Twenty-five coins are being separated into piles in the following manner. First, they are randomly divided into two groups. Then, any one of the existing groups is again divided into two groups, and so on until each group consists of one coin. During each division of a group into two, the product of the number of coins in the two resulting groups is recorded. What could be the sum of all the recorded numbers?


Simplify the following expression:
\[
\frac{\sin ^{2} 4 \alpha + 4 \sin ^{4} 2 \alpha - 4 \sin ^{2} 2 \alpha \cos ^{2} 2 \alpha}{4 - \sin ^{2} 4 \alpha - 4 \sin ^{2} 2 \alpha}
\]

Solution: \boxed{\tan^4 2\alpha}


Solve the inequality:

$$
\log _{3+\sin x-\cos x}\left(3-\frac{\cos 2 x}{\cos x+\sin x}\right) \geq e^{\sqrt{x}}
$$


Let the set \( M = \{1, 2, \cdots, 10\} \),
\[ A = \{(x, y, z) \mid x, y, z \in M, \text{ and } 9 \mid (x^3 + y^3 + z^3) \} . \]

The number of elements in the set \( A \) is \(\quad\).


From a sequence of natural numbers, all numbers that are squares or cubes of integers have been removed. Which number remains in the 100th position?


Identical coins are arranged on a table in the shape of a hexagon. If they are arranged such that the side of the hexagon is made up of 2 coins, 7 coins are needed, and if the side consists of 3 coins, a total of 19 coins is required. How many coins are needed to construct a hexagon with a side made up of 10 coins?


Given a triangular pyramid \( S-ABC \) whose base \( ABC \) is an isosceles right triangle with \( AB \) as the hypotenuse, and satisfying \( SA = SB = SC = 2 \) and \( AB = 2 \), assume that the four points \( S, A, B, C \) are all on the surface of a sphere centered at point \( O \). Find the distance from point \( O \) to the plane \( ABC \).


Find the sum of all three-digit natural numbers that do not contain the digits 0 or 5.


Given \(a\) is a digit from 1 to 9, if the repeating decimal \(0.1\mathrm{a} = \frac{1}{\mathrm{a}}\), find \(a\).


Find all positive integers $n > 3$ such that there exist $n$ points $A_{1}, A_{2}, \cdots, A_{n}$ in the plane and real numbers $r_{1}, r_{2}, \cdots, r_{n}$ satisfying the following conditions:
(1) Any 3 points among $A_{1}, A_{2}, \cdots, A_{n}$ are not collinear;
(2) For each triplet of points $\left\{A_{i}, A_{j}, A_{k}\right\}$ (where $1 \leq i < j < k \leq n$), the area of triangle $\triangle A_{i} A_{j} A_{k}$, denoted by $S_{ijk}$, is equal to $r_{i} + r_{j} + r_{k}$.

(Note: This is a problem from the 36th IMO in 1995)


Determine the largest value of \( a \) such that \( a \) satisfies the equations \( a^2 - bc - 8a + 7 = 0 \) and \( b^2 + c^2 + bc - 6a + 6 = 0 \) for some real numbers \( b \) and \( c \).


Find the mass of the plate $D$ with surface density $\mu = 16 x + \frac{9 y^2}{2}$, bounded by the curves

$$x = \frac{1}{4}, \quad y = 0, \quad y^2 = 16 x \quad (y \geq 0)$$


In a bag, there are 10 white balls, 5 black balls, and 4 blue balls. All the balls are identical except for color. To ensure that there are at least 2 balls of each color among the balls drawn, what is the minimum value of $n$ for the number of balls drawn?


Given that \( x, y, z \) are positive real numbers such that \( x + y + z = 1 \), find the minimum value of the function \( f(x, y, z) = \frac{3x^{2} - x}{1 + x^{2}} + \frac{3y^{2} - y}{1 + y^{2}} + \frac{3z^{2} - z}{1 + z^{2}} \), and provide a proof.


In a regular hexagon $A B C D E F$, the diagonals $A C$ and $C E$ are divided by points $M$ and $N$ respectively in the following ratios: $\frac{A M}{A C} = \frac{C N}{C E} = r$. If points $B$, $M$, and $N$ are collinear, determine the ratio $r$.


Determine the number of pairs of integers \((a, b)\) such that \(1 \leq a \leq 30\), \(3 \leq b \leq 30\) and \(a\) is divisible by both \(b\) and \(b-2\).


In a certain year, a certain date was never a Sunday in any month. Determine that date.


A tetrahedron. There are 6 rods of different lengths, and it is known that no matter how they are ordered, they can form a tetrahedron (triangular pyramid). How many different tetrahedrons can be formed with these rods?


Given that \( P \) is a moving point on side \( AF \) of a regular hexagon \( ABCDEF \) with side length 1, find the minimum value of \(\overrightarrow{PD} \cdot \overrightarrow{PE}\).


The integers from \(1\) to \(n\) are written in increasing order from left to right on a blackboard. David and Goliath play the following game: starting with David, the two players alternate erasing any two consecutive numbers and replacing them with their sum or product. Play continues until only one number on the board remains. If it is odd, David wins, but if it is even, Goliath wins. Find the 2011th smallest positive integer greater than \(1\) for which David can guarantee victory.


In the royal dining hall, there are three tables with three identical pies. For lunch, the king invites six princes to his table. At the second table, there can be 12 to 18 courtiers, and at the third table, there can be 10 to 20 knights. Each pie is divided into equal pieces according to the number of people seated at the table. There is a rule at the court: the amount of food a knight receives combined with the amount a courtier receives equals the amount received by the king. Determine the maximum possible number of knights the king can invite for lunch that day. How many courtiers will be seated at their table in this case?


\(ABCD\) is a square with side length 9. Let \(P\) be a point on \(AB\) such that \(AP: PB = 7:2\). Using \(C\) as the center and \(CB\) as the radius, a quarter circle is drawn inside the square. The tangent from \(P\) meets the circle at \(E\) and \(AD\) at \(Q\). The segments \(CE\) and \(DB\) meet at \(K\), while \(AK\) and \(PQ\) meet at \(M\). Find the length of \(AM\).


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


When the natural number \( n \) is divided by 8, the remainder is 6. What is the remainder when \( n^2 \) is divided by 32?


Vasya thought of three natural numbers with a sum of 1003. After calculating their product, Vasya noticed that it ends in $N$ zeros. What is the maximum possible value of $N$?


Along a straight alley, 100 lanterns are placed at equal intervals, numbered sequentially from 1 to 100. Simultaneously, Petya and Vasya start walking towards each other from opposite ends of the alley with different constant speeds (Petya starts from the first lantern, and Vasya starts from the hundredth lantern). When Petya reached the 22nd lantern, Vasya was at the 88th lantern. At which lantern will they meet? If the meeting occurs between two lanterns, indicate the smaller number of the two.


Suppose that \( f(x) \) is a function defined for every real number \( x \) with \( 0 \leq x \leq 1 \) with the properties that

- \( f(1-x)=1-f(x) \) for all real numbers \( x \) with \( 0 \leq x \leq 1 \),
- \( f\left(\frac{1}{3} x\right)=\frac{1}{2} f(x) \) for all real numbers \( x \) with \( 0 \leq x \leq 1 \), and
- \( f(a) \leq f(b) \) for all real numbers \( 0 \leq a \leq b \leq 1 \).

What is the value of \( f\left(\frac{6}{7}\right) \) ?

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


Find the Cauchy principal value of the integral \( I = \int_{-\infty}^{\infty} \sin(2x) \, dx \).


A positive integer \( N \) and \( N^2 \) end with the same sequence of digits \(\overline{abcd}\), where \( a \) is a non-zero digit. Find \(\overline{abc}\).


Among all polynomials \( P(x) \) with integer coefficients for which \( P(-10)=145 \) and \( P(9)=164 \), compute the smallest possible value of \( |P(0)| \).


In the Cartesian coordinate plane \( xOy \), circle \(\Omega\) passes through the points \((0,0)\), \((2,4)\), and \((3,3)\). The maximum distance from a point on circle \(\Omega\) to the origin is \(\underline{\quad}\).


Given \(0 < a < b\), draw two lines \(l\) and \(m\) through two fixed points \(A(a, 0)\) and \(B(b, 0)\), respectively, such that they intersect the parabola \(y^2 = x\) at four distinct points. When these four points lie on a common circle, find the locus of the intersection point \(P\) of the two lines \(l\) and \(m\).

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


In the given triangle \( \triangle ABC \) with an area of 36, point \( D \) is on \( AB \) such that \( BD = 2AD \), and point \( E \) is on \( DC \) such that \( DE = 2EC \). What is the area of \( \triangle BEC \)?


Express 33 as the sum of \(n\) consecutive natural numbers. When \(n\) is maximized, replace all "+" signs in the resulting sum with "×". What is the product of the terms in this expression?


Given points \( A(4,0) \) and \( B(2,2) \) are inside the ellipse \( \frac{x^{2}}{25}+\frac{y^{2}}{9}=1 \), and \( M \) is a point on the ellipse, find the maximum value of \( |MA| + |MB| \).

Solution: \boxed{10 + 2 \sqrt{10}}


Let \( n \) be the second smallest integer that can be written as the sum of two positive cubes in two different ways. Compute \( n \).


We have a lighter and two cords (not necessarily identical). When either cord is lit, it takes one hour to burn completely. How can you measure forty-five minutes?

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


Real numbers \( x \) and \( y \) satisfy \( 4x^2 - 5xy + 4y^2 = 5 \). Let \( s = x^2 + y^2 \). Find the value of \( \frac{1}{s_{\max}} \).


In the number A, the digits are in ascending order (from left to right). What is the sum of the digits of the number \( 9 \cdot A \) ?


In triangle \( \triangle ABC \), if \( \frac{\cos A}{\sin B} + \frac{\cos B}{\sin A} = 2 \), and the perimeter of \( \triangle ABC \) is 12, find the maximum possible value of its area.

Solution: \boxed{36(3 - 2\sqrt{2})}


In a tetrahedron \(ABCD\), the angle between faces \(ABC\) and \(BCD\) is \(30^{\circ}\), the area of \(\triangle ABC\) is \(120\), the area of \(\triangle BCD\) is \(80\), and \(BC = 10\). Find the volume of this tetrahedron.


In Lhota, there was an election for the mayor. Two candidates ran: Mr. Schopný and his wife, Dr. Schopná. The village had three polling stations. In the first and second stations, Dr. Schopná received more votes. The vote ratios were $7:5$ in the first station and $5:3$ in the second station. In the third polling station, the ratio was $3:7$ in favor of Mr. Schopný. The election ended in a tie, with both candidates receiving the same number of votes. In what ratio were the valid votes cast in each polling station if we know that the same number of people cast valid votes in the first and second polling stations?

Solution: \boxed{24 : 24 : 25}


Each of the 10 people is either a knight, who always tells the truth, or a liar, who always lies. Each of them has thought of a natural number. Then the first said, "My number is greater than 1," the second said, "My number is greater than 2," and so on until the tenth said, "My number is greater than 10." Afterwards, in a different order, they each said one sentence: "My number is less than 1," "My number is less than 2," and so on until "My number is less than 10." What is the maximum number of knights that could be among these 10 people?


After walking so much that his feet get really tired, the beaver staggers so that, at each step, his coordinates change by either $(+1,+1)$ or $(+1,-1)$. Now he walks from $(0,0)$ to $(8,0)$ without ever going below the $x$-axis. How many such paths are there?


4n points are arranged around a circle and are colored alternately yellow and blue. The yellow points are divided into pairs, with each pair connected by a yellow line segment. Similarly, the blue points are divided into pairs and each pair is connected by a blue line segment. At most two segments meet at any point inside the circle. Show that there are at least n points of intersection between a yellow segment and a blue segment.

Solution: \boxed{n}


Vanya decided to give Masha a bouquet of an odd number of flowers, consisting of yellow and red tulips, so that the count of one color differs from the count of the other by exactly one. Yellow tulips cost 50 rubles each, and red tulips cost 31 rubles each. What is the maximum number of tulips he can buy for Masha's birthday bouquet without spending more than 600 rubles?


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


A traveler visited a village where each person either always tells the truth or always lies. The villagers stood in a circle, and each told the traveler whether the neighbor to their right was truthful. Based on these reports, the traveler was able to determine exactly what proportion of the villagers are liars. Determine what this proportion is.


Let the function \( f(x) = \frac{1}{2} + \log_2 \frac{x}{1-x} \) and \( S_n = \sum_{i=1}^{n-1} f\left(\frac{i}{n}\right) \), where \( n \in \mathbf{N}^{*} \) and \( n \geq 2 \). Find \( S_n \).


Calculate the areas of the regions bounded by the curves given in polar coordinates.

$$
r=\cos 2 \phi
$$


Find the solution of the equation \( y^{\prime} - y = \cos x - \sin x \), which satisfies the condition that \( y \) is bounded as \( x \rightarrow +\infty \).

Solution: \boxed{y = \sin x}


In the isosceles triangle \(ABC\), \(\angle ACB = \angle ABC = 40^\circ\). On the ray \(AC\), mark off the segment \(AD\) such that \(AD = BC\). What is the measure of \(\angle BDC\)?


Consider a list of six numbers. When the largest number is removed from the list, the average is decreased by 1. When the smallest number is removed, the average is increased by 1. When both the largest and the smallest numbers are removed, the average of the remaining four numbers is 20. Find the product of the largest and the smallest numbers.


In a table tennis match between players A and B, they follow the rules that: the winner of each game gets 1 point and the loser gets 0 points. The match stops when one player has 2 more points than the other or after a maximum of 6 games. Let the probability that A wins each game be $\frac{3}{4}$ and the probability that B wins each game be $\frac{1}{4}$, and assume the outcomes of each game are independent. Find the expected number of games $\xi$ played when the match stops, denoted as $\mathrm{E} \xi$.


A number was multiplied by the first digit and resulted in 494, by the second digit and resulted in 988, and by the third digit and resulted in 1729. Find this number.


Find the largest natural number \( n \) with the following property: for any odd prime number \( p \) less than \( n \), the difference \( n - p \) is also a prime number.


The absolute value of a number \( x \) is equal to the distance from 0 to \( x \) along a number line and is written as \( |x| \). For example, \( |8|=8, |-3|=3 \), and \( |0|=0 \). For how many pairs \( (a, b) \) of integers is \( |a|+|b| \leq 10 \)?


In a grove, there are four types of trees: birches, spruces, pines, and aspens. There are 100 trees in total. It is known that among any 85 trees, there are trees of all four types. What is the smallest number of any trees in this grove that must include trees of at least three types?


Find all positive integer solutions \((x, y, z, t)\) to the equation \(2^{y} + 2^{z} \times 5^{t} - 5^{x} = 1\).

Solution: \boxed{(2, 4, 1, 1)}


Determine the radius of the sphere that touches the faces of the unit cube passing through vertex $A$ and the edges passing through vertex $B$.

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


Four boys, \( A, B, C, \) and \( D \) made three statements each about the same number \( x \). We know that each of them has at least one statement that is true, but also at least one statement that is false. Determine whether \( x \) can be identified. The statements are:

\( A \):

1. The reciprocal of \( x \) is not less than 1.
2. The decimal representation of \( x \) does not contain the digit 6.
3. The cube of \( x \) is less than 221.

\( B \):

4. \( x \) is an even number.
5. \( x \) is a prime number.
6. \( x \) is an integer multiple of 5.

\( C \):

7. \( x \) cannot be expressed as a ratio of two integers.
8. \( x \) is less than 6.
9. \( x \) is a square of a natural number.

\( D \):

10. \( x \) is greater than 20.
11. \( x \) is positive, and its base-10 logarithm is at least 2.
12. \( x \) is not less than 10.


Find the smallest natural number \( n \) for which:
a) \( n! \) is divisible by 2016;
b) \( n! \) is divisible by \( 2016^{10} \).

(Note that \( n! = 1 \cdot 2 \cdot 3 \cdot \ldots \cdot n \)).


Given $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are three natural numbers, and the least common multiple (LCM) of $\mathrm{a}$ and $\mathrm{b}$ is $60$, the LCM of $\mathrm{a}$ and $\mathrm{c}$ is $270$, find the LCM of $\mathrm{b}$ and $\mathrm{c}$.


In decimal notation, the square of a number with more than one digit has its tens digit equal to 7. What is the units digit of this square number?


The lateral faces of a pentagonal pyramid \( S A B C D E \) are acute-angled triangles. We will call a lateral edge of the pyramid good if it is equal to the height of the opposite lateral face, drawn from the apex of the pyramid. For example, edge \( S A \) is good if it is equal to the height of triangle \( S C D \), drawn from vertex \( S \). What is the maximum number of good edges that the pyramid can have?


Masha has 2 kg of "Lastochka" candies, 3 kg of "Truffle" candies, 4 kg of "Ptichye moloko" candies, and 5 kg of "Citron" candies. What is the maximum number of New Year gifts she can make if each gift must contain 3 different types of candies, 100 grams of each type?


Let \( A = \cos^2 10^\circ + \cos^2 50^\circ - \sin 40^\circ \sin 80^\circ \). Determine the value of \( 100A \).


As shown in the figure, the side length of square $ABCD$ is 8 centimeters. Point $E$ is on the extension of $AB$, and quadrilateral $BEFG$ is a square. Find the area of triangle $AFC$.


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


A number appears on a computer screen. It is known that if \( x \) appears on the screen, the number \( x^{2}-2x+1 \) appears immediately after. If the first number to appear is 2, what is the 2020th number to appear?


Standa and Jana received two three-digit numbers. Standa placed a decimal point after the first digit of the first number and after the second digit of the second number, then added the resulting decimal numbers and got a result of 50.13. Jana placed a decimal point after the second digit of the first number and after the first digit of the second number, then added the resulting decimal numbers and got a result of 34.02.

Determine the sum of the original three-digit numbers.


Given a rectangle with a length of 8 and a width of 4, it is folded along a diagonal and pressed flat. Find the area of the overlapping part (grey triangle).


A regular hexagon and an equilateral triangle have the same perimeter. What is the ratio of their areas?


A person is waiting at the $A$ HÉV station. They get bored of waiting and start moving towards the next $B$ HÉV station. When they have traveled $1 / 3$ of the distance between $A$ and $B$, they see a train approaching $A$ station at a speed of $30 \mathrm{~km/h}$. If they run at full speed either towards $A$ or $B$ station, they can just catch the train. What is the maximum speed at which they can run?

Solution: \boxed{10 \text{ km/h}}


Suppose positive real numbers \( x, y, z \) satisfy \( x y z = 1 \). Find the maximum value of \( f(x, y, z) = (1 - yz + z)(1 - zx + x)(1 - xy + y) \) and the corresponding values of \( x, y, z \).


A wolf saw a roe deer a few meters away and started chasing it along a straight forest path. The wolf's jump is $22\%$ shorter than the roe deer's jump. Both animals jump at a constant speed. All jumps of the roe deer are of the same length, and the wolf's jumps are also equal to each other. There is an interval of time during which both the wolf and the roe deer make some integer number of jumps. Each time it turns out that the wolf has made $t\%$ more jumps than the roe deer. Find the largest integer value of $t$ for which the wolf cannot catch up with the roe deer.


Removing the jokers from a standard deck of cards, we have 52 cards remaining. If five cards are drawn at random from this deck, what is the probability that at least two of the cards have the same rank (number or letter $K$, $Q$, $J$, $A$)? Calculate this probability to two decimal places.


A farmer presented 6 types of sour cream in containers of \(9, 13, 17, 19, 20, \text{ and } 38\) liters at the market. On the first day, he sold the sour cream from three containers entirely, and on the second day, he sold the contents of two more containers completely. The volume of sour cream sold on the first day was twice the volume sold on the second day. Which containers were emptied on the first day? Indicate the maximum possible total volume of sour cream sold on the first day in the answer.


Given a triangle ABC with sides \(a\), \(b\), and \(c\) satisfying \(a^2 + b^2 + 3c^2 = 7\), what is the maximum area of triangle ABC?


Calculate the definite integral:

$$
\int_{-\pi / 2}^{0} 2^{8} \cdot \sin^2(x) \cos^6(x) \, dx
$$

Solution: \boxed{5 \pi}


Find all natural numbers \(a\) for which the number

$$
\frac{a+1+\sqrt{a^{5}+2 a^{2}+1}}{a^{2}+1}
$$

is also a natural number.

Solution: \boxed{a = 1}


Given 99 positive numbers arranged in a circle, it is found that for any four consecutive numbers, the sum of the first two numbers in the clockwise direction is equal to the product of the last two numbers in the clockwise direction. What can be the sum of all 99 numbers placed in a circle?


In a country with 20 cities, an airline wants to organize bidirectional flights between them such that it is possible to reach any city from any other city with no more than $\mathrm{k}$ layovers. The number of flight routes should not exceed four. What is the minimum value of $\mathrm{k}$ for which this is possible?


Solve the inequality \(\frac{\sqrt{\frac{x}{\gamma}+(\alpha+2)}-\frac{x}{\gamma}-\alpha}{x^{2}+a x+b} \geqslant 0\).

Indicate the number of integer roots of this inequality in the answer. If there are no integer roots or there are infinitely many roots, indicate the number 0 on the answer sheet.

Given:
\[
\alpha = 3, \gamma = 1, a = -15, b = 54.
\]


Let $p$ be a prime number and $q$ be a prime divisor of $p^{p-1}+\ldots+ p+1$. Show that $q \equiv 1 (\bmod p)$.

Solution: \boxed{q \equiv 1 \pmod{p}}


How many three-digit positive integers are there such that the three digits of every integer, taken from left to right, form an arithmetic sequence?


A right triangle is circumscribed around a circle with a radius of 4, and its hypotenuse is 26. Find the perimeter of the triangle.


In a right triangle $ABC$, the leg $AC = 15$ and the leg $BC = 20$. On the hypotenuse $AB$, the segment $AD$ is 4 units long, and the point $D$ is connected to $C$. Find $CD$.


Find the area of a triangle given that two of its sides are 1 and $\sqrt{13}$, and the median to the third side is 2.


What is the product

\[
\log_{3} 2 \cdot \log_{4} 3 \cdot \log_{5} 4 \ldots \log_{10} 9
\]

if it is known that \(\lg 2=0,3010\)?


Solution: \boxed{2 x \sqrt[3]{2 x}}


In a carriage, any group of $m \ (m \geq 3)$ passengers have only one common friend (where if A is a friend of B, then B is also a friend of A; no one is their own friend). What is the maximum number of friends that the person with the most friends in this carriage can have?

Solution: \boxed{m}


The mid-term exam results of Chinese, Mathematics, and English for a certain class are as follows: There are 18 students who have achieved excellence in at least one course; 9 students have achieved excellence in Chinese, 11 in Mathematics, and 8 in English; 5 students have achieved excellence in both Chinese and Mathematics, 3 in both Mathematics and English, and 4 in both Chinese and English.
(1) How many students have achieved excellence in at least one of the courses, Chinese or Mathematics?
(2) How many students have achieved excellence in all three courses, Chinese, Mathematics, and English?


The numbers \(2^{2021}\) and \(5^{2021}\) are written out one after the other. How many digits were written in total?


Let the set \( T = \{0, 1, \dots, 6\} \),
$$
M = \left\{\left.\frac{a_1}{7}+\frac{a_2}{7^2}+\frac{a_3}{7^3}+\frac{a_4}{7^4} \right\rvert\, a_i \in T, i=1,2,3,4\right\}.
$$
If the elements of the set \( M \) are arranged in decreasing order, what is the 2015th number?


In triangle \(ABC\), \(AB = 20\), \(BC = 21\), and \(CA = 29\). Point \(M\) is on side \(AB\) with \(\frac{AM}{MB}=\frac{3}{2}\), while point \(N\) is on side \(BC\) with \(\frac{CN}{NB}=2\). Points \(P\) and \(Q\) are on side \(AC\) such that line \(MP\) is parallel to \(BC\) and line \(NQ\) is parallel to \(AB\). Suppose \(MP\) and \(NQ\) intersect at point \(R\). Find the area of triangle \(PQR\).


A circle with radius \( R \) is inscribed in an isosceles trapezoid. The upper base of the trapezoid is half of its height. Find the area of the trapezoid.


In the city where Absent-Minded Scientist lives, phone numbers consist of 7 digits. The Scientist easily remembers a phone number if it is a palindrome, which means it reads the same forwards and backwards. For example, the number 4435344 is easily remembered by the Scientist because it is a palindrome. However, the number 3723627 is not a palindrome, so the Scientist finds it hard to remember. Find the probability that a new random acquaintance's phone number will be easily remembered by the Scientist.


30 tigers and 30 foxes are divided into 20 groups, each with 3 animals. Tigers always tell the truth, and foxes always lie. When asked, "Is there a fox in your group?", 39 out of the 60 animals answered "No." How many groups consist entirely of tigers?


The vertices of triangle \( \triangle ABC \) are \( A(0,0), B(0,420), C(560,0) \). A die has its six faces marked with \( A, A, B, B, C, C \). A point \(P_{1}=(k, m)\) is chosen inside \( \triangle ABC \), and subsequent points \( P_{2}, P_{3}, P_{4}, \cdots \) are generated according to the following rule: If \( P_{n} \) is already chosen, roll the die to get a label \( L, L \in \{A, B, C\} \). Then \( P_{n+1} \) is the midpoint of \( P_{n} \) and \( L \). Given \( P_{7}=(14,92) \), find the value of \( k+m \).


Points \( A_{1}, A_{2}, A_{3}, A_{4}, A_{5}, A_{6} \) divide a circle of radius 1 into six equal parts. From \( A_{1} \), a ray \( l_{1} \) is drawn in the direction of \( A_{2} \); from \( A_{2} \), a ray \( l_{2} \) is drawn in the direction of \( A_{3} \), and so on, from \( A_{6} \), a ray \( l_{6} \) is drawn in the direction of \( A_{1} \). From a point \( B_{1} \), taken on the ray \( l_{1} \), a perpendicular is dropped onto ray \( l_{6} \), from the base of this perpendicular another perpendicular is dropped onto \( l_{5} \), and so on. The base of the sixth perpendicular coincides with \( B_{1} \). Find the segment \( B_{1} A_{1} \).


In a class of 45 students, all students participate in a tug-of-war. Among the remaining three events, each student participates in at least one event. It is known that 39 students participate in the shuttlecock kicking event and 28 students participate in the basketball shooting event. How many students participate in all three events?


Students of Class 4(1) are lined up for a field trip. Xiaohong is the 15th person from the front of the line, and Xiaoming is the 20th person from the end of the line. There are 3 people between Xiaohong and Xiaoming. How many students are there in Class 4(1) in total?


Consider the following three arithmetic sequences:

\[2, 11, 20, 29, \ldots\]

\[7, 15, 23, 31, \ldots\]

\[8, 19, 30, 41, \ldots\]

Determine the smallest (positive) common term of these three sequences!


Find a six-digit number that starts with the digit 1 and such that if this digit is moved to the end, the resulting number is three times the original number.


The sheriff believes that if he catches a number of bandits on a given day that is a prime number, then he is lucky. On Monday and Tuesday, the sheriff was lucky. Starting from Wednesday, the number of bandits he caught was equal to the sum of the number caught the day before yesterday and twice the number caught the day before. What is the maximum number of consecutive days the sheriff could be lucky this week? Justify your answer.


There are 7 clean sheets of paper on a table. Every minute, Vasya selects any 4 out of them and draws one star on each of the chosen sheets. Vasya wants each sheet to end up with a different number of stars (meaning no two sheets have the same number of stars). What is the minimum total number of stars Vasya will have to draw to achieve this?


If the positive real numbers \( x \) and \( y \) satisfy the equation \( x^3 + y^3 + 3xy = 1 \), then find the minimum value of \( \left(x + \frac{1}{x}\right)^3 + \left(y + \frac{1}{y}\right)^3 \).


You are trapped in ancient Japan, and a giant enemy crab is approaching! You must defeat it by cutting off its two claws and six legs and attacking its weak point for massive damage. You cannot cut off any of its claws until you cut off at least three of its legs, and you cannot attack its weak point until you have cut off all of its claws and legs. In how many ways can you defeat the giant enemy crab? (Note that the legs are distinguishable, as are the claws.)


We intersected the unit cube $ABCDEFGH$ with a plane such that it intersects the edges $AB$ and $AD$ at the internal points $P$ and $Q$, respectively, both at a distance $x$ from $A$, and it intersects the edge $BF$ at the point $R$. What is the distance $BR$ if $\angle QPR = 120^\circ$?


Given points \( A(1, 3), B(5, 4), C(3, 7), D(7, 1), E(10, 2), F(8, 6) \). If line \( l \) is parallel to line \( DF \), and the shortest distance from line \( l \) to any point inside (including the boundary) of triangle \( ABC \) is equal to the shortest distance from line \( l \) to any point inside (including the boundary) of triangle \( DEF \), find the equation of line \( l \).

Solution: \boxed{10x - 2y - 55 = 0}


What is the least 6-digit natural number that is divisible by 198?


PQR Entertainment wishes to divide their popular idol group PRIME, which consists of seven members, into three sub-units - PRIME-P, PRIME-Q, and PRIME-R - with each of these sub-units consisting of either two or three members. In how many different ways can they do this, if each member must belong to exactly one sub-unit?


Suppose that \( \mathbf{G} = 10^{100} \). (\( \mathbf{G} \) is known as a googol.) How many times does the digit 9 occur in the integer equal to \( \mathbf{G} - 1009^{2} \)?


The bisector of angle $B A D$ of the right trapezoid $A B C D$ (with bases $A D$ and $B C$, and $\angle B A D=90^{\circ}$) intersects the lateral side $C D$ at point $E$. Find the ratio $C E: E D$ if $A D+B C=A B$.


King Arthur has two equally wise advisors – Merlin and Percival. Each of them finds the correct answer to any question with a probability \( p \), or an incorrect answer with a probability \( q = 1-p \).

If both advisors give the same answer, the king listens to them. If they give opposite answers, the king makes a decision by flipping a coin.

One day, Arthur wondered whether he needed two advisors or if one would suffice. He called the advisors and said:

- I think the probability of making the right decisions won't decrease if I only keep one advisor and follow him. If this is correct, I should fire one of you. If not, I will leave everything as it is. Answer me, should I fire one of you?

- Whom exactly are you going to fire, King Arthur? – asked the advisors.
- If I decide to fire one of you, I will choose by drawing lots, flipping a coin.

The advisors left to think about the answer. The advisors, it is to be repeated, are equally wise, but not equally honest.

Percival is very honest and will try to give the correct answer, even if he risks being fired. But Merlin, honest in all other matters, in this situation decides to give an answer that maximizes the probability of not being fired. What is the probability that Merlin will be fired?


Four friends came back from fishing. Each pair of them counted the sum of their catches. They obtained six numbers: $7, 9, 14, 14, 19, 21$. How many fish were caught in total?


What is the smallest number of tetrahedrons into which a cube can be partitioned?


Two positive numbers \(a\) and \(b\), with \(a > b\), are such that twice their sum is equal to three times their difference. What is the ratio \(a : b\)?

Solution: \boxed{5 : 1}


On a beautiful Sunday spring morning, the father of a family went for a walk with his sons.

- Have you noticed, - he said to them, - that the age of the oldest of you is equal to the sum of the ages of your two other brothers?
- Yes. And we also noticed - they answered in unison - that the product of our ages and your age equals the sum of the cube of the number of your sons, multiplied by a thousand, and ten times the square of this number.

Can you determine from this conversation the age of the father at the time of the birth of the second son?


In right triangle \( \triangle ABC \) with \( \angle C = 90^{\circ} \) and \( AB = 1 \), point \( E \) is the midpoint of side \( AB \), and \( CD \) is the altitude from \( C \) to \( AB \). Find the maximum value of \( (\overrightarrow{CA} \cdot \overrightarrow{CD}) \cdot (\overrightarrow{CA} \cdot \overrightarrow{CE}) \).


From the consecutive natural numbers \(1, 2, 3, \ldots, 2014\), select \(n\) numbers such that any two of the selected numbers do not include one number being seven times another. Find the maximum value of \(n\) and provide a justification.


There are 200 computers in a computer center, some of which are connected by cables, with a total of 345 cables used. We define a "cluster" as a set of computers such that a signal can travel from any computer in the set to all other computers in the set, possibly through intermediate computers. Initially, all the computers formed a single cluster. One night, a malicious hacker cut several cables, resulting in the formation of 8 clusters. Find the maximum possible number of cables that were cut.


Inside triangle \( ABC \), a point \( D \) is chosen such that \(\angle BAD = 60^\circ\) and \(\angle ABC = \angle BCD = 30^\circ\). It is known that \( AB = 15 \) and \( CD = 8 \). Find the length of segment \( AD \). If necessary, round the answer to 0.01 or write the answer as a common fraction.


In rectangle \(ABCD\), a point \(E\) is marked on the extension of side \(CD\) beyond point \(D\). The bisector of angle \(ABC\) intersects side \(AD\) at point \(K\), and the bisector of angle \(ADE\) intersects the extension of side \(AB\) at point \(M\). Find \(BC\) if \(MK = 8\) and \(AB = 3\).


The diagonals of a convex quadrilateral \(ABCD\) intersect at point \(E\). It is known that the area of each of the triangles \(ABE\) and \(DCE\) is equal to 1, and the area of the entire quadrilateral does not exceed 4. Given that \(AD = 3\), find the side \(BC\).


Given a geometric sequence $\{a_{n}\}$ that satisfies $\lim_{n \rightarrow +\infty} \sum_{i=1}^{n} a_{4} = 4$ and $\lim_{n \rightarrow +\infty} \sum_{i=1}^{n} a_{i}^{2} = 8$, determine the common ratio $q$.


If \( x_1, x_2, x_3, x_4, \) and \( x_5 \) are positive integers that satisfy \( x_1 + x_2 + x_3 + x_4 + x_5 = x_1 x_2 x_3 x_4 x_5 \), that is the sum is the product, find the maximum value of \( x_5 \).


Let \( c \in \mathbf{C} \). The polynomial
\[ p(x) = \left(x^{2}-2x+2\right)\left(x^{2}-cx+4\right)\left(x^{2}-4x+8\right) \]
has exactly four distinct complex roots. Then \( |c| = \ ? \)
(A) 2
(B) \( \sqrt{6} \)
(C) \( 2\sqrt{2} \)
(D) 3
(E) \( \sqrt{10} \)


There are \( n \) pieces of paper, each containing 3 different positive integers no greater than \( n \). Any two pieces of paper share exactly one common number. Find the sum of all the numbers written on these pieces of paper.


In a tournament with 25 chess players, each having different skill levels and where the stronger player always wins:

What is the minimum number of matches required to determine the two strongest players?


If the value of the expression $(\square + 121 \times 3.125) \div 121$ is approximately 3.38, what natural number should be placed in $\square$?


Given an integer sequence $a_{1}, a_{2}, \cdots, a_{10}$ satisfying $a_{10}=3 a_{1}$, $a_{2}+a_{8}=2 a_{5}$, and
$$
a_{i+1} \in\left\{1+a_{i}, 2+a_{i}\right\}, i=1,2, \cdots, 9,
$$
find the number of such sequences.


At 8:30 AM, a helicopter was over point $A$. After flying in a straight line for $s$ kilometers, the helicopter found itself over point $B$. After hovering over point $B$ for 5 minutes, the helicopter set off on the return course along the same route. The helicopter returned to point $A$ at 10:35 AM. The helicopter flew with the wind from $A$ to $B$ and against the wind on the return. The wind speed remained constant the entire time. Determine the wind speed, given that the helicopter's own speed was also constant and equal to $\mathrm{v}$ km/h in the absence of wind.

Under what conditions among the given quantities does the problem have a solution?

Solution: \boxed{\sqrt{v(v - s)}}


Find the volume of the body $\Omega$ bounded by the surfaces

$$
z=\frac{9}{2} \sqrt{x^{2}+y^{2}}, \quad z=\frac{11}{2}-x^{2}-y^{2}
$$

Solution: \boxed{V = 2\pi}


Several hundred years ago, Columbus discovered the Americas. The four digits of the year he discovered the new continent are all different and their sum is 16. If you add 1 to the tens digit, it will be exactly 5 times the units digit. In which year did Columbus discover the Americas?


In a \(3 \times 3\) table, the numbers 1, 2, and 3 are placed such that each number occurs only once in each row and only once in each column. What is the value of \(X + Y\)?

(A) 3
(B) 2
(C) 5
(D) 6
(E) 4


Given integers \( m \) and \( n \) such that \( m, n \in \{1, 2, \cdots, 1981\} \) and

\[
\left(n^{2} - m n - m^{2}\right)^{2} = 1,
\]

find the maximum value of \( m^{2} + n^{2} \).


The points $A, B, C, D$ are consecutive vertices of a regular polygon, and the following equation holds:

$$
\frac{1}{A B}=\frac{1}{A C}+\frac{1}{A D}
$$

How many sides does the polygon have?


Given a plane's first (second) trace line and the angle that this plane forms with the second (first) projection plane, find the plane's second (first) trace line, assuming that the intersection point of the given trace line with the projection axis lies outside the paper's boundary. How many solutions does this problem have?


For a natural number $N$, if at least five out of the nine natural numbers $1-9$ can divide $N$, then $N$ is called a "five-divisible number". What is the smallest "five-divisible number" greater than 2000?


Let \( p(x) = a_{n} x^{n} + a_{n-1} x^{n-1} + \ldots + a_{0} \), where each \( a_{i} \) is either 1 or -1. Let \( r \) be a root of \( p \). If \( |r| > \frac{15}{8} \), what is the minimum possible value of \( n \)?


Given a regular tetrahedron \(ABCD\) with edge length 2, there is a point \(P\) on edge \(AB\) such that \(AP < 1\). A cut is made through point \(P\) perpendicular to the edge \(AB\) and continues through the faces but stops at a certain point. When the cut stops, the length of the cut on face \(ABD\) is \(PM = 1\) and the length of the cut on face \(ABC\) is \(PN = \frac{2}{3}\). Find the length \(MN\).

Solution: \boxed{MN = 1}


A circle is tangent to the extensions of two sides \(AB\) and \(AD\) of a square \(ABCD\) with side length \(2\sqrt{3} \text{ cm}\). From point \(C\), two tangents are drawn to this circle. Find the radius of the circle given that the angle between the tangents is \(30^{\circ}\), and it is known that \(\sin 15^{\circ} = \frac{\sqrt{3}-1}{2\sqrt{2}}\).


Point \( O \) is the center of the circle circumscribed around triangle \( ABC \) with sides \( AB = 8 \) and \( AC = 5 \). Find the length of side \( BC \) if the length of the vector \(\overrightarrow{OA} + 3 \overrightarrow{OB} - 4 \overrightarrow{OC}\) is 10.


Martin decided to spend all his savings on sweets. He found out that he could buy three cream puffs and $3 \mathrm{dl}$ of Kofola, or $18 \mathrm{dkg}$ of yogurt raisins, or $12 \mathrm{dkg}$ of yogurt raisins and half a liter of Kofola. In the end, he bought one cream puff and $6 \mathrm{dl}$ of Kofola. How many grams of yogurt raisins does he have left over?


Find the smallest positive integer $n$ that satisfies the following two properties:

1. $n$ has exactly 144 distinct positive divisors.
2. Among the positive divisors of $n$, there are ten consecutive integers.


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

Solution: \boxed{=}


Two faces of a tetrahedron are equilateral triangles with side lengths of one unit, and the other two faces are isosceles right triangles. What is the volume of the tetrahedron?


Find the smallest real number $\lambda$ such that
\[
\left(a_{1}^n + \prod_{i=1}^{n} a_{i}, a_{2}^n + \prod_{i=1}^{n} a_{i}, \ldots, a_{n}^n + \prod_{i=1}^{n} a_{i}\right) \leq \lambda\left(a_{1}, a_{2}, \ldots, a_{n}\right)^n
\]
holds for any positive odd number $n$ and any $n$ positive integers $a_{1}, a_{2}, \ldots, a_{n}$.


Given \( x, y \in [0,+\infty) \) and satisfying \( x^{3} + y^{3} + 3xy = 1 \).
Find the maximum value of \( x^{2} y \).


Given that the first pocket contains 2 white balls, 3 red balls, and 5 yellow balls, and the second pocket contains 2 white balls, 4 red balls, and 4 yellow balls, if one ball is drawn from each pocket, what is the probability that the colors of the drawn balls are different?


The diagonal of a rectangular parallelepiped is 13, and the diagonals of its side faces are $4 \sqrt{10}$ and $3 \sqrt{17}$.

Find its volume.


The newly admitted freshmen at the university were divided into study groups so that each group had an equal number of students. Due to a reduction in the number of specializations, the number of groups decreased by 9, and all freshmen were redistributed into groups; the groups again became equal in number, with each having less than 30 students. It is known that there were 2376 freshmen in total. How many groups are there now?

*Answer: 99.*


The sum of four consecutive primes is itself prime. What is the largest of the four primes?
A) 37
B) 29
C) 19
D) 13
E) 7


In an acute-angled triangle \( ABC \), points \( D, E \), and \( F \) are the feet of the perpendiculars from \( A, B \), and \( C \) onto \( BC, AC \), and \( AB \), respectively. Suppose \(\sin A = \frac{3}{5}\) and \( BC = 39 \). Find the length of \( AH \), where \( H \) is the intersection of \( AD \) with \( BE \).


A coach decided to award 12 students who ran the distance in the best time. Each of them needs to be awarded with a "gold," "silver," or "bronze" medal. All three types of medals must be used (at least one of each), and a student who finished earlier cannot be awarded a less valuable medal than the one who finished later.

How many ways can the coach distribute the medals (assuming all runners have different times)?


Given 10 positive integers, the sums of any 9 of them take exactly 9 different values: 86, 87, 88, 89, 90, 91, 93, 94, 95. After arranging these 10 positive integers in descending order, find the sum of the 3rd and the 7th numbers.


Find all solutions to the numerical puzzle

$$
\mathrm{AX}+\mathrm{YX}=\mathrm{YPA}
$$

(different letters correspond to different digits, and identical letters correspond to identical digits).

Solution: \boxed{89 + 19 = 108}


Let \( S(n) \) be the sum of the digits in the decimal representation of the number \( n \). Find \( S\left(S\left(S\left(S\left(2017^{2017}\right)\right)\right)\right) \).


X is a set with n elements. Show that we cannot find more than \(2^{n-1}\) subsets of X such that every pair of subsets has non-empty intersection.


In the multiplication problem shown in the figure, \( A, B, C, D, E, F, G, H, I \) each represent a different single-digit number. What is the five-digit number represented by "FIGAA"?

```
   ABC
×   DC
-----
   BEA
------
 FIGAA
```


Two circles of the same radius 9 intersect at points \(A\) and \(B\). A point \(C\) is chosen on the first circle, and a point \(D\) is chosen on the second circle. It turns out that point \(B\) lies on the segment \(CD\), and \(\angle CAD = 90^\circ\). On the perpendicular to \(CD\) passing through point \(B\), a point \(F\) is chosen such that \(BF = BD\) (points \(A\) and \(F\) are on opposite sides of the line \(CD\)). Find the length of the segment \(CF\).


In the expression $(1+x+y)^{20}$, after expanding the brackets but not combining like terms, how many terms will there be?


The sequence $\left\{a_{n}\right\}$ satisfies $a_{1}=\frac{1}{3}$, and for any $n \in \mathbf{N}^{*}, a_{n+1}=a_{n}^{2}+a_{n}$. Determine the integer part of $\sum_{n=1}^{2016} \frac{1}{a_{n}+1}$.


Divide the sides of a unit square \(ABCD\) into 5 equal parts. Let \(D'\) denote the second division point from \(A\) on side \(AB\), and similarly, let the second division points from \(B\) on side \(BC\), from \(C\) on side \(CD\), and from \(D\) on side \(DA\) be \(A'\), \(B'\), and \(C'\) respectively. The lines \(AA'\), \(BB'\), \(CC'\), and \(DD'\) form a quadrilateral.

What is the area of this quadrilateral?


A group of \( n \) students doing an art project used red, blue, and yellow paint. Every student used at least one color, and some students used more than one color.

- The yellow paint was used by a total of 46 students.
- The red paint was used by a total of 69 students.
- The blue paint was used by a total of 104 students.
- Exactly 14 students used both yellow and blue and did not use red.
- Exactly 13 students used both yellow and red and did not use blue.
- Exactly 19 students used both blue and red and did not use yellow.
- Exactly 16 students used all three colors.

What is the value of \( n \)?


Find the shortest distance from the line \(3x + 4y = 25\) to the circle \(x^2 + y^2 = 6x - 8y\).


If the real numbers \( x \) and \( y \) satisfy \(\frac{x + y}{1 - xy} = \sqrt{5}\), then what is the value of \(\frac{|1 - xy|}{\sqrt{1 + x^2} \sqrt{1 + y^2}}?\)


In quadrilateral \(ABCD\), it is known that \(AB = BC\) and \(\angle ABC = \angle ADC = 90^{\circ}\). From vertex \(B\), a perpendicular \(BH\) is dropped to side \(AD\). Find the area of quadrilateral \(ABCD\) given that \(BH = h\).


Let the functions \( f(\alpha, x) \) and \( g(\alpha) \) be defined as
\[ f(\alpha, x)=\frac{\left(\frac{x}{2}\right)^{\alpha}}{x-1} \]
\[ g(\alpha)=\left.\frac{d^{4} f}{d x^{4}}\right|_{x=2} \]

Then \( g(\alpha) \) is a polynomial in \( \alpha \). Find the leading coefficient of \( g(\alpha) \).


Find all finite non-empty sets $S$ consisting of positive integers that satisfy the condition: If $m, n \in S$, then $\frac{m+n}{(m,n)} \in S$ (where $m$ and $n$ do not need to be distinct).

Solution: \boxed{S = \{2\}}


How many positive integers divide at least one of \( 10^{40} \) and \( 20^{30} \)?


A positive integer is called mystical if it has at least two digits and every pair of two consecutive digits, read from left to right, forms a perfect square. For example, 364 is a mystical integer because 36 and 64 are both perfect squares, but 325 is not mystical because 32 is not a perfect square. What is the largest mystical integer?


Let $a \geq 2$ be a real number, and let $x_1$ and $x_2$ be the roots of the equation $x^{2} - a x + 1 = 0$. Define $S_{n} = x_{1}^{n} + x_{2}^{n}$ for $n = 1, 2, \cdots$.

(1) Determine the monotonicity of the sequence $\left\{\frac{S_{n}}{S_{n+1}}\right\}$ and provide a proof.

(2) Find all real numbers $a$ such that for any positive integer $n$, certain conditions are satisfied.

Solution: \boxed{a = 2}


Find all strictly positive integers \( (a, b, p) \) with \( p \) being a prime such that \( 2^{a} + p^{b} = 19^{a} \).

Solution: \boxed{(1, 1, 17)}


The lateral edges of a triangular pyramid are pairwise perpendicular and equal to \(a\), \(b\), and \(c\). Find the volume of the pyramid.

Solution: \boxed{\frac{1}{6} abc}


Given natural numbers \( m \) and \( n \). There are two piles on the table: the first pile contains \( n \) stones, and the second pile contains \( m \) stones. Petya and Vasya play the following game. Petya starts. In one move, a player can break one of the available piles on the table into several smaller piles. The player who cannot make a move loses. For which values of \( m \) and \( n \) can Petya ensure victory regardless of Vasya's play?

Solution: \boxed{m \neq n}


A car departed from point A to point B, and with some delay, a second car followed. When the first car had traveled half the distance, the second car had traveled $26 \frac{1}{4}$ km. When the second car had traveled half the distance, the first car had traveled $31 \frac{1}{5}$ km. After overtaking the first car, the second car reached point B, immediately turned back, and after driving 2 km, met the first car. Find the distance between points A and B. Give your answer as a number without units.


The number \(27,000,001\) has exactly four prime factors. Find their sum.


Let \( \triangle ABC \) have side lengths \(a, b, c\) with \( b > \max \{a, c\} \). Given three nonzero real numbers \( x_0, y_0, z_0 \) that satisfy the line \( ax + by + c = 0 \) passing through the point \( \left(\frac{z_0}{x_0}, \frac{2 y_0}{x_0}\right) \), and the point \( \left(\frac{z_0}{y_0}, \frac{x_0}{y_0}\right) \) lying on the ellipse \( x^2 + \frac{y^2}{4} = 1 \), find the maximum value of \( \tan B \cdot \cot C \).


The ant starts at point A and travels $1+\frac{1}{10}$ cm north, then $2+\frac{2}{10}$ cm west, then $3+\frac{3}{10}$ cm south, then $4+\frac{4}{10}$ cm east, then $5+\frac{5}{10}$ cm north, then $6+\frac{6}{10}$ cm west, and so on. After 1000 steps, the ant reaches point B. Find the straight-line distance between points A and B in centimeters. Record the square of the found distance as your answer.


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


How many four-digit numbers contain the digit 9 followed immediately by the digit 5?


A two-digit integer between 10 and 99, inclusive, is chosen at random. Each possible integer is equally likely to be chosen. What is the probability that its tens digit is a multiple of its units (ones) digit?


Two players, A and B, play a game: A coin is tossed repeatedly. The game ends when either the total number of heads (or tails) reaches 5. If the total number of heads reaches 5 by the end of the game, player A wins; otherwise, player B wins. What is the probability that the game will conclude in fewer than 9 tosses?


In a triangle, the points \( a \) (a > 0) denote the incenter, and also the points of tangency of the inscribed circle with the sides respectively. Using no more than three lines with a straightedge, construct a segment of length \( a - c \).

Solution: \boxed{a - c}


Let \( n \) be an integer greater than 2, and \( a_n \) be the largest \( n \)-digit number that is neither the sum nor the difference of two perfect squares.
(1) Find \( a_n \) (expressed as a function of \( n \)).
(2) Find the smallest value of \( n \) such that the sum of the squares of the digits of \( a_n \) is a perfect square.

Solution: \boxed{n = 66}


Ladybugs gathered in a meadow. If a ladybug has six spots on its back, it always tells the truth, and if it has four spots, it always lies. There were no other types of ladybugs in the meadow. The first ladybug said, "Each of us has the same number of spots on our back." The second said, "Altogether, there are 30 spots on our backs." The third contradicted, "No, altogether there are 26 spots on our backs." "Among these three, exactly one is telling the truth," stated each of the remaining ladybugs. How many ladybugs gathered in the meadow?


How many solutions does the cryptarithm \(\frac{B+O+C+b+M+O+\breve{U}}{K+J+A+C+C}=\frac{22}{29}\) have, where different letters represent different digits, the same letters represent the same digits, and it is known that the digit 0 is not used?


Find the cosine of the angle between the slant height and the diagonal of the base of a regular four-sided pyramid where the lateral edge is equal to the side of the base.


In a tennis tournament, 512 schoolchildren participate. For a win, 1 point is awarded, for a loss, 0 points. Before each round, pairs are drawn from participants with an equal number of points (those who do not have a pair are awarded a point without playing). The tournament ends as soon as a sole leader is determined. How many schoolchildren will finish the tournament with 6 points?


Given sets \(A, B, C\) (not necessarily distinct) satisfying \(A \cup B \cup C = \{1, 2, 3, \dots, 11\}\), find the number of ordered triples \((A, B, C)\) satisfying this condition.


Given that one root of the equation \( x^2 - 4x + b = 0 \) is the opposite of a root of the equation \( x^2 + 4x - b = 0 \), find the positive root of the equation \( x^2 + bx - 4 = 0 \).


A segment connecting the centers of two intersecting circles is divided by their common chord into segments equal to 5 and 2. Find the length of the common chord, given that the radii of the circles are in the ratio \(4: 3\).


Vasya built a pyramid of balls as follows: there is 1 ball at the top of the pyramid, 4 balls in the second layer from the top, and so on. The balls lie along the border and inside the pyramid. Find the total number of balls lying in the third and fifth layers from the top.


The distance between docks \( A \) and \( B \) is covered by a motor ship downstream in 5 hours and upstream in 6 hours. How many hours will it take for a raft to float downstream over this distance?

Solution: \boxed{60 \text{ hours}}


Let \( n \) be the answer to this problem. An urn contains white and black balls. There are \( n \) white balls and at least two balls of each color in the urn. Two balls are randomly drawn from the urn without replacement. Find the probability, in percent, that the first ball drawn is white and the second is black.


Three people \(A\), \(B\), and \(C\) play a game of passing a basketball from one to another. Find the number of ways of passing the ball starting with \(A\) and reaching \(A\) again on the 11th pass. For example, one possible sequence of passing is:
\[ A \rightarrow B \rightarrow A \rightarrow B \rightarrow C \rightarrow A \rightarrow B \rightarrow C \rightarrow B \rightarrow C \rightarrow B \rightarrow A . \]


The set $\{[x] + [2x] + [3x] \mid x \in \mathbb{R}\} \mid \{x \mid 1 \leq x \leq 100, x \in \mathbb{Z}\}$ has how many elements, where $[x]$ denotes the greatest integer less than or equal to $x$.


Calculate

$$
\left|\begin{array}{rrr}
2 & -1 & 1 \\
3 & 2 & 2 \\
1 & -2 & 1
\end{array}\right|
$$


A researcher receives a container for analysis holding approximately 150 samples of oil. Each sample has specific characteristics regarding sulfur content - either low-sulfur or high-sulfur - and density - either light or heavy. The relative frequency (statistical probability) that a randomly chosen sample is heavy oil is $\frac{2}{11}$. The relative frequency that a randomly chosen sample is light oil with low sulfur content is $\frac{7}{13}$. How many samples of high-sulfur oil are there in the container if there are no low-sulfur samples among the heavy oil samples?


Consider the parabola consisting of the points \((x, y)\) in the real plane satisfying
\[
(y + x) = (y - x)^2 + 3(y - x) + 3.
\]
Find the minimum possible value of \(y\).


What is the greatest possible value of the ratio of a three-digit number to the sum of its digits?


Fill integers $1, 2, 3, \cdots, 2016^{2}$ into a $2016 \times 2016$ grid, with each cell containing one number, and all numbers being different. Connect the centers of any two cells with a vector, with the direction from the smaller number to the bigger number. If the sum of numbers in each row and each column of the grid is equal, find the sum of all such vectors.


In a company, employees have a combined monthly salary of $10,000. A kind manager proposes to triple the salaries of those earning up to $500, and to increase the salaries of others by $1,000, resulting in a total salary of $24,000. A strict manager proposes to reduce the salaries of those earning more than $500 to $500, while leaving others' salaries unchanged. What will the total salary be in this case?


A natural number is considered "cool" when each of its digits is greater than any of the other digits to its left. For example, 3479 is cool, while 2231 is not. How many cool numbers are there between 3000 and 8000?


Schoolboy Alexey told his parents that he is already an adult and can manage his finances independently. His mother suggested using a duplicate bank card from her account. For participation in a charitable Christmas program, Alexey wants to buy 40 "Joy" chocolate bars and donate them to an orphanage. However, the bank, where Alexey's parents are clients, has implemented a new system to protect against unauthorized card payments. The protection system analyzes the root mean square (RMS) value of expenses for the last 3 purchases (S) using the formula \(S=\sqrt{\frac{x_{1}^{2}+x_{2}^{2}+x_{3}^{2}}{3}}\), where \(x_{1}, x_{2}\), and \(x_{3}\) are the costs of the previous purchases, and compares the value of \(S\) with the cost of the current purchase. If the cost of the current payment exceeds the value \(S\) by 3 times, the bank blocks the payment and requires additional verification (e.g., a call from mom to the call center). In the last month, payments made on the card were only for cellphone bills in the amount of 300 rubles each. Into how many minimum number of receipts should Alexey split the purchase so that he can buy all 40 "Joy" chocolate bars at a cost of 50 rubles each?


I am thinking of a four-digit number, where each digit is different. When I omit the last two digits of this number, I get a prime number. Similarly, I get a prime number if I omit the second and fourth digits, and also if I omit the middle two digits. However, the number I am thinking of is not a prime number itself - it can be divided by three without a remainder. There are several numbers with these properties, but mine is the largest of them. Which number am I thinking of?


Given the real number \( x \), \([x] \) denotes the integer part that does not exceed \( x \). Find the positive integer \( n \) that satisfies:

\[
\left[\log _{2} 1\right] + \left[\log _{2} 2\right] + \left[\log _{2} 3\right] + \cdots + \left[\log _{2} n\right] = 1994
\]


In a regular tetrahedron \(ABCD\), \(AO \perp\) plane \(BCD\) at the foot \(O\). Let \(M\) be a point on segment \(AO\) such that \(\angle BMC = 90^\circ\). Then, find \(\frac{AM}{MO} = \quad\).


How many nondecreasing sequences \(a_{1}, a_{2}, \ldots, a_{10}\) are composed entirely of at most three distinct numbers from the set \(\{1,2, \ldots, 9\}\)? For example, the sequences \(1,1,1,2,2,2,3,3,3,3\) and \(2,2,2,2,5,5,5,5,5,5\) are both allowed.


Determine the number of ordered pairs of integers \((m, n)\) for which \(m n \geq 0\) and \(m^{3}+n^{3}+99 m n=33^{3}\).


Let \( a, b, c, d, e \) be positive integers whose sum is 2018. Let \( M = \max (a+b, b+c, c+d, d+e) \). Find the smallest possible value of \( M \).


Find the minimum value of the function \( f(x, y) = \frac{2015(x+y)}{\sqrt{2015 x^{2} + 2015 y^{2}}} \) and specify all pairs \((x, y)\) where it is attained.


In $\triangle ABC$, \(AB = 5\), \(AC = 4\), and \(\overrightarrow{AB} \cdot \overrightarrow{AC} = 12\). Let \(P\) be a point on the plane of \(\triangle ABC\). Find the minimum value of \(\overrightarrow{PA} \cdot (\overrightarrow{PB} + \overrightarrow{PC})\).


Calculate the area of one petal of the curve $\rho = \sin^2 \varphi$.


An arbitrary point \( E \) inside the square \( ABCD \) with side length 1 is connected by line segments to its vertices. Points \( P, Q, F, \) and \( T \) are the points of intersection of the medians of triangles \( BCE, CDE, DAE, \) and \( ABE \) respectively. Find the area of the quadrilateral \( PQFT \).


In triangle $A B C$, the sides are given as: $A B=4, A C=3, B C=\sqrt{37}$. Point $P$ is the midpoint of side $A B$, and point $Q$ is on side $A C$ at a distance of 1 from point $C$. Find $P Q$.


A pedestrian departed from point \( A \) to point \( B \). After walking 8 km, a second pedestrian left point \( A \) following the first pedestrian. When the second pedestrian had walked 15 km, the first pedestrian was halfway to point \( B \), and both pedestrians arrived at point \( B \) simultaneously. What is the distance between points \( A \) and \( B \)?


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


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


Find all pairs of two-digit natural numbers whose arithmetic mean is $25/24$ times greater than their geometric mean. In the answer, indicate the largest of the arithmetic means for all such pairs.


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


In how many ways can we place two bishops of the same color on a chessboard such that they are on different rows, columns, and squares of distinct colors?


Given the cryptarithm: REKA + KARE = ABVAD. Each identical letter corresponds to the same digit, and different letters correspond to different digits. Find the value of the letter B.


Does there exist a two-variable polynomial \(P(x, y)\) with real number coefficients such that \(P(x, y)\) is positive exactly when \(x\) and \(y\) are both positive?

Solution: \boxed{\text{No}}


In the regular tetrahedron \(ABCD\), the midpoint \(M\) is taken on the altitude \(AH\), connecting \(BM\) and \(CM\), then \(\angle BMC =\)


How many triangles exist such that the lengths of the sides are integers not greater than 10?


The sequence \(a_1, a_2, a_3, \ldots\) of positive integers is determined by its first two members and the rule \( a_{n+2} = \frac{a_{n+1} + a_n}{\text{gcd}(a_n, a_{n+1})} \). For which values of \(a_1\) and \(a_2\) is this sequence bounded?

Solution: \boxed{a_1 = a_2 = 2}


A right-angled triangle has sides of integer length. One of its sides has length 20. List all the different possible hypotenuses of such triangles and find the sum of all the numbers in this list.


Choose 4 different numbers from $1, 2, 3, 4, 5$ and fill them into the 4 squares in the expression $\square+\square > \square+\square$. There are $\qquad$ different ways to do this to make the expression true. (Hint: $1+5>2+3$ and $5+1>2+3$ are considered different ways.)


Find all functions \( f: \mathbf{R} \rightarrow \mathbf{R} \) such that for all \( x, y \in \mathbf{R} \), the following holds:
\[ f(1 + xy) - f(x + y) = f(x) f(y), \]
and \( f(-1) \neq 0 \).

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


Find all integer values that the expression

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

can take, where \( p \) and \( q \) are prime numbers.


On the ground, there are three points: \( A \), \( B \), and \( C \). A frog is at point \( P \) on the ground, which is 0.27 meters away from point \( C \). The frog's first jump is a "symmetric jump" from \( P \) to the symmetric point \( P_{1} \) with respect to point \( A \). The second jump is a symmetric jump from \( P_{1} \) with respect to point \( B \) to reach \( P_{2} \). The third jump is a symmetric jump from \( P_{2} \) with respect to point \( C \) to reach \( P_{3} \). The fourth jump is a symmetric jump from \( P_{3} \) with respect to point \( A \) to reach \( P_{4} \), and so on, jumping in this way continuously. After the 2009th symmetric jump, the frog reaches \( P_{2009} \). What is the distance between \( P_{2009} \) and the starting point \( P \) in centimeters?


Calculator ACH-2016 can perform two operations: taking the cube root and taking the tangent. Initially, the number \(2^{-243}\) was entered into the calculator. What is the minimum number of operations required to obtain a number greater than 1?


Solve the equation

\[ x^3 = 3y^3 + 9z^3 \]

in non-negative integers \(x\), \(y\), and \(z\).

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


On a line, we have four points $A, B, C,$ and $D$ arranged in that order such that $AB = CD$. E is a point outside the line such that $CE = DE$. Show that $\angle CED = 2 \angle AEB$ if and only if $AC = EC$.

Solution: \boxed{AC = EC}


A cube with a side length of 2 is inscribed in a sphere. A second cube, with faces parallel to the first, is inscribed between the sphere and one face of the first cube. What is the length of a side of the smaller cube?


Three workers are digging a hole. They take turns working, with each one working as long as it takes the other two to dig half the hole. Working in this way, they finished digging the hole. How many times faster would they have finished the job if they had worked simultaneously?


A real number \( x \) is chosen uniformly at random from the interval \( (0, 10) \). Compute the probability that \( \sqrt{x}, \sqrt{x+7} \), and \( \sqrt{10-x} \) are the side lengths of a non-degenerate triangle.


Given the sequence of positive integers \(\left\{a_{n}\right\}\) defined by \(a_{0}=m\) and \(a_{n+1}=a_{n}^{5}+487\) for \(n \geqslant 0\), find the value of \(m\) such that the number of perfect squares in the sequence \(\left\{a_{n}\right\}\) is maximized.


Find the smallest positive integer \( n \) such that a cube with side length \( n \) can be divided into 1996 smaller cubes, each with side length a positive integer.


In the interval \([0, \pi]\), the number of solutions to the trigonometric equation \(\cos 7x = \cos 5x\) is $\qquad$.


$A A_1$ is the altitude of the acute-angled triangle $ABC$, $H$ is the orthocenter, and $O$ is the circumcenter of the triangle $ABC$. Find $OH$, given that $AH=3$, $A_1H=2$, and the radius of the circumcircle is 4.


In triangle \(ABC\), a median \(AM\) is drawn. Circle \(\alpha\) passes through point \(A\), touches line \(BC\) at point \(M\), and intersects sides \(AB\) and \(AC\) at points \(D\) and \(E\) respectively. On arc \(AD\) that does not contain point \(E\), a point \(F\) is chosen such that \(\angle BFE = 72^\circ\). It is found that \(\angle DEF = \angle ABC\). Find \(\angle CME\).


The numbers \(1, 2, \ldots, 2016\) are divided into pairs such that the product of the numbers in each pair does not exceed a certain natural number \(N\). What is the smallest \(N\) for which this is possible?


Let the function \( f(x) \) be defined on \( \mathbb{R} \), and for any \( x \), the condition \( f(x+2) + f(x) = x \) holds. It is also known that \( f(x) = x^3 \) on the interval \( (-2, 0] \). Find \( f(2012) \).


In the figure, the area of triangle $\triangle \mathrm{ABC}$ is 60. Points $\mathrm{E}$ and $\mathrm{F}$ are on $\mathrm{AB}$ and $\mathrm{AC}$, respectively, satisfying $\mathrm{AB}=3 \mathrm{AE}$ and $\mathrm{AC}=3 \mathrm{AF}$. Point $D$ is a moving point on segment $BC$. Let the area of $\triangle FBD$ be $S_{1}$, and the area of $\triangle EDC$ be $S_{2}$. Find the maximum value of $S_{1} \times S_{2}$.


Scientists found a fragment of an ancient mechanics manuscript. It was a piece of a book where the first page was numbered 435, and the last page was numbered with the same digits, but in some different order. How many sheets did this fragment contain?


As shown in the figure, the right triangle \( \triangle ABC \) has all three vertices on the given parabola \( x^{2}=2py \) (where \( p > 0 \)), and the hypotenuse \( AB \) is parallel to the \( x \)-axis. Find the height \( |CD| \) from the vertex \( C \) to the hypotenuse.


The sequence of functions is defined by the formulas

\[ f_{0}(x)=3 \sin x, \quad f_{n+1}(x)=\frac{9}{3-f_{n}(x)} \]

for any integer \( n \geq 0 \). Find \( f_{2023}\left(\frac{\pi}{6}\right) \).


Given \( x, y, z \in \mathbf{Z} \) such that \(\left\{\begin{array}{l}x+y+z=3 \\ x^{3}+y^{3}+z^{3}=3\end{array}\right.\), find the set of all possible values of \( x^{2}+y^{2}+z^{2} \).

Solution: \boxed{\{3, 57\}}


Gavrila found out that the front tires of a car last for 24,000 km, and the rear tires last for 36,000 km. Therefore, he decided to swap them at some point to maximize the total distance the car can travel. Find this maximum possible distance (in km).


If \( P \) is an arbitrary point in the interior of the equilateral triangle \( ABC \), find the probability that the area of \( \triangle ABP \) is greater than each of the areas of \( \triangle ACP \) and \( \triangle BCP \).


The numbers from 1 to 8 are placed at the vertices of a cube such that the sum of the numbers at any three vertices lying on the same face is at least 10. What is the minimum possible sum of the numbers at the vertices of one face?


On the segment $OA$ of length $L$ on the number line $Ox$, two points, $B(x)$ and $C(y)$, are randomly placed. Find the probability that from the three resulting segments a triangle can be formed.


A math competition problem: The probabilities that A, B, and C solve the problem independently are $\frac{1}{a}$, $\frac{1}{b}$, and $\frac{1}{c}$ respectively, where $a$, $b$, and $c$ are all single-digit numbers. If A, B, and C attempt the problem independently and the probability that exactly one of them solves the problem is $\frac{7}{15}$, then the probability that none of them solves the problem is $\qquad$.


Maria received an enormous chocolate bar as an Easter gift. She decides to divide it into pieces to eat it gradually. On the first day, she divides it into 10 pieces and eats only one of them. On the second day, she divides one of the pieces left from the previous day into 10 pieces and eats only one of them. On the third day, she does the same, i.e., divides one of the pieces left from the previous day into 10 others and eats only one of them. She continues repeating this procedure until the next year's Easter.

a) How many pieces will she have at the end of the third day?

b) Is it possible for her to have exactly 2014 pieces on any given day?

Solution: \boxed{\text{No}}


Let \(ABC\) be a triangle with integral side lengths such that \(\angle A = 3 \angle B\). Find the minimum value of its perimeter.


In a certain logarithm system, $\log 450$ is 1.1639 more than $\log 40$. What is the base of the system and what are the logarithms of the two numbers in this system?


Katya correctly solves a problem with a probability of $4 / 5$, and the magic pen solves a problem correctly without Katya's help with a probability of $1 / 2$. In a test containing 20 problems, solving 13 correctly is enough to get a "good" grade. How many problems does Katya need to solve on her own and how many should she leave to the magic pen to ensure that the expected number of correct answers is at least 13?


Several points, including points \( A \) and \( B \), are marked on a line. We consider all possible segments with endpoints at the marked points. Vasya counted that point \( A \) is inside 40 of these segments, and point \( B \) is inside 42 segments. How many points were marked? (The endpoints of a segment are not considered its internal points.)


The intervals on which the function \( y = \left\{
\begin{array}{cc}
2x + 3, & x \leq 0, \\
x + 3, & 0 < x \leq 1, \\
-x + 5, & x > 1
\end{array}
\right. \) is monotonically increasing.

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


The sum of two natural numbers is 2013. If the last two digits of one of them are crossed out, 1 is added to the resulting number, and the result is then multiplied by 5, you get the other number. Find these numbers. Enter the largest of them in the provided field.


For all real numbers \( x \) and \( y \) satisfying \( |x| + |y| \leq 1 \), the inequality \(\left|2x - 3y + \frac{3}{2}\right| + |y - 1| + |2y - x - 3| \leq a\) always holds. What is the minimum value of the real number \( a \)?


Triangle \(ABC\) has side lengths \(AB=231\), \(BC=160\), and \(AC=281\). Point \(D\) is constructed on the opposite side of line \(AC\) as point \(B\) such that \(AD=178\) and \(CD=153\). Compute the distance from \(B\) to the midpoint of segment \(AD\).


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


In the rectangular prism \(A B C D - A_{1} B_{1} C_{1} D_{1}\), \(A B = A A_{1} = 2\), \(A D = 2 \sqrt{3}\). Point \(M\) lies within plane \(B A_{1} C_{1}\). Find the minimum value of \(\overrightarrow{M A} \cdot \overrightarrow{M C}\).


We consider 2021 lines in the plane, no two of which are parallel and no three of which are concurrent. Let E be the set of their intersection points. We want to assign a color to each point in E such that any two points on the same line, whose connecting segment contains no other point of E, have different colors. What is the minimum number of colors needed to achieve such a coloring?


Find the angle $\Theta$ between the gradients of the functions

$$
u=\sqrt{x^{2}+y^{2}} \text{ and } v=x+y+2\sqrt{xy}
$$

at the point $M_{0}(1, I)$.


At the points of intersection of the graph of the function \( y = \frac{20x^{2} - 16x + 1}{5x - 2} \) with the \( O_x \)-axis, tangents to this graph were drawn. Find the angles of inclination of these lines to the \( O_x \)-axis.


In triangle \( \triangle ABC \), the angle bisectors of \( \angle C \) and \( \angle A \) intersect at point \( O \). Given that \( AC + AO = BC \) and \( \angle B = 25^\circ \), find the measure of \( \angle ACB \).


Two circles touch externally. A tangent to the first circle passes through the center of the second circle. The distance from the point of tangency to the center of the second circle is three times the radius of the second circle. By what factor is the circumference of the first circle greater than the circumference of the second circle?


Find the smallest natural number \( n \) such that the sum of the digits of each of the numbers \( n \) and \( n+1 \) is divisible by 17.


Viewers rate a movie with an integer score from 0 to 10. At any given moment, the rating of the movie is computed as the sum of all scores divided by their quantity. At a certain moment \( T \), the rating was an integer, and then with each subsequent voter, the rating decreased by one unit. What is the maximum number of viewers who could have voted after moment \( T \)?


Let the sets \( M = \{u \mid u = 12m + 8n + 4l, \; m, n, l \in \mathbf{Z}\} \) and \( N = \{u \mid u = 20p + 16q + 12r, \; p, q, r \in \mathbf{Z}\} \). Determine the relationship between \( M \) and \( N \).

Solution: \boxed{M = N}


There is a magical tree with 123 fruits. On the first day, 1 fruit falls from the tree. From the second day onwards, the number of fruits falling each day increases by 1 compared to the previous day. However, if the number of fruits on the tree is less than the number of fruits that should fall on a given day, the falling process restarts from 1 fruit on that day and a new cycle begins. Following this pattern, on which day will all the fruits have fallen from the tree?

Solution: \boxed{17 \text{ days}}


An acute angle of $60^{\circ}$ contains two circles that are externally tangent to each other. The radius of the smaller circle is $r$. Find the radius of the larger circle.


In a big box, there is a medium box, and inside the medium box, there is a small box. 100 balls are placed in the boxes. Let $n$ be the number of balls in the big box but not in the medium box, and $m$ be the number of balls in the medium box but not in the small box. Using $m$ and $n$, determine the number of balls in the small box.

Solution: \boxed{100 - n - m}


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

\[
\left\{
\begin{array}{l}
x^{2}+x y+y^{2}=147 \\
y^{2}+y z+z^{2}=16 \\
z^{2}+x z+x^{2}=163
\end{array}\right.
\]

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


Find the value of the parameter \( p \) for which the equation \( p x^{2} = |x-1| \) has exactly three solutions.


Let $\alpha$ and $\beta$ be a pair of complex conjugates. If $|\alpha - \beta| = 2 \sqrt{3}$ and $\frac{\alpha}{\beta^{2}}$ is a real number, then $|\alpha| = \qquad$.


In an isosceles triangle \(ABC\) with base \(AC\), point \(D\) divides side \(BC\) in the ratio \(3:1\) starting from vertex \(B\), and point \(E\) is the midpoint of segment \(AD\). It is known that \(BE = \sqrt{7}\) and \(CE = 3\). Find the radius of the circumcircle of triangle \(ABC\).


Determine the remainder when
\[
\sum_{i=0}^{2015}\left\lfloor\frac{2^{i}}{25}\right\rfloor
\]
is divided by 100, where \(\lfloor x\rfloor\) denotes the largest integer not greater than \(x\).


Let \( a, b \), and \( c \) be real numbers. Consider the system of simultaneous equations in variables \( x \) and \( y \):
\[
\begin{aligned}
a x + b y &= c - 1 \\
(a+5) x + (b+3) y &= c + 1
\end{aligned}
\]

Determine the value(s) of \( c \) in terms of \( a \) such that the system always has a solution for any \( a \) and \( b \).

Solution: \boxed{\frac{2a + 5}{5}}


There are three piles of stones. Each time A moves one stone from one pile to another, A earns a reward from B. The reward is equal to the difference in the number of stones between the pile to which A moves the stone and the pile from which A takes the stone. If this difference is negative, A must pay B that amount instead (if A doesn’t have enough money, A can owe it temporarily). At a certain moment, all the stones are in the piles where they initially started. Determine the maximum possible amount of money A can have by that moment.


What is the minimum force required to press a cube with a volume of $10 \mathrm{~cm}^{3}$, floating in water, so that it is completely submerged? The density of the cube's material is $700 \mathrm{~kg/m}^{3}$, and the density of water is $1000 \mathrm{~kg/m}^{3}$. Give the answer in SI units. Take the acceleration due to gravity as $10 \mathrm{~m/s}^{2}$.

Solution: \boxed{0.03 \, \text{N}}


There are 10 numbers written on a circle, and their sum equals 100. It is known that the sum of any three consecutive numbers is at least 29.

What is the smallest number \( A \) such that in any such set of numbers, each number does not exceed \( A \)?


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


Given that \( f(x) \) is an odd function on \( \mathbf{R} \) and \( f(1) = 1 \), with the condition that for any \( x < 0 \), \( f\left(\frac{x}{x-1}\right) = x f(x) \). Find the value of \( f(1) f\left(\frac{1}{100}\right) + f\left(\frac{1}{2}\right) f\left(\frac{1}{99}\right) + f\left(\frac{1}{3}\right) f\left(\frac{1}{98}\right) + \cdots + f\left(\frac{1}{50}\right) f\left(\frac{1}{51}\right) \).


Given the linear function \( y = ax + b \) and the hyperbolic function \( y = \frac{k}{x} \) (where \( k > 0 \)) intersect at points \( A \) and \( B \), with \( O \) being the origin. If the triangle \( \triangle OAB \) is an equilateral triangle with an area of \( \frac{2\sqrt{3}}{3} \), find the value of \( k \).


The dwarfs went to work, and Snow White is bored. She placed a pile of 36 stones on the table. Every minute, Snow White splits one of the existing piles into two and then adds a new stone to one of them. After a while, she has seven piles, each with an equal number of stones. How many stones are there in each pile?


In Vegetable Village it costs 75 pence to buy 2 potatoes, 3 carrots, and 2 leeks at the Outdoor Market, whereas it costs 88 pence to buy 2 potatoes, 3 carrots, and 2 leeks at the Indoor Market. To buy a potato, a carrot, and a leek costs 6 pence more at the Indoor Market than it does at the Outdoor Market. What is the difference, in pence, between the cost of buying a carrot at the Outdoor Market and the cost of buying a carrot at the Indoor Market?


A laboratory has flasks of two different sizes (volume $V$ and volume $V/3$) with a total of 100 flasks, with at least 2 flasks of each size. A technician randomly selects two flasks in sequence, filling the first one with a 70% salt solution and the second with a 40% salt solution. The contents of these two flasks are then mixed in one dish, and the percentage of salt in the mixture is determined. What is the minimum number of large flasks $N$ needed to ensure that the event "the percentage of salt in the dish is between 50% and 60% inclusive" occurs less frequently than the event "when two fair coins are flipped, one shows heads and the other shows tails (in any order)"? Justify your answer.


Given a prime number \( p \) and a positive integer \( n \) (\( p \geq n \geq 3 \)), the set \( A \) consists of different sequences of length \( n \) formed from the elements of the set \(\{1, 2, \cdots, p\}\) (such that not all elements in a sequence are the same). If, for any two sequences \((x_{1}, x_{2}, \cdots, x_{n})\) and \((y_{1}, y_{2}, \cdots, y_{n})\) in the set \( A \), there exist three different positive integers \( k, l, m \) such that \( x_{k} \neq y_{k}, x_{l} \neq y_{l}, x_{m} \neq y_{m} \), find the maximum number of elements in the set \( A \).


Gustave has 15 steel bars of masses \(1 \mathrm{~kg}, 2 \mathrm{~kg}, 3 \mathrm{~kg}, \ldots, 14 \mathrm{~kg}, 15 \mathrm{~kg}\). He also has 3 bags labelled \(A, B, C\). He places two steel bars in each bag so that the total mass in each bag is equal to \(M \mathrm{~kg}\). How many different values of \(M\) are possible?


Let $s$ be the set of all rational numbers $r$ that satisfy the following conditions: $0<r<1$, and $r$ can be represented as a repeating decimal of the form $0.abcabcabc\cdots = 0.\dot{a}b\dot{c}$, where the digits $a, b, c$ do not necessarily have to be distinct. How many different numerators are there when the elements of $s$ are written as reduced fractions?


Triangle \(ABC\) has a right angle at \(B\), with \(AB = 3\) and \(BC = 4\). If \(D\) and \(E\) are points on \(AC\) and \(BC\), respectively, such that \(CD = DE = \frac{5}{3}\), find the perimeter of quadrilateral \(ABED\).


Olya drew $N$ different lines on a plane, any two of which intersect. It turned out that among any 15 lines, there are always two that form an angle of $60^{\circ}$ between them. What is the largest possible value of $N$ for which this is possible?


Shan solves the simultaneous equations
$$
xy = 15 \text { and } (2x - y)^4 = 1
$$
where \(x\) and \(y\) are real numbers. She calculates \(z\), the sum of the squares of all the \(y\)-values in her solutions. What is the value of \(z\)?


On the board, the number \( x = 9999 \) is written in a numeral system with an even base \( r \). Vasya found out that the \( r \)-ary representation of \( x^2 \) is an eight-digit palindrome, in which the sum of the second and third digits is 24. (A palindrome is a number that reads the same from left to right and right to left). For which \( r \) is this possible?

Solution: \boxed{r = 26}


The witch Gingema cast a spell on a wall clock so that the minute hand moves in the correct direction for five minutes, then three minutes in the opposite direction, then five minutes in the correct direction again, and so on. How many minutes will the hand show after 2022 minutes, given that it pointed exactly to 12 o'clock at the beginning of the five-minute interval of correct movement?


Given a three-digit number \( N \), the remainders when \( N \) is divided by 3, 7, and 11 are 1, 3, and 8, respectively. Find the maximum value of \( N \).


In triangle \(ABC\), we have \(AB = BC = 5\) and \(CA = 8\). What is the area of the region consisting of all points inside the triangle which are closer to \(AB\) than to \(AC\)?


Find the maximum value of the expression \( ab + bc + ac + abc \) given that \( a + b + c = 12 \) (where \( a, b, \) and \( c \) are non-negative numbers).


Given two linear functions $f(x)$ and $g(x)$ such that the graphs of $y=f(x)$ and $y=g(x)$ are parallel lines, not parallel to the coordinate axes. Find the minimum value of the function $(g(x))^{2}+f(x)$, if the minimum value of the function $(f(x))^{2}+g(x)$ is -6.


Let the line \( L: lx + my + n = 0 \) intersect the parabola \( y^2 = 4ax \) at points \( P \) and \( Q \). \( F \) is the focus of the parabola. The line segments \( PF \) and \( QF \) intersect the parabola at points \( R \) and \( S \). Determine the equation of the line \( RS \).

Solution: \boxed{nx - may + la^2 = 0}


Altitudes \( BE \) and \( CF \) of acute triangle \( ABC \) intersect at \( H \). Suppose that the altitudes of triangle \( EHF \) concur on line \( BC \). If \( AB=3 \) and \( AC=4 \), then \( BC^2=\frac{a}{b} \), where \(a\) and \(b\) are relatively prime positive integers. Compute \(100a+b\).


Determine all positive integers \( n \) such that:

\[ 5^{n-1} + 3^{n-1} \mid 5^n + 3^n \]


Find the probability that a randomly chosen number on the interval $[0 ; 5]$ is a solution to the equation $\sin (x+|x-\pi|)+2 \sin ^{2}(x-|x|)=0$.


Let the set \( A = \{1, 2, 3, 4, 5, 6\} \) and the mapping \( f: A \rightarrow A \). If the triple composition \( f \cdot f \cdot f \) is an identity mapping, how many such functions \( f \) are there?


A positive integer \( n \) is said to be good if \( 3n \) is a re-ordering of the digits of \( n \) when they are expressed in decimal notation. Find a four-digit good integer which is divisible by 11.


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 that \( A \), \( B \), and \( C \) are any three non-collinear points on a plane, and point \( O \) is inside \( \triangle ABC \) such that:
\[
\angle AOB = \angle BOC = \angle COA = 120^\circ.
\]
Find the maximum value of \( \frac{OA + OB + OC}{AB + BC + CA} \).


Given that the sequence $\left\{a_{n}\right\}$ is an arithmetic sequence with the first term and common difference both being positive, and that $a_{2}$, $a_{5}$, and $a_{9}$ form a geometric sequence in order, find the smallest positive integer value of $k$ such that $a_{1}+a_{2}+\cdots+a_{k}>100 a_{1}$.


Several non-intersecting line segments are placed on a plane. Is it always possible to connect their endpoints with additional line segments so that a closed, non-self-intersecting polygonal chain is formed?

Solution: \boxed{\text{No}}


A production team in a factory is manufacturing a batch of parts. Initially, when each worker is on their own original position, the task can be completed in 9 hours. If the positions of workers $A$ and $B$ are swapped, and other workers' efficiency remains the same, the task can be completed one hour earlier. Similarly, if the positions of workers $C$ and $D$ are swapped, the task can also be completed one hour earlier. How many minutes earlier can the task be completed if the positions of $A$ and $B$ as well as $C$ and $D$ are swapped at the same time, assuming other workers' efficiency remains unchanged?

Solution: \boxed{108 \text{ minutes}}


Given \( x \in [0, 3] \), find the maximum value of \( \frac{\sqrt{2 x^3 + 7 x^2 + 6 x}}{x^2 + 4 x + 3} \).


In triangle \(ABC\), point \(K\) is taken on side \(AB\) such that \(AK: BK = 1: 2\), and point \(L\) is taken on side \(BC\) such that \(CL: BL = 2: 1\). Let \(Q\) be the point of intersection of lines \(AL\) and \(CK\). Find the area of triangle \(ABC\) if the area of triangle \(BQC\) is 1.


Compute the definite integral:

$$
\int_{0}^{5} x^{2} \cdot \sqrt{25-x^{2}} \, dx
$$

Solution: \boxed{\frac{625 \pi}{16}}


In a sequence of 37 numbers, the sum of every six consecutive numbers is 29. The first number is 5. What could the last number be?


A cylinder with a base diameter of 12 is cut by a plane that forms a $30^{\circ}$ angle with the base, forming an ellipse $S$. What is the eccentricity of the ellipse $S$?


Find all positive integer solutions \((x, y, z, n)\) that satisfy the equation \(x^{2n+1} - y^{2n+1} = xyz + 2^{2n+1}\), where \(n \geq 2\) and \(z \leq 5 \cdot 2^{2n}\).

Solution: \boxed{(3, 1, 70, 2)}


Robot Petya displays three three-digit numbers every minute, which sum up to 2019. Robot Vasya swaps the first and last digits of each of these numbers and then sums the resulting numbers. What is the maximum sum that Vasya can obtain?


In a sequence, all natural numbers from 1 to 2017 inclusive were written down. How many times was the digit 7 written?


a) What is the maximum number of squares on an $8 \times 8$ board that can be colored black so that in every "corner" of three squares, at least one square remains uncolored?

b) What is the minimum number of squares on an $8 \times 8$ board that must be colored black so that in every "corner" of three squares, at least one square is black?


Rumcajs teaches Cipísek to write numbers. They started from one and wrote consecutive natural numbers. Cipísek pleaded to stop, and Rumcajs promised that they would stop writing when Cipísek had written a total of 35 zeros. What is the last number Cipísek writes?


There are 16 distinct points on a circle. Determine the number of different ways to draw 8 nonintersecting line segments connecting pairs of points so that each of the 16 points is connected to exactly one other point.


The set of positive odd numbers $\{1, 3, 5, \cdots\}$ is arranged in ascending order and grouped by the $n$th group having $(2n-1)$ odd numbers as follows:
$$
\begin{array}{l}
\{1\}, \quad \{3,5,7\}, \quad \{9,11,13,15,17\}, \cdots \\
\text{ (First group) (Second group) (Third group) } \\
\end{array}
$$

In which group does the number 1991 appear?


All lateral edges of the pyramid are equal to $b$, and the height is $h$. Find the radius of the circle circumscribed around the base.

Solution: \boxed{\sqrt{b^2 - h^2}}


Six natural numbers (with possible repetitions) are written on the faces of a cube, such that the numbers on adjacent faces differ by more than 1. What is the smallest possible sum of these six numbers?


If there is a positive integer \( m \) such that the factorial of \( m \) has exactly \( n \) trailing zeros, then the positive integer \( n \) is called a "factorial trailing number." How many positive integers less than 1992 are non-"factorial trailing numbers"?


Write the equation of a plane that passes through point \( A \) and is perpendicular to vector \( \overrightarrow{BC} \).

\( A(-3, 6, 4) \)

\( B(8, -3, 5) \)

\( C(10, -3, 7) \)

Solution: \boxed{x + z - 1 = 0}


As shown in the figure, square \(ABCD\) and rectangle \(BEFG\) share a common vertex \(B\). The side length of square \(ABCD\) is \(6 \, \text{cm}\) and the length of rectangle \(BEFG\) is \(9 \, \text{cm}\). Find the width of the rectangle in centimeters.

Solution: \boxed{4 \text{ cm}}


Twenty-five coins are divided into piles in the following way. First, they are randomly split into two groups. Then any of the existing groups is split into two groups again, and this process continues until each group consists of just one coin. Each time a group is split into two groups, the product of the number of coins in the two resulting groups is recorded. What could the sum of all the recorded numbers be?


Points $A_{1}$ and $C_{1}$ are located on the sides $BC$ and $AB$ of triangle $ABC$. Segments $AA_{1}$ and $CC_{1}$ intersect at point $M$.

In what ratio does line $BM$ divide side $AC$, if $AC_{1}: C_{1}B = 2: 3$ and $BA_{1}: A_{1}C = 1: 2$?


A three-digit natural number was written on the board. We wrote down all other three-digit numbers that can be obtained by rearranging its digits. Therefore, there were three new numbers in addition to the original number on the board. The sum of the smallest two of all four numbers is 1088. What digits does the original number contain?

Solution: \boxed{0, 5, 8}


Solution: \boxed{2017! + 2018}


A point is randomly thrown on the segment [3, 8] and let \( k \) be the resulting value. Find the probability that the roots of the equation \((k^{2}-2k-3)x^{2}+(3k-5)x+2=0\) satisfy the condition \( x_{1} \leq 2x_{2} \).


Can the square of any natural number begin with 1983 nines?

Solution: \boxed{Yes}


The center of circle $k$ is $O$. Points $A, B, C, D$ are consecutive points on the circumference of $k$ such that $\angle AOB = \angle BOC = \angle COD = \alpha < 60^\circ$. Point $E$ is the projection of $D$ onto the diameter $AO$. Point $F$ is the trisection point of segment $DE$ closer to $E$. Finally, $G$ is the intersection point of $AO$ and $BF$. What is the measure of angle $OGD$ as $\alpha \rightarrow 60^\circ$?


Find the limit of the numerical sequence:

$$
\lim _{n \rightarrow \infty}\left(\frac{2 n^{2}+2 n+3}{2 n^{2}+2 n+1}\right)^{3 n^{2}-7}
$$


At a joint conference of the Parties of Liars and Truth-lovers, 32 people were elected to the presidium and seated in four rows of 8 people each. During the break, each member of the presidium claimed that among their neighbors there are representatives of both parties. It is known that liars always lie, and truth-lovers always tell the truth. What is the minimum number of liars in the presidium for the described situation to be possible? (Two members of the presidium are neighbors if one of them is seated to the left, right, in front, or behind the other).


In the triangular prism \(P-ABC\), \(\triangle ABC\) is an equilateral triangle with side length \(2\sqrt{3}\), \(PB = PC = \sqrt{5}\), and the dihedral angle \(P-BC-A\) is \(45^\circ\). Find the surface area of the circumscribed sphere around the triangular prism \(P-ABC\).


Several points were marked on a line. After that, a point was added between each pair of neighboring points. This operation was repeated three times, and as a result, there were 65 points on the line. How many points were there initially?


Point \( O \) is the center of the circle circumscribed around triangle \( ABC \) with sides \( BC = 5 \) and \( AB = 4 \). Find the length of side \( AC \) if the length of the vector \( 3 \overrightarrow{OA} - 4 \overrightarrow{OB} + \overrightarrow{OC} \) is 10.


A group of 101 Dalmatians participate in an election, where they each vote independently for either candidate \( A \) or \( B \) with equal probability. If \( X \) Dalmatians voted for the winning candidate, the expected value of \( X^{2} \) can be expressed as \( \frac{a}{b} \) for positive integers \( a, b \) with \( \operatorname{gcd}(a, b)=1 \). Find the unique positive integer \( k \leq 103 \) such that \( 103 \mid a - b k \).


Consider a fixed circle $\mathscr{C}$ passing through a point $A$ and a fixed line $\mathscr{D}$ passing through a point $B$. A variable circle passing through $A$ and $B$ intersects the circle $\mathscr{C}$ at $C$ and the line $\mathscr{D}$ at $E$. The line $CE$ intersects $\mathscr{C}$ at $K$. Show that the point $K$ is fixed.

Solution: \boxed{K \text{ is fixed}}


How many even six-digit numbers exist where the same digits are not next to each other?


Let \(\pi\) be a permutation of the numbers from 2 through 2012. Find the largest possible value of \(\log_{2} \pi(2) \cdot \log_{3} \pi(3) \cdots \log_{2012} \pi(2012)\).


A five-digit number \(abcde\) satisfies:
\[ a < b, \, b > c > d, \, d < e, \, \text{and} \, a > d, \, b > e. \]

For example, 34 201, 49 412. If the digit order's pattern follows a variation similar to the monotonicity of a sine function over one period, then the five-digit number is said to follow the "sine rule." Find the total number of five-digit numbers that follow the sine rule.

Note: Please disregard any references or examples provided within the original problem if they are not part of the actual problem statement.


For an integer \( n>3 \), we use \( n ? \) to represent the product of all prime numbers less than \( n \) (called " \( n \)-question mark"). Solve the equation \( n ? = 2n + 16 \).

Solution: \boxed{n = 7}


Given a prime number \( p \) and a positive integer \( n \) where \( p \geq n \geq 3 \). The set \( A \) consists of sequences of length \( n \) with elements taken from the set \(\{1, 2, \cdots, p\}\). If for any two sequences \((x_{1}, x_{2}, \cdots, x_{n})\) and \((y_{1}, y_{2}, \cdots, y_{n})\) in set \( A \), there exist three distinct positive integers \( k, l, m \) such that \( x_{k} \neq y_{k}, x_{l} \neq y_{l}, x_{m} \neq y_{m} \), find the maximum number of elements in set \( A \).


We rotate the square \(ABCD\) with a side length of 1 around its vertex \(C\) by \(90^\circ\). What area does side \(AB\) sweep out?


Cat food is sold in large and small packages (with more food in the large package than in the small one). One large package and four small packages are enough to feed a cat for exactly two weeks. Is one large package and three small packages necessarily enough to feed the cat for 11 days?

Solution: \boxed{Yes}


Show that if the element \(a_{ik}\) of an \(n\)-th order determinant is equal to the smaller of the numbers \(i\) and \(k\) (or the common value of \(i\) and \(k\) when \(i = k\)), then the value of the determinant is 1.


Find all solutions in positive integers to the following system of equations:
\[ a + b = \gcd(a, b)^2 \]
\[ b + c = \gcd(b, c)^2 \]
\[ c + a = \gcd(c, a)^2 \]

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


1. Two people, A and B, are playing a game: A writes two rows, each with 10 numbers, arranged such that they satisfy the following rule: If $b$ is below $a$ and $d$ is below $c$, then $a+d=b+c$. After knowing this rule, B wants to determine all the written numbers. B can ask A questions like "What is the number at the third position in the first row?" or "What is the number at the ninth position in the second row?" How many such questions does B need to ask to determine all the numbers?
2. In an $m \times n$ grid filled with numbers, any rectangle formed by selecting two rows and two columns has the property that the sum of the numbers at two opposite corners equals the sum of the numbers at the other two opposite corners. Show that if some numbers are erased and the remaining numbers allow reconstruction of the erased numbers, at least $m+n-1$ numbers must remain.

Solution: \boxed{m + n - 1}