 1 \leq k \leq n \). What is the maximum number of \( k \)-element subsets of the set \(\{1, 2, \ldots, n\}\) such that for any two of these subsets, one is composed of the \( k \) smallest elements of their union?

Solution: \boxed{n - k + 1}


In trapezoid \(ABCD\), the bases \(AD\) and \(BC\) are equal to 8 and 18, respectively. It is known that the circumcircle of triangle \(ABD\) is tangent to lines \(BC\) and \(CD\). Find the perimeter of the trapezoid.


The committee consists of a chairperson, a deputy chairperson, and five more members. In how many ways can the members of the committee distribute the duties of the chairperson and the deputy chairperson among themselves?


Given \( n \) points in the plane, show that it is possible to choose at least \( \sqrt{n} \) of them such that they do not form an equilateral triangle.

Solution: \boxed{\sqrt{n}}


In the polynomial \((1-z)^{b_{1}} \cdot (1-z^{2})^{b_{2}} \cdot (1-z^{3})^{b_{3}} \cdots (1-z^{32})^{b_{32}}\), where \(b_{i} (i=1,2, \cdots, 32)\) are positive integers, this polynomial has the following remarkable property: after expanding it and removing the terms with \(z\) of degree higher than 32, exactly \(1-2z\) remains. Determine \(b_{32}\) (the answer can be expressed as the difference of two powers of 2).

Solution: \boxed{2^{27} - 2^{11}}


On a birthday card printed with April 29, a child inserts two positive integers $x$ and $y$ between the 4 and the 29, forming a five-digit number $\overline{4 x y 29}$. This number is the square of the integer $T$ that corresponds to the child's birthday: $\overline{4 x y 29} = T^{2}$. What is the integer $T$ that corresponds to the child's birthday?


Two people are tossing a coin: one tossed it 10 times, and the other tossed it 11 times.

What is the probability that the second person's coin landed on heads more times than the first person's coin?


A box contains two white socks, three blue socks, and four grey socks. Three of the socks have holes in them, but Rachel does not know what colour these socks are. She takes one sock at a time from the box without looking. How many socks must she take to be certain she has a pair of socks of the same colour without holes?


Given the sequence $\left\{a_{n}\right\}$ with the partial sum $S_{n}=2 a_{n}-1$ 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 partial sum of the sequence $\left\{b_{n}\right\}$.

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


In a $3 \times 3$ grid (each cell is a $1 \times 1$ square), place two identical chess pieces, with at most one piece per cell. There are ___ different ways to arrange the pieces (if two arrangements can overlap by rotation, they are considered the same arrangement).


Susie thinks of a positive integer \( n \). She notices that, when she divides 2023 by \( n \), she is left with a remainder of 43. Find how many possible values of \( n \) there are.


There is a uniformly growing grassland. If 20 cows are grazed, they will just finish eating all the grass in 60 days. If 30 cows are grazed, they will just finish eating all the grass in 35 days. Now, 6 cows are grazing on the grassland. After a month, 10 more cows are added. How many more days will it take for all the grass to be eaten?


Grain warehouses A and B each originally stored a certain number of full bags of grain. If 90 bags are transferred from warehouse A to warehouse B, the number of bags in warehouse B will be twice the number in warehouse A. If an unspecified number of bags are transferred from warehouse B to warehouse A, the number of bags in warehouse A will be six times the number in warehouse B. What is the minimum number of bags originally stored in warehouse A?


Let \( N \geqslant 2 \) be a natural number. What is the sum of all fractions of the form \( \frac{1}{mn} \), where \( m \) and \( n \) are coprime natural numbers such that \( 1 \leqslant m < n \leqslant N \) and \( m+n > N \)?


Find all \( x \in [1,2) \) such that for any positive integer \( n \), the value of \( \left\lfloor 2^n x \right\rfloor \mod 4 \) is either 1 or 2.


The last two digits of $\left[\frac{10^{93}}{10^{31}+3}\right]$ are $\qquad$ (where [x] denotes the greatest integer less than or equal to $x$).


In a middle school math competition, there were three problems labeled A, B, and C. Among the 25 students who participated, each student solved at least one problem. Among the students who did not solve problem A, the number who solved problem B was twice the number of those who solved problem C. The number of students who solved only problem A was one more than the number of students who solved A along with any other problem. Among the students who solved only one problem, half did not solve problem A. How many students solved only problem B?


Different positive 3-digit integers are formed from the five digits \(1, 2, 3, 5, 7\), and repetitions of the digits are allowed. As an example, such positive 3-digit integers include 352, 577, 111, etc. Find the sum of all the distinct positive 3-digit integers formed in this way.


On the radius \( AO \) of a circle with center \( O \), a point \( M \) is selected. On one side of \( AO \) on the circle, points \( B \) and \( C \) are chosen such that \( \angle AMB = \angle OMC = \alpha \). Find the length of \( BC \) if the radius of the circle is 10 and \( \sin \alpha = \frac{\sqrt{21}}{5} \).


Solution: \boxed{a_n = n^2 - 1}


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


Find the smallest positive integer \( m \) such that the equation regarding \( x, y, \) and \( z \):
\[ 2^x + 3^y - 5^z = 2m \]
has no positive integer solutions.


Let \( ABC \) be a right triangle with hypotenuse \( AC \). Let \( B' \) be the reflection of point \( B \) across \( AC \), and let \( C' \) be the reflection of \( C \) across \( AB' \). Find the ratio of \([BCB']\) to \([BC'B']\).


To determine the number of positive divisors of a number, you simply factor it as powers of distinct primes and multiply the successors of the exponents. For example, \(2016=2^{5} \cdot 3^{2} \cdot 5^{1}\) has \((5+1)(2+1)(1+1)=36\) positive divisors. Consider the number \(n=2^{7} \cdot 3^{4}\).

a) Determine the number of positive divisors of \(n^{2}\).

b) How many divisors of \(n^{2}\) are greater than \(n\)?

c) How many divisors of \(n^{2}\) are greater than \(n\) and are not multiples of \(n\)?


Find all natural numbers that are 5 times greater than their last digit.


Given that \(O\) is the origin of coordinates, the curve \(C_1: x^2 - y^2 = 1\) intersects the curve \(C_2: y^2 = 2px\) at points \(M\) and \(N\). If the circumcircle of \(\triangle OMN\) passes through the point \(P\left(\frac{7}{2}, 0\right)\), then the equation of the curve \(C_2\) is ____.

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


Distinct points are placed on a circle. Each pair of points is joined with a line segment. If 8 distinct points are placed on a circle, how many line segments would there be?
(A) 27
(B) 36
(C) 21
(D) 56
(E) 28


There are exactly 120 ways to color five cells in a \(5 \times 5\) grid such that each row and each column has exactly one colored cell.

There are exactly 96 ways to color five cells in a \(5 \times 5\) grid, excluding a corner cell, such that each row and each column has exactly one colored cell.

How many ways are there to color five cells in a \(5 \times 5\) grid, excluding two corner cells, such that each row and each column has exactly one colored cell?


Find the maximum value of the expression \(\cos (x-y)\), given that \(\sin x - \sin y = \frac{3}{4}\).


In a singing competition, the participants were Rooster, Crow, and Cuckoo. Each jury member voted for one of the three singers. The Woodpecker counted that there were 59 judges, with a total of 15 votes for Rooster and Crow combined, 18 votes for Crow and Cuckoo combined, and 20 votes for Cuckoo and Rooster combined. The Woodpecker is bad at counting, but each of the four numbers he named differs from the correct number by no more than 13. How many judges voted for Crow?


There are 29 students in a class: some are honor students who always tell the truth, and some are troublemakers who always lie.

All the students in this class sat at a round table.

- Several students said: "There is exactly one troublemaker next to me."
- All other students said: "There are exactly two troublemakers next to me."

What is the minimum number of troublemakers that can be in the class?


a) How many four-digit numbers have a sum of their digits that is even?

b) A two-digit number with distinct and non-zero digits is called beautiful if the tens digit is greater than the units digit. How many beautiful numbers exist?

c) How many even four-digit numbers can we form using the digits $0,1,2,3,4,5$ without repeating any digit?

d) What is the average of all five-digit numbers that can be formed using each of the digits $1,3,5,7,$ and $8$ exactly once?


Let \( S \) be a subset of the set \(\{1, 2, 3, \ldots, 2015\} \) such that for any two elements \( a, b \in S \), the difference \( a - b \) does not divide the sum \( a + b \). Find the maximum possible size of \( S \).


Let \( p_{i} \) be the \(i^{\text {th }}\) prime number; for example, \( p_{1}=2, p_{2}=3 \), and \( p_{3}=5 \). For each prime number, construct the point \( Q_{i}\left(p_{i}, 0\right) \). Suppose \( A \) has coordinates \((0,2)\). Determine the sum of the areas of the triangles \(\triangle A Q_{1} Q_{2}, \triangle A Q_{2} Q_{3}, \triangle A Q_{3} Q_{4}, \triangle A Q_{4} Q_{5}, \triangle A Q_{5} Q_{6}, \) and \(\triangle A Q_{6} Q_{7}\).


Given two distinct lines \(OA\) and \(OB\), take any point on \(OA\) and draw a perpendicular to \(OB\). Then, from the foot of this perpendicular, draw another perpendicular to \(OA\), and from the second foot, draw a perpendicular to \(OB\), and so on, indefinitely. If the lengths of the initial two perpendicular segments are \(a\) and \(b\) (\(a \neq b\)), what is the sum of these perpendicular segments as the process continues indefinitely? The limit of this sum as the perpendiculars increase indefinitely is:

(A) \(\frac{b}{a-b}\)

(B) \(\frac{a}{a-b}\)

(C) \(\frac{ab}{a-b}\)

(D) \(\frac{b^{2}}{a-b}\)

(E) \(\frac{a^{2}}{a-b}\)

Solution: \boxed{\frac{a^2}{a - b}}


Given a triangle \(ABC\). An incircle is inscribed in it, touching the sides \(AB\), \(AC\), and \(BC\) at the points \(C_1\), \(B_1\), \(A_1\) respectively. Find the radius of the excircle \(w\), which touches the side \(AB\) at the point \(D\), the extension of the side \(BC\) at the point \(E\), and the extension of the side \(AC\) at the point \(G\). It is known that \(CE = 6\), the radius of the incircle is 1, and \(CB_1 = 1\).


How many square columns are there where the edge length measured in cm is an integer, and the surface area measured in $\mathrm{cm}^{2}$ is equal to the volume measured in $\mathrm{cm}^{3}$?


Given that $x$ is a four-digit number and the sum of its digits is $y$. If the value of $\frac{x}{y}$ is minimized, what is $x$?


You are given an unlimited supply of red, blue, and yellow cards to form a hand. Each card has a point value and your score is the sum of the point values of those cards. The point values are as follows: the value of each red card is 1, the value of each blue card is equal to twice the number of red cards, and the value of each yellow card is equal to three times the number of blue cards. What is the maximum score you can get with fifteen cards?


Let the set \( M = \{1, 2, \cdots, 1000\} \). For any non-empty subset \( X \) of \( M \), let \( a_X \) represent the sum of the largest and smallest numbers in \( X \). What is the arithmetic mean of all such \( a_X \)?


Let \(ABC\) be a triangle with \(AB = 2\), \(CA = 3\), and \(BC = 4\). Let \(D\) be the point diametrically opposite \(A\) on the circumcircle of \(\triangle ABC\), and let \(E\) lie on line \(AD\) such that \(D\) is the midpoint of \(\overline{AE}\). Line \(l\) passes through \(E\) perpendicular to \(\overline{AE}\), and \(F\) and \(G\) are the intersections of the extensions of \(\overline{AB}\) and \(\overline{AC}\) with \(l\). Compute \(FG\).


We define \( a \star b = a \times a - b \times b \). Find the value of \( 3 \star 2 + 4 \star 3 + 5 \star 4 + \cdots + 20 \star 19 \).


Let \(ABC\) be a triangle and \(O\) be the center of its circumcircle. Let \(P\) and \(Q\) be points on the sides \(AC\) and \(AB\), respectively. Let \(K, L\), and \(M\) be the midpoints of the segments \(BP, CQ\), and \(PQ\), respectively, and let \(\Gamma\) be the circle passing through \(K, L\), and \(M\). Suppose that the line \(PQ\) is tangent to the circle \(\Gamma\).

Show that \(OP = OQ\).

Solution: \boxed{OP = OQ}


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


Define: \(\triangle a = a + (a + 1) + (a + 2) + \cdots + (2a - 2) + (2a - 1)\). For example: \(\triangle 5 = 5 + 6 + 7 + 8 + 9\). What is the result of \(\triangle 1 + \triangle 2 + \triangle 3 + \cdots + \triangle 19 + \triangle 20\)?


Find the sum of all four-digit natural numbers composed of the digits 3, 6, and 9.


Before the Christmas concert, students offered 60 handmade items for sale, allowing each customer to choose their own price. The entire proceeds were for charity. At the start of the concert, students calculated the average revenue per sold item, which resulted in an exact whole number. Since they had not yet sold all 60 items, they continued to offer them after the concert. Seven more items were bought for a total of 2505 Kč, increasing the average revenue per sold item to exactly 130 Kč. How many items remained unsold?


Two people, A and B, depart simultaneously from points A and B, respectively, and travel towards each other, meeting at point C. If at the start, person A increases their speed by $\frac{1}{4}$ and person B increases their speed by 10 km/h, and they still meet at point C, determine the original speed of person B in km/h.

Solution: \boxed{40 \text{ km/h}}


Given $f(x)$ is a function defined on $\mathbf{R}$, and for any real number $x$, it satisfies $2 f(x) + f\left(x^{2}-1\right) = 1$. Find the value of $f(\sqrt{2})$.


A square with a side of 75 mm was cut by two parallel cuts into three rectangles. It turned out that the perimeter of one of these rectangles is half the sum of the perimeters of the other two. What is the perimeter of this rectangle? Provide the answer in centimeters.

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


The Absent-Minded Scientist commutes to work on weekdays using the Ring Line of the Moscow Metro from "Taganskaya" station to "Kievskaya" station, and back in the evening. When he enters the station, he boards the first arriving train. It is known that in both directions, trains run at approximately equal intervals. The train traveling via the northern route (through "Belorusskaya") takes 17 minutes to go from "Kievskaya" to "Taganskaya" or vice versa, while the train traveling via the southern route (through "Paveletskaya") takes 11 minutes.

According to his long-term observations, the train traveling counterclockwise arrives at "Kievskaya" on average 1 minute and 15 seconds after the clockwise train arrives. The same is true for "Taganskaya." Moreover, the scientist notes that his average travel time from home to work is 1 minute less than his average travel time from work to home. Find the expected interval between trains traveling in one direction.

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


Calculate the limit of the function:

$$\lim _{x \rightarrow 0} \frac{5^{2 x}-2^{3 x}}{\sin x+\sin x^{2}}$$

Solution: \boxed{\ln \frac{25}{8}}


A light ray falls at an angle $\alpha=60^{\circ}$ on the front surface of a plane-parallel glass plate. The refractive index of the glass is $n=1.6$. By what angle from the direction of the incident ray is the ray, reflected from the back surface of the plate and exiting back through the front surface, deviated?


Let \( x \) be the number of 7th graders who together scored \( n \) points in the tournament. The number of 8th graders participating in the tournament is \( 10 \times x \), and they scored a total of \( 4.5 \times n \) points. Thus, the total number of students participating in the tournament is \( 11 \times x \), and they scored \( 5.5 \times n \) points collectively. The total number of points scored by all participants is equal to the number of games played. We obtain the equation:

$$
\begin{gathered}
5.5 \times n = \frac{11 \times x \times (11x - 1)}{2} \\
n = x \times (11x - 1)
\end{gathered}
$$

Each 7th grader played \( 11x - 1 \) games (since there are \( 11x \) participants in total). Therefore, \( x \) 7th graders could collectively score \( n \) points, i.e., \( x \times (11x - 1) \) points, only if each of them won all their games. This is possible only when \( x = 1 \) (since two 7th graders cannot simultaneously win against each other).


Given a rectangle \(ABCD\). A circle intersects the side \(AB\) at points \(K\) and \(L\). Find the length of segment \(MN\) if \(AK = 10\), \(KL = 17\), \(DN = 7\).


Find all integers $n$ such that $n^{4} + 6n^{3} + 11n^{2} + 3n + 31$ is a perfect square.

Solution: \boxed{n = 10}


Find all functions \( g: \mathbb{N} \rightarrow \mathbb{N} \) such that \( (g(n)+m)(g(m)+n) \) is a perfect square for all \( n, m \).

Solution: \boxed{g(n) = n + c}


A circle on a plane divides the plane into 2 parts. A circle and a line can divide the plane into a maximum of 4 parts. A circle and 2 lines can divide the plane into a maximum of 8 parts. A circle and 5 lines can divide the plane into a maximum of how many parts?


In the pyramid $ABCD$, the edges are given as follows: $AB = 7$, $BC = 8$, $CD = 4$. Find the edge $DA$ given that the lines $AC$ and $BD$ are perpendicular.


Given a cube \( A B C D A_{1} B_{1} C_{1} D_{1} \) with edge length 1. A line \( l \) passes through the point \( E \), the midpoint of edge \( C_{1} D_{1} \), and intersects the lines \( A D_{1} \) and \( A_{1} B \). Find the distance from point \( E \) to the point of intersection of line \( l \) with the line \( A_{1} B \).


Integrate the equation

$$
\left(y^{3}-2 x y\right) dx+\left(3 x y^{2}-x^{2}\right) dy=0
$$

Solution: \boxed{y^3 x - x^2 y = C}


Find the area of a triangle if two of its medians are equal to $\frac{15}{7}$ and $\sqrt{21}$, and the cosine of the angle between them is $\frac{2}{5}$.


A parallelepiped \( ABCD A_{1} B_{1} C_{1} D_{1} \) is given. A point \( X \) is chosen on edge \( A_{1} D_{1} \) and a point \( Y \) is chosen on edge \( BC \). It is known that \( A_{1}X = 5 \), \( BY = 3 \), and \( B_{1} C_{1} = 14 \). The plane \( C_{1}XY \) intersects the ray \( DA \) at point \( Z \). Find \( DZ \).


A novice gardener planted daisies, buttercups, and marguerites in their garden. When they sprouted, it turned out that there were 5 times more daisies than non-daisies and 5 times fewer buttercups than non-buttercups. What fraction of the sprouted plants are marguerites?


Find all real polynomials \( p(x) \) such that \( 1 + p(x) \equiv \frac{p(x-1) + p(x+1)}{2} \).

Solution: \boxed{p(x) = x^2 + bx + c}


What is the three-digit (integer) number that is 12 times the sum of its digits?


Let \( ABC \) be a triangle, and let \( M \) be the midpoint of side \( AB \). If \( AB \) is 17 units long and \( CM \) is 8 units long, find the maximum possible value of the area of \( ABC \).


Given a triangle \(ABC\) such that the median, altitude, and angle bisector from \(A\) divide the angle \(\widehat{A}\) into four equal angles \(\alpha\). Express all the angles in the figure in terms of \(\alpha\) and calculate \(\alpha\).


In an $11 \times 11$ square, the central cell is colored black. Maxim found the largest rectangular grid area that is entirely within the square and does not contain the black cell. How many cells are in that rectangle?


As shown in the figure, the area of rectangle $ABCD$ is $56 \mathrm{~cm}^{2}$. Given that $BE=3 \mathrm{~cm}$ and $DF=2 \mathrm{~cm}$, please find the area of triangle $AEF$.

Solution: \boxed{25 \text{ cm}^2}


Figure 0 consists of a square with side length 18. For each integer \(n \geq 0\), Figure \(n+1\) consists of Figure \(n\) with the addition of two new squares constructed on each of the squares that were added in Figure \(n\). The side length of the squares added in Figure \(n+1\) is \(\frac{2}{3}\) of the side length of the smallest square(s) in Figure \(n\). Define \(A_{n}\) to be the area of Figure \(n\) for each integer \(n \geq 0\). What is the smallest positive integer \(M\) with the property that \(A_{n}<M\) for all integers \(n \geq 0\)?


Given constants \( a \) and \( b \) with \( a \neq 0 \), the function \( f(x) = \frac{x}{a x + 3} \) satisfies \( f(2) = 1 \) and \( f(x) = x \) has a unique solution for \( x \).

1. Find the values of \( a \) and \( b \).
2. Given \( x_1 = 1 \) and \( x_n = f(x_{n-1}) \) for \( n = 2, 3, \ldots \), find the general formula for the sequence \( \{x_n\} \).

Solution: \boxed{x_n = \frac{2}{n+1}}


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


The sequence \(\left\{\alpha_{n}\right\}\) is an arithmetic sequence with common difference \(\beta\). The sequence \(\left\{\sin \alpha_{n}\right\}\) is a geometric sequence with common ratio \(q\). Given \(\alpha_{1}\) and \(\beta\) are real numbers, find \(q\).

Solution: \boxed{\pm 1}


A number \( A \) consisting of eight non-zero digits is added to a seven-digit number consisting of identical digits, resulting in an eight-digit number \( B \). It turns out that \( B \) can be obtained by permuting some of the digits of \( A \). What digit can \( A \) start with if the last digit of \( B \) is 5?


Find those prime numbers $p$ for which the number $p^{2}+11$ has exactly 6 positive divisors.


Let \( a_{1}=1, a_{2}=2 \) and for all \( n \geq 2 \), \[ a_{n+1}=\frac{2 n}{n+1} a_{n}-\frac{n-1}{n+1} a_{n-1}. \] It is known that \( a_{n} > 2 + \frac{2009}{2010} \) for all \( n \geq m \), where \( m \) is a positive integer. Find the least value of \( m \).


Given a cube \( ABCD A_1 B_1 C_1 D_1 \) with an edge length of 1. A sphere passes through vertices \( A \) and \( C \) and the midpoints \( F \) and \( E \) of edges \( B_1 C_1 \) and \( C_1 D_1 \) respectively. Find the radius \( R \) of this sphere.


Calculate: \(\left(2 \frac{2}{3} \times\left(\frac{1}{3}-\frac{1}{11}\right) \div\left(\frac{1}{11}+\frac{1}{5}\right)\right) \div \frac{8}{27} = 7 \underline{1}\).

Solution: \boxed{7 \frac{1}{2}}


Find a five-digit number that has the following property: when multiplied by 9, the result is a number represented by the same digits but in reverse order.


In triangle \( ABC \) with sides \( AB = c \), \( BC = a \), and \( AC = b \), a median \( BM \) is drawn. Incircles are inscribed in triangles \( ABM \) and \( BCM \). Find the distance between the points where these incircles touch the median \( BM \).


The base of the rectangular parallelepiped $A B C D A_{1} B_{1} C_{1} D_{1}$ is a square $A B C D$. Find the maximum possible angle between the line $B D_{1}$ and the plane $B D C_{1}$.

Solution: \boxed{\arcsin \frac{1}{3}}


Person A and person B start from locations A and B respectively, walking towards each other at speeds of 65 meters per minute and 55 meters per minute simultaneously. They meet after 10 minutes. What is the distance between A and B in meters? Also, what is the distance of the meeting point from the midpoint between A and B in meters?


A fair coin is flipped ten times. Let $\frac{i}{j}$ be the probability that heads do not appear consecutively, where $i$ and $j$ are coprime. Find $i + j$.


In a sequence of positive integers, an inversion is a pair of positions where the element at the leftmost position is greater than the element at the rightmost position. For example, the sequence 2, 5, 3, 1, 3 has 5 inversions: between the first and the fourth position, between the second position and all the positions to its right, and finally between the third and the fourth position. Among all sequences of positive integers whose elements sum to $n$, what is the maximum possible number of inversions if

a) $n=7$ ?

b) $n=2019$ ?

Note: The sequences of positive integers considered in this problem can have more than 5 elements.


Point \( D \) lies on the side \( AC \) of triangle \( ABC \). A circle with diameter \( BD \) intersects sides \( AB \) and \( BC \) at points \( P \) and \( T \) respectively. Points \( M \) and \( N \) are the midpoints of the segments \( AD \) and \( CD \) respectively. It is known that \( PM \| TN \).

a) Find the angle \( ABC \).

b) Additionally, it is known that \( MP = 1 \), \( NT = \frac{3}{2} \), and \( BD = \sqrt{5} \). Find the area of triangle \( ABC \).


Three sisters, whose average age is 10, all have different ages. The average age of one pair of the sisters is 11, while the average age of a different pair is 12. What is the age of the eldest sister?


Suppose \( m \) and \( n \) are integers with \( 0 < m < n \). Let \( P = (m, n) \), \( Q = (n, m) \), and \( O = (0,0) \). For how many pairs of \( m \) and \( n \) will the area of triangle \( OPQ \) be equal to 2024?


How many zeros are at the end of the product \( s(1) \cdot s(2) \cdot \ldots \cdot s(100) \), where \( s(n) \) denotes the sum of the digits of the natural number \( n \)?


There are 5 different lines on a plane, forming $m$ intersection points. How many different values can $m$ take?


In square \( ABCD \), \( E \) and \( F \) are the midpoints of \( AB \) and \( BC \) respectively. \( AF \) and \( CE \) intersect at \( G \), and \( AF \) and \( DE \) intersect at \( H \). Determine the ratio \( AH : HG : GF \).


Mat is digging a hole. Pat asks him how deep the hole will be. Mat responds with a riddle: "I am $90 \mathrm{~cm}$ tall and I have currently dug half the hole. When I finish digging the entire hole, the top of my head will be as far below the ground as it is above the ground now." How deep will the hole be when finished?

Solution: \boxed{120 \text{ cm}}


Find the minimum value of
$$
\max (a+b+c, b+c+d, c+d+e, d+e+f, e+f+g)
$$
subject to the constraints
(i) \( a, b, c, d, e, f, g \geq 0 \),
(ii) \( a+b+c+d+e+f+g=1 \).


Given the point \( P(-2,5) \) lies on the circle \(\odot C: x^{2}+y^{2}-2x-2y-23=0\), and the line \( l: 3x+4y+8=0 \) intersects \(\odot C\) at points \( A \) and \( B \). Find \(\overrightarrow{AB} \cdot \overrightarrow{BC}\).


Simplify the expression:
\[ \frac{\sqrt{x-2 \sqrt{2}}}{\sqrt{x^{2}-4 x \sqrt{2}+8}} - \frac{\sqrt{x+2 \sqrt{2}}}{\sqrt{x^{2}+4 x \sqrt{2}+8}} \]
given \( x = 3 \).


A and B are taking turns shooting with a six-shot revolver that has only one bullet. They randomly spin the cylinder before each shot. A starts the game. Find the probability that the gun will fire while A is holding it.


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


23. Two friends, Marco and Ian, are talking about their ages. Ian says, "My age is a zero of a polynomial with integer coefficients."
Having seen the polynomial \( p(x) \) Ian was talking about, Marco exclaims, "You mean, you are seven years old? Oops, sorry I miscalculated! \( p(7) = 77 \) and not zero."
"Yes, I am older than that," Ian's agreeing reply.
Then Marco mentioned a certain number, but realizes after a while that he was wrong again because the value of the polynomial at that number is 85.
Ian sighs, "I am even older than that number."
Determine Ian's age.


Point \( P \) is located on side \( AB \) of square \( ABCD \) such that \( AP:PB = 1:2 \). Point \( Q \) lies on side \( BC \) of the square and divides it in the ratio \( BQ:QC = 2 \). Lines \( DP \) and \( AQ \) intersect at point \( E \). Find the ratio of lengths \( PE:ED \).


From the same number of squares with sides 1, 2, and 3, form a square of the smallest possible size.


Point \( P(4, -5) \) is outside the circle \( x^{2} + y^{2} = 4 \). Let \( PA \) and \( PB \) be the tangents to the circle from point \( P \). What is the equation of the line containing the points of tangency \( A \) and \( B \)?

Solution: \boxed{4x - 5y = 4}


There are \( n \) lines in general position on a plane. How many triangles are formed by these lines?


A thin diverging lens, with an optical power \( D_p = -6 \) diopters, is illuminated by a light beam with a diameter of \( d_1 = 5 \) cm. On a screen parallel to the lens, a bright spot with a diameter of \( d_2 = 20 \) cm is observed. After replacing the thin diverging lens with a thin converging lens, the size of the spot on the screen remains unchanged. Determine the optical power \( D_c \) of the converging lens.

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


On a line, the points \( 0, \pm 1, \pm 2, \pm 3, \ldots \) are marked.

A particle, upon reaching point \( n \), moves to point \( n+1 \) with probability \( \frac{1}{2} \) and to point \( n-1 \) with probability \( \frac{1}{2} \) in one unit of time. Initially, the particle is at point 0. Find:

a) the probability \( x \) that the particle will ever be at point 1;

b) the probability \( y \) that the particle will ever be at point -1;

c) the probability \( z \) that the particle will ever return to point 0 (i.e., be at point 0 at a moment distinct from the initial one).


There are 52 marbles in five bags in total. No two bags contain the same number of marbles, and some bags may even be empty. It is possible to redistribute the marbles from any (non-empty) bag to the remaining four bags such that they all contain an equal number of marbles.

a) Find a distribution of marbles into the bags that satisfies all the given properties.

b) Show that in any distribution with the given properties, there is exactly one bag containing 12 marbles.


Find the mass of the body $\Omega$ with density $\mu = 20z$, bounded by the surfaces

$$
z = \sqrt{1 - x^{2} - y^{2}}, \quad z = \sqrt{\frac{x^{2} + y^{2}}{4}}
$$

Solution: \boxed{m = 4\pi}


Eighty percent of dissatisfied customers leave angry reviews about a certain online store. Among satisfied customers, only fifteen percent leave positive reviews. This store has earned 60 angry reviews and 20 positive reviews. Using this data, estimate the probability that the next customer will be satisfied with the service in this online store.


Two cars covered the same distance. The speed of the first car was constant and three times less than the initial speed of the second car. The second car traveled the first half of the journey without changing speed, then its speed was suddenly halved, then traveled with constant speed for another quarter of the journey, and halved its speed again for the next eighth part of the journey, and so on. After the eighth decrease in speed, the car did not change its speed until the end of the trip. By how many times did the second car take more time to complete the entire journey than the first car?


Let \(a, b, c, d\) be positive integers such that the least common multiple (L.C.M.) of any three of them is \(3^{3} \times 7^{5}\). How many different sets of \((a, b, c, d)\) are possible if the order of the numbers is taken into consideration?


Given a cube $ABCD A_1B_1C_1D_1$ with $O, E, F, G$ being the midpoints of $BD$, $BB_1$, $A_1D_1$, and $D_1C_1$ respectively, and $AB = 1$. Find the volume of the tetrahedron $OEFG$.


In a notebook, all irreducible fractions with a numerator of 15 that are greater than $\frac{1}{16}$ and less than $\frac{1}{15}$ are written. How many such fractions are written in the notebook?


Let \( f(x) = a \cos(x + 1) + b \cos(x + 2) + c \cos(x + 3) \), where \( a, b, c \) are real. Given that \( f(x) \) has at least two zeros in the interval \( (0, \pi) \), find all its real zeros.

Solution: \boxed{\mathbb{R}}


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

\[ r = \frac{1}{2} + \cos \phi \]


Let \( f(x) \) be a function defined on \( \mathbf{R} \). Given that \( f(0)=2008 \), and for any \( x \in \mathbf{R} \), it satisfies \( f(x+2)-f(x) \leq 3 \cdot 2^{x} \) and \( f(x+6)-f(x) \geq 63 \cdot 2^{x} \), find \( f(2008) \).

Solution: \boxed{2^{2008} + 2007}


Find the sum of the squares of the natural divisors of the number 1800. (For example, the sum of the squares of the natural divisors of the number 4 is \(1^{2}+2^{2}+4^{2}=21\)).


The number written on the board is allowed to be multiplied by 5 or to rearrange its digits (zero cannot be placed at the first position). Is it possible to obtain a 150-digit number $5222 \ldots 2223$ from the number 1 in this way?

Solution: \boxed{\text{No}}


In how many ways can we arrange 7 white balls and 5 black balls in a line so that there is at least one white ball between any two black balls?


The lateral edges of a triangular pyramid are mutually perpendicular, and the sides of the base are $\sqrt{85}$, $\sqrt{58}$, and $\sqrt{45}$. The center of the sphere, which touches all the lateral faces, lies on the base of the pyramid. Find the radius of this sphere.


Calculate \(\sin (\alpha-\beta)\) if \(\sin \alpha - \cos \beta = \frac{3}{4}\) and \(\cos \alpha + \sin \beta = -\frac{2}{5}\).


Given a trapezoid \(ABCD\) and a point \(M\) on the side \(AB\) such that \(DM \perp AB\). It is found that \(MC = CD\). Find the length of the upper base \(BC\), if \(AD = d\).


Given three points \(A, B, C\) on a plane and three angles \(\angle D, \angle E, \angle F\), each less than \(180^{\circ}\) and summing to \(360^{\circ}\), use a ruler and protractor to construct a point \(O\) on the plane such that \(\angle A O B = \angle D\), \(\angle B O C = \angle E\), and \(\angle C O A = \angle F\). You may use the protractor to measure and set angles.

Solution: \boxed{O}


With eight small cubes each with an edge length of $1 \text{ cm}$ stacked into a three-dimensional shape, and given its top view as shown in the diagram, how many distinct stacking methods are there (considering rotations that result in the same shape as one method)?


The length of a rectangle is 25% greater than its width. A straight cut, parallel to the shorter side, splits this rectangle into a square and a rectangular strip. By what percentage is the perimeter of the square greater than the perimeter of the strip?


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


The non-zero real numbers \(a\) and \(b\) satisfy the equation

$$
\frac{a^{2} b^{2}}{a^{4}-2 b^{4}}=1
$$

Find, with reasoning, all the values taken by the expression

$$
\frac{a^{2}-b^{2}}{a^{2}+b^{2}}
$$


Find the non-negative integer-valued functions $f$ defined on the non-negative integers that have the following two properties:

(i) $\quad f(1)>0$;

(ii) $\quad f\left(m^{2}+n^{2}\right)=f^{2}(m)+f^{2}(n)$

for any non-negative integers $m$ and $n$.

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


In the tetrahedron \( P-ABC \), \(\angle APB = \angle BPC = \angle CPA = 90^\circ\). Point \( D \) is inside the base \( ABC \), and \(\angle APD = 45^\circ\), \(\angle BPD = 60^\circ\). Find the cosine of \(\angle CPD\).


Mom made homemade currant juice and poured it into bottles. She had two types of bottles: small bottles with a volume of \(500 \ \mathrm{ml}\) and large bottles with a volume of \(750 \ \mathrm{ml}\). In the end, she had 12 small empty bottles, and the remaining bottles were completely filled. Then Mom realized that she could have poured the juice in such a way that only large bottles would be left empty and all others would be completely filled.

How many empty bottles would she have in that case?


Let's call a number "wonderful" if it has exactly 3 different odd natural divisors (and any number of even ones). How many "wonderful" two-digit numbers exist?


Vasya wrote a set of distinct natural numbers on the board, each of which does not exceed 2023. It turned out that for any two written numbers \(a\) and \(b\), the number \(a + b\) is not divisible by the number \(a - b\). What is the maximum number of numbers Vasya might have written?


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


The cells of a $9 \times 9$ board are painted in black and white in a checkerboard pattern. How many ways are there to place 9 rooks on cells of the same color on the board such that no two rooks attack each other? (A rook attacks any cell that is in the same row or column as it.)


Find all functions \( f \) defined on all real numbers and taking real values such that

\[ f(f(y)) + f(x - y) = f(x f(y) - x) \]

for all real numbers \( x, y \).

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


Given that \( f(x) \) is an odd function defined on \( \mathbf{R} \), and the function \( y = f(x+1) \) is an even function. When \( -1 \leq x \leq 0 \), \( f(x) = x^3 \). Find \( f\left( \frac{9}{2} \right) \).


Given a right trapezoid \(ABCD\) with bases \(AB\) and \(CD\), and right angles at \(A\) and \(D\). Given that the shorter diagonal \(BD\) is perpendicular to the side \(BC\), determine the smallest possible value for the ratio \(CD / AD\).


A ball, sliding on a smooth horizontal surface, catches up with a block that was moving on the same surface. The speed of the ball is perpendicular to the face of the block it hits. The mass of the ball is much smaller than the mass of the block. After an elastic collision, the ball slides on the surface in the opposite direction with a speed that is 2 times less than its initial speed.

Find the ratio of the velocities of the ball and the block before the collision.


Given that in triangle \( \triangle ABC \), the incenter is \( I \), and it satisfies \( 2 \overrightarrow{IA} + 5 \overrightarrow{IB} + 6 \overrightarrow{IC} = \overrightarrow{0} \), determine the value of \( \cos \angle B \).


Alice draws three cards from a standard 52-card deck with replacement. Ace through 10 are worth 1 to 10 points respectively, and the face cards King, Queen, and Jack are each worth 10 points. The probability that the sum of the point values of the cards drawn is a multiple of 10 can be written as \(\frac{m}{n}\), where \(m, n\) are positive integers and \(\operatorname{gcd}(m, n)=1\). Find \(100m+n\).


Obtain the equation of a line on the plane, given that the line passes through the point $P(2, 3)$ and is perpendicular to the vector $\vec{n} = \{A, B\} = \{4, -1\}$.

Solution: \boxed{4x - y - 5 = 0}


Find the numerical value of the expression

$$
\frac{1}{a^{2}+1}+\frac{1}{b^{2}+1}+\frac{2}{ab+1}
$$

if it is known that \(a\) is not equal to \(b\) and the sum of the first two terms is equal to the third term.


A right isosceles triangle is inscribed in a triangle with a base of 30 and a height of 10 such that its hypotenuse is parallel to the base of the given triangle, and the vertex of the right angle lies on this base. Find the hypotenuse.


Find all pairs \((p, q)\) of prime numbers for which the difference of their fifth powers is also a prime number.

Solution: \boxed{(3, 2)}


Let the sequence \(\{a_n\}\) be defined as:
\[a_1 = 1, \quad a_{n+1} = a_n + \frac{1}{a_n} \quad (n = 1, 2, \ldots)\]

Find the integer part of \(a_{2019}\).


Is it possible to evenly divide 13 identical rectangular cakes among six children so that each cake is either not cut at all, cut into two equal parts, or cut into three equal parts?

Solution: \boxed{Yes}


On the New Year's table, there are 4 glasses in a row: the first and third contain orange juice, and the second and fourth are empty. While waiting for guests, Valya absent-mindedly and randomly pours juice from one glass to another. Each time, she can take a full glass and pour all its contents into one of the two empty glasses.

Find the expected number of pourings for the first time when the situation is reversed: the first and third glasses are empty, and the second and fourth glasses are full.


From the condition, it follows that the quadrilateral ABCD is inscribed. Then $\mathrm{MD} \cdot \mathrm{MC}=\mathrm{MA} \cdot \mathrm{MB}=(3+2) \cdot 3=15$


Select several numbers from $1, 2, 3, \cdots, 9, 10$ so that every number among $1, 2, 3, \cdots, 19, 20$ can be expressed as either one of the selected numbers or the sum of two selected numbers (which can be the same). What is the minimum number of selections needed?


Given a function \( f: \mathbf{R} \rightarrow \mathbf{R} \) such that for any real numbers \( x, y, z \), it holds that \( f(xy) + f(xz) - 2f(x)f(yz) \geq \frac{1}{2} \). Find the value of \([1 \cdot f(1)] + [2 \cdot f(2)] + \cdots + [2022 \cdot f(2022)]\), where \([x]\) denotes the greatest integer less than or equal to \( x \).


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

$$
f(f(n)) \leq \frac{f(n) + n}{2}
$$

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


\( p(z) \) and \( q(z) \) are complex polynomials with the same set of roots (but possibly different multiplicities). \( p(z) + 1 \) and \( q(z) + 1 \) also have the same set of roots. Show that \( p(z) \equiv q(z) \).

Solution: \boxed{p(z) \equiv q(z)}


Calculate the value of the square of the expression

$$
1+\frac{x}{2}-\frac{x^{2}}{8}+\frac{x^{3}}{16}-\frac{5 x^{4}}{128}
$$

to 6 decimal places for $x= \pm 0.1, \pm 0.05, \pm 0.01$.


In a competition, the full score is 100 points. Among the participating students, the highest score is 83 points and the lowest score is 30 points (all scores are integers). There are a total of 8000 students participating. At least how many students have the same score?


How many integers \( n \) are there within the range 1 to 1000, such that \( n^{1993} + 1 \) and \( n^{1994} + 1 \) are coprime?


We consider non-decreasing paths on the integer grid $\mathbb{Z}_{+}^{2}=\{(i, j) \mid i, j=0,1,2, \ldots\}$, which start at the point $(0, 0)$ and end at the point $(n, n)$ while remaining below or touching the "diagonal" (i.e., paths passing through the points in the set $\{(i, j) \in \mathbb{Z}_{+}^{2} \mid 0 \leqslant j \leqslant i \leqslant n\}$).

Show that the number of such paths is equal to $C_{n+1}$, where

$$
C_{n}=\frac{1}{n+1} \binom{2n}{n}, \quad n \geqslant 1
$$

are the Catalan numbers. (Sometimes the Catalan numbers are denoted as $c_{n}=C_{n+1}$, $n \geqslant 0$.)


Two people are playing "Easter egg battle." In front of them is a large basket of eggs. They randomly pick one egg each and hit them against each other. One of the eggs breaks, the defeated player takes a new egg, and the winner keeps their egg for the next round (the outcome of each round depends only on which egg has the stronger shell; the winning egg retains its strength). It is known that the first player won the first ten rounds. What is the probability that they will also win the next round?


The center of the upper base of a cube with an edge length of \(a\) is connected to the midpoints of the sides of the lower base, which are also connected in sequential order. Calculate the total surface area of the resulting pyramid.


If $A$ is the number of edges, $F$ is the number of faces, and $S$ is the number of vertices of a closed polyhedron, then show that always:

$$
A + 2 = F + S
$$

Solution: \boxed{A + 2 = F + S}


In the quadrilateral \(A B C D\), \(A B = B C = m\), \(\angle A B C = \angle A D C = 120^{\circ}\). Find \(B D\).

Solution: \boxed{m}


Find the value of
$$
\sum_{1 \leq a<b<c} \frac{1}{2^{a} 3^{b} 5^{c}}
$$
(i.e. the sum of \(\frac{1}{2^{a} 3^{b} 5^{c}}\) over all triples of positive integers \((a, b, c)\) satisfying \(a < b < c\)).


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

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


In $\triangle ABC$, $\angle BAC = 60^\circ$. The angle bisector of $\angle BAC$, line segment $AD$, intersects $BC$ at $D$ and satisfies $\overrightarrow{AD} = \frac{1}{4} \overrightarrow{AC} + t \overrightarrow{AB}$. Given that $AB = 8$, find the length of $AD$.


Along a road, 10 lamp posts were placed at equal intervals, and the distance between the extreme posts was $k$ meters. Along another road, 100 lamp posts were placed at the same intervals, and the distance between the extreme posts was $m$ meters. Find the ratio $m: k$.


In triangle \(ABC\), angle \(C\) is three times the size of angle \(A\). Point \(D\) is chosen on the side \(AB\) such that \(BD = BC\).

Find \(CD\), given that \(AD = 4\).

Solution: \boxed{CD = 4}


As shown in the figure, the small square $EFGH$ is inside the larger square $ABCD$. The total shaded area is 124 square centimeters. Points $E$ and $H$ are on the side $AD$, and $O$ is the midpoint of line segment $CF$. Find the area of the quadrilateral $BOGF$ in square centimeters.


If organisms do not die but only divide, then the population will certainly never die out.

The conditions are satisfied by the function whose graph is highlighted in the image.

$$
x(p)=\left\{\begin{array}{l}
1, \text { if } 0 \leq p \leq \frac{1}{2} \\
\frac{q}{p}, \text { if } \frac{1}{2}<p \leq 1
\end{array}\right.
$$

In our case $p=0.6>\frac{1}{2}$, therefore, $x=\frac{q}{p}=\frac{2}{3}$.


In $\triangle ABC$, $\overrightarrow{GA} + \overrightarrow{GB} + \overrightarrow{GC} = 0$ and $\overrightarrow{GA} \cdot \overrightarrow{GB} = 0$. Find $\frac{(\tan A + \tan B) \tan C}{\tan A \tan B} = \quad$.


Find the sum of all primes \( p \) for which there exists a prime \( q \) such that \( p^{2} + p q + q^{2} \) is a square.


Split the 2019 natural numbers $1,2,3,4, \ldots ,2019$ into 20 groups such that the average of each group is equal. What is the average of each group?


If \(\log _{2}\left(\log _{4} P\right)=\log _{4}\left(\log _{2} P\right)\) and \(P \neq 1\), find the value of \(P\).


How many five-digit numbers are exactly divisible by 6, 7, 8, and 9?


Find the number $n$ of sides of a convex $n$-gon if each of its interior angles is at least $143^{\circ}$ and at most $146^{\circ}$.


If \( A = 2011^{2011} \), and \( B = (1 \times 2 \times \cdots \times 2011)^2 \), then \( A \) $\qquad$ B. (Fill in “ $>$ ”, “＝” or “＜”)

Solution: \boxed{<}


Let $A$ and $B$ be the vertices on the major axis of the ellipse $\Gamma$, $E$ and $F$ be the foci of $\Gamma$, $|AB|=4$, and $|AF|=2+\sqrt{3}$. Point $P$ is on $\Gamma$ such that $|PE||PF|=2$. Find the area of the triangle $\triangle PEF$.


Given that \(p, q, \frac{2q-1}{p}, \frac{2p-1}{q} \in \mathbf{Z}\), and \(p > 1\), \(q > 1\), find the value of \(p + q\).


Let \( p \) and \( q \) be positive integers such that \(\frac{72}{487}<\frac{p}{q}<\frac{18}{121}\). Find the smallest possible value of \( q \).


Find the number of first-type circular permutations that can be formed using 2 $a$'s, 2 $b$'s, and 2 $c$'s.


Three motorcyclists start simultaneously from the same point on a circular track in the same direction. The first motorcyclist overtakes the second for the first time after 4.5 laps from the start, and 30 minutes before that, he overtakes the third motorcyclist for the first time. The second motorcyclist overtakes the third motorcyclist for the first time three hours after the start. How many laps per hour does the first motorcyclist complete?


The sequence \(\left\{a_{n}\right\}\) is defined by \(a_{1}=1.5\) and \(a_{n}=\frac{1}{n^{2}-1}\) for \(n \in \mathbb{N}, n>1\). Are there any values of \(n\) such that the sum of the first \(n\) terms of this sequence differs from 2.25 by less than 0.01? If yes, find the smallest such value of \(n\).

Solution: \boxed{n = 100}


Let \( x_{1}, x_{2}, \ldots, x_{n} \) be real numbers with absolute values less than 1. What is the minimum value of \( n \) such that

\[
\left| x_{1} \right| + \left| x_{2} \right| + \ldots + \left| x_{n} \right| = 1989 + \left| x_{1} + x_{2} + \ldots + x_{n} \right|
\]


In trapezoid \(ABCD\), segments \(BC\) and \(AD\) are the bases. Given that \(BC = 9.5\), \(AD = 20\), \(AB = 5\), and \(CD = 8.5\), find its area.


How many positive integers \( n \) less than 2007 can we find such that \(\left\lfloor \frac{n}{2} \right\rfloor + \left\lfloor \frac{n}{3} \right\rfloor + \left\lfloor \frac{n}{6} \right\rfloor = n\), where \(\left\lfloor x \right\rfloor\) is the greatest integer less than or equal to \( x \)? (For example, \(\left\lfloor 2.5 \right\rfloor = 2\), \(\left\lfloor 5 \right\rfloor = 5\), \(\left\lfloor -2.5 \right\rfloor = -3\), etc.)


Given seven points in the plane, some of them are connected by segments so that:
(i) among any three of the given points, two are connected by a segment;
(ii) the number of segments is minimal.
How many segments does a figure satisfying (i) and (ii) contain? Give an example of such a figure.


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


A circle touches one side of a right angle with vertex $O$ and intersects the other side at points $A$ and $B$. Find the radius of the circle if $O A = a$ and $O B = b$.

Solution: \boxed{\frac{a + b}{2}}


Alex and Bob have 30 matches. Alex picks up somewhere between one and six matches (inclusive), then Bob picks up somewhere between one and six matches, and so on. The player who picks up the last match wins. How many matches should Alex pick up at the beginning to guarantee that he will be able to win?


If the real numbers \( x \) and \( y \) satisfy
\[
x - 3 \sqrt{x + 1} = 3 \sqrt{y + 2} - y,
\]
then the maximum value of \( x + y \) is \(\quad\).

Solution: \boxed{9 + 3 \sqrt{15}}


On the extensions of the sides \(AB\), \(BC\), \(CD\), and \(DA\) of the convex quadrilateral \(ABCD\), points \(B_1\), \(C_1\), \(D_1\), and \(A_1\) are taken respectively such that \(BB_1 = AB\), \(CC_1 = BC\), \(DD_1 = CD\), and \(AA_1 = DA\). By how many times is the area of quadrilateral \(A_1B_1C_1D_1\) greater than the area of quadrilateral \(ABCD\)? (10 points)


The bases \( AB \) and \( CD \) of the trapezoid \( ABCD \) are equal to 65 and 31 respectively, and its lateral sides are mutually perpendicular. Find the dot product of the vectors \( \overrightarrow{AC} \) and \( \overrightarrow{BD} \).


Let \( a_{1}, a_{2}, \ldots \) be an arithmetic sequence and \( b_{1}, b_{2}, \ldots \) be a geometric sequence. Suppose that \( a_{1} b_{1}=20 \), \( a_{2} b_{2}=19 \), and \( a_{3} b_{3}=14 \). Find the greatest possible value of \( a_{4} b_{4} \).


How many circles of radius 1 are needed to cover a square with a side length of 2?


Two cars covered the same distance. The speed of the first car was constant and three times less than the initial speed of the second car. The second car traveled the first half of the journey without changing speed, then its speed was suddenly halved, then traveled with constant speed for another quarter of the journey, and halved its speed again for the next eighth part of the journey, and so on. After the eighth decrease in speed, the car did not change its speed until the end of the trip. By how many times did the second car take more time to complete the entire journey than the first car?


An integer \( n \) is chosen uniformly at random from the set \( \{1, 2, 3, \ldots, 2023!\} \). Compute the probability that
$$
\operatorname{gcd}(n^n + 50, n + 1) = 1
$$


A group consisting of 6 young men and 6 young women was randomly paired up. Find the probability that at least one pair consists of two young women. Round your answer to two decimal places.


A three-digit number is 56 times greater than its last digit. By how many times is it greater than its first digit? Justify your answer.


How many paths are there on $\mathbb{Z}^{2}$ starting from $(0,0)$, making steps of $+(1,0)$ or $+(0,1)$, and ending at $(m, n)$, where $m, n \geq 0$?

Solution: \boxed{\binom{m+n}{m}}


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


Given that \( A_{n} \) and \( B_{n} \) are the sums of the first \( n \) terms of the arithmetic sequences \( \{a_{n}\} \) and \( \{b_{n}\} \) respectively, and that \(\frac{A_{n}}{B_{n}}=\frac{5n-3}{n+9} \), find \(\frac{a_{8}}{b_{8}}\).


It is required to decipher what numbers are encoded, given that the same digits are replaced by the same letters:

$$
\text{KIS} + \text{KSI} = \text{ISK}.
$$

Solution: \boxed{495 + 459 = 954}


a) The lengths of the edges \( A B, A C, A D \), and \( B C \) of an orthocentric tetrahedron are respectively 5, 7, 8, and 6 cm. Find the lengths of the remaining two edges.
b) Is the tetrahedron \( A B C D \) orthocentric if \( A B = 8 \) cm, \( B C = 12 \) cm, and \( D C = 6 \) cm?

Solution: \boxed{\text{No}}


Find the number of eight-digit integers comprising the eight digits from 1 to 8 such that \( (i+1) \) does not immediately follow \( i \) for all \( i \) that runs from 1 to 7.


Given \( n \), how many ways can we write \( n \) as a sum of one or more positive integers \( a_1 \leq a_2 \leq \cdots \leq a_k \) with \( a_k - a_1 = 0 \) or \( 1 \)?

Solution: \boxed{n}


Let \( D \) be the set of divisors of 100. Let \( Z \) be the set of integers between 1 and 100, inclusive. Mark chooses an element \( d \) of \( D \) and an element \( z \) of \( Z \) uniformly at random. What is the probability that \( d \) divides \( z \)?


There are 29 students in a class: some are honor students who always tell the truth, and some are troublemakers who always lie.

All the students in this class sat at a round table.

- Several students said: "There is exactly one troublemaker next to me."
- All other students said: "There are exactly two troublemakers next to me."

What is the minimum number of troublemakers that can be in the class?


In which number systems is there a number written with two identical digits whose square is written with four identical digits?


The numbers \( p \) and \( q \) are chosen such that the parabola \( y = p x - x^2 \) intersects the hyperbola \( x y = q \) at three distinct points \( A, B, \) and \( C \). The sum of the squares of the sides of triangle \( ABC \) is 324, and the intersection point of its medians is at a distance of 2 from the origin. Find the product \( pq \).


In how many different ways can the digits $0,1,2,3,4,5,6$ form a seven-digit number that is divisible by 4? (The number cannot start with 0.)


Let $[x]$ denote the largest integer not greater than the real number $x$. Define $A=\left[\frac{7}{8}\right]+\left[\frac{7^{2}}{8}\right]+\cdots+\left[\frac{7^{2016}}{8}\right]$. Find the remainder when $A$ is divided by 50.


In triangle $K L M$, the lengths of the sides are $8$, $3\sqrt{17}$, and $13$. Find the area of the figure consisting of those points $P$ inside the triangle $K L M$ for which the condition $P K^{2} + P L^{2} + P M^{2} \leq 145$ is satisfied.


Compute the sum of all integers \(1 \leq a \leq 10\) with the following property: there exist integers \(p\) and \(q\) such that \(p, q, p^{2}+a\) and \(q^{2}+a\) are all distinct prime numbers.


Let \( S = \{1, 2, 3, 4, \ldots, 16\} \). Each of the following subsets of \( S \):
\[ \{6\},\{1, 2, 3\}, \{5, 7, 9, 10, 11, 12\}, \{1, 2, 3, 4, 5, 6, 7, 8, 9\} \]
has the property that the sum of all its elements is a multiple of 3. Find the total number of non-empty subsets \( A \) of \( S \) such that the sum of all elements in \( A \) is a multiple of 3.


Which integers from 1 to 80000 (inclusive) are more numerous and by how many: those containing only even digits or those containing only odd digits?


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


The smallest of three consecutive natural numbers is a multiple of 9, the middle number is a multiple of 8, and the largest number is a multiple of 7. What is the smallest possible sum of these three numbers?


Find the point $M^{\prime}$ that is symmetric to point $M$ relative to the line.

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

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


Among the numbers $1^{2}, 2^{2}, 3^{2}, \cdots, 95^{2}$, how many of these 95 numbers have an odd tens digit?


Let \( r \) be a positive integer. Show that if a graph \( G \) has no cycles of length at most \( 2r \), then it has at most \(|V|^{2016}\) cycles of length exactly \( 2016r \), where \(|V|\) denotes the number of vertices in the graph \( G \).


A kilogram of beef with bones costs 78 rubles, a kilogram of boneless beef costs 90 rubles, and a kilogram of bones costs 15 rubles. How many grams of bones are in a kilogram of beef?

Solution: \boxed{160 \text{ grams}}


On Namek, each hour has 100 minutes. On the planet's clocks, the hour hand completes one full circle in 20 hours and the minute hand completes one full circle in 100 minutes. How many hours have passed on Namek from 0:00 until the next time the hour hand and the minute hand overlap?


A permutation of \(\{1, \ldots, n\}\) is chosen at random. How many fixed points does it have on average?


For all real $x$ and $y$, the equality $f\left(x^{2}+y\right)=f(x)+f\left(y^{2}\right)$ holds. Find $f(-1)$.


In the Cartesian coordinate system, there is an ellipse with foci at $(9,20)$ and $(49,55)$, and it is tangent to the $x$-axis. What is the length of the major axis of the ellipse?


Given \( a, b > 0 \), determine the solutions \( f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+} \) such that \( f(f(x)) + a f(x) = b(a + b) x \).

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


Given a right triangle \( ABC \) with a right angle at vertex \( C \) and leg lengths in the ratio of \( 1:3 \). Points \( K \) and \( L \) are the centers of squares that share one side with leg \( AC \) and \( BC \) respectively, and these squares do not overlap with triangle \( ABC \). Point \( M \) is the midpoint of the hypotenuse \( AB \).

a) Justify that point \( C \) lies on the line segment \( KL \).

b) Calculate the ratio of the areas of triangles \( ABC \) and \( KLM \).

(J. Švrček)


The probability that exactly two out of three red, yellow, and blue balls randomly placed into five different boxes $A, B, C, D, E$ end up in the same box is $\qquad$ .


During the university entrance exams, each applicant is assigned a cover code consisting of five digits. The exams were organized by a careful but superstitious professor who decided to exclude from all possible codes (i.e., 00000 to 99999) those that contained the number 13, that is, the digit 3 immediately following the digit 1. How many codes did the professor have to exclude?


Find the smallest integer \( n > 1 \) such that \(\frac{1^2 + 2^2 + 3^2 + \ldots + n^2}{n}\) is a square.


Шоссе Долгое пересекается с улицей Узкой и с улицей Тихой. На обоих перекрёстках стоят светофоры. Первый светофор $x$ секунд разрешает движение по шоссе, а полминуты - по ул. Узкой. Второй светофор две минуты разрешает движение по шоссе, а $x$ секунд - по ул. Тихой. Светофоры работают независимо друг от друга. При каком значении $x$ вероятность проехать по Долгому шоссе оба перекрестка, не останавливаясь на светофорах, будет наибольшей? Чему равна эта наибольшая вероятность?


A square with an integer side length was cut into 2020 smaller squares. It is known that the areas of 2019 of these squares are 1, while the area of the 2020th square is not equal to 1. Find all possible values that the area of the 2020th square can take. Provide the smallest of these possible area values in the answer.


How many four-digit palindromes \( a b b a \) have the property that the two-digit integer \( a b \) and the two-digit integer \( b a \) are both prime numbers? (For example, 2332 does not have this property, since 23 is prime but 32 is not.)


Given \((a + b i)^2 = 3 + 4i\), where \(a, b \in \mathbf{R}\), and \(i\) is the imaginary unit, find the value of \(a^2 + b^2\).


A trapezoid \(AEFG\) (\(EF \parallel AG\)) is positioned inside a square \(ABCD\) with a side length of 14, such that points \(E\), \(F\), and \(G\) lie on sides \(AB\), \(BC\), and \(CD\) respectively. The diagonals \(AF\) and \(EG\) are perpendicular, and \(EG = 10\sqrt{2}\). Find the perimeter of the trapezoid.


The number of proper subsets of the set
\[
\left\{x \left\lvert\,-1 \leqslant \log _{\frac{1}{x}} 10<-\frac{1}{2}\right., x \in \mathbf{N}\right\}
\]
is ____.

Solution: \boxed{2^{90} - 1}


Let $D$ be a point inside triangle $\triangle ABC$ such that $AB = DC$, $\angle DCA = 24^\circ$, $\angle DAC = 31^\circ$, and $\angle ABC = 55^\circ$. Find $\angle DAB$.


Let \( M \) be the centroid of \( \triangle ABC \), and \( AM = 3 \), \( BM = 4 \), \( CM = 5 \). Find the area of \( \triangle ABC \).


It is known that the numbers \( x, y, z \) form an arithmetic progression in the given order with a common difference \( \alpha = \arccos \left(-\frac{3}{7}\right) \), and the numbers \( \frac{1}{\cos x}, \frac{7}{\cos y}, \frac{1}{\cos z} \) also form an arithmetic progression in the given order. Find \( \cos^{2} y \).


Inside a grid rectangle with a perimeter of 50 cells, a rectangular hole with a perimeter of 32 cells is cut out along the cell borders (the hole does not contain boundary cells). If you cut this figure along all the horizontal grid lines, you get 20 strips that are 1 cell wide. How many strips will you get if you cut it along all the vertical grid lines instead? (A $1 \times 1$ square is also considered a strip!)


There are $N$ natural numbers written on a board, where $N \geq 5$. It is known that the sum of all the numbers is 80, and the sum of any five of them is no more than 19. What is the smallest possible value of $N$?


Quadrilateral $F$ under the mapping $f:(x, y) \rightarrow (x-1, y+2)$ results in quadrilateral $F'$. The area of quadrilateral $F$ is 6. What is the area of quadrilateral $F'$?


Let \( R \) be the radius of the sphere circumscribed around a right square pyramid, \( r \) be the radius of the sphere inscribed in the pyramid, and \( d \) be the distance between the centers of these spheres. Show that

\[
d^{2} + (R + r)^{2} = 2 R^{2}
\]

What can be inferred about the maximum value of the ratio \( r / R \)? (Cf. F. 2096.)

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


Suppose \(\alpha, \beta \in(0, \pi / 2)\). If \(\tan \beta = \frac{\cot \alpha - 1}{\cot \alpha + 1}\), find \(\alpha + \beta\).


The city has the shape of a \(5 \times 5\) square. What is the minimum length a route can have if it needs to go through each street of this city and return to the original location? (Each street can be traversed any number of times.)


Calculate the flux of the vector field \( \mathbf{a} = y \mathbf{i} + z \mathbf{j} + x \mathbf{k} \) through the closed surface bounded by the cylinder \( x^{2} + y^{2} = R^{2} \) and the planes \( z = x \) and \( z = 0 \) (for \( z \geq 0 \)).


90 students arrived at the camp. It is known that among any 10 students, there are necessarily two friends. A group of students is said to form a chain of friendships if the children in the group can be numbered from 1 to \( k \) such that all students can be divided into no more than 9 groups, each of which forms a chain of friendships. (A group of one student also forms a chain of friendships.)


The sequence $\left\{a_{n}\right\}$ is defined by the conditions $a_{1}=0$ and $a_{n+1}=\sum_{k=1}^{n}\left(a_{k}+1\right)$ for $n \geqslant 1$. Find an explicit formula for this sequence.

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


Kolya and Zhenya agreed to meet in the subway during the first hour of the afternoon. Kolya arrives at the meeting place between noon and 1 PM, waits for 10 minutes, and leaves. Zhenya does the same.

a) What is the probability that they will meet?

b) How does the probability of meeting change if Zhenya decides to arrive earlier than 12:30 PM, while Kolya still arrives between noon and 1 PM?

c) How does the probability of meeting change if Zhenya decides to arrive at any random time between 12:00 PM and 12:50 PM, while Kolya still arrives between 12:00 PM and 1:00 PM?


The numbers \(a\) and \(b\) are such that the polynomial \(x^{4} - x^{3} + x^{2} + a x + b\) is the square of some other polynomial. Find \(b\).


When $x^{2}$ was added to the quadratic polynomial $f(x)$, its minimum value increased by 1. When $x^{2}$ was subtracted from it, its minimum value decreased by 3. How will the minimum value of $f(x)$ change if $2x^{2}$ is added to it?


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


Find all values of the parameter \( a \) for which the equation \( a x^{2}+\sin^{2} x=a^{2}-a \) has a unique solution.

Solution: \boxed{a = 1}


Let $a$ and $b$ be two real numbers. Show that $a^{2} + b^{2} \geq 2ab$.

Solution: \boxed{a^2 + b^2 \geq 2ab}


What is the smallest three-digit number \( K \) which can be written as \( K = a^b + b^a \), where both \( a \) and \( b \) are one-digit positive integers?


Given the following table filled with signs:

$$
\begin{aligned}
& ++-+ \\
& --++ \\
& ++++ \\
& +-+-
\end{aligned}
$$

A move consists of choosing a row or a column and changing the signs within it. Is it possible to achieve a table filled with '+' signs?

Solution: \boxed{\text{No}}


Find the smallest positive integer \( n \) such that \( 107n \) has the same last two digits as \( n \).


Let \( x > 0 \). Show that

\[ x + \frac{1}{x} \geq 2 \]

For which values of \( x \) does equality hold?

Solution: \boxed{x = 1}


If \(\cos ^{4} \theta + \sin ^{4} \theta + (\cos \theta \cdot \sin \theta)^{4} + \frac{1}{\cos ^{4} \theta + \sin ^{4} \theta} = \frac{41}{16}\), find the value of \(\sin ^{2} \theta\).


Maxim came up with a new method for dividing numbers by a two-digit number \( N \). To divide an arbitrary number \( A \) by the number \( N \), you need to do the following steps:
1) Divide \( A \) by the sum of the digits of \( N \);
2) Divide \( A \) by the product of the digits of \( N \);
3) Subtract the second result from the first.

For which numbers \( N \) will Maxim’s method give the correct result? (20 points)


Given \( n = p \cdot q \cdot r \cdot s \), where \( p, q, r, s \) are distinct primes such that:

1. \( s = p + r \)
2. \( p(p + q + r + s) = r(s - q) \)
3. \( qs = 1 + qr + s \)

Find \( n \).


For the four-digit number $\overline{a b c d}$, if $a > b$, $b < c$, and $c > d$, it is classified as a $P$ type number. If $a < b$, $b > c$, and $c < d$, it is classified as a $Q$ type number. What is the difference between the total number of $P$ type numbers and $Q$ type numbers?


For a soccer match, three types of tickets (A, B, and C) were sold, totaling 400 tickets. The prices for type A, type B, and type C tickets are 50 yuan, 40 yuan, and 30 yuan respectively. The total revenue from ticket sales is 15,500 yuan. Additionally, the number of type B and type C tickets sold is the same. How many tickets of each type were sold?


For what value of \( p \) is the ratio of the roots of the equation \( x^{2} + p x - 16 = 0 \) equal to \( -4 \)?

Solution: \boxed{\pm 6}


Find the value of the function \( f(x) \) at the point \( x_{0} = 1000 \), given that \( f(0) = 1 \) and for any \( x \), the equality \( f(x+2) = f(x) + 4x + 2 \) holds.


The center of a cube is reflected off all its faces, and the 6 points obtained, together with the vertices of the cube, determine 14 vertices of a polyhedron, all 12 faces of which are congruent rhombuses. (This polyhedron is called a rhombic dodecahedron—a shape naturally occurring in garnet single crystals.) Calculate the surface area and volume of the constructed polyhedron if the edge of the original cube has a length of \( a \).


From one point, a perpendicular and two oblique lines are drawn to a given straight line.

Find the length of the perpendicular if the oblique lines are 41 and 50, and their projections onto the given straight line are in the ratio \(3: 10\).


Given that \( \tan \left(\frac{\pi}{4}+\alpha\right)=2 \), find the value of \( \frac{1}{2 \sin \alpha \cos \alpha+\cos ^{2} \alpha} \).


Calculate the limit of the function:

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


Let \( p \) be an integer such that both roots of the equation
\[ 5x^2 - 5px + (66p - 1) = 0 \]
are positive integers. Find the value of \( p \).


Find the smallest six-digit number that is divisible by 11, where the sum of the first and fourth digits is equal to the sum of the second and fifth digits, and equal to the sum of the third and sixth digits.


Indicate the integer closest to the number: \(\sqrt{2012-\sqrt{2013 \cdot 2011}}+\sqrt{2010-\sqrt{2011 \cdot 2009}}+\ldots+\sqrt{2-\sqrt{3 \cdot 1}}\).


We place 9 integers in the cells of a $3 \times 3$ grid in such a way that the sum of the numbers in any row or column is always odd. What values can the number of even cells in such a configuration take?

Solution: \boxed{0, 4, 6}


The sides of a regular 18-sided polygon inscribed in a unit circle have length \(a\). Show that

$$
a^{3}=3a-1
$$

Solution: \boxed{a^3 = 3a - 1}


Beginner millionaire Bill buys a bouquet of 7 roses for $20. Then, he can sell a bouquet of 5 roses for $20 per bouquet. How many bouquets does he need to buy to "earn" a difference of $1000?


Let's call a natural number a "snail" if its representation consists of the representations of three consecutive natural numbers, concatenated in some order: for example, 312 or 121413. "Snail" numbers can sometimes be squares of natural numbers: for example, $324=18^{2}$ or $576=24^{2}$. Find a four-digit "snail" number that is the square of some natural number.


Find the number of 11-digit positive integers such that the digits from left to right are non-decreasing. (For example, 12345678999, 55555555555, 23345557889.)


Calculate the limit of the function:

$$
\lim _{x \rightarrow 1}\left(\frac{x^{2}+2 x-3}{x^{2}+4 x-5}\right)^{\frac{1}{2-x}}
$$


In the Cartesian coordinate plane, circles \( C_{1} \) and \( C_{2} \) intersect at points \( P \) and \( Q \). The coordinates of point \( P \) are \( (3, 2) \), and the product of the radii of the two circles is \( \frac{13}{2} \). If both circles are tangent to the line \( l: y = kx \) and the x-axis, find the equation of the line \( l \).

Solution: \boxed{y = 2 \sqrt{2} x}


Given a triangle \( \triangle ABC \) with \( BC=a \), \( AC=b \), and \( AB=c \), if \( \angle A + \angle C = 2 \angle B \), find the largest positive integer \( n \) such that \( a^n + c^n \leq 2b^n \) holds for any such triangle.


The midsegment of a trapezoid divides it into two quadrilaterals. The difference in the perimeters of these two quadrilaterals is 24, and the ratio of their areas is $\frac{20}{17}$. Given that the height of the trapezoid is 2, what is the area of this trapezoid?


Solve the equation \(2 x \log x + x - 1 = 0\) in the set of real numbers.

Solution: \boxed{x = 1}


In how many ways can the cells of a \(4 \times 4\) table be filled in with the digits \(1,2, \ldots, 9\) so that each of the 4-digit numbers formed by the columns is divisible by each of the 4-digit numbers formed by the rows?


How many 7-digit positive integers are made up of the digits 0 and 1 only, and are divisible by 6?


The total \( T \) is obtained as the sum of the integers from 2006 to 2036 inclusive. What is the sum of all the prime factors of \( T \)?


In the cube \(ABCDE_{1}B_{1}C_{1}D_{1}\) as shown in Figure 7-10, find the degree of the dihedral angle \(A-BD_{1}-A_{1}\).


The area of an isosceles trapezoid is 100, and its diagonals are mutually perpendicular. Find the height of this trapezoid.


The base of the quadrangular pyramid \( M A B C D \) is a parallelogram \( A B C D \). Given that \( \overline{D K} = \overline{K M} \) and \(\overline{B P} = 0.25 \overline{B M}\), the point \( X \) is the intersection of the line \( M C \) and the plane \( A K P \). Find the ratio \( M X: X C \).

Solution: \boxed{M X : X C = 3 : 4}


In the diagram, \( A B C D E F \) is a regular hexagon with side length 2. Points \( E \) and \( F \) are on the \( x \)-axis and points \( A, B, C, \) and \( D \) lie on a parabola. What is the distance between the two \( x \)-intercepts of the parabola?


Show that \( 3 < \pi < 4 \) without using approximate values of \(\pi\).

Solution: \boxed{3 < \pi < 4}


Given that both \( p \) and \( q \) are prime numbers, and that \( 7p + q \) and \( 2q + 11 \) are also prime numbers, find the value of \( p^q + q^p \).


The probability of event \( A \) occurring in each of 484 independent trials is 0.75. Find the probability that the relative frequency of event \( A \) deviates from its probability by no more than 0.03 in absolute value.


Find the volume of the solid obtained by rotating around the $x$-axis the curvilinear trapezoid bounded by the parabola $y^{2}=2x$, the line $x=3$, and the $x$-axis.


The product of all the positive integer divisors of \( 6^{16} \) equals \( 6^k \) for some integer \( k \). Determine the value of \( k \).


In a chess tournament, 12 people participated. After the tournament concluded, each participant created 12 lists. The first list includes only themselves, the second one includes themselves and those they defeated, the third includes everyone from the second list and those they defeated, and so on. The 12th list includes everyone from the 11th list and those they defeated. It is known that for any participant in the tournament, there is a person in their 12th list who was not in their 11th list. How many drawn games were played in the tournament?


The numbers \(a, b, c, d\) belong to the interval \([-8.5, 8.5]\). Find the maximum value of the expression \(a + 2b + c + 2d - ab - bc - cd - da\).


\( \arcsin \frac{1}{\sqrt{10}} + \arcsin \frac{1}{\sqrt{26}} + \arcsin \frac{1}{\sqrt{50}} + \arcsin \frac{1}{\sqrt{65}} = \)


Determine all pairs \((a, n)\) of positive integers such that

$$
3^n = a^2 - 16
$$

Solution: \boxed{(5, 2)}


The Fibonacci sequence is defined as follows: \( F_{0}=0, F_{1}=1 \), and \( F_{n}=F_{n-1}+F_{n-2} \) for all integers \( n \geq 2 \). Find the smallest positive integer \( m \) such that \( F_{m} \equiv 0 \pmod{127} \) and \( F_{m+1} \equiv 1 \pmod{127} \).


Select three different digits from $0, 1, \cdots, 9$ to form a four-digit number (one of the digits may appear twice), such as 5224. How many such four-digit numbers are there?


On the sphere \(\left\{(x, y, z) \mid x^{2}+y^{2}+z^{2}=1\right\}\), given \(n (n \geq 2)\) points \(A_{1}, A_{2}, \ldots, A_{n}\), find the maximum possible value of \(\sum_{1 \leq i < j \leq n} \left| A_i A_j \right|^2\).


We call an odd six-digit number "just cool" if it consists of digits that are prime numbers and no two identical digits stand next to each other. How many "just cool" numbers exist?


8. Andrey likes all numbers that are not divisible by 3, and Tanya likes all numbers that do not contain digits that are divisible by 3.

a) How many four-digit numbers are liked by both Andrey and Tanya?

b) Find the total sum of the digits of all such four-digit numbers.


Find the limit of the function:
1) \(f(x) = x^3 - 5x^2 + 2x + 4 \) as \( x \rightarrow -3 \).
2) \(\varphi(t) = t \sqrt{t^2 - 20} - \lg \left( t + \sqrt{t^2 - 20} \right) \) as \( t \rightarrow 6 \).


How many zeros are at the end of the product $s(1) \cdot s(2) \cdot \ldots \cdot s(100)$, where $s(n)$ denotes the sum of the digits of the natural number $n$?


If $\log_{4}(x + 2y) + $\log_{4}(x - 2y) = 1, what is the minimum value of $|x| - |y|$?


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


A dandelion blossoms in the morning, blooms yellow for three days, turns white on the morning of the fourth day, and by the evening of the fifth day, its seeds disperse. On Monday afternoon, there were 20 yellow and 14 white dandelions in the meadow, and on Wednesday, there were 15 yellow and 11 white. How many white dandelions will be in the meadow on Saturday?


Find the digits $x, y, z$ that satisfy the equation $\overline{x 5} \cdot \overline{3 y z}=7850$.

Solution: \boxed{x = 2, y = 1, z = 4}


It is known that for certain natural numbers \(a, b\), the number \(N=\frac{a^{2}+b^{2}}{a b-1}\) is also a natural number. Find all possible values of \(N\).


A boy and a girl. Assume that the birth of a girl and a boy are equally likely. It is known that a certain family has two children.

a) What is the probability that they have one boy and one girl?

b) Additionally, it is known that one of the children is a boy. What is the probability now that the family has one boy and one girl?

c) Additionally, it is known that the boy was born on a Monday. What is the probability now that the family has one boy and one girl?


An even power of a certain number is a four-digit number with the first digit as 3 and the last digit as 5. Find this number.


Find the two-digit number that is equal to the sum of the cube of its tens digit and the square of its units digit.


Remove all perfect squares and perfect cubes from the set
$$
A=\left\{n \mid n \leqslant 10000, n \in \mathbf{Z}_{+}\right\}
$$
and arrange the remaining elements in ascending order. What is the 2014th element of this sequence?


Given that $f(x)$ is a function defined on $\mathbf{R}$. It is known that $f(1)=1$ and for any $x \in \mathbf{R}$, $f(x+5) \geqslant f(x)+5$ and $f(x+1) \leqslant f(x)+1$. If $g(x)=f(x)+1-x$, then $g(2002)=$ $\qquad$


Find the number of four-digit numbers in which all digits are different, the first digit is divisible by 2, and the sum of the first and last digits is divisible by 3.


Write the number \( 123456789101112 \cdots 19941995 \) on the blackboard to form an integer \( N_1 \). Erase the digits in the even positions of \( N_1 \) to form \( N_2 \). Then, erase the digits in the odd positions of \( N_2 \) to form \( N_3 \). Continue this process of erasing digits in even and odd positions alternately until only one digit remains on the blackboard. Determine this final digit. (Note: Positions are counted from left to right; for example, in \( 12345 \), \( 1 \) is in the first position, \( 2 \) is in the second position, and so on.) (Australian Mathematics Olympiad, 1995)


If one side of square \( ABCD \) lies on the line \( y = 2x - 17 \) and the other two vertices are on the parabola \( y = x^2 \), what is the minimum possible area of the square?


A polynomial \( P \) is of the form \( \pm x^{6} \pm x^{5} \pm x^{4} \pm x^{3} \pm x^{2} \pm x \pm 1 \). Given that \( P(2)=27 \), what is \( P(3) \)?


Calculate the limit of the numerical sequence:

$$
\lim _{n \rightarrow \infty} \frac{\sqrt{3 n-1}-\sqrt[3]{125 n^{3}+n}}{\sqrt[5]{n}-n}
$$


Let \( x \) be a real number such that \( x^{3} + 4x = 8 \). Determine the value of \( x^{7} + 64x^{2} \).


The robot vacuum cleaner is programmed to move on the floor according to the law:

$$\left\{\begin{array}{l} x = t(t-6)^2 \\ y = 0, \quad 0 \leq t \leq 7 \\ y = (t-7)^2, \quad t \geq 7 \end{array}\right.$$

where the axes are chosen parallel to the walls and the movement starts from the origin. Time $t$ is measured in minutes, and coordinates are measured in meters. Find the distance traveled by the robot in the first 7 minutes and the absolute change in the velocity vector during the eighth minute.


Given a set of numbers \(\{-1, -2, -3, \ldots, -26\}\), write down all possible subsets of this set that contain at least 2 numbers. Calculate the product of all numbers for each written subset. What is the sum of all these products?


Find the number of five-digit numbers that contain at least one digit 8 in their decimal representation.


For what value of \( x \) is the fourth term in the expansion of \( \left(\sqrt{2^{x-1}}+\sqrt[3]{2^{-x}}\right)^{m} \) 20 times greater than \( m \), if the binomial coefficient of the fourth term is related to the binomial coefficient of the second term as 5:1?

Solution: \boxed{x = 4}


One of the simplest multicellular organisms, the algae Volvox, is a spherical shell composed mainly of heptagonal, hexagonal, and pentagonal cells (i.e., cells having seven, six, or five neighboring cells; three cells meet at each "vertex"). There are some specimens that have quadrilateral and octagonal cells as well, but biologists have noticed that if there are no such "non-standard" cells (with fewer than five and more than seven sides), then there are always 12 more pentagonal cells than heptagonal cells. Can you explain this fact?

Solution: \boxed{f_5 - f_7 = 12}


Let real numbers \( s \) and \( t \) satisfy \( 19s^2 + 99s + 1 = 0 \) and \( t^2 + 99t + 19 = 0 \), respectively, with \( st \neq 1 \). Find the value of \(\frac{st + 4s + 1}{t}\).


Given the sequence \(\{a_n\}\) with the sum of its first \(n\) terms denoted by \(S_n\), let \(T_n = \frac{S_1 + S_2 + \cdots + S_n}{n}\). \(T_n\) is called the "mean" of the sequence \(a_1, a_2, \cdots, a_n\). It is known that the "mean" of the sequence \(a_1, a_2, \cdots, a_{1005}\) is 2012. Determine the "mean" of the sequence \(-1, a_1, a_2, \cdots, a_{1005}\).


Given a positive integer \( a \), define a sequence of integers \( x_1, x_2, x_3, \ldots \) such that \( x_1 = a \) and \( x_{n+1} = 2x_n + 1 \). Let \( y_n = 2^{x_n} - 1 \). Determine the largest possible integer \( k \) such that for some positive integer \( a \), the numbers \( y_1, y_2, \ldots, y_k \) are all prime numbers.


Given that the function \( f(x) \) satisfies the equation \( 2 f(x) + x^{2} f\left(\frac{1}{x}\right) = \frac{3 x^{3} - x^{2} + 4 x + 3}{x + 1} \) and \( g(x) = \frac{5}{x + 1} \), determine the minimum value of \( f(x) + g(x) \).


There are 9 cards, each with a digit from 1 to 9 written on it (one digit per card). Three cards are randomly chosen and placed in a sequence. What is the probability that the resulting three-digit number is divisible by 3?


Solve the equation in integers:

a) \(xy + 3x - 5y = -3\);

b) \(x - y = \frac{x}{y}\).

Solution: \boxed{(4, 2)}


Given points \( A(1, -1, 0) \) and \( B(2, 0, 1) \) are on skew lines \( a \) and \( b \) respectively, and the direction vectors of lines \( a \) and \( b \) are \(\boldsymbol{a} = (1, 0, -1)\) and \(\boldsymbol{b} = (1, -1, 1)\) respectively. Find the distance between the two skew lines \( a \) and \( b \).


Given \(0 < \theta < \pi\) and \(x + \frac{1}{x} = 2 \cos \theta\), express \(x^n + \frac{1}{x^n}\) in terms of \(n\) and \(\theta\).

Solution: \boxed{2 \cos n \theta}


In a convex quadrilateral \(ABCD\), the diagonals \(AC\) and \(DB\) are perpendicular to sides \(DC\) and \(AB\) respectively. From point \(B\), a perpendicular is dropped to side \(AD\), intersecting \(AC\) at point \(O\). Find \(AO\) if \(AB = 4\) and \(OC = 6\).


For what smallest natural number $n$ is each of the fractions

$$
\frac{5}{n+6}, \quad \frac{6}{n+7}, \frac{7}{n+8}, \ldots, \quad \frac{24}{n+25}
$$

in its simplest form?


Along a straight alley, 400 lampposts are placed at equal intervals and are numbered consecutively from 1 to 400. At the same time, Alla and Boris start walking towards each other from opposite ends of the alley at different constant speeds (Alla from the first lamppost and Boris from the four-hundredth lamppost). When Alla is at the 55th lamppost, Boris is at the 321st lamppost. At which lamppost will they meet? If the meeting occurs between two lampposts, specify the smaller of the two numbers.


How many ways can the letters in the word MATEMATIKA be arranged so that the two letters "A" are never next to each other?


A cube-shaped container with an edge length of $12 \mathrm{~cm}$ was filled with liquid up to $\frac{5}{8}$ of its volume, and then slightly tilted along one of its edges. The figure shows the cross-section of the container and the liquid level inside. We know that the length of segment $LC$ is exactly twice the length of segment $KB$. Determine the length of segment $LC$.

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


Using the four digits $2, 0, 1, 5$, how many three-digit numbers can be formed without repeating any digits, such that these three-digit numbers are multiples of 5?


In the quadrilateral pyramid \( P-ABCD \), \( BC \parallel AD \), \( AD \perp AB \), \( AB=2\sqrt{3} \), \( AD=6 \), \( BC=4 \), \( PA = PB = PD = 4\sqrt{3} \). Find the surface area of the circumscribed sphere of the triangular pyramid \( P-BCD \).


Points \( M, N, \) and \( K \) are located on the lateral edges \( A A_{1}, B B_{1}, \) and \( C C_{1} \) of the triangular prism \( A B C A_{1} B_{1} C_{1} \) such that \( \frac{A M}{A A_{1}} = \frac{5}{6}, \frac{B N}{B B_{1}} = \frac{6}{7}, \) and \( \frac{C K}{C C_{1}} = \frac{2}{3} \). Point \( P \) belongs to the prism. Find the maximum possible volume of the pyramid \( M N K P \), given that the volume of the prism is 35.


Zhang Bing was born in 1953. In a certain year before this year, his age was a multiple of 9 and was also equal to the sum of the digits of that year. How old was he in that year?


A two-digit number is approximated to 3.8. What is the maximum possible value of this number, and what is the minimum possible value of this number?

Solution: \boxed{3.84 \text{, } 3.75}


Let \( a \) and \( b \) be real numbers. Find the minimum value of \( \max \{|a+b|, |a-b|, |1-b|\} \).


Given $n$ new students, where among any 3 students, at least 2 know each other, and among any 4 students, at least 2 do not know each other, find the maximum value of $n$.


Calculate the limit of the function:

\[
\lim _{x \rightarrow \frac{\pi}{6}} \frac{2 \sin ^{2} x+\sin x-1}{2 \sin ^{2} x-3 \sin x+1}
\]


Solve the inequality

$$
\sqrt{6 x-13}-\sqrt{3 x^{2}-13 x+13} \geqslant 3 x^{2}-19 x+26
$$

In the answer, provide the sum of all integer values of $x$ that satisfy the inequality.


A rectangle $ABCD$ is given. A circle intersects the side $AB$ at points $K$ and $L$, and intersects the side $CD$ at points $M$ and $N$. Find the length of segment $MN$ if $AK = 10$, $KL = 17$, and $DN = 7$.


31 cars simultaneously started from the same point on a circular track: the first car at a speed of 61 km/h, the second at 62 km/h, and so on up to the 31st car at 91 km/h. The track is narrow, and if one car overtakes another, they collide and both crash out of the race. Eventually, one car remains. What is its speed?

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


In rectangle \(ABCD\), point \(E\) is the midpoint of side \(CD\). On side \(BC\), point \(F\) is chosen such that angle \(AEF\) is a right angle. Find the length of segment \(FC\) if \(AF = 9\) and \(BF = 6\).


Determine \( m \) such that the expression

\[
4x^2 - 6x + m
\]

is divisible by \( (x - 3) \) without remainder.


The system

$$
\begin{aligned}
& (x - y)\left( x^2 - y^2 \right) = 160 \\
& (x + y)\left( x^2 + y^2 \right) = 580
\end{aligned}
$$

has a pair of solutions $x_{1} = 3, y_{1} = 7$. What other pair of solutions should this system have?

Solution: \boxed{(7, 3)}


A motorcyclist left point A for point B, and at the same time, a pedestrian left point B for point A. When they met, the motorcyclist took the pedestrian on his motorcycle to point A and then immediately went back to point B. As a result, the pedestrian reached point A 4 times faster than if he had walked the entire distance. How many times faster would the motorcyclist have arrived at point B if he didn't have to return?


There were 21 consecutive natural numbers written on the board. When one of the numbers was erased, the sum of the remaining numbers became 2017. Which number was erased?


Find the maximum value of the expression \( x + y \), where \( x \) and \( y \) are integer solutions of the equation \( 3x^{2} + 5y^{2} = 345 \).


You walk a spiraling maze on the Cartesian plane as follows: starting at \((0,0)\) and the first five stops are at \(A(1,0)\), \(B(1,1)\), \(C(0,1)\), \(D(-1,1)\), and \(E(-1,0)\). Your ninth stop is at the point \((2,-1)\). What is the \(x\)-coordinate of the point which you would arrive at on your 2005th stop?


Define the operation: \( a \odot b = \frac{(a \times 2 + b)}{2} \). What is \( (4 \odot 6) \odot 8 \)?


Solution: \boxed{x = 1}


If the cotangents of the three interior angles \(A, B, C\) of triangle \(\triangle ABC\), denoted as \(\cot A, \cot B, \cot C\), form an arithmetic sequence, then the maximum value of angle \(B\) is \(\frac{\pi}{3}\).


If \( 20 \times 21 \times 22 \times \ldots \times 2020 = 26^{k} \times m \), where \( m \) is an integer, what is the maximum value of \( k \)?


Let \( A \) be the 221-digit number where every digit is 9. What is the sum of the digits of \( A^{2} \)?


How many ordered pairs \((a, b)\), with \(a\) and \(b\) being positive integers, satisfy

\[ a + b + \gcd(a, b) = 33? \]


How many positive integers less than 2019 are divisible by either 18 or 21, but not both?


Let \( f(n) = \sum_{k=2}^{\infty} \frac{1}{k^n \cdot k!} \). Calculate \( \sum_{n=2}^{\infty} f(n) \).

Solution: \boxed{3 - e}


The base of a pyramid is an isosceles triangle with a base of 6 and a height of 9. Each lateral edge is 13. Find the volume of the pyramid.


How many solutions in integers does the equation \( x^{2} + y^{2} = 6x + 2y + 15 \) have?


Calculate the limit of the function:

\[
\lim_{x \rightarrow 0} \frac{e^{5x} - e^{3x}}{\sin 2x - \sin x}
\]


Given that $\alpha^{2005} + \beta^{2005}$ can be expressed as a polynomial in two variables $\alpha + \beta$ and $\alpha \beta$, find the sum of the coefficients of this polynomial.


A line parallel to side \( AB \) of triangle \( ABC \) intersects side \( BC \) at point \( M \) and side \( AC \) at point \( N \). The area of triangle \( MCN \) is twice the area of trapezoid \( ABMN \). Find the ratio \( CM:MB \).

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


As shown in the diagram, in square $ABCD$, extend $BA$ to $G$ such that $AG = BD$. What is the measure of $\angle BCG$ in degrees?


Give a formula that maps each integer to an even number in a one-to-one fashion.

Solution: \boxed{m = 2n}


The minimum value of the function \( f(x) = \sqrt{x^{2} + 3x + 3} + \sqrt{x^{2} - 3x + 3} \) for \( x \in \mathbb{R} \) is \(\qquad\).

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


A ticket contains six digits \(a, b, c, d, e, f\). This ticket is said to be "lucky" if \(a + b + c = d + e + f\). How many lucky tickets are there (including the ticket 000000)?


Find the number of digits in the decimal representation of the number \(2^{120}\), given that the decimal representation of the number \(2^{200}\) contains 61 digits.


Find the natural number that has six natural divisors (including one and the number itself), two of which are prime, and the sum of all its natural divisors is equal to 78.


Natural numbers \(a_{1}, a_{2}, \ldots, a_{49}\) satisfy the equation \(a_{1} + a_{2} + \ldots + a_{49} = 540\). What is the greatest possible value of their greatest common divisor (GCD)?