In a family of 4 people, if Masha's scholarship is doubled, the total income of the family will increase by 5%. If instead, the mother's salary is doubled, the total income will increase by 15%. If the father's salary is doubled, the total income will increase by 25%. By what percentage will the total income of the family increase if the grandfather's pension is doubled?


Petya has seven cards with the digits 2, 2, 3, 4, 5, 6, 8. He wants to use all the cards to form the largest natural number that is divisible by 12. What number should he get?


Pyramid \( E A R L Y \) is placed in \((x, y, z)\) coordinates so that \( E=(10,10,0) \), \( A=(10,-10,0) \), \( R=(-10,-10,0) \), \( L=(-10,10,0) \), and \( Y=(0,0,10) \). Tunnels are drilled through the pyramid in such a way that one can move from \((x, y, z)\) to any of the 9 points \((x, y, z-1)\), \((x \pm 1, y, z-1)\), \((x, y \pm 1, z-1)\), \((x \pm 1, y \pm 1, z-1)\). Sean starts at \( Y \) and moves randomly down to the base of the pyramid, choosing each of the possible paths with probability \(\frac{1}{9}\) each time. What is the probability that he ends up at the point \((8,9,0)\)?


A Christmas garland hanging along the school corridor consists of red and blue bulbs. Next to each red bulb, there is necessarily at least one blue bulb. What is the maximum number of red bulbs that can be in this garland if there are a total of 50 bulbs?


For which values of the parameter \(a\) does the equation \(\frac{\log_{a} x}{\log_{a} 2} + \frac{\log_{x} (2a - x)}{\log_{x} 2} = \frac{1}{\log_{a^2 - 1} 2}\) have (1) a solution? (2) exactly one solution?


Find all real numbers \(a\) such that \(a+\frac{2}{3}\) and \(\frac{1}{a}-\frac{3}{4}\) are integers.


The product of three prime numbers. There is a number that is the product of three prime factors whose sum of squares is equal to 2331. There are 7560 numbers (including 1) less than this number and coprime with it. The sum of all the divisors of this number (including 1 and the number itself) is 10560. Find this number.


Let \( P \) be any interior point of an equilateral triangle \( ABC \). Let \( D, E, \) and \( F \) be the feet of the perpendiculars dropped from \( P \) to the sides \( BC, CA, \) and \( AB \) respectively. Determine the value of the ratio

$$
\frac{PD + PE + PF}{BD + CE + AF}.
$$


How many natural numbers not exceeding 500 are not divisible by 2, 3, or 5?


There are \( n \) vectors in space such that any pair of them forms an obtuse angle. What is the maximum possible value of \( n \)?


If Alex does not sing on Saturday, then she has a 70% chance of singing on Sunday; however, to rest her voice, she never sings on both days. If Alex has a 50% chance of singing on Sunday, find the probability that she sings on Saturday.


Place all terms of the arithmetic sequence \(2, 6, 10, 14, \cdots, 2006\) closely together to form a large number \(A = 261014 \cdots 2006\). Find the remainder when \(A\) is divided by 9.


Given the regular hexagon \( A_{1} A_{2} A_{3} A_{4} A_{5} A_{6} \). Show that the diagonals \( A_{1} A_{3}, A_{3} A_{5}, A_{5} A_{1} \) and \( A_{2} A_{4}, A_{4} A_{6}, A_{6} A_{2} \) define a regular hexagon. How does the area of this smaller hexagon compare to the area of the given hexagon?


Fill the numbers from 1 to 9 in the spaces of the given diagram (with 2 and 3 already filled), so that the number on the left is greater than the number on the right, and the number above is greater than the number below. How many different ways are there to fill the diagram?


What is the limit of the following sequence?

$$
a_{n}=\frac{\sum_{j=0}^{\infty}\binom{n}{2 j} \cdot 2^{j}}{\sum_{j=0}^{\infty}\binom{n}{2 j+1} \cdot 2^{j}}
$$


A student did not notice the multiplication sign between two three-digit numbers and wrote a six-digit number, which turned out to be seven times greater than their product. Find these numbers.


A farmer bought two sheep in 1996. That year, he didn't have any offspring. The first sheep gave birth to one lamb every three years, and the second sheep gave birth to one lamb every two years. All the sheep born each year also gave birth to one lamb annually. How many sheep will the farmer have in the year 2000?


Sean enters a classroom in the Memorial Hall and sees a 1 followed by 2020 0's on the blackboard. As he is early for class, he decides to go through the digits from right to left and independently erase the \(n\)th digit from the left with probability \(\frac{n-1}{n}\). (In particular, the 1 is never erased.) Compute the expected value of the number formed from the remaining digits when viewed as a base-3 number. (For example, if the remaining number on the board is 1000, then its value is 27.)


First-grader Petya was arranging his available chips into the outline of an equilateral triangle such that each side, including the vertices, contains the same number of chips. Then, using the same chips, he managed to arrange them into the outline of a square in the same manner. How many chips does Petya have if the side of the square contains 2 fewer chips than the side of the triangle?


Given the set $P$ consisting of positive integers not exceeding 2012, explicitly $P = \{1,2,3, \cdots, 2012\}$. The set $A$ is a subset of $P$, where $|A|$ denotes the number of elements in set $A$, and $S(A)$ denotes the sum of all elements in set $A$.

1. If the difference between any two numbers in set $A$ is not a multiple of 101, find the maximum value of $|A|$.
2. If the difference between any two numbers in set $A$ is not a multiple of 101, and the sum of any two numbers is also not a multiple of 101, find the maximum value of $|A|$.
3. If the difference between any two numbers in set $A$ is not a multiple of 101, the sum of any two numbers is also not a multiple of 101, and $S(A) = 2012$, find the maximum value of $|A|$.


Suppose \( n \) is a natural number. For any real numbers \( x, y, z \), the inequality \(\left(x^{2} + y^{2} + z^{2}\right) \leqslant n \left(x^{4} + y^{4} + z^{4}\right)\) always holds. Then the minimum value of \( n \) is \(\qquad\)


One day, Papa Smurf conducted an assessment for 45 Smurfs in the Smurf Village. After the assessment, he found that the average score of the top 25 Smurfs was 93 points, and the average score of the bottom 25 Smurfs was 89 points. By how many points does the total score of the top 20 Smurfs exceed that of the bottom 20 Smurfs?


If 14! is divisible by \(6^k\), where \(k\) is an integer, find the largest possible value of \(k\).


In a movie theater, five friends took seats numbered 1 through 5 (where seat number 1 is the leftmost). During the movie, Anya went to get popcorn. When she returned, she found that Varya had moved three seats to the right, Galya had moved one seat to the left, and Diana and Ella had switched seats, leaving the end seat for Anya. In which seat was Anya sitting before she stood up?


In the figure below, $ABCD$ is a trapezoid and its diagonals $AC$ and $BD$ are perpendicular. Additionally, $BC=10$ and $AD=30$.

a) Determine the ratio between the segments $BE$ and $ED$.

b) Find the length of the segments $EC$, $AE$, and $ED$ in terms of the length of $BE=x$.

c) If $AE \cdot EC=108$, determine the value of $BE \cdot ED$.


A rectangle with a perimeter of 100 cm was divided into 70 identical smaller rectangles by six vertical cuts and nine horizontal cuts. What is the perimeter of each smaller rectangle if the total length of all cuts equals 405 cm?

Solution: \boxed{13 \text{ cm}}


If a die is rolled, event \( A = \{1, 2, 3\} \) consists of rolling one of the faces 1, 2, or 3. Similarly, event \( B = \{1, 2, 4\} \) consists of rolling one of the faces 1, 2, or 4.

The die is rolled 10 times. It is known that event \( A \) occurred exactly 6 times.

a) Find the probability that under this condition, event \( B \) did not occur at all.

b) Find the expected value of the random variable \( X \), which represents the number of occurrences of event \( B \).


Given \( n \in \mathbf{N}, n > 4 \), and the set \( A = \{1, 2, \cdots, n\} \). Suppose there exists a positive integer \( m \) and sets \( A_1, A_2, \cdots, A_m \) with the following properties:
1. \( \bigcup_{i=1}^{m} A_i = A \);
2. \( |A_i| = 4 \) for \( i=1, 2, \cdots, m \);
3. Let \( X_1, X_2, \cdots, X_{\mathrm{C}_n^2} \) be all the 2-element subsets of \( A \). For every \( X_k \) \((k=1, 2, \cdots, \mathrm{C}_n^2)\), there exists a unique \( j_k \in\{1, 2, \cdots, m\} \) such that \( X_k \subseteq A_{j_k} \).

Find the smallest value of \( n \).


A point is chosen randomly with uniform distribution in the interior of a circle with a radius of 1. What is its expected distance from the center of the circle?


There is a deck of 1024 cards, each of which has a set of different digits from 0 to 9 written on them, and all sets are different (`in particular, there is also an empty card`). We call a set of cards complete if each digit from 0 to 9 appears exactly once on them.

Find all natural numbers \( k \) for which there exists a set of \( k \) cards with the following condition: among them, it is impossible to choose a complete set, but adding any card from the deck would violate this condition.


Let the set \( S = \{1, 2, \cdots, 280\} \). Find the smallest positive integer \( n \) such that any \( n \)-element subset \( T \) of \( S \) contains 5 numbers that are pairwise coprime.


Every day, Ivan Ivanovich is driven to work by a company car. One day, Ivan Ivanovich decided to walk and left his house an hour earlier than usual. On the way, he met the company car, and he completed the remaining distance in the car. As a result, he arrived to work 10 minutes earlier than usual. How long did Ivan Ivanovich walk?

Solution: \boxed{55 \text{ minutes}}


An electronic watch shows 6:20:25 at 6 hours, 20 minutes, and 25 seconds. Within the one-hour period from 5:00:00 to 5:59:59, how many instances are there where all five digits of the time displayed are different?


There are two cylinders with a volume ratio of 5:8. Their lateral surfaces can be unfolded into identical rectangles. If the length and width of this rectangle are both increased by 6, the area increases by 114. What is the area of this rectangle?


Find the smallest natural number that can be represented in the form $13x + 73y$ in three different ways, where $x$ and $y$ are natural numbers.


A polynomial of degree 3n has the value 2 at 0, 3, 6, ... , 3n, the value 1 at 1, 4, 7, ... , 3n-2, and the value 0 at 2, 5, 8, ..., 3n-1. Its value at 3n+1 is 730. What is n?


Given an equilateral triangle \( \triangle ABC \) with each side having a length of 2. If \(\overrightarrow{AP} = \frac{1}{3}(\overrightarrow{AB} + \overrightarrow{AC})\) and \(\overrightarrow{AQ} = \overrightarrow{AP} + \frac{1}{2} \overrightarrow{BC}\), find the area of \(\triangle APQ\).


In triangle \(ABC\), the three sides are given as \(AB = 26\), \(BC = 30\), and \(AC = 28\). Find the part of the area of this triangle that is enclosed between the altitude and the angle bisector drawn from vertex \(B\).


Zeroes are written in all cells of a $5 \times 5$ board. We can take an arbitrary cell and increase by 1 the number in this cell and all cells having a common side with it. Is it possible to obtain the number 2012 in all cells simultaneously?

Solution: \boxed{\text{No}}


Find the smallest natural number $n$ such that $\sin n^{\circ} = \sin (2016n^{\circ})$.


( Fixed points and limits) Find all functions from $\mathbb{R}_{+}^{*}$ to $\mathbb{R}_{+}^{*}$ such that for all $x, y > 0$, \( f(x f(y)) = y f(x) \) and \( \lim _{x \rightarrow+\infty} f(x) = 0 \).

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


Is it possible to tile a $23 \times 23$ chessboard with $2 \times 2$ and $3 \times 3$ squares? With a $1 \times 1$ square and $2 \times 2$ and $3 \times 3$ squares?

Solution: \boxed{\text{Yes}}


Find all positive integers \( n \) less than 1000 such that the cube of the sum of the digits of \( n \) equals \( n^2 \).

Solution: \boxed{1 \text{ and } 27}


A semiprime is a number that is a product of two prime numbers. How many semiprime numbers less than 2018 can be expressed as \( x^{3}-1 \) for some natural number \( x \)?


The circles OAB, OBC, and OCA have equal radius \( r \). Show that the circle ABC also has radius \( r \).

Solution: \boxed{r}


The polynomial \( x^n + n x^{n-1} + a_2 x^{n-2} + \cdots + a_0 \) has \( n \) roots whose 16th powers have sum \( n \). Find the roots.


There are knights who always tell the truth and liars who always lie living on an island. One day, 35 islanders sat down at 7 tables, with 5 people at each table. Each of these 35 islanders was asked: "Are there more than three tables with at least 3 knights sitting at them?"

(a) What is the maximum number of islanders who could have answered "Yes"?

(b) What is the maximum number of islanders who could have answered "No"?


Given that the graph of the function \( y = f(x) \) is symmetric with respect to the point \( (1,1) \) and the line \( x + y = 0 \), if \( f(x) = \log_{2}(x + 1) \) when \( x \in (0,1) \), find the value of \( f\left(\log_{2} 10\right) \).


Find \(\sin^4 \alpha + \cos^4 \alpha\), given that \(\sin \alpha - \cos \alpha = \frac{1}{2}\).


Determine the number of pairs of integers, \((a, b)\), with \(1 \leq a \leq 100\) so that the line with equation \(b=ax-4y\) passes through point \((r, 0)\), where \(r\) is a real number with \(0 \leq r \leq 3\), and passes through point \((s, 4)\), where \(s\) is a real number with \(2 \leq s \leq 4\).


16. Variance of the number of matches. A deck of playing cards is laid out on a table (for example, in a row). On top of each card, a card from another deck is placed. Some cards may match. Find:

a) the expected number of matches;

b) the variance of the number of matches.


Let \( a, b, c \) be positive numbers such that \( a + b + c = \lambda \). If the inequality
\[ \frac{1}{a(1 + \lambda b)} + \frac{1}{b(1 + \lambda c)} + \frac{1}{c(1 + \lambda a)} \geq \frac{27}{4} \]
always holds, find the range of values for \( \lambda \).

Solution: \boxed{(0, 1]}


Count how many 8-digit numbers there are that contain exactly four nines as digits.


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}


Quadrilateral \(ABCD\) is inscribed in a circle with center \(O\). Two circles \(\Omega_1\) and \(\Omega_2\) of equal radii with centers \(O_1\) and \(O_2\) are inscribed in angles \(ABC\) and \(ADC\) respectively, with the first circle touching side \(BC\) at point \(K\), and the second circle touching side \(AD\) at point \(T\).

a) Find the radius of circle \(\Omega_1\) if \(BK = 3\sqrt{3}\), \(DT = \sqrt{3}\).

b) Additionally, it is known that point \(O_1\) is the center of the circle circumscribed around triangle \(BOC\). Find the angle \(BDC\).


Find the functions \( f: \mathbb{Q} \rightarrow \mathbb{Q} \) such that \( f(1) = 2 \) and for all \( x, y \in \mathbb{Q} \): \( f(xy) = f(x) f(y) - f(x + y) + 1 \).

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


Show from the previous problem that the square of the hypotenuse is equal to the sum of the squares of the legs.

Solution: \boxed{c^2 = a^2 + b^2}


In the number \(2 * 0 * 1 * 6 * 0 * 2 *\), replace each of the 6 asterisks with any of the digits \(0, 1, 2, 3, 4, 5, 6, 7, 8\) (digits may be repeated) so that the resulting 12-digit number is divisible by 45. In how many ways can this be done?


The numbers \( x_{1}, \ldots, x_{n}, y_{1}, \ldots, y_{n} \) satisfy the condition \( x_{1}^{2}+\ldots+x_{n}^{2}+y_{1}^{2}+\ldots+y_{n}^{2} \leq 2 \). Find the maximum value of the expression

$$
A=\left(2\left(x_{1}+\ldots+x_{n}\right)-y_{1}-\ldots-y_{n}\right) \cdot \left(x_{1}+\ldots+x_{n}+2\left(y_{1}+\ldots+y_{n}\right)\right)
$$


In a right triangle, the bisector of an acute angle divides the opposite leg into segments of lengths 4 cm and 5 cm. Determine the area of the triangle.

Solution: \boxed{54 \text{ cm}^2}


Angelica wants to choose a three-digit code for her suitcase lock. To make it easier to remember, Angelica wants all the digits in her code to be in non-decreasing order. How many different possible codes does Angelica have to choose from?


Solve the following system of equations. It has a solution if and only if each term equals zero:

$$
\left\{\begin{array}{c}
3 x^{2}+8 x-3=0 \\
3 x^{4}+2 x^{3}-10 x^{2}+30 x-9=0
\end{array}\right.
$$


Each cell of a \(50 \times 50\) square contains a number equal to the count of \(1 \times 16\) rectangles (both vertical and horizontal) for which this cell is an endpoint. How many cells contain numbers that are greater than or equal to 3?


Anton, Vasya, Sasha, and Dima were driving from city A to city B, each taking turns at the wheel. The entire journey was made at a constant speed.

Anton drove the car for half the time Vasya did, and Sasha drove for as long as Anton and Dima together. Dima was at the wheel for only one-tenth of the distance. What fraction of the distance did Vasya drive? Provide your answer as a decimal.


Explain a method to multiply two numbers using only addition, division by 2, and multiplication by 2, as illustrated by the following example of multiplying 97 by 23. The method involves the following steps:

1. Write the two numbers at the top of two columns.
2. In the first column, repeatedly divide the number by 2 (discarding the remainder) until you reach 1.
3. In the second column, repeatedly multiply the number by 2 the same number of times.

For example:

| 97 | 23 |
| ---: | :---: |
| 48 | 46 |
| 24 | 92 |
| 12 | 184 |
| 6 | 368 |
| 3 | 736 |
| 1 | 1472 |
| | 2231 |

Next, cross out the entries in the second column corresponding to even numbers in the first column, and sum the remaining numbers:

The numbers not crossed out are 23, 736, and 1472. Adding these together gives the product 2231.

Why does this method work?


Calculate the volume of the tetrahedron with vertices at the points $A_{1}, A_{2}, A_{3}, A_{4}$ and find its height dropped from the vertex $A_{4}$ to the face $A_{1} A_{2} A_{3}$.

$A_{1}(5, 2, 0)$

$A_{2}(2, 5, 0)$

$A_{3}(1, 2, 4)$

$A_{4}(-1, 1, 1)$


Calculate: \( 2013 \div (25 \times 52 - 46 \times 15) \times 10 = \)


Five athletes came to practice with their own balls, and upon leaving, each took someone else's ball. How many ways is this possible?


Determine whether there exists an infinite number of lines \( l_{1}, l_{2}, \cdots, l_{n}, \cdots \) on the coordinate plane that satisfy the following conditions:
1. The point \( (1, 1) \) lies on \( l_{n} \) for \( n = 1, 2, \cdots \).
2. \( k_{n+1} = a_{n} - b_{n} \), where \( k_{n+1} \) is the slope of \( l_{n+1} \), and \( a_{n}, b_{n} \) are the intercepts of \( l_{n} \) on the \( x \)-axis and \( y \)-axis respectively, for \( n = 1, 2, 3, \cdots \).
3. \( k_{n} \cdot k_{n+1} \geq 0 \) for \( n = 1, 2, 3, \cdots \).

Solution: \boxed{\text{No}}


How many ways are there to win tic-tac-toe in \(\mathbb{R}^{n}\)? That is, how many lines pass through three of the lattice points \((a_{1}, \ldots, a_{n})\) in \(\mathbb{R}^{n}\) with each coordinate \(a_{i}\) in \(\{1,2,3\}\)? Express your answer in terms of \(n\).

Solution: \boxed{\frac{5^n - 3^n}{2}}


The non-zero numbers \( a, b, \) and \( c \) are such that the doubled roots of the quadratic polynomial \( x^{2}+a x+b \) are the roots of the polynomial \( x^{2}+b x+c \). What can the ratio \( a / c \) equal?


In the plane $\alpha$, two perpendicular lines are drawn. Line $l$ forms angles of $45^{\circ}$ and $60^{\circ}$ with them. Find the angle that line $l$ makes with the plane $\alpha$.


Where are the points \( M(x, y) \) located if \( x = y \) (the first coordinate is equal to the second)?

Solution: \boxed{y = x}


Let \( P \) be an arbitrary point on the graph of the function \( y = x + \frac{2}{x} \) (where \( x > 0 \)). From point \( P \), perpendiculars are drawn to the line \( y = x \) and the \( y \)-axis, with the foots of these perpendiculars being points \( A \) and \( B \) respectively. Determine the value of \( \overrightarrow{PA} \cdot \overrightarrow{PB} \).


Let the set \( A = \{ 1, 2, \cdots, n \} \). Let \( S_n \) denote the sum of all elements in the non-empty proper subsets of \( A \), and \( B_n \) denote the number of subsets of \( A \). Find the limit \(\lim_{{n \to \infty}} \frac{S_n}{n^2 B_n}\).


Let \( p(x) = x^2 - x + 1 \). Let \(\alpha\) be a root of \( p(p(p(p(x)))) \). Find the value of
\[
(p(\alpha) - 1) p(\alpha) p(p(\alpha)) p(p(p(\alpha)))
\]


\(D\) is the point of tangency of the incircle with side \(BC\). \(N\) is an arbitrary point on segment \(ID\). The perpendicular to \(ID\) at point \(N\) intersects the circumcircle. \(O_1\) is the center of the circumcircle of \(\triangle XIY\). Find the product \(OO_1 \cdot IN\).

Solution: \boxed{Rr}


We take a \(6 \times 6\) chessboard, which has six rows and columns, and indicate its squares by \((i, j)\) for \(1 \leq i, j \leq 6\). The \(k\)-th northeast diagonal consists of the six squares satisfying \(i - j \equiv k \pmod{6}\); hence there are six such diagonals.

Determine if it is possible to fill the entire chessboard with the numbers \(1, 2, \ldots, 36\) (each exactly once) such that each row, each column, and each of the six northeast diagonals has the same sum.

Solution: \boxed{\text{No}}


Given a sequence $\left\{a_{n}\right\}$ with 9 terms, where $a_{1}=a_{9}=1$, and for each $i \in \{1,2, \cdots, 8\}$, $\frac{a_{i+1}}{a_{i}} \in \left\{2,1,-\frac{1}{2}\right\}$, determine the number of such sequences.


Given that $\alpha, \beta \in \mathbf{R}$, the lines $\frac{x}{\sin \alpha+\sin \beta}+\frac{y}{\sin \alpha+\cos \beta}=1$ and $\frac{x}{\cos \alpha+\sin \beta}+\frac{y}{\cos \alpha+\cos \beta}=1$ intersect on the line $y=-x$, find the value of $\sin \alpha+\cos \alpha+\sin \beta+\cos \beta$.


In a right triangle \(ABC\) with a right angle at \(A\), the altitude \(AH\) is drawn. A circle passing through points \(A\) and \(H\) intersects the legs \(AB\) and \(AC\) at points \(X\) and \(Y\) respectively. Find the length of segment \(AC\), given that \(AX = 5\), \(AY = 6\), and \(AB = 9\).


Fill the first eight positive integers in a $2 \times 4$ table, one number per cell, such that each row's four numbers increase from left to right, and each column's two numbers increase from bottom to top. How many different ways can this be done?


A triangle has an angle of \(70^{\circ}\). How can you construct an angle of \(40^{\circ}\) using it?


Write an \( n \)-digit number using the digits 1 and 2, such that no two consecutive digits are both 1. Denote the number of such \( n \)-digit numbers as \( f(n) \). Find \( f(10) \).


Given that the equation for \(x\), \(x^{2} - 4|x| + 3 = t\), has exactly three real roots. Find the value of \(t\).


On the right is an equation. Nine Chinese characters represent the numbers 1 to 9. Different characters represent different numbers. What is the possible maximum value of the equation:

盼 × 望 + 树 × 翠绿 + 天空 × 湛蓝


For \( x > 0 \), let \( f(x) = \frac{4^x}{4^x + 2} \). Determine the value of the sum

$$
\sum_{k=1}^{1290} f\left(\frac{k}{1291}\right)
$$


Let \( ABC \) be an isosceles triangle with \( A \) as the vertex angle. Let \( M \) be the midpoint of the segment \( [BC] \). Let \( D \) be the reflection of point \( M \) over the segment \( [AC] \). Let \( x \) be the angle \( \widehat{BAC} \). Determine, as a function of \( x \), the value of the angle \( \widehat{MDC} \).


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

\[
\begin{cases}
x^2 + xy + y^2 = 108 \\
y^2 + yz + z^2 = 9 \\
z^2 + xz + x^2 = 117
\end{cases}
\]

Find the value of the expression \( xy + yz + xz \).


In triangle \( ABC \), the measure of angle \( A \) is twice the measure of angle \( B \), and the lengths of the sides opposite these angles are 12 cm and 8 cm, respectively. Find the length of the third side of the triangle.

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


Solve the equation among positive numbers:

$$
\frac{x \cdot 2014^{\frac{1}{x}}+\frac{1}{x} \cdot 2014^{x}}{2}=2014
$$


The set
$$
A=\{\sqrt[n]{n} \mid n \in \mathbf{N} \text{ and } 1 \leq n \leq 2020\}
$$

has the largest element as $\qquad$ .


Let \( F \) be the number of integral solutions of \( x^{2}+y^{2}+z^{2}+w^{2}=3(x+y+z+w) \). Find the value of \( F \).


Find the number of different monic quadratic polynomials (i.e., with the leading coefficient equal to 1) with integer coefficients such that they have two different roots which are powers of 5 with natural exponents, and their coefficients do not exceed in absolute value $125^{48}$.


Majka examined multi-digit numbers in which odd and even digits alternate regularly. Those that start with an odd digit, she called "funny," and those that start with an even digit, she called "cheerful" (for example, the number 32387 is funny, the number 4529 is cheerful).

Majka created one three-digit funny number and one three-digit cheerful number, using six different digits without including 0. The sum of these two numbers was 1617. The product of these two numbers ended with the digits 40.

Determine Majka's numbers and calculate their product.


Let \( ABC \) be a triangle. The midpoints of the sides \( BC \), \( AC \), and \( AB \) are denoted by \( D \), \( E \), and \( F \) respectively.

The two medians \( AD \) and \( BE \) are perpendicular to each other and their lengths are \(\overline{AD} = 18\) and \(\overline{BE} = 13.5\).

Calculate the length of the third median \( CF \) of this triangle.


Given that \( S_n \) and \( T_n \) are the sums of the first \( n \) terms of the arithmetic sequences \( \{a_n\} \) and \( \{b_n\} \), respectively, and that
\[
\frac{S_n}{T_n} = \frac{2n + 1}{4n - 2} \quad (n = 1, 2, \ldots),
\]
find the value of
\[
\frac{a_{10}}{b_3 + b_{18}} + \frac{a_{11}}{b_6 + b_{15}}.
\]


The perimeter of a triangle is 28, and the midpoints of its sides are connected by segments. Find the perimeter of the resulting triangle.


Inside triangle \( ABC \) with angles \(\angle A = 50^\circ\), \(\angle B = 60^\circ\), \(\angle C = 70^\circ\), a point \( M \) is chosen such that \(\angle AMB = 110^\circ\) and \(\angle BMC = 130^\circ\). Find \(\angle MBC\).


Call an integer \( n > 1 \) radical if \( 2^n - 1 \) is prime. What is the 20th smallest radical number?


Calculate the volumes of solids generated by rotating the region bounded by the graphs of the functions about the x-axis.

$$
y = \sin^2(x), x = \frac{\pi}{2}, y = 0
$$


A person flips a coin, where the probability of heads up and tails up is $\frac{1}{2}$ each. Construct a sequence $\left\{a_{n}\right\}$ such that
$$
a_{n}=\left\{
\begin{array}{ll}
1, & \text{if the } n \text{th flip is heads;} \\
-1, & \text{if the } n \text{th flip is tails.}
\end{array}
\right.
$$
Let $S_{n}=a_{1}+a_{2}+\cdots+a_{n}$. Find the probability that $S_{2} \neq 0$ and $S_{8}=2$. Provide your answer in its simplest fractional form.


How many different four-digit numbers divisible by 15 can be formed using the digits $0, 1, 3, 5, 6$ without repetition of digits?


Given non-negative real numbers \( a, b, c, x, y, z \) that satisfy \( a + b + c = x + y + z = 1 \), find the minimum value of \( \left(a - x^{2}\right)\left(b - y^{2}\right)\left(c - z^{2}\right) \).


Vojta began writing the number of the current school year 2019202020192020... continuously. When he wrote 2020 digits, he got bored.

How many twos did he write?

Hint: How many twos would Vojta write if he wrote only 20 digits?


What is the smallest positive integer that is divisible by 111 and has the last four digits as 2004?


In a small town, the police are looking for a wanderer. There is a four in five chance that he is in one of the eight bars in the town, with no preference for any particular one. Two officers visited seven bars but did not find the wanderer. What are the chances of finding him in the eighth bar?


A bridge over a river connects two different regions of a country. At one point, one of the regions repainted its section of the bridge. If the newly painted section were 1.2 times larger, it would constitute exactly half of the entire bridge. What fraction of the bridge still needs to be painted for it to be exactly half painted?


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?


On the coordinate plane, the points \(A(0, 2)\), \(B(1, 7)\), \(C(10, 7)\), and \(D(7, 1)\) are given. Find the area of the pentagon \(A B C D E\), where \(E\) is the intersection point of the lines \(A C\) and \(B D\).


At the tourist base, the number of two-room cottages is twice the number of one-room cottages. The number of three-room cottages is a multiple of the number of one-room cottages. If the number of three-room cottages is tripled, it will be 25 more than the number of two-room cottages. How many cottages are there in total at the tourist base, given that there are at least 70 cottages?


Let \( M \) be a finite set of numbers. It is known that among any three elements of this set, there exist two whose sum also belongs to \( M \).

What is the maximum possible number of elements in \( M \)?


Given a square \(ABCD\) with side length \(a\), vertex \(A\) lies in plane \(\beta\), and the other vertices are on the same side of plane \(\beta\). The distances from points \(B\) and \(D\) to plane \(\beta\) are 1 and 2, respectively. If the dihedral angle between plane \(ABCD\) and plane \(\beta\) is 30 degrees, then \(a =\)?


Find the number of pairs of integers \( (x, y) \) that satisfy the condition \( x^{2} + 6xy + 5y^{2} = 10^{100} \).


Find the maximum value of the function \( f(x) = \lg 2 \cdot \lg 5 - \lg 2x \cdot \lg 5x \).


Luna has an infinite supply of red, blue, orange, and green socks. She wants to arrange 2012 socks in a line such that no red sock is adjacent to a blue sock and no orange sock is adjacent to a green sock. How many ways can she do this?

Solution: \boxed{4 \cdot 3^{2011}}


When \( s \) and \( t \) range over all real numbers, the expression
$$
(s+5-3|\cos t|)^{2}+(s-2|\sin t|)^{2}
$$
achieves a minimum value of \(\qquad\).


Let the sets \( A \supseteq A_{1} \supseteq A_{2} \). If \( A_{2} \sim A \), then \( A_{1} \sim A \).

Solution: \boxed{A_1 \sim A}


All possible non-empty subsets are taken from the set of numbers $1,2,3, \ldots, n$. For each subset, the reciprocal of the product of all its numbers is taken. Find the sum of all such reciprocals.

Solution: \boxed{n}


On an island, there are knights, liars, and followers; each person knows who is who. All 2018 island residents were lined up and each was asked to answer "Yes" or "No" to the question: "Are there more knights than liars on the island?" The residents responded one by one in such a way that the others could hear. Knights always told the truth, liars always lied. Each follower answered the same as the majority of the preceding respondents, and if the "Yes" and "No" answers were split equally, they could give either answer. It turned out that there were exactly 1009 "Yes" answers. What is the maximum number of followers that could be among the island residents?


In a meadow, ladybugs have gathered. If a ladybug has six spots on its back, it always tells the truth. If it has four spots, it always lies. There are 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, "Together we have 30 spots in total." The third disagreed, saying, "Altogether, we have 26 spots on our backs." "Of these three, exactly one told the truth," declared each of the remaining ladybugs. How many ladybugs were there in total in the meadow?


Inside the triangle \(ABC\), there are points \(P\) and \(Q\) such that point \(P\) is at distances 6, 7, and 12 from lines \(AB\), \(BC\), and \(CA\) respectively, and point \(Q\) is at distances 10, 9, and 4 from lines \(AB\), \(BC\), and \(CA\) respectively. Find the radius of the inscribed circle of triangle \(ABC\).


Natural numbers \( x_{1}, x_{2}, \ldots, x_{13} \) are such that \( \frac{1}{x_{1}} + \frac{1}{x_{2}} + \ldots + \frac{1}{x_{13}} = 2 \). What is the minimum value of the sum of these numbers?


Find the integers \(a\), \(b\), and \(c\) such that they satisfy the condition:
\[ a^{2} + b^{2} + c^{2} + 3 < a b + 3 b + 2 c. \]

Solution: \boxed{a = 1, b = 2, c = 1}


Calculate the definite integral:

$$
\int_{0}^{\frac{\pi}{2}}\left(1 - 5x^{2}\right) \sin x \, dx
$$

Solution: \boxed{11 - 5\pi}


On the base \( AC \) of an isosceles triangle \( ABC \) (\( AB = BC \)), a point \( M \) is marked. It is known that \( AM = 7 \), \( MB = 3 \), and \(\angle BMC = 60^{\circ}\). Find the length of the segment \( AC \).


The tadpoles of the Triassic Discoglossus have five legs each, while the tadpoles of the Saber-toothed Frog have several tails (all having the same number). A Jurassic Park staff member scooped up several tadpoles along with water. It turned out that the total caught had 100 legs and 64 tails. How many tails does each Saber-toothed Frog tadpole have, if all five-legged tadpoles have one tail, and all multi-tailed tadpoles have four legs?


For what maximum \( a \) is the inequality \(\frac{\sqrt[3]{\operatorname{tg} x}-\sqrt[3]{\operatorname{ctg} x}}{\sqrt[3]{\sin x}+\sqrt[3]{\cos x}}>\frac{a}{2}\) satisfied for all permissible \( x \in \left(\frac{3 \pi}{2}, 2 \pi\right) \)? If necessary, round your answer to the nearest hundredth.


Find the smallest positive period \( T \) of the function \( f(x) = \frac{2 \sin x + 1}{3 \sin x - 5} \).


A number \( \mathrm{A} \) is a prime number, and \( \mathrm{A}+14 \), \( \mathrm{A}+18 \), \( \mathrm{A}+32 \), \( \mathrm{A}+36 \) are also prime numbers. What is the value of \( \mathrm{A} \)?


In the final of the giraffe beauty contest, two giraffes, Tall and Spotted, reached the finals. There are 135 voters divided into 5 districts, with each district divided into 9 precincts, and each precinct having 3 voters. The voters in each precinct choose the winner by majority vote; in a district, the giraffe that wins in the majority of precincts wins the district; finally, the giraffe that wins in the majority of the districts is declared the winner of the final. The giraffe Tall won. What is the minimum number of voters who could have voted for Tall?


Find the sum of the first twelve terms of an arithmetic sequence if its fifth term $a_{5} = 1$ and its seventeenth term $a_{17} = 18$.


Three points \( A \), \( B \), and \( C \) are randomly selected on the unit circle. Find the probability that the side lengths of triangle \( \triangle ABC \) do not exceed \( \sqrt{3} \).


Given any 4-digit positive integer \( x \) not ending in '0', we can reverse the digits to obtain another 4-digit integer \( y \). For example, if \( x \) is 1234, then \( y \) is 4321. How many possible 4-digit integers \( x \) are there if \( y - x = 3177 \)?


Given that \( g(x) = \sqrt{x(1-x)} \) is a function defined on the interval \([0,1]\), find the area enclosed by the graph of the function \( x g(x) \) and the x-axis.


Find the number of natural numbers \( k \) not exceeding 353500 such that \( k^{2} + k \) is divisible by 505.


Forty-two cards are labeled with the natural numbers 1 through 42 and randomly shuffled into a stack. One by one, cards are taken off the top of the stack until a card labeled with a prime number is removed. How many cards are removed on average?


Is it possible in a calendar year that no Sunday falls on the seventh day of any month?

Solution: \boxed{No}


Let \( k \in \mathbb{N}^* \). Suppose that all positive integers are colored using \( k \) different colors, and there exists a function \( f: \mathbb{Z}^+ \rightarrow \mathbb{Z}^+ \) satisfying:
1. For positive integers \( m \) and \( n \) of the same color (they can be the same), \( f(m+n) = f(m) + f(n) \);
2. There exist positive integers \( m \) and \( n \) (they can be the same) such that \( f(m+n) \neq f(m) + f(n) \).

Find the minimum value of \( k \).


The height of a right triangle, dropped to the hypotenuse, is $h$.

What is the minimum length that the median, drawn from the vertex of the larger acute angle, can have?


From the numbers $1,2,3, \cdots, 10$, 3 numbers are drawn at random. What is the probability that at least two of these numbers are consecutive integers?


Let \(\{a_{n}\}\) be a geometric sequence with each term greater than 1, then the value of \(\lg a_{1} \lg a_{2012} \sum_{i=1}^{20111} \frac{1}{\lg a_{i} \lg a_{i+1}}\) is ________ .


Find the number of points in the plane \(xOy\) that have natural coordinates \((x, y)\) and lie on the parabola \(y = -\frac{x^2}{4} + 5x + 39\).


Find the minimum value of the function \( y = \sqrt{x^2 + 2x + 2} + \sqrt{x^2 - 2x + 2} \).


In the regular hexagon \( A B C D E F \), two of the diagonals, \( F C \) and \( B D \), intersect at \( G \). The ratio of the area of quadrilateral FEDG to the area of \( \triangle B C G \) is:
(A) \( 3 \sqrt{3}: 1 \)
(B) \( 4: 1 \)
(C) \( 6: 1 \)
(D) \( 2 \sqrt{3}: 1 \)
(E) \( 5: 1 \)

Solution: \boxed{E \text{ (5:1)}}


Around a circle with a radius of 1, a rhombus and a triangle are described. Two sides of the triangle are parallel to the diagonals of the rhombus, and the third side is parallel to one of the sides of the rhombus and is equal to 5. Find the side of the rhombus.


Ali Baba went to a cave where there are gold and diamonds. He had with him one large bag. It is known that a full bag of gold weighs 200 kg, and if the bag is filled only with diamonds, it weighs 40 kg (the empty bag weighs nothing). A kilogram of gold costs 20 dinars, and a kilogram of diamonds costs 60 dinars. What is the maximum amount of money Ali Baba can earn if he can carry no more than 100 kg in this bag? Provide justification for your answer.

Solution: \boxed{3000 \text{ dinars}}


Around the circle, a pentagon is described, the lengths of the sides of which are integers, and the first and third sides are equal to 1. Into which segments does the point of tangency divide the second side?


Replace the asterisk $(*)$ in the expression $\left(x^{3}-2\right)^{2}+\left(x^{2}+*\right)^{2}$ with a monomial such that, after squaring and combining like terms, the resulting expression has four terms.


A point \( P \) lies inside a regular hexagon \( A B C D E F \). The distances from \( P \) to the sides \( A B, B C, C D, D E, E F \), and \( F A \) are respectively \( 1, 2, 5, 7, 6 \), and \( x \). Find \( x \).


A quadrilateral \(ABCD\), whose diagonals are perpendicular to each other, is inscribed in a circle with center \(O\).

Find the distance from point \(O\) to side \(AB\), given that \(CD = 8\).


Five identical balls are moving in one direction along a straight line, and five other identical balls are moving in the opposite direction. The speeds of all the balls are the same. When any two balls collide, they rebound in opposite directions with the same speed they had before the collision. How many collisions will occur in total between the balls?


Grass in a given area grows equally fast and dense. It is known that 70 cows would eat all the grass in 24 days, and 30 cows would eat it in 60 days. How many cows would eat all the grass in 96 days? (Assume that the cows eat the grass uniformly).


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

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

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


As shown in the figure, identical equilateral triangles are joined downward progressively to form larger equilateral triangles. The number of vertices (counting overlapping vertices just once) of the smallest triangles in sequence are $3, 6, 10, 15, 21, \cdots$. What is the 9th term in this sequence?


There are 20 cards numbered $1, 2, \cdots, 19, 20$. They are placed in a box, and 4 people each draw one card. The two people who draw the smaller numbers are placed in one group, and the two who draw the larger numbers are placed in another group. If two of the people draw the numbers 5 and 14, what is the probability that these two people are in the same group?


A three-digit number, when a decimal point is added in an appropriate place, becomes a decimal. This decimal is 201.6 less than the original three-digit number. What is the original three-digit number?


Let $\mathcal{O}$ be a regular octahedron. How many lines are there such that a rotation of at most $180^{\circ}$ around these lines maps $\mathcal{O}$ onto itself?


Several oranges (not necessarily of equal mass) were picked from a tree. On weighing them, it turned out that the mass of any three oranges taken together is less than 5% of the total mass of the remaining oranges. What is the minimum number of oranges that could have been picked?


Let \( x, y, z \) be distinct positive integers such that \( xyz \mid (xy-1)(yz-1)(zx-1) \). Find all possible values of \( x, y, z \).


Find all values of \( x \) for which the greater of the numbers \( \sqrt{\frac{x}{2}} \) and \( \operatorname{tg} x \) is not greater than 1. Provide the total length of the intervals on the number line that satisfy this condition, rounding the result to the nearest hundredth if necessary.


If the real and imaginary parts of both $\frac{z}{20}$ and $\frac{20}{z}$ are between 0 and 1 (exclusive), find the area of the figure formed by the corresponding points of such complex numbers $z$.


What is the maximum number of months in a year that can have five Sundays?


The circle, which has its center on the hypotenuse $AB$ of the right triangle $ABC$, touches the two legs $AC$ and $BC$ at points $E$ and $D$ respectively.

Find the angle $ABC$, given that $AE = 1$ and $BD = 3$.


What are the last two digits of the number

$$
8 + 88 + 888 + \cdots + \overbrace{88 \cdots 88}^{2008} ?
$$


What is the maximum area that a quadrilateral with side lengths of 1, 4, 7, and 8 can have?


From cities $A$ and $B$, which are 240 km apart, two cars simultaneously start driving towards each other. One car travels at 60 km/h and the other at 80 km/h. How far from the point $C$, located at the midpoint between $A$ and $B$, will the cars meet? Give the answer in kilometers, rounding to the nearest hundredth if necessary.


Given the set \( A = \{ a \mid -1 \leqslant a \leqslant 2 \} \), find the area of the plane region \( B = \{ (x, y) \mid x, y \in A, x + y \geqslant 0 \} \).


After walking \( \frac{2}{5} \) of the length of a narrow bridge, a pedestrian noticed a car approaching the bridge from behind. He then walked back and met the car at the beginning of the bridge. If the pedestrian had continued walking forward, the car would have caught up with him at the end of the bridge. Find the ratio of the speed of the car to the speed of the pedestrian.


Find the smallest positive period of the function \( f(x)=\sin x \sin \frac{1}{2} x \sin \frac{1}{3} x \).


There were seven boxes. Some of these boxes were filled with seven additional boxes each (not nested within one another) and so on. As a result, there were 10 non-empty boxes.

How many boxes are there in total?


In a $3 \times 4$ table, 12 numbers are arranged such that all seven sums of these numbers in the rows and columns of the table are distinct. What is the maximum number of numbers in this table that can be equal to zero?


At the end of a day of activities, an amusement park collected 100 reais from the tickets of 100 people. We know that each adult had to pay 3 reais to enter, each youth 2 reais, and each child 30 cents. What is the smallest number of adults that entered the park that day?


If integers \( a \) and \( b \) are neither relatively prime nor does one divide the other, then \( a \) and \( b \) are called a "union" pair. Let \( A \) be an \( n \)-element subset of the set \( M = \{1, 2, \ldots, 2017\} \), such that any two numbers in \( A \) are a "union" pair. Determine the maximum value of \( n \).


Given the real numbers \( a, b, c \) satisfy \( a + b + c = 6 \), \( ab + bc + ca = 5 \), and \( abc = 1 \), determine the value of \( \frac{1}{a^3} + \frac{1}{b^3} + \frac{1}{c^3} \).


The parabola \( y = ax^2 + bx + 1 \) has parameters \( a \) and \( b \) satisfying \( 8a^2 + 4ab = b^3 \). Find the equation of the locus of the vertex \((s, t)\) of the parabola as \( a \) and \( b \) vary.

Solution: \boxed{s t = 1}


Ksyusha runs twice as fast as she walks (both speeds are constant).

On Tuesday, when she left home for school, she first walked, and then, when she realized she was late, she started running. The distance Ksyusha walked was twice the distance she ran. As a result, she reached the school from home in exactly 30 minutes.

On Wednesday, Ksyusha left home even later, so she had to run twice the distance she walked. How many minutes did it take her to get from home to school on Wednesday?


In an acute-angled triangle, two altitudes are 3 and \(2 \sqrt{2}\), and their intersection point divides the third altitude in the ratio 5:1, counting from the vertex of the triangle. Find the area of the triangle.


A pedestrian and a cyclist started simultaneously in the same direction from a point on a circular track. The speed of the cyclist is 55% greater than the speed of the pedestrian, which results in the cyclist overtaking the pedestrian from time to time. How many different points on the track will these overtakes occur?


Masha braided her dolls' hair: half of the dolls got one braid each, a quarter of the dolls got two braids each, and the remaining quarter of the dolls got four braids each. She used one ribbon for each braid. How many dolls does Masha have if she needed 24 ribbons in total?


Given a regular triangular pyramid $S-ABC$ with a height $SO = 3$ and a base side length of 6. From point $A$, a perpendicular is drawn to the opposite face $SBC$, with the foot of the perpendicular being $O'$. On $AO'$, find a point $P$ such that $\frac{AP}{PO'} = 8$. Find the area of the cross-section passing through point $P$ and parallel to the base.


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


Find the number of 6-digit positive integers \(\overline{abcdef}\) satisfying the following two conditions:
(a) Each digit is non-zero.
(b) \(a \times b + c \times d + e \times f\) is even.


Three cones are placed on a table, standing on their bases and touching each other. The radii of their bases are \(2r\), \(3r\), and \(10r\). A truncated cone (frustum) is placed on the table with its smaller base downward, sharing a slant height with each of the other cones. Find \(r\), if the radius of the smaller base of the truncated cone is 15.


Determine all natural numbers \( n \geq 2 \) for which there exist two permutations \((a_{1}, a_{2}, \ldots, a_{n})\) and \((b_{1}, b_{2}, \ldots, b_{n})\) of the numbers \(1, 2, \ldots, n\) such that \((a_{1}+b_{1}, a_{2}+b_{2}, \ldots, a_{n}+b_{n})\) are consecutive natural numbers.

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


Starting from her house to get to school, Julia needs to walk eight blocks to the right and five blocks up, as indicated in the given diagram.

She knows that there are many different ways to take the house-to-school route, always following the shortest path. Since she is very curious, she would like to always take different routes. How many such routes exist from Julia's house to the school?


The sum of one hundred numbers is 1000. The largest of them was doubled, and another number was decreased by 10. It turned out that the sum did not change. Find the smallest of the original numbers.


Given that $a$ and $b$ are positive integers satisfying $\frac{1}{a} - \frac{1}{b} = \frac{1}{2018}$, find the number of all positive integer pairs $(a, b)$.


The function \( f \) has the following properties:

1. Its domain is all real numbers.
2. It is an odd function, i.e., \( f(-x) = -f(x) \) for every real number \( x \).
3. \( f(2x-3) - 2f(3x-10) + f(x-3) = 28 - 6x \) for every real number \( x \).

Determine the value of \( f(4) \).


Let \( x \) and \( y \) be real numbers. Determine the maximum value of \( \frac{2x + \sqrt{2}y}{2x^4 + 4y^4 + 9} \).


Add together all natural numbers less than 1980 for which the sum of their digits is even!


Find all real numbers \( x, y, z \) such that

\[
\begin{aligned}
x^{2} y + y^{2} z + z^{2} &= 0, \\
z^{3} + z^{2} y + z y^{3} + x^{2} y &= \frac{1}{4}\left(x^{4} + y^{4}\right).
\end{aligned}
\]

Solution: \boxed{x = y = z = 0}


Convex quadrilateral \(ABCD\) has sides \(AB = BC = 7\), \(CD = 5\), and \(AD = 3\). Given additionally that \( \angle ABC = 60^{\circ} \), find \(BD\).


Mice built an underground house consisting of chambers and tunnels:

- Each tunnel leads from one chamber to another (i.e., none are dead ends).
- From each chamber, exactly three tunnels lead to three different chambers.
- From each chamber, it is possible to reach any other chamber through tunnels.
- There is exactly one tunnel such that, if it is filled in, the house will be divided into two separate parts.

What is the minimum number of chambers the mice's house could have? Draw a possible configuration of how the chambers could be connected.


The coefficients of the quadratic polynomials \(f(x) = x^2 + mx + n\) and \(p(x) = x^2 + kx + l\) satisfy the condition \(k > m > n > l > 0\). Is it possible for \(f(x)\) and \(p(x)\) to have a common root?

Solution: \boxed{\text{No}}


Fisherman Vasya caught several fish. He placed the three largest fish, which constitute 35% of the total weight of the catch, in the refrigerator. He gave the three smallest fish, which constitute 5/13 of the weight of the remaining fish, to the cat. Vasya ate all the rest of the caught fish himself. How many fish did Vasya catch?


Two signals, $A$ and $B$, are transmitted over a communication line with probabilities of 0.72 and 0.28, respectively. Due to noise, $\frac{1}{6}$ of $A$ signals are distorted and received as $B$ signals, and $\frac{1}{7}$ of transmitted $B$ signals are received as $A$ signals.

a) Determine the probability that an $A$ signal will be received at the receiving end.

b) Given that an $A$ signal is received, what is the probability that it was originally transmitted as an $A$ signal?


Let \( N \) be the set of non-negative integers. The function \( f: N \to N \) satisfies \( f(a+b) = f(f(a)+b) \) for all \( a \), \( b \) and \( f(a+b) = f(a) + f(b) \) for \( a+b < 10 \). Also \( f(10) = 1 \). How many three-digit numbers \( n \) satisfy \( f(n) = f(N) \), where \( N \) is the "tower" 2, 3, 4, 5, in other words, it is \( 2^{a} \), where \( a = 3^{b} \), where \( b = 4^{5} \)?


One material particle entered the opening of a pipe, and after 6.8 minutes, a second particle entered the same opening. Upon entering the pipe, each particle immediately began linear motion along the pipe: the first particle moved uniformly at a speed of 5 meters per minute, while the second particle covered 3 meters in the first minute and in each subsequent minute covered 0.5 meters more than in the previous minute. How many minutes will it take for the second particle to catch up with the first?


You have a set of n biased coins. The mth coin has a probability of \( \frac{1}{2m+1} \) of landing heads (for \( m = 1, 2, ... , n \)) and the results for each coin are independent. What is the probability that if each coin is tossed once, you get an odd number of heads?


Lisa considers the number

$$
x=\frac{1}{1^{1}}+\frac{1}{2^{2}}+\cdots+\frac{1}{100^{100}} .
$$

Lisa wants to know what $x$ is when rounded to the nearest integer. Help her determine its value.


Given that \( x \geq 1 \), \( y \geq 1 \), and \(\lg ^{2} x+\lg ^{2} y=\lg 10 x^{2}+\lg 10 y^{2}\), find the maximum value of \( u=\lg x y \).

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


Given \( x, y \geq 1, x + y = 8 \), find the maximum value of \( \left|\sqrt{x - \frac{1}{y}} + \sqrt{y - \frac{1}{x}} \right| \).


In the decimal system, a $(n+1)$-digit number $A=\overline{a_{n} a_{n-1} \ldots a_{1} a_{0}}$ is called the reverse of $A*=\overline{a_{0} a_{1} \ldots a_{n}}$. (Thus, the reverse of 759 is 957, and the reverse of 980 is 89.)

Find the four-digit numbers that "reverse" when multiplied by 9, meaning $9A = A^*$.


Given a quadratic polynomial \( f(x) \) such that the equation \( (f(x))^3 - f(x) = 0 \) has exactly three solutions. Find the ordinate of the vertex of the polynomial \( f(x) \).


Let the set \( S = \{1, 2, \cdots, 3n\} \), where \( n \) is a positive integer. Let \( T \) be a subset of \( S \) such that for any \( x, y, z \in T \) (where \( x, y, z \) can be the same), the condition \( x + y + z \notin T \) is satisfied. Find the maximum number of elements in such a subset \( T \).


Points \( A, B, C, D \) are marked on a sheet of paper. A recognition device can perform two types of operations with absolute accuracy: a) measure the distance between two given points in centimeters; b) compare two given numbers. What is the minimum number of operations this device needs to perform to definitively determine whether quadrilateral \( A B C D \) is a square?


Find the maximum possible value of the GCD \((x + 2015y, y + 2015x)\), given that \(x\) and \(y\) are coprime numbers.


Solution: \boxed{n = 2}


Vasya wrote all natural numbers from 104 to 203 in the cells of a $10 \times 10$ table. He calculated the product of numbers in each row of the table and obtained a set of ten numbers. Then he calculated the product of numbers in each column of the table and also obtained a set of ten numbers. Could the resulting sets be identical?

Solution: \boxed{\text{No}}


The three midlines of a triangle divide it into four parts. The area of one of them is \( S \). Find the area of the given triangle.


Let \( A_{n} \) be the area outside a regular \( n \)-gon of side length 1 but inside its circumscribed circle, and let \( B_{n} \) be the area inside the \( n \)-gon but outside its inscribed circle. Find the limit as \( n \) tends to infinity of \(\frac{A_{n}}{B_{n}}\).


Given a point $P$ on the inscribed circle of a square $ABCD$, considering the angles $\angle APC = \alpha$ and $\angle BPD = \beta$, find the value of $\tan^2 \alpha + $\tan^2 \beta$.


A set of real numbers is called "simple" if it contains elements such that \( x + y = z \). Given the set \( \{1, 2, \cdots, 2n+1\} \), find the maximum number of elements in one of its simple subsets.


All natural numbers whose digit sum is equal to 5 are arranged in ascending order. Which number is in the 125th position?


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


An urn initially contains one white ball. Another ball, either white or black (with equal probabilities), is added to the urn. After that, a ball is randomly drawn from the urn, and it turns out to be white. What is the conditional probability that the remaining ball in the urn is also white?


Given an ellipse $C: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$ with left and right foci $F_{1}$ and $F_{2}$ respectively, and upper and lower vertices $B_{1}$ and $B_{2}$ respectively, the right vertex is $A$. The line $A B_{1}$ intersects $B_{2} F_{1}$ at point D. If $2\left|A B_{1}\right|=3\left|B_{1} D\right|$, determine the eccentricity of ellipse $C$.


Does there exist a natural number \( n \) such that

\[ n + S(n) = 125 \]

where \( S(n) \) is the sum of the digits of \( n \)?


In the rhombus \(ABCD\), the measure of angle \(B\) is \(40^\circ\). \(E\) is the midpoint of \(BC\), and \(F\) is the foot of the perpendicular dropped from \(A\) to \(DE\). Find the measure of angle \(DFC\).


Determine the volume of a regular quadrangular prism if its diagonal forms an angle of $30^{\circ}$ with the plane of one of its lateral faces, and the side of its base is equal to $a$.

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


On a rectangular sheet of paper, a picture in the shape of a "cross" formed by two rectangles $ABCD$ and $EFGH$ is drawn, with their sides parallel to the edges of the sheet. It is known that $AB=9$, $BC=5$, $EF=3$, and $FG=10$. Find the area of the quadrilateral $AFCH$.


Find all positive integers \( n \) with the following property: the \( k \) positive divisors of \( n \) have a permutation \(\left( d_{1}, d_{2}, \ldots, d_{k} \right)\) such that for every \( i = 1, 2, \ldots, k \), the number \( d_{1} + \cdots + d_{i} \) is a perfect square.

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


In the number \(2 * 0 * 1 * 6 * 0 * 2 *\), replace each of the 6 asterisks with any digit from the set \(\{0,1,2,3,4,5,6,7,8\}\) (digits may repeat) so that the resulting 12-digit number is divisible by 45. How many ways can this be done?


Find all real numbers \( x \) that satisfy the equation \(\lg (x+1)=\frac{1}{2} \log _{3} x\).


Let \( \triangle ABC \) be a triangle with \( AB = 16 \) and \( AC = 5 \). Suppose the bisectors of angles \( \angle ABC \) and \( \angle BCA \) meet at point \( P \) in the triangle's interior. Given that \( AP = 4 \), compute \( BC \).


In triangle \( \triangle ABC \), given \( AB = 4 \), \( AC = 3 \), and \( P \) is a point on the perpendicular bisector of \( BC \), find \( \overrightarrow{BC} \cdot \overrightarrow{AP} \).


In a trapezoid, the midline is equal to 7, the height is $\frac{15 \sqrt{3}}{7}$, and the angle between the diagonals opposite the base is $120^{\circ}$.

Find the diagonals of the trapezoid.


In the number $2 * 0 * 1 * 6 * 02 *$, each of the 5 asterisks should be replaced with any of the digits $0, 2, 4, 7, 8, 9$ (the digits may repeat) such that the resulting 11-digit number is divisible by 12. How many ways can this be done?


Find all natural numbers \( N \) such that the remainder when 2017 is divided by \( N \) is 17. Indicate the number of such \( N \).


Given acute angles \(\alpha\) and \(\beta\) that satisfy the condition: \(\frac{\sin^{4} \alpha}{\cos^{2} \beta} + \frac{\cos^{4} \alpha}{\sin^{2} \beta} = 1\), find \(\alpha + \beta\).


Find any four distinct natural numbers with the following property: if you add the product of any two of them to the product of the other two numbers, the result will be a prime number.

Solution: \boxed{1, 2, 3, 5}


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


Given the polynomial

\[ P(x) = a_{2n} x^{2n} + a_{2n-1} x^{2n-1} + \ldots + a_{1} x + a_{0} \]

where each coefficient \( a_i \) belongs to the interval \([100, 101]\). For what minimum \( n \) can such a polynomial have a real root?


Given a linear function \( f(x) \). It is known that the distance between the points of intersection of the graphs \( y = x^2 - 1 \) and \( y = f(x) + 1 \) is \( 3\sqrt{10} \), and the distance between the points of intersection of the graphs \( y = x^2 \) and \( y = f(x) + 3 \) is \( 3\sqrt{14} \). Find the distance between the points of intersection of the graphs \( y = x^2 \) and \( y = f(x) \).

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


The number 4567 is tripled. The ones digit (units digit) in the resulting number is:
(A) 5
(B) 6
(C) 7
(D) 3
(E) 1


Given the function \( f(x) \) that satisfies: for any non-negative real number \( x \), \( f(x) = f(1) \cdot x + \frac{f(2)}{x} - 1 \). Then the minimum value of \( f(x) \) on \( (0, +\infty) \) is \(\quad\).

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


If a non-negative integer \( m \) and the sum of its digits are both multiples of 6, then \( m \) is called a "Liuhe number." Find the number of Liuhe numbers less than 2012.


The angle bisector of angle \( A \) of triangle \( ABC \) intersects side \( BC \) at point \( D \). A circle with radius 35, whose center lies on the line \( BC \), passes through points \( A \) and \( D \). It is known that \( AB^2 - AC^2 = 216 \), and the area of triangle \( ABC \) is \( 90 \sqrt{3} \). Find the radius of the circumcircle of triangle \( ABC \).

Solution: \boxed{7 \sqrt{3}}


The smallest positive period of the function \( y = \sin x - 2 \cos 2x + 4 \sin 4x \) is ______.


Given that $1 \le k$ is a natural number, and $\frac{1001 \cdot 1002 \cdot \cdots \cdot 2005 \cdot 2006}{11^{k}}$ is an integer, what is the maximum value of $k$?


What is the probability that when rolling 3 dice:

a) Exactly one die shows a 6?

b) At least one die shows a 6?

c) At most one die shows a 6?


We call a pair of natural numbers $(a, p)$ good if the number $a^{3} + p^{3}$ is divisible by $a^{2} - p^{2}$, and $a > p$.

(a) (1 point) Give any possible value of $a$ for which the pair $(a, 11)$ is good.

(b) (3 points) Find the number of good pairs, where $p$ is a prime number less than 16.


On the extension of side $AD$ of rectangle $ABCD$ beyond point $D$, point $E$ is taken such that $DE = 0.5 AD$ and $\angle BEC = 30^\circ$.

Find the ratio of the sides of rectangle $ABCD$.


Calculate the circumference of a circle with radius \( R \) using a definite integral.

Solution: \boxed{2 \pi R}


As shown in Figure 1, two containers with vertices facing down are vertically placed and contain equal masses of liquid. The radii of the liquid surface circles are 3 and 6, respectively. A small ball with a radius of 1 is placed in each container, fully submerged in the liquid and resting at the bottom without causing any overflow. What is the ratio of the height increase of the liquid surface in the narrow-mouthed container to the wide-mouthed container?

(A) \(1: 1\)
(B) \(47: 43\)
(C) \(2: 1\)
(D) \(40: 13\)
(E) \(4: 1\)

Solution: \boxed{4: 1}


Michael picks a random subset of the complex numbers $\left\{1, \omega, \omega^{2}, \ldots, \omega^{2017}\right\}$ where $\omega$ is a primitive $2018^{\text {th }}$ root of unity and all subsets are equally likely to be chosen. If the sum of the elements in his subset is $S$, what is the expected value of $|S|^{2}$? (The sum of the elements of the empty set is 0.)


Among all the factors of \(11! \) (where \(11! = 11 \times 10 \times \ldots \times 1\)), the largest factor that can be expressed as \(6k + 1\) (where \(k\) is a natural number) is what?


What is the smallest number of weights needed to be able to measure any number of grams from 1 to 100 on a balance scale if the weights can be placed only on one side of the scale?


Given \( x_1, x_2, \cdots, x_{1993} \) that satisfy
\[
\begin{array}{l}
\left|x_1 - x_2\right| + \left|x_2 - x_3\right| + \cdots + \left|x_{1992} - x_{1993}\right| = 1993, \\
y_k = \frac{x_1 + x_2 + \cdots + x_k}{k}, \quad (k=1, 2, \cdots, 1993)
\end{array}
\]

What is the maximum possible value of \( \left|y_1 - y_2\right| + \left|y_2 - y_3\right| + \cdots + \left|y_{1992} - y_{1993}\right| \)?


There are 26 bricks, and two brothers are vying to take them. The younger brother arrives first and arranges the bricks. The elder brother arrives and thinks the younger brother has taken too many, so he takes half of the bricks from the younger brother. The younger brother, feeling confident, takes half of the bricks from the elder brother. The elder brother doesn't allow this, so the younger brother has to give 5 bricks to the elder brother. In the end, the elder brother ends up with 2 more bricks than the younger brother. How many bricks did the younger brother originally plan to take?


As shown in the figure, given that the three vertices of the triangle \( \triangle ABC \) are on the ellipse \( \frac{x^{2}}{12}+\frac{y^{2}}{4}=1 \) with the coordinate origin \( O \) being the centroid of \( \triangle ABC \). Find the area of \( \triangle ABC \).


Determine all composite positive integers \( n \) with the following property: If \( 1 = d_1 < d_2 < \ldots < d_k = n \) are all the positive divisors of \( n \), then

\[
\left(d_2 - d_1\right) : \left(d_3 - d_2\right) : \cdots : \left(d_k - d_{k-1}\right) = 1 : 2 : \cdots : (k-1).
\]


In triangle \(ABC\), a point \(D\) is marked on side \(BC\) such that \(BD: DC = 1:3\). Additionally, points \(E\) and \(K\) are marked on side \(AC\) such that \(E\) lies between \(A\) and \(K\). Segment \(AD\) intersects segments \(BE\) and \(BK\) at points \(M\) and \(N\) respectively, and it is given that \(BM: ME = 7:5\) and \(BN: NK = 2:3\). Find the ratio \(MN: AD\).


Given the set \( A = \{1, 2, 3\} \) and a function \( f: A \rightarrow A \) which satisfies \( f(f(x)) \geq x \) for any \( x \in A \), how many such functions \( f \) exist?


There are 7 different positive integers arranged in ascending order to form an arithmetic sequence. It is known that the average of the first three numbers is 20, and the average of the last three numbers is 24. Find the average of the middle three numbers.


A circle passes through the vertices $A$ and $C$ of triangle $ABC$, intersecting side $AB$ at point $E$ and side $BC$ at point $F$. The angle $AEC$ is 5 times the angle $BAF$, and the angle $ABC$ is $72^{\circ}$. Find the radius of the circle if $AC=6$.


The first term of a sequence is 2, the second term is 3, and each subsequent term is formed such that each term is 1 less than the product of its two neighbors. What is the sum of the first 1095 terms of the sequence?


Find the greatest common divisor of all five-digit numbers formed using the digits 1, 2, 3, 4, and 5 without repetition.


Dima claims to have come up with a quadratic equation that has two negative roots. The next day, Dima rearranged the coefficients of this equation and claims that now the equation has two positive roots. Could Dima be right?

Solution: \boxed{\text{No}}


A convoy of cars is moving on a highway at a speed of 80 km/h with a distance of 10 meters between the cars. Upon passing a speed limit sign, all cars reduce their speed to 60 km/h. What will be the distance between the cars in the convoy after passing the sign?

Solution: \boxed{7.5 \text{ m}}


The image depicts a top-down view of a three-layered pyramid made of 14 identical cubes. Each cube is assigned a natural number in such a way that the numbers corresponding to the cubes in the bottom layer are all different, and the number on any other cube is the sum of the numbers on the four adjacent cubes from the layer below.

Determine the smallest number divisible by four that can be assigned to the topmost cube.


Determine the volume of the square truncated pyramid if its height is 6, the side of the lower base is 4, and the side of the upper base is 2.


A turtle crawled out of its home and crawled in a straight line at a constant speed of 5 m/hour. After an hour, it turned $90^{\circ}$ (either right or left) and continued crawling, then crawled for another hour, then turned again $90^{\circ}$ (either right or left), and so on. It crawled for a total of 11 hours, turning $90^{\circ}$ at the end of each hour. What is the minimum distance it could be from its home?

Solution: \boxed{5 \text{ m}}


Let \(\mathrm{P}\) be the set \(\{1, 2, 3, \ldots, 14, 15\}\). Consider \(\mathrm{A} = \{a_{1}, a_{2}, a_{3}\}\), a subset of \(\mathrm{P}\) where \(a_{1} < a_{2} < a_{3}\) and \(a_{1} + 6 \leq a_{2} + 3 \leq a_{3}\). How many such subsets are there in \(\mathrm{P}\)?


There are 18 visitor tickets to be distributed among four classes A, B, C, and D. Class A must receive at least 1 ticket but no more than 5 tickets. Class B must receive at least 1 ticket but no more than 6 tickets. Class C must receive at least 2 tickets but no more than 7 tickets. Class D must receive at least 4 tickets but no more than 10 tickets. How many different distribution methods are there?


Given an acute triangle $ABC$, let $M$ be the orthocenter (the intersection point of its altitudes). Find angle $A$ if it is known that $AM = BC$.


a) Given a line \( l \) and two points \( A \) and \( B \) on the same side of this line, find the point \( X \) on the line \( l \) such that the sum of distances from \( X \) to points \( A \) and \( B \) is as small as possible.

b) Given a line \( l \) and two points \( A \) and \( B \) on opposite sides of it, find the point \( X \) on the line \( l \) such that the absolute value of the difference of distances from \( X \) to points \( A \) and \( B \) is as large as possible.

Solution: \boxed{X}


Suppose you forgot one digit of the phone number you need and you are dialing it at random. What is the probability that you will need to make no more than two calls?


Using 2 red beads, 2 blue beads, and 2 purple beads to form a bracelet as shown in the figure, how many different bracelets can be formed?


Find the number of non-negative integer solutions to the equation \( x_1 + x_2 + x_3 = 14 \) under the conditions \( x_1 \leq 8 \), \( x_2 \leq 8 \), and \( x_3 \leq 8 \).


For $a, b \in \mathbf{R}$, we define
$$
\max \{a, b\}=\left\{\begin{array}{ll}
a, & a \geqslant b; \\
b, & a < b,
\end{array}\right.
$$

The function $f(x) = \max \left\{2^{-x}, -|x-1| + 2\right\}$. Determine the range of real values $a$ for which the equation $f(x) = a$ has three roots.

Solution: \boxed{(1, 2)}


There are four people, A, B, C, and D, each holding a different number of cards.

A says: "I have 16 more cards than C."
B says: "D has 6 more cards than C."
C says: "A has 9 more cards than D."
D says: "If A gives me 2 cards, then I will have 3 times as many cards as C."

We know that the person with the fewest cards is lying, and the others are telling the truth. How many cards does D have?


Two people play a math game. Player A selects a group of single-digit integers \( x_{1}, x_{2}, \cdots, x_{n} \) as the secret numbers, which can be positive or negative. Player B can ask questions about the sum \( a_{1} x_{1} + a_{2} x_{2} + \cdots + a_{n} x_{n} \), where \( a_{1}, a_{2}, \cdots, a_{n} \) can be any set of numbers. Determine the minimum number of questions Player B needs to ask in order to guess the secret numbers.


Let \(\alpha\) and \(\beta\) be real numbers. If for any real numbers \(x, y, z\), the inequality \(a(xy + yz + zx) \leq M \leq \beta (x^2 + y^2 + z^2)\) always holds, where \(M = \sqrt{x^2 + xy + y^2} \cdot \sqrt{y^2 + yz + z^2} + \sqrt{y^2 + yz + z^2} \cdot \sqrt{z^2 + zx + x^2} + \sqrt{z^2 + zx + x^2} \cdot \sqrt{x^2 + xy + y^2}\), find the maximum value of \(\alpha\) and the minimum value of \(\beta\).


$M$ is a subset of $\{1,2,3, \ldots, 15\}$ such that the product of any three distinct elements of $M$ is not a square. Determine the maximum number of elements in $M$.


Set \( A = \{1, 2, \cdots, n\} \). If there exist nonempty sets \( B \) and \( C \) such that \( B \cap C = \emptyset \), \( B \cup C = A \), and the sum of the squares of the elements in \( B \) is \( M \), and the sum of the squares of the elements in \( C \) is \( N \), and \( M - N = 2016 \), find the smallest value of \( n \).


Let \( a_{1}, a_{2}, \cdots, a_{n}, \cdots \) be a non-decreasing sequence of positive integers. For \( m \geqslant 1 \), define \( b_{m}=\min \left\{n ; a_{n} \geqslant m\right\} \), i.e., \( b_{m} \) is the smallest value of \( n \) such that \( a_{n} \geqslant m \). Given that \( a_{19}=85 \), find the maximum value of
\[ a_{1}+a_{2}+\cdots+a_{19}+b_{1}+b_{2}+\cdots+b_{85}. \]


Let \( ABCD \) be a quadrilateral such that \(\angle ABC = \angle CDA = 90^\circ\), and \( BC = 7 \). Let \( E \) and \( F \) be on \( BD \) such that \( AE \) and \( CF \) are perpendicular to \( BD \). Suppose that \( BE = 3 \). Determine the product of the smallest and largest possible lengths of \( DF \).


Given that the graph of the function \( f(x)=\log_{a}(2+ax) \) and the graph of the function \( g(x)=\log_{\frac{1}{a}}(a+2x) \) (where \( a > 0 \) and \( a \neq 1 \)) are symmetric with respect to the line \( y=b \) (where \( b \) is a constant), find the value of \( a + b \).


Find all positive integers \( n \) such that \( n^{4} - n^{3} + 3n^{2} + 5 \) is a perfect square.


Let \( f(x)=x^{2}+px+q \) where \( p \) and \( q \) are some coefficients. By what minimum amount can the maximum value of the function \( g(x)=|f(x)| \) differ from the minimum value of this function on the interval \([2, 6]\)?


Find the number of integer points \((x, y)\) that satisfy the equation \(\frac{1}{|x|} + \frac{1}{|y|} = \frac{1}{2017}\).


Given that \(\tan (3 \alpha - 2 \beta) = \frac{1}{2}\) and \(\tan (5 \alpha - 4 \beta) = \frac{1}{4}\), find the value of \(\tan \alpha\).


P, Q, R are adjacent vertices of a regular 9-gon with center O. M is the midpoint of the segment joining O to the midpoint of QR and N is the midpoint of PQ. Find the ∠ONM.


In how many ways can four married couples sit around a circular table such that no man sits next to his wife?


The vertices of a regular 222-gon are colored red and blue. We will call a side monochromatic if its vertices are the same color, and dichromatic if its vertices are different colors. Is it possible to color the vertices so that the number of monochromatic and dichromatic sides are equal?

Solution: \boxed{\text{No}}


There are 111 lamps, each with its own switch. It is allowed to flip 13 of them at the same time. Initially, some lamps are on and some are off.

a) Is it possible to turn off all the lamps?

b) How many flips are required to do this if all the lamps were initially on?


Kayla went for a walk every day last week. Each day, she walked half as far as she did the day before. If she walked 8 kilometres on Monday last week, how many kilometres did she walk on Friday last week?
(A) 0.25
(B) 4
(C) 1
(D) 2
(E) 0.5


What is the exact value of \( 2^{\log _{6} 18} \cdot 3^{\log _{6} 3} \)?


Mr. Wang drives from his home to location $A$. On the way there, he drives the first $\frac{1}{2}$ of the distance at a speed of 50 km/h and increases his speed by $20\%$ for the remaining distance. On the way back, he drives the first $\frac{1}{3}$ of the distance at a speed of 50 km/h and increases his speed by $32\%$ for the remaining distance. The return trip takes 31 minutes less than the trip to $A$. What is the distance in kilometers between Mr. Wang's home and location $A$?


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


Compute the sum of all 2-digit prime numbers \( p \) such that there exists a prime number \( q \) for which \( 100q + p \) is a perfect square.


Anna randomly picked five integers from the following list:
$$
53, 62, 66, 68, 71, 82, 89
$$
and discovered that the average value of the five integers she picked is an integer. If two of the integers she picked were 62 and 89, find the sum of the remaining three integers.


Mrs. Lígia has a square-shaped plot. She decides to divide it into five regions: four rectangles and one square, as illustrated in the figure below:

In the figure above:

- The central square has an area of \(64 \, \text{m}^2\);
- The longer sides of the four rectangles have the same length;
- The five regions have the same perimeter.

Determine the area of Mrs. Lígia's plot.

Solution: \boxed{256 \text{ m}^2}


The area of triangle \( ABC \) is 1. On the rays \( AB \), \( BC \), and \( CA \), points \( B' \), \( C' \), and \( A' \) are marked respectively, such that:

\[ BB' = AB, \quad CC' = 2BC, \quad AA' = 3CA \]

Calculate the area of triangle \( A'B'C' \).


Solve the boundary value problem \(x^{2} y^{\prime \prime} + 2x y^{\prime} - 6y = 0\), given \(y(1) = 1\), and \(y(x)\) is bounded as \(x \rightarrow 0\).

Solution: \boxed{y(x) = x^2}


Let \( t \) be a positive number greater than zero.

Quadrilateral \(ABCD\) has vertices \(A(0,3), B(0,k), C(t, 10)\), and \(D(t, 0)\), where \(k>3\) and \(t>0\). The area of quadrilateral \(ABCD\) is 50 square units. What is the value of \(k\)?


Solve over the positive real numbers the functional equation:
$$
f(x y+f(x))=x f(y)+2 .
$$

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


A circle passing through the vertex \( P \) of triangle \( PQR \) touches side \( QR \) at point \( F \) and intersects sides \( PQ \) and \( PR \) at points \( M \) and \( N \), respectively, different from vertex \( P \). Find the ratio \( QF : FR \) if it is known that the length of side \( PQ \) is 1.5 times the length of side \( PR \), and the ratio \( QM : RN = 1 : 6 \).


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

$$
\forall x, y \in \mathbb{R}, \quad f(2 f(x)+f(y))=2 x+f(y)
$$

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


The maximum value of the function
$$
f(x)=\sqrt{3} \sin 2 x+2 \sin x+4 \sqrt{3} \cos x
$$
is .


The letters of the word MATHEMATICS are rearranged to form distinct strings of the same 11 letters. What proportion of these strings do not contain the string MATH?


How many five-digit positive integers can be formed by arranging the digits 1, 1, 2, 3, 4 so that the two 1s are not next to each other?


Given positive real numbers \( a, b, \) and \( c \) that satisfy \( 9a + 4b = abc \), find the minimum value of \( a + b + c \).


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


We placed 6 different dominoes in a closed chain on the table. The total number of points on the dominoes is $D$. What is the smallest possible value of $D$? (The number of points on each side of the dominoes ranges from 0 to 6, and the number of points must be the same on touching sides of the dominoes.)


Attach a single digit to the left and right of the eight-digit number 20222023 so that the resulting 10-digit number is divisible by 72. (Specify all possible solutions.)


Zach has twelve identical-looking chocolate eggs. Exactly three of the eggs contain a special prize inside. Zach randomly gives three of the twelve eggs to each of Vince, Wendy, Xin, and Yolanda. What is the probability that only one child will receive an egg that contains a special prize (that is, that all three special prizes go to the same child)?


As shown in the figure, in the unit cube \( ABCD-A_1B_1C_1D_1 \), points \( E, F, G \) are the midpoints of edges \( AA_1, C_1D_1, D_1A_1 \) respectively. Find the distance from point \( B_1 \) to the plane containing \( EFG \).


All three-digit numbers from 100 to 999 are written consecutively without spaces. Kostya underlined $k$ consecutive digits in this sequence, and Andrey underlined another $k$ consecutive digits in this sequence. The $k$-digit numbers underlined by the boys are equal. What is the largest possible value of $k$ for which this could happen?


Show that for any strictly positive real numbers \(a, b, c\), the following inequality holds:

\[
\frac{a}{\sqrt{a^{2}+8 b c}}+\frac{b}{\sqrt{b^{2}+8 c a}}+\frac{c}{\sqrt{c^{2}+8 a b}} \geqslant 1
\]


For each positive integer \( n \), define \( s(n) \) to equal the sum of the digits of \( n \). For example, \( s(2023)=2+0+2+3 \). The number of integers \( n \) with \( 100 \leq n \leq 999 \) and \( 7 \leq s(n) \leq 11 \) is \( S \). What is the integer formed by the rightmost two digits of \( S \)?


Let \( S = \{1, 2, \cdots, 10\} \). If a subset \( T \) of \( S \) has at least 2 elements and the absolute difference between any two elements in \( T \) is greater than 1, then \( T \) is said to have property \( P \). Find the number of different subsets of \( S \) that have property \( P \).


Kolya found a fun activity: he rearranges the digits of the number 2015, after which he puts a multiplication sign between any two digits and calculates the value of the resulting expression. For example: \(150 \cdot 2 = 300\), or \(10 \cdot 25 = 250\). What is the largest number he can get as a result of such a calculation?


Given that person A traveled from point X to point Y in an integer number of hours, and the number of kilometers A walks per hour is equal to the amount of time taken. Person B walks from point X to point Y at a rate of 2 kilometers per hour, taking a 1-hour rest after every 4 kilometers. B took a total of 11 hours for the trip. What is the distance between points X and Y in kilometers?

Solution: \boxed{16 \text{ km}}


Suppose that \( a, b, c, d, \) and \( e \) are consecutive positive integers with \( a < b < c < d < e \). If \( a^2 + b^2 + c^2 = d^2 + e^2 \), what is the value of \( a \)?


Given a real polynomial \( P \) and a real number \( a \), what is the remainder when \( P \) is divided by \( (X-a)? \)

Solution: \boxed{P(a)}


A circle with a radius of 3 is inscribed in a right trapezoid, where the shorter base is 4. Find the length of the longer base of the trapezoid.


You are given 16 pieces of paper numbered \(16, 15, \ldots, 2, 1\) in that order. You want to put them in the order \(1, 2, \ldots, 15, 16\) switching only two adjacent pieces of paper at a time. What is the minimum number of switches necessary?


Given \( a > b > 2 \), and \( a + b, a - b, ab, \frac{b}{a} \) form a geometric sequence in some order, find \( a \).

Solution: \boxed{7 + 5 \sqrt{2}}


A triangle \( \triangle ABC \) has its vertices lying on a circle \( C \) of radius 1, with \( \angle BAC = 60^\circ \). A circle with center \( I \) is inscribed in \( \triangle ABC \). The line \( AI \) meets circle \( C \) again at \( D \). Find the length of the segment \( ID \).


Given an isosceles triangle \( ABC \) (\( AB = BC \)). Point \( E \) is marked on ray \( BA \) beyond point \( A \), and point \( D \) is marked on side \( BC \). It is known that
$$ \angle ADC = \angle AEC = 60^\circ, \quad AD = CE = 13. $$
Find the length of segment \( AE \) if \( DC = 9 \).


Let \(AMOL\) be a quadrilateral with \(AM = 10\), \(MO = 11\), and \(OL = 12\). Given that the perpendicular bisectors of sides \(AM\) and \(OL\) intersect at the midpoint of segment \(AO\), find the length of side \(LA\).


Triangle \( \triangle ABC \) is a right-angled triangle with \( \angle ABC = 90^\circ \). A circle \( C_1 \) is drawn with \( AB \) as its diameter, and another circle \( C_2 \) is drawn with \( BC \) as its diameter. The circles \( C_1 \) and \( C_2 \) intersect at points \( B \) and \( P \). If \( AB = 5 \) cm, \( BC = 12 \) cm, and \( BP = x \) cm, find the value of \( \frac{2400}{x} \).


Find all functions that are continuous on the entire number line and satisfy the identity \(3 f(x+y) = f(x) f(y)\) and the condition \(f(1) = 12\).

Solution: \boxed{f(x) = 3 \cdot 4^x}


Given 2414 cards, each with a unique natural number from 1 to 2414. We need to choose two cards such that the sum of the numbers on them is divisible by 100. In how many ways can this be done?


A number line starts at -55 and ends at 55. If we start at -55, what percentage of the way along is the number 5.5?


One mole of an ideal gas undergoes a closed cycle in which:

$1-2$ - isobaric process, where the volume increases by 4 times;

$2-3$ - isothermal process, where the pressure increases;

$3-1$ - process in which the gas is compressed according to the law $T=\gamma V^{2}$.

Determine by what factor the volume in state 3 exceeds the initial volume in state 1.


What is the maximum number of pawns that can be placed on a chessboard (no more than one pawn per square), if:

1) A pawn cannot be placed on the e4 square;
2) No two pawns can be placed on squares that are symmetric with respect to the e4 square?


Consider each permutation of the nine digits \(1, 2, \ldots, 9\) as a nine-digit number. How many of these nine-digit numbers are divisible by 11?


The number of ordered pairs \((a, b)\) that satisfy the equation \((a+b \mathrm{i})^{6}=a-b \mathrm{i}\) where \(a, b \in \mathbf{R}\) and \(\mathrm{i}^{2}=-1\) is ___.


(a) Determine the largest possible value $M$ that $x + y + z$ can take if $x$, $y$, and $z$ are positive real numbers satisfying

$$
16 x y z = (x + y)^2 (x + z)^2
$$

(b) Show that there are infinitely many triples $(x, y, z)$ of positive rational numbers such that

$$
16 x y z = (x + y)^2 (x + z)^2 \text{ and } x + y + z = M
$$

Solution: \boxed{M = 4}


In the expression \((x+y+z)^{2030}+(x-y-z)^{2030}\), the parentheses were expanded, and like terms were collected. How many monomials of the form \(x^{a} y^{b} z^{c}\) have a nonzero coefficient?


Seven natives from several tribes are sitting around a campfire in a circle. Each tells their neighbor to the left: "There are no members of my tribe among the other five." It is known that natives lie to strangers and tell the truth to their own people. How many tribes are represented at the campfire?


Place point \( C \) at the origin of the Cartesian coordinate system and point \( B \) on the x-axis, as shown in the figure. Then points \( C, B, A \) will have coordinates \( C(0,0), B(2,0), A(1, \sqrt{3}) \).

The locus of points \( K(x, y) \) such that \( B K: C K = 1:2 \) represents a circle. Indeed,

\[
\frac{B K^{2}}{C K^{2}} = \frac{(x-2)^{2}+y^{2}}{x^{2}+y^{2}} = \frac{1}{4}
\]

which is equivalent to

\[
\left(x - \frac{8}{3} \right)^{2} + y^{2} = \frac{16}{9}.
\]

Similarly, the locus of points \( M \) such that \( A M: M C = 1:2 \) is a circle, given by the equation

\[
\left(x - \frac{4}{3}\right)^{2} + \left( y - \frac{4 \sqrt{3}}{3} \right)^{2} = \frac{16}{9}.
\]

To maximize the distance between point \( K \) (lying on circle (1)) and point \( M \) (on circle (2)), these points should be placed on a line passing through the centers of the circles.

Thus, the sought maximum distance is equal to the distance between the centers of circles (1), resulting in \( M K_{\text {max}} = \frac{8}{3} + \frac{4}{3} + \frac{4}{3} = \frac{16}{3} \).


There are 5 distinct lines on a plane, and these 5 lines form \( n \) intersections. How many different values can \( n \) have?


Given that \(\frac{1+\sin x}{\cos x}=\frac{22}{7}\) and \(\frac{1+\cos x}{\sin x}=\frac{m}{n}\), where \(\frac{m}{n}\) is in simplest form, find \(m+n\).


Let \( S(x) \) denote the sum of the digits of a positive integer \( x \). Find the maximum possible value of \( S(x+2019) - S(x) \).


In a $3 \times 3$ table, natural numbers (not necessarily distinct) are placed such that the sums in all rows and columns are different. What is the minimum value that the sum of the numbers in the table can take?


The base of a pyramid with vertex \( P \) is a quadrilateral \( ABCD \), where the sum of angles \( A \) and \( D \) is five times the sum of angles \( B \) and \( C \). Find the angle between the planes of the faces \( PAB \) and \( PCD \) if they are both perpendicular to the base.


The side of the rhombus $ABCD$ is 6. The distance between the centers of the circumcircles of triangles $ABC$ and $BCD$ is 8. Find the radii of these circumcircles.

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


Find all solutions to the equation \( x^{2} - [x] = 1 \). Here, \([x]\) is the integer part of \(x\), which is the greatest integer less than or equal to the given number. For example, \([2.9] = 2\); \([-2.9] = -3\).


Your national football coach brought a squad of 18 players to the 2010 World Cup, consisting of 3 goalkeepers, 5 defenders, 5 midfielders, and 5 strikers. Midfielders are versatile enough to play as both defenders and midfielders, while the other players can only play in their designated positions. How many possible teams of 1 goalkeeper, 4 defenders, 4 midfielders, and 2 strikers can the coach field?


A circle of radius 1 is inscribed in triangle \(ABC\), where \(\cos \widehat{ABC} = 0.8\). This circle touches the midline of triangle \(ABC\) that is parallel to side \(AC\). Find the length of side \(AC\).


As shown in the diagram, in the cube $A B C D-A_{1} B_{1} C_{1} D_{1}$, $M$ and $N$ are the midpoints of edges $A_{1} B_{1}$ and $B B_{1}$, respectively. A cross-section of the cube passes through points $D, M,$ and $N$, and the cross-section forms a polygon $\Gamma$. Find the cosine of the interior angle of $\Gamma$ at vertex $D$.


a) \( p, p+10, p+14 \) are prime numbers. Find \( p \).

b) \( p, 2p+1, 4p+1 \) are prime numbers. Find \( p \).


The lateral edges of a triangular pyramid are pairwise perpendicular, and the sides of the base are $\sqrt{61}, \sqrt{52}$, and $\sqrt{41}$. The center of a sphere that touches all the lateral faces lies on the base of the pyramid. Find the radius of this sphere.


Let \( a, b, c, d \) be 4 distinct nonzero integers such that \( a + b + c + d = 0 \) and the number \( M = (bc - ad)(ac - bd)(ab - cd) \) lies strictly between 96100 and 98000. Determine the value of \( M \).


On Qingming Festival, students take a bus to the Martyrs' Cemetery. If the bus increases its speed by one-fifth after traveling for 1 hour at its original speed, it can arrive 20 minutes earlier than the scheduled time. If the bus first travels 72 km at its original speed and then increases its speed by one-third, it can arrive 30 minutes earlier than the scheduled time. How many kilometers is it from the school to the Martyrs' Cemetery?

Solution: \boxed{216 \text{ km}}


Gina's running app tracked her average rate in minutes per kilometre. After starting the app, Gina stood still for 15 seconds and then ran at a constant rate of 7 minutes per kilometre for the rest of the run. How many kilometres did Gina run between when her app showed her average rate as 7 minutes 30 seconds per kilometre and when it showed 7 minutes 5 seconds per kilometre?

Solution: \boxed{2.5 \text{ km}}


As shown in the figure, three circles overlap to create seven sections. Fill each section with the integers from $1$ to $7$ such that the sum of the four numbers in each circle is equal. What is the maximum possible sum of the numbers in each circle?


There are 5 integers written on a board. By summing them pairwise, the following set of 10 numbers is obtained: \( 5, 9, 10, 11, 12, 16, 16, 17, 21, 23 \). Determine which numbers are written on the board. Provide the product of these numbers in your answer.


Density is defined as the ratio of mass to volume. There are two cubes. The second cube is made from a material with twice the density of the first, and the side length of the second cube is 100% greater than the side length of the first. By what percentage is the mass of the second cube greater than the mass of the first?


The password consists of four different digits whose sum equals 27. How many possible passwords are there?


A certain operation is performed on a positive integer: if it is even, divide it by 2; if it is odd, add 1. This process continues until the number becomes 1. How many integers become 1 after exactly 10 operations?


We draw the diagonals of the convex quadrilateral $ABCD$, then find the centroids of the 4 triangles formed. What fraction of the area of quadrilateral $ABCD$ is the area of the quadrilateral determined by the 4 centroids?


Let cube \( ABCD-A_1B_1C_1D_1 \) have an edge length of 1. Point \( M \) lies on the ray \( AA_1 \) and point \( N \) lies on the ray \( BC \). Given that segment \( MN \) intersects edge \( C_1D_1 \), find the minimum length of segment \( MN \).


As shown in the figure, in rectangle \(ABCD\), \(AB=4\), \(BC=6\), points \(E, F, G, H\) lie on \(AB, BC, CD, DA\) respectively, and the ratios \(AE:EB=3:1\), \(BF:FC=2:1\), \(DG:GC=1:3\), \(AH:HD=1:2\). Point \(P\) lies on \(HF\), and the area of quadrilateral \(AEPH\) is 5. Find the area of quadrilateral \(PFCG\).


Solve the system of equations:
\[
\left\{
\begin{array}{l}
x^{2} y - x y^{2} - 3 x + 3 y + 1 = 0, \\
x^{3} y - x y^{3} - 3 x^{2} + 3 y^{2} + 3 = 0.
\end{array}
\right.
\]

Solution: \boxed{(2, 1)}


From the consecutive natural numbers \(1, 2, 3, \ldots, 2014\), select \(n\) numbers such that any two selected numbers do not have one number being 5 times the other. Find the maximum value of \(n\) and explain the reasoning.


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


The three roots of the polynomial \( x^{3} - 14x^{2} + Bx - 84 \) are the sides of a right triangle. Find \( B \).


Compute the definite integral:

$$
\int_{0}^{2 \pi}\left(3 x^{2}+5\right) \cos 2 x \, dx
$$


You need to cut a wooden cube with an edge length of 40 cm into 64 smaller cubes with an edge length of 10 cm. This can be easily done with nine cuts if you do not move the cut pieces relative to each other. By repositioning the cut pieces after each cut, how much can you reduce the number of cuts?


At a rectangular plaza measuring 500 meters in length and 300 meters in width, flower pots are currently placed every 2.5 meters around the perimeter. They now want to change the spacing to every 2 meters, while keeping the flower pots at the four corners of the plaza unchanged. Determine how many additional flower pots are needed and how many flower pots do not need to be moved.


The faces of a cube are labeled with six distinct natural numbers from 6 to 11. The cube is rolled twice. The first time, the sum of the numbers on the four lateral faces is 36, and the second time, the sum is 33. What number is written on the face opposite to the face with the number 10? Justify your answer.


For a permutation \(\sigma\) of \(1,2, \ldots, 7\), a transposition is a swapping of two elements. (For instance, we could apply a transposition to the permutation \(3,7,1,4,5,6,2\) and get \(3,7,6,4,5,1,2\) by swapping the 1 and the 6.) Let \(f(\sigma)\) be the minimum number of transpositions necessary to turn \(\sigma\) into the permutation \(1,2,3,4,5,6,7\). Find the sum of \(f(\sigma)\) over all permutations \(\sigma\) of \(1,2, \ldots, 7\).


A painting is hung on a wall such that its bottom edge is $a$ meters and its top edge is $b$ meters above the observer's eye level. At what distance $x$ from the wall should the observer stand so that the painting is seen at the largest possible angle?

Solution: \boxed{x = \sqrt{ab}}


Find the minimum value for \(a, b > 0\) of the expression

$$
\frac{(3 a b - 6 b + a(1 - a))^{2} + \left(9 b^{2} + 2 a + 3 b(1 - a)\right)^{2}}{a^{2} + 9 b^{2}}
$$


A three-digit number when multiplied by 7 results in a product where the digits following the thousands place are 638. What is this three-digit number?


Find the smallest positive integer \( n \) such that \(\underbrace{2^{2 \cdot \cdot}}_{n} > 3^{3^{3^{3}}}\). (The notation \(\underbrace{2^{2 \cdot \cdot}}_{n}\) is used to denote a power tower with \( n \) 2's. For example, \(\underbrace{2^{2^{2}}}_{n}\) with \( n=4 \) would equal \( 2^{2^{2^{2}}} \).)


On the ceiling of a rectangular classroom measuring $10 \text{ m}$ in length, two lamps emitting conical light beams with an opening angle of $90^{\circ}$ are installed. The first lamp is located at the center of the ceiling and illuminates a circle on the floor with a diameter of $6 \text{ m}$. The second lamp is adjusted such that the illuminated area along the length of the classroom spans a $10 \text{ m}$ section, but the light does not reach the two opposite walls. How far apart are the two lamps?

Solution: \boxed{4 \text{ méter}}


Let $ABCD$ be a square and $S$ be a point inside the square such that the triangle $ABS$ is equilateral. Determine the angle $\widehat{DSC}$.


Peter has 5 cages with rabbits (the cages are arranged in a row). It is known that there is at least one rabbit in each cage. We will call two rabbits neighbors if they are either in the same cage or in adjacent cages. It turned out that each rabbit has either 3 or 7 neighbors. How many rabbits are in the central cage?


Alvita is planning a garden patio to be made from identical square paving stones laid out in a rectangle measuring $x$ stones by $y$ stones. She finds that when she adds a border of width one stone around the patio, the area of the border is equal to the original area of the patio. How many possible values for $x$ are there?


Consider the quadratic polynomial \( P(x) = a x^2 + b x + c \) which has distinct positive roots. Vasya wrote four numbers on the board: the roots of \( P(x) \) and the roots of the polynomial \( Q(x) = c x^2 + b x + a \). What is the smallest integer value that the sum of these four numbers can have?


At the end of the term, Vovochka wrote down his current singing grades in a row and placed multiplication signs between some of them. The product of the resulting numbers turned out to be 2007. What grade does Vovochka have for the term in singing? (The singing teacher does not give "kol" grades.)


Given an integer \( n > 4 \), the coefficients of the terms \( x^{n-4} \) and \( xy \) in the expansion of \( (x + 2 \sqrt{y} - 1)^n \) are equal. Find the value of \( n \).


What is the sum of all four-digit numbers that each use the digits 2, 4, 6, and 8 exactly once?


Let \( A, B, C \), and \( D \) be points on a circle such that \( AB = 11 \) and \( CD = 19 \). Point \( P \) is on segment \( AB \) with \( AP = 6 \), and \( Q \) is on segment \( CD \) with \( CQ = 7 \). The line through \( P \) and \( Q \) intersects the circle at \( X \) and \( Y \). If \( PQ = 27 \), find \( XY \).


In the sequence $\left\{a_{n}\right\}$, for $1 \leqslant n \leqslant 5$, we have $a_{n}=n^{2}$. Additionally, for all positive integers $n$, the following holds: $a_{n+5}+a_{n+1}=a_{n+4}+a_{n}$. Find the value of $a_{2023}$.


Suppose there is a bag containing one red, one yellow, and one blue ball. Each time a ball is drawn from the bag (the probability of drawing any particular color ball is the same), the color is noted, and then the ball is replaced into the bag. This process continues until two consecutive red balls are drawn. Let $\xi$ denote the number of draws at that point. What is the expected value of $\xi$?


Anton thought of a three-digit number, and Alex is trying to guess it. Alex successively guessed the numbers 109, 704, and 124. Anton observed that each of these numbers matches the thought number exactly in one digit place. What number did Anton think of?


Find all positive integers \(m\) and \(n\) such that the inequality
\[
[(m+n) \alpha] + [(m+n) \beta] \geq [m \alpha] + [m \beta] + [n(\alpha + \beta)]
\]
holds for any real numbers \(\alpha\) and \(\beta\). Here, \([x]\) denotes the integer part of the real number \(x\).

Solution: \boxed{m = n}


A quadrilateral is divided by its diagonals into four triangles. The areas of three of them are 10, 20, and 30, each being less than the area of the fourth triangle. Find the area of the given quadrilateral.


A square $ABCD$ has an area equal to $4 \sqrt{3}$. Its four vertices are located on a circle $\Gamma$. Suppose that the circle $\Gamma$ is tangent to the three sides $[MN],[NP]$, and $[PM]$ of an equilateral triangle $MNP$. Determine the area of the triangle $MNP$.


Let \([x]\) be the largest integer not greater than \(x\). If \(a=\left[(\sqrt{3}-\sqrt{2})^{2009}\right]+16\), find the value of \(a\).