54. How many five-digit numbers exist, in which

a) the digit 5 appears exactly once?

b) the digit 5 appears at most once?

c) the digit 5 appears at least once?


Every day at noon, a mail steamer leaves from Le Havre to New York, and at the same time, another steamer from the same company leaves New York for Le Havre. Each of these steamers takes exactly seven days to complete their journey, and they travel the same route.

How many steamers from the same company will a steamer traveling from Le Havre to New York meet on its way?


In the set of real numbers $\mathbf{R}$, define the operation $*$ satisfying:
(i) $x * 0 = 1$ (for any $x \in \mathbf{R}$).
(ii) $(x * y) * z = (z * x y) + z$ (for any $x, y, z \in $\mathbf{R}$).
Find $31 * 32$.


Petya bought himself football shorts in the store.

- If he had bought the shorts with a T-shirt, the cost of the purchase would have been twice as much.
- If he had bought the shorts with boots, the cost of the purchase would have been five times as much.
- If he had bought the shorts with shin guards, the cost of the purchase would have been three times as much.

By how many times more would the cost of the purchase be if Petya bought the shorts, T-shirt, boots, and shin guards?


Let \( c \in \mathbb{R} \) be a non-zero constant such that \( f(x+1) = f(x) + c \) and \( f\left(x^2\right) = f(x)^2 \) for all real \( x \). Show that \( c = 1 \) and \( f \) is the identity function.

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


Let $k$ be a positive integer. For any real number $x$, the real number $|x|$ is the absolute value of $x$, which is $x$ if $x$ is positive, and $-x$ if $x$ is negative. Find the number of triplets $(x, y, z)$ where $x, y, z$ are integers such that $x+y+z=0$ and $|x|+|y|+|z|=2k$.


The quadratic polynomial \( f(x) = a x^{2} + b x + c \) has exactly one root, and the quadratic polynomial \( f(3x + 2) - 2f(2x - 1) \) also has exactly one root. Find the root of the polynomial \( f(x) \).


The radius of a sphere that touches all the edges of a regular tetrahedron is 1. Find the edge length of the tetrahedron.

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


Xiao Qiao spent 660 yuan to buy a certain number of 4-yuan and 8-yuan Shanghai Expo stamps. If the number of 8-yuan stamps is 30 more than the number of 4-yuan stamps, how many stamps did Xiao Qiao buy in total?


The elder brother has 25% more money than the younger brother. What fraction of his money should the elder brother give to the younger brother so that they have an equal amount of money?


Person A and Person B start walking towards each other from points $A$ and $B$ respectively, which are 10 kilometers apart. If they start at the same time, they will meet at a point 1 kilometer away from the midpoint of $A$ and $B$. If Person A starts 5 minutes later than Person B, they will meet exactly at the midpoint of $A$ and $B$. Determine how long Person A has walked in minutes in this scenario.


Xiaoming started adding the page numbers of a book from page 1, sequentially until the end, and obtained a sum of 4979. Later, he discovered that one sheet (which includes two consecutive pages) was missing from the book. How many pages did the book originally have?


In some cells of a strip of size \(1 \times 2021\), one chip is placed in each cell. In each empty cell, a number is recorded equal to the absolute value of the difference in the number of chips to the left and right of that cell. It is known that all recorded numbers are distinct and non-zero. What is the minimum number of chips that can be placed in the cells?


In each cell of a $2017 \times 2017$ square, both diagonals are drawn. Does there exist a closed path composed of these diagonals that does not traverse any diagonal more than once and visits every cell of the square (i.e., includes at least one diagonal from each cell)?

Solution: \boxed{\text{No}}


Find the rational number that is the value of the expression

$$
\cos ^{6}(3 \pi / 16)+\cos ^{6}(11 \pi / 16)+3 \sqrt{2} / 16
$$


Two distinct numbers \( x \) and \( y \) (not necessarily integers) are such that \( x^{2} - 2000x = y^{2} - 2000y \). Find the sum of the numbers \( x \) and \( y \).


Rahid has a large number of cubic building blocks. Each block has sides of length \(4 \text{ cm}\), \(6 \text{ cm}\), or \(10 \text{ cm}\). Rahid makes little towers built from three blocks stacked on top of each other. How many different heights of tower can he make?


A circle with a radius of 2 touches another circle externally at point \( A \). The common tangent to both circles, drawn through point \( A \), intersects another common tangent to the circles at point \( B \). Find the radius of the second circle if \( AB = 4 \).


Petrov and Vasechkin were repairing a fence. Each had to nail a certain number of boards (the same number). Petrov hammered either two nails into some boards or three nails into the remaining boards. Vasechkin hammered either three nails into some boards or five nails into the remaining boards. Find out how many boards each of them nailed, given that Petrov hammered in 87 nails and Vasechkin hammered in 94 nails.


Find the smallest positive integer that, when multiplied by 1999, results in a number whose last 4 digits are 2001.


In how many ways can 4 paints be chosen from 7 different ones?


Find the non-negative integer values of \( x \) that satisfy the inequality \(\frac{x+3}{x^{2}-4} - \frac{1}{x+2} < \frac{2x}{2x - x^{2}}\).


Given a pyramid \( S A B C \) with a triangular base \( A B C \) that is equilateral, and all the plane angles at the vertex \( S \) are equal to \( \alpha \). What is the smallest \( \alpha \) for which we can claim that this pyramid is regular?


Find the coordinates of point \( A \), equidistant from points \( B \) and \( C \).

\( A(0, 0, z) \)

\( B(-6, 7, 5) \)

\( C(8, -4, 3) \)

Solution: \boxed{(0, 0, 5.25)}


Let \( ABC \) be a triangle and \( D \) the foot of the angle bisector from \( A \). Denote \( O \) (resp. \( O_1 \), resp. \( O_2 \)) as the circumcenter of \( \triangle ABC \) (resp. \( \triangle ABD \), resp. \( \triangle ADC \)). Show that \( OO_1 = OO_2 \).

Solution: \boxed{OO_1 = OO_2}


An isosceles trapezoid \(A B C D\) with bases \(A B\) and \(D C\) has an inscribed circle with center at point \(O\). Find the area of the trapezoid if \(O B = b\) and \(O C = c\).


On a board, the digit 1 is written $n$ times. An operation consists of choosing two numbers $a$ and $b$ written on the board, erasing them, and writing $\frac{a+b}{4}$ instead. Show that the number written on the board after $n-1$ steps is greater than or equal to $\frac{1}{n}$.


Find the intersection point of the line and the plane.

\[
\frac{x-1}{6}=\frac{y-3}{1}=\frac{z+5}{3}
\]

\[
3x - 2y + 5z - 3 = 0
\]

Solution: \boxed{(7, 4, -2)}


A student "illegally" simplified the exponents but got the correct answer:

\[
\frac{43^{3}+17^{3}}{43^{3}+26^{3}}=\frac{43+17}{43+26}=\frac{20}{23}
\]

Explain the reason for this "coincidence."


The curve given by the equation \( y = 2^p x^2 + 5px - 2^{p^2} \) intersects the \( Ox \) axis at points \( A \) and \( B \), and the \( Oy \) axis at point \( C \). Find the sum of all values of the parameter \( p \) for which the center of the circle circumscribed around triangle \( ABC \) lies on the \( Ox \) axis.


Given sets \( A, B, C \) (not necessarily distinct) such that \( A \cup B \cup C = \{1, 2, 3, \ldots, 11\} \), find the number of ordered pairs \((A, B, C)\) that satisfy this condition.


A swimming pool contains 200 cubic meters of water, while another contains 112 cubic meters. Faucets are turned on to fill both pools. After how many hours will the amount of water in both pools be the same, if the second pool is being filled with 22 cubic meters of water per hour more than the first pool?

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


Line segment \( PQ \) has endpoints at \( P(6,-2) \) and \( Q(-3,10) \). Point \( R(a, b) \) is a point on \( PQ \) such that the distance from \( P \) to \( R \) is one-third the distance from \( P \) to \( Q \). What is \( b-a \) ?


The points of intersection of the graphs of four functions given by the formulas \( y = kx + b \), \( y = kx - b \), \( y = mx + b \), and \( y = mx - b \) are vertices of a quadrilateral. Find the coordinates of the intersection point of its diagonals.

Solution: \boxed{(0, 0)}


The lines containing the bisectors of the exterior angles of a triangle with angle measures of 42 and 59 degrees intersect pairwise to form a new triangle. Find the degree measure of its largest angle.


Let \( K(x) \) be the number of such irreducible fractions \(\frac{a}{b}\) such that \( a < x \) and \( b < x \) ( \( a \) and \( b \) are natural numbers). For example, compute the sum \( K(100) + K\left( \frac{100}{2} \right) + K\left( \frac{100}{3} \right) + \ldots + K\left( \frac{100}{99} \right) + K\left( \frac{100}{100} \right) \).


Find the smallest natural number that, when increased by 1, would be divisible by 4, 6, 10, and 12.


The altitudes of a scalene acute triangle \(ABC\) intersect at point \(H\). \(O\) is the circumcenter of triangle \(BHC\). The incenter \(I\) of triangle \(ABC\) lies on the segment \(OA\). Find the angle \(BAC\).


In a right triangle $ABC$ with legs $AB = 3$ and $BC = 4$, a circle is drawn through the midpoints of sides $AB$ and $AC$, touching the leg $BC$. Find the length of the segment of the hypotenuse $AC$ that lies inside this circle.


On Qingming Festival, students took a car to visit the Martyrs' Cemetery. If the car increased its speed by one-fifth after traveling for 1 hour, they could arrive 10 minutes earlier than the scheduled time. If the car traveled 60 kilometers at the original speed before increasing its speed by one-third, they could arrive 20 minutes earlier than the scheduled time. How many kilometers is it from the school to the Martyrs' Cemetery?


In triangle \( ABC \) with \( AB = 8 \) and \( AC = 10 \), the incenter \( I \) is reflected across side \( AB \) to point \( X \) and across side \( AC \) to point \( Y \). Given that segment \( XY \) bisects \( AI \), compute \( BC^2 \). (The incenter \( I \) is the center of the inscribed circle of triangle \( ABC \).)


Ivan is twice as old as Peter was when Ivan was as old as Peter is now. When Peter is as old as Ivan is now, the sum of their ages will be 54 years. How old is Peter?


There are 3 identical white balls, 4 identical red balls, and 5 identical yellow balls. These balls are to be placed into 3 different boxes, allowing some boxes to contain balls of only some colors. How many different ways can this be done? (Answer in numbers).


In the three words "尽心尽力", "力可拔山", and "山穷水尽", each Chinese character represents a digit from 1 to 8. Identical characters represent the same digit, and different characters represent different digits. If the sum of the digits represented by the characters in each word is 19, and "尽" > "山" > "力", what is the maximum possible value of "水"?


Junior and Carlson ate a barrel of jam and a basket of cookies, starting and finishing at the same time. Initially, Junior ate the cookies and Carlson ate the jam, then (at some point) they switched. Carlson ate both the jam and the cookies three times faster than Junior. What fraction of the jam did Carlson eat, given that they ate an equal amount of cookies?


Replace the letters A, B, C, D, E with digits (different letters correspond to different digits) so that the difference of the three-digit numbers ABC - DEF takes the smallest possible positive value. In the answer, state the value of this difference.


Each of two sides of a triangle is divided into six equal parts; the corresponding division points are connected by line segments. Find the sum of the lengths of these segments if the third side of the triangle is equal to 12.


The chord \( AB \) of a sphere with a radius of 1 has a length of 1 and is located at an angle of \( 60^\circ \) to the diameter \( CD \) of this sphere. It is known that \( AC = \sqrt{2} \) and \( AC < BC \). Find the length of the segment \( BD \).


Consider \(A \in \mathcal{M}_{2020}(\mathbb{C})\) such that
\[ A + A^{\times} = I_{2020} \]
\[ A \cdot A^{\times} = I_{2020} \]
where \(A^{\times}\) is the adjugate matrix of \(A\), i.e., the matrix whose elements are \(a_{ij} = (-1)^{i+j} d_{ji}\), where \(d_{ji}\) is the determinant obtained from \(A\), eliminating the row \(j\) and the column \(i\).

Find the maximum number of matrices verifying these conditions such that any two of them are not similar.


Using "学" and "习" to represent two different digits, the four-digit number "学学学学" and "羽习习习习" have a product that is a seven-digit number. And the units digit and the million's place digit of the product are the same as the digit represented by "学". How many different two-digit numbers can "学习" represent?


Calculate the definite integral:

$$
\int_{0}^{\sin 1} \frac{(\arcsin x)^{2}+1}{\sqrt{1-x^{2}}} \, dx
$$


In the Cartesian plane \( xOy \), let point \( P \) be in the first quadrant, and points \( A \) and \( B \) be on the \( Ox \) and \( Oy \) axes respectively, such that \( PA = PB = 2 \). What is the maximum area of the convex quadrilateral \( PAOB \)?

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


At 8:08 AM, Xiaoming departs from home on a bicycle. Eight minutes later, his father begins chasing him on a motorcycle. His father catches up to him 4 kilometers away from home, then immediately returns home. Upon arriving home, the father immediately heads out again to chase Xiaoming and catches up to him again exactly 8 kilometers from home. What time is it when the father catches up to Xiaoming the second time?


Show that \(a b\left(a^{4}-b^{4}\right)\) is divisible by 30 if \(a\) and \(b\) are integers.


Ira paid for a book with 5-kopek coins, and Marina paid for the same book with 3-kopek coins. Together, they deposited more than 10 but fewer than 20 coins in the cash register. How much does the book cost?

Solution: \boxed{30 \text{ kopeks}}


What is the largest perfect square that can be written as the product of three different one-digit positive integers?


Find the maximum value of the expression \((\sin 2x + \sin y + \sin 3z)(\cos 2x + \cos y + \cos 3z)\). (15 points)


In this subtraction problem, \( P, Q, R, S, T \) represent single digits. What is the value of \( P + Q + R + S + T \)?

\[
\begin{array}{rrrrr}
7 & Q & 2 & S & T \\
-P & 3 & R & 9 & 6 \\
\hline
2 & 2 & 2 & 2 & 2
\end{array}
\]


Merlin has two $100 \times 100$ grids. One grid is blank and hangs on a cliff at the entrance of the cave. The other grid has a number in each cell and hangs on the wall inside the cave. You can specify a square subgrid of any size (from $1 \times 1$ to $100 \times 100$) in the blank grid and pay Merlin a shilling to find out the sum of all the numbers in the corresponding subgrid in the second grid. What is the minimum number of shillings needed to find the sum of all the numbers along the diagonal of the second grid?


Kelvin the frog lives in a pond with an infinite number of lily pads, numbered \(0,1,2,3,\) and so forth. Kelvin starts on lily pad 0 and jumps from pad to pad in the following manner: when on lily pad \(i\), he will jump to lily pad \((i+k)\) with probability \(\frac{1}{2^{k}}\) for \(k>0\). What is the probability that Kelvin lands on lily pad 2019 at some point in his journey?


The hares are cutting the log again, but now both ends of the log are fixed. Ten middle logs fell, and the two end ones remained fixed. How many cuts did the hares make?


What is the maximum number of kings that can be placed on a chessboard so that no two of them attack each other?

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


\( x, y > 0 \). Let \( S \) denote the smallest of the numbers \( x, \frac{1}{y}, y + \frac{1}{x} \). What is the maximum value that \( S \) can take?


For some real number \( c \), the graphs of the equation \( y = |x - 20| + |x + 18| \) and the line \( y = x + c \) intersect at exactly one point. What is \( c \)?


$ABCD$ is a trapezium with $AB//DC$ and $AB > DC$. $E$ is a point on $AB$ such that $AE = DC$. $AC$ meets $DE$ and $DB$ at $F$ and $G$ respectively. Find the value of $\frac{AB}{CD}$ for which $\frac{\text{Area of } \triangle DFG}{\text{Area of trapezium } ABCD}$ is maximum.


When the voltage increases, a break in the electrical circuit may occur due to the failure of one of the three series-connected elements; the probabilities of element failure are 0.2, 0.3, and 0.4, respectively. Determine the probability that the circuit will not break.


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


In the decimal notation of the six-digit number \(a\), there are no zeros and the sum of its digits is 18. Find the sum of all the different numbers obtained by cyclic permutations of its digits. In a cyclic permutation, all digits of the number, except the last one, are shifted one place to the right, and the last digit is moved to the first place.


There were no more than 70 mushrooms in the basket, among which 52% were white. If you throw out the three smallest mushrooms, the white mushrooms will become half of the total.

How many mushrooms are in the basket?


Find all prime numbers \( p \) such that the number \( p^{2} + 11 \) has exactly six distinct divisors (including 1 and the number itself).


Calculate the minimum of $\sqrt{(x-1)^{2}+y^{2}}+\sqrt{(x+1)^{2}+y^{2}}+|2-y|$ for $x, y \in \mathbb{R}$.

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


If \( A^2 + B^2 + C^2 = AB + BC + CA = 3 \) and \( a = A^2 \), find the value of \( a \).

Given that \( n \) and \( b \) are integers satisfying the equation \( 29n + 42b = a \). If \( 5 < b < 10 \), find the value of \( b \).

If \(\frac{\sqrt{3}-\sqrt{5}+\sqrt{7}}{\sqrt{3}+\sqrt{5}+\sqrt{7}}=\frac{c\sqrt{21}-18\sqrt{15}-2\sqrt{35}+b}{59}\), find the value of \( c \).

If \( c \) has \( d \) positive factors, find the value of \( d \).


In quadrilateral \(ABCD\), we have \(AB=5\), \(BC=6\), \(CD=5\), \(DA=4\), and \(\angle ABC=90^\circ\). Let \(AC\) and \(BD\) meet at \(E\). Compute \(\frac{BE}{ED}\).


Given non-negative real numbers \( a_{1}, a_{2}, \cdots, a_{n} \) satisfying
\[ a_{1} + a_{2} + \cdots + a_{n} = 1, \]
find the minimum value of \( S = \sum_{i=1}^{n} \frac{a_{i}}{2 - a_{i}} \).

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


For which values of the parameter \( p \) will the equation

\[
\sqrt[3]{x} + \sqrt[3]{2 - x} = p
\]

have exactly one solution?

Solution: \boxed{p = 2}


There are \( n \) expressions of the form \({ }^{*} \chi^{2} + *_{X} + * = 0\) written on the board (\( n \) is an odd number). Two players play a game where they take turns. On each turn, one is allowed to replace one of the asterisks with a non-zero number. After \( 3n \) moves, \( n \) quadratic equations are completed. The first player aims to maximize the number of these equations that have no roots, while the second player tries to prevent this. What is the maximum number of equations without roots that the first player can achieve regardless of the second player's strategy?


Let \( O \) be the center of the circle inscribed in a rhombus \( ABCD \). Points \( E, F, G, H \) are chosen on sides \( AB, BC, CD \), and \( DA \) respectively so that \( EF \) and \( GH \) are tangent to the inscribed circle. Show that \( EH \) and \( FG \) are parallel.

Solution: \boxed{EH \parallel FG}


\( \textbf{Al} \) has a rectangle of integer side lengths \( a \) and \( b \), and area 1000. What is the smallest perimeter it could have?


The number of elements in the set $\left\{(x, y) \mid x^{2}+y^{2} \leq 100 \text{ and } x, y \in \mathbf{Z}\right\}$


After calculating the number \(8^{2021}\), compute the sum of its digits and record the result. Then, in the new recorded number, compute the sum of its digits and record the result again. Repeat these actions until a single-digit number is obtained. Find this single-digit number.


A truck and a passenger car are moving in the same direction on adjacent lanes at speeds of 65 km/h and 85 km/h, respectively. What distance will be between them 3 minutes after they are side by side?

Solution: \boxed{1 \, \text{km}}


Compute, for every integer \( n \geq 3 \), the greatest common divisor (GCD) of \( n^3 - 6n^2 + 11n - 6 \) and \( n^2 - 4n + 4 \).

Solution: \boxed{n - 2}


Find the length of the vector: a) \(\vec{a} = (5, 12)\); b) \(\vec{b} = (7, -1)\).

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


In rectangle \(ABCD\), \(AB=1\) and \(BC=2\). Point \(E\) is on side \(BC\), and point \(F\) is on side \(CD\). If the length of \(|\overrightarrow{EF}| = 1\), find the minimum value of \(\overrightarrow{AE} \cdot \overrightarrow{AF}\).

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


In the Cartesian coordinate system $xOy$, if the line $l$ bisects the circle $x^{2}+y^{2}-2x-4y=0$ and does not pass through the fourth quadrant, what is the range of values for the slope of $l$?

Solution: \boxed{[0, 2]}


Find the maximum value of the function \( f(x) = 6 \sin x + 8 \cos x \).


An equilateral triangle with a side length of 3 is divided into nine triangular cells, as shown in the figure. Initially, zeros are written in these cells. In one move, you can choose two numbers located in adjacent cells and either add one to both or subtract one from both. Petya wants to make several moves such that, after these moves, the cells contain consecutive natural numbers \( n, n+1, \ldots, n+8 \) in some order. For which \( n \) can he achieve this?


Submit an integer \( x \) as your answer to this problem. The number of points you receive will be \(\max (0, 8 - |8x - 100|)\). (Non-integer answers will be given 0 points.)


The function \( f(n) \) is defined for all positive integers \( n \) and takes on non-negative integer values such that \( f(2)=0 \), \( f(3)>0 \), and \( f(9999)=3333 \). Also, for all \( m, n \):
$$
f(m+n)-f(m)-f(n)=0 \quad \text{or} \quad 1.
$$

Determine \( f(2005) \).


In a draft of an arithmetic workbook, there was an example in which a given number needed to be multiplied by 3 and 4 subtracted from the result. In the print, an error occurred: instead of the multiplication sign, a division sign was placed, and instead of the minus sign, a plus sign was used. Nevertheless, the final result did not change. What was the original example intended to be placed in the workbook?

Solution: \boxed{3 \cdot 3 - 4}


The formula to convert Celsius to Fahrenheit is
$$
F = 1.8 \cdot C + 32.
$$

In Celsius, it is $10^{\circ}$ warmer in New York right now than in Boston. In Fahrenheit, how much warmer is it in New York than in Boston?


A cube with a side length of 6 is cut into several smaller cubes with side lengths that are integers. If the total surface area of these smaller cubes is twice the surface area of the original large cube, what is the side length of the smaller cubes?


Around the campfire, eight natives from four tribes are sitting in a circle. Each native tells their neighbor to the left: "If you don't count us, there is no one from my tribe here." It is known that natives lie to outsiders and tell the truth to their own. How many natives can there be from each tribe?


A hare is jumping in one direction on a strip divided into cells. In one jump, it can move either one cell or two cells. How many ways can the hare get from the 1st cell to the 12th cell?


Find all natural numbers $n$ such that the number $2^{n} + n^{2} + 25$ is a cube of a prime number.


Let \( p, q, r \) be positive real numbers and \( n \in \mathbb{N} \). Show that if \( pqr=1 \), then

\[
\frac{1}{p^{n}+q^{n}+1}+\frac{1}{q^{n}+r^{n}+1}+\frac{1}{r^{n}+p^{n}+1} \leq 1
\]


Find a three-digit number that is divisible by 45, given that the difference between this number and its reversed version is 297.


Given the line \( l: 2x + y = 10 \), construct a line \( l^{\prime} \) passing through the point \((-10,0)\) such that \( l^{\prime} \) is perpendicular to \( l \). What are the coordinates of the intersection point of \( l^{\prime} \) and \( l \)?

Solution: \boxed{(2, 6)}


The parallelogram \(ABCD\) is such that \(\angle B < 90^\circ\) and \(AB < BC\). Points \(E\) and \(F\) are chosen on the circumcircle \(\omega\) of triangle \(ABC\) such that the tangents to \(\omega\) at these points pass through \(D\). It is given that \(\angle EDA = \angle FDC\). Find the angle \(ABC\).


In triangle \( \triangle ABC \), the incircle \( \odot O \) is tangent to \( BC \), \( CA \), and \( AB \) at points \( D \), \( E \), and \( F \) respectively. Connect \( AD \) and let it intersect the incircle \( \odot O \) again at point \( P \). Connect \( BP \) and \( CP \). The necessary and sufficient condition for \(\angle BPC = 90^\circ\) is \( AE + AP = PD \).

Solution: \boxed{AE + AP = PD}


For any two different real numbers \( x \) and \( y \), we define \( D(x, y) \) to be the unique integer \( d \) satisfying \( 2^{d} \leq |x-y| < 2^{d+1} \). Given a set of real numbers \(\mathcal{F}\), and an element \( x \in \mathcal{F} \), we say that the scales of \( x \) in \(\mathcal{F}\) are the values of \( D(x, y) \) for \( y \in \mathcal{F} \) with \( x \neq y \).

Let \( k \) be a given positive integer. Suppose that each member \( x \) of \(\mathcal{F}\) has at most \( k \) different scales in \(\mathcal{F}\). What is the maximum possible size of \(\mathcal{F}\)?


Given 50 numbers, it is known that there are exactly 500 negative pairwise products among them. Determine the number of zeros among these numbers.


The distances from a point \(M\), lying inside triangle \(ABC\), to its sides \(AC\) and \(BC\) are 2 cm and 4 cm, respectively. Calculate the distance from point \(M\) to the line \(AB\), given that \(AB = 10 \text{ cm}\), \(BC = 17 \text{ cm}\), and \(AC = 21 \text{ cm}\).

Solution: \boxed{5.8 \ \text{cm}}


Let \(A_{n}\) and \(B_{n}\) be the sums of the first \(n\) terms of two arithmetic sequences \(\{a_n\}\) and \(\{b_n\}\), respectively. If \(\frac{A_{n}}{B_{n}} = \frac{5n - 3}{n + 9}\), find the value of \(\frac{a_6}{b_3}\).


A square piece of paper has 1965 holes pierced in its interior, and none of these holes are collinear with any combination of the square's vertices. A number of non-intersecting line segments are drawn, with their endpoints being these holes or the square's vertices, to partition the square into triangles, ensuring that there are no additional holes on the edges or interior of these triangles. How many line segments were drawn in total? How many triangles were obtained?


Given the positive real numbers \( a \) and \( b \) that satisfy \( a^b = (8a)^{9b} \), find the value of \( \log_a (4a^2) \).


Points \( A \) and \( B \) lie in the plane \( \alpha \). Let \( M \) be a point in space such that \( AM = 2 \), \( BM = 5 \), and the orthogonal projection of segment \( BM \) onto the plane \( \alpha \) is three times the orthogonal projection of segment \( AM \) onto this plane. Find the distance from point \( M \) to the plane \( \alpha \).


There are several bowls on a table, each containing some grapes. Different bowls may have different numbers of grapes. If 8 more grapes are added to each of 12 bowls, the average number of grapes in all the bowls increases by 6. How many bowls are on the table?


Let $[x]$ denote the greatest integer less than or equal to $x$. Find the value of $\sum_{k=1}^{2008}\left[\frac{2008 k}{2009}\right]$.


A cold drink shop launched a "half price for the second cup of summer drinks" promotion. Xiao Gang bought 2 cups of drinks for a total of 13.5 yuan. What is the original price of one cup of the drink?


What is the maximum number of spheres with radius 7 that can simultaneously, without intersecting, touch a sphere with a radius of 3?


Define the sequence \(\{a_n\}\) as \(a_n = n^3 + 4\), where \(n \in \mathbb{N}_+\). Let \(d_n = (a_n, a_{n+1})\) which is the greatest common divisor of \(a_n\) and \(a_{n+1}\). What is the maximum value of \(d_n\)?


A line segment is divided into four parts by three randomly selected points. What is the probability that these four parts can form the four sides of a quadrilateral?


Let \( m \) and \( n \) both be positive integers such that \( 24m = n^4 \). What is the minimum value of \( m \)?


Andrey was driving to the airport in a neighboring city. After driving for an hour at a speed of 60 km/h, he realized that if he did not change his speed, he would be 20 minutes late. So, he sharply increased his speed, and as a result, he covered the remaining distance at an average speed of 90 km/h, arriving at the airport 20 minutes earlier than initially planned. What is the distance from Andrey's home to the airport?

Solution: \boxed{180 \text{ km}}


Pentagon \( A B C D E \) is inscribed in circle \( \omega \). Diagonal \( A C \) is the diameter of the circle \( \omega \). Find \(\angle B E C\) if \(\angle A D B = 20^\circ\).


Determine the distinct digits \(A\) and \(B\) such that the following multiplication holds: \(A \cdot \overline{AB} = \overline{BAA}\) (in decimal system).

Solution: \boxed{A = 8, B = 6}


Numbers \( a \), \( b \), and \( c \) satisfy the equation \( \sqrt{a} = \sqrt{b} + \sqrt{c} \). Find \( a \) if \( b = 52 - 30 \sqrt{3} \) and \( c = a - 2 \).

Solution: \boxed{a = 27}


Let \( n \) be a positive integer, \( [x] \) be the greatest integer less than or equal to the real number \( x \), and \( \{x\}=x-[x] \).
(1) Find all positive integers \( n \) satisfying \( \sum_{k=1}^{2013}\left[\frac{k n}{2013}\right]=2013+n \);
(2) Find all positive integers \( n \) that make \( \sum_{k=1}^{2013}\left\{\frac{k n}{2013}\right\} \) attain its maximum value, and determine this maximum value.


A stationery store is running a promotion: each notebook has a sticker, and for every 5 stickers, a customer can receive another notebook (also with a sticker). Fifth-grader Katya believes she needs to buy as many notebooks as possible before the new semester. Each notebook costs 4 rubles, and Katya has 150 rubles. How many notebooks will Katya get?


How many 9-digit numbers that are divisible by 5 can be formed by permuting the digits of the number 377353752?


Petya can draw only 4 things: a sun, a ball, a tomato, and a banana. Today he drew several things, including exactly 15 yellow items, 18 round items, and 13 edible items. What is the maximum number of balls he could have drawn?

Petya believes that all tomatoes are round and red, all balls are round and can be of any color, and all bananas are yellow and not round.


A three-digit number begins with the digit 4. If this digit is moved to the end of the number, the resulting number is 0.75 times the original number. Find the original number.


Dmitry has socks in his closet: 10 pairs of blue socks, 22 pairs of black socks, and 12 pairs of white socks. Dmitry bought several more pairs of black socks and found that now the black socks account for $2 / 3$ of the total number of socks. How many pairs of black socks did Dmitry buy?


Arnaldo and Bernaldo decide to play a game with a limited number of turns. Arnaldo writes the number 1 on the board on his first turn. Then, Bernaldo writes 2 or 4 on the board. After that, Arnaldo writes 3 or 9 on the board. They continue playing alternately, following the rule that on turn $n$, the player writes $n$ or $n^{2}$ on the board. Arnaldo wins the game if, after the final turn, the sum of the numbers on the board is divisible by 3. If the sum is not divisible by 3, then Bernaldo wins.

a) Assume the game ends on turn 15. Show that Bernaldo can guarantee victory.

b) Assume the game ends on turn 7. In this case, which player can always guarantee victory regardless of how their opponent plays? How should they play to win?

Solution: \boxed{Arnaldo}


Compute the limit of the numerical sequence:

$$
\lim _{n \rightarrow \infty} \frac{(n+7)^{3}-(n+2)^{3}}{(3 n+2)^{2}+(4 n+1)^{2}}
$$


Let us call a number remarkable if it can be represented as the sum of 2023 addends (not necessarily distinct), each of which is a natural composite number. Find the largest integer that is not remarkable.


If the graph of the function \( f(x) \) is symmetric about the points \( (a, b) \) and \( (m, n) \) (where \( a \neq m \)), then for every \( x \) in the domain of the function \( f(x) \), the sequence \( \left\{ y_{k} \right\} \) formed by \( y_{k} = f[x + 2k(a - m)] \) (\( k \in \mathbf{Z} \)) is an arithmetic sequence with a common difference of \( 2(b - n) \).


Given real numbers \( x \) and \( y \) satisfy \( x^2 + y^2 = 20 \). Find the maximum value of \( xy + 8x + y \).


In the center of a circular field stands the geologists' house. From it, 6 straight roads extend, dividing the field into 6 equal sectors. Two geologists set off on a journey from their house at a speed of 5 km/h along randomly chosen roads. Determine the probability that the distance between them after one hour will be more than 8 km.


Determine the sum of the real numbers \( x \) for which \(\frac{2 x}{x^{2}+5 x+3}+\frac{3 x}{x^{2}+x+3}=1\).


Let \( f(x) \) be a function defined on \(\mathbf{R}\). Given that \( f(x) + x^{2} \) is an odd function and \( f(x) + 2^{x} \) is an even function, find the value of \( f(1) \).


In triangle \(ABC\), \(D\) is an internal point on side \(AB\). The incircle of triangle \(ABC\) touches side \(AB\) at point \(P\), while the incircles of triangles \(ADC\) and \(DBC\) touch side \(DC\) at points \(Q\) and \(R\), respectively.

Show that \(DP = QR\).

Solution: \boxed{DP = QR}


Emily's broken clock runs backwards at five times the speed of a regular clock. How many times will it display the correct time in the next 24 hours? Note that it is an analog clock that only displays the numerical time, not AM or PM. The clock updates continuously.


Given that a line passes through the intersection of the lines \(3x + 2y - 5 = 0\) and \(2x + 3y - 5 = 0\), and has an intercept of -5 on the y-axis, what is the equation of this line?

Solution: \boxed{6x - y - 5 = 0}


There were 100 cards lying face up with the white side on the table. Each card has one white side and one black side. Kostya flipped 50 cards, then Tanya flipped 60 cards, and after that Olya flipped 70 cards. As a result, all 100 cards ended up lying with the black side up. How many cards were flipped three times?


A magician has a set of $16^{2}$ unique cards. Each card has one red side and one blue side; on each card, there is a natural number between 1 and 16 written on both sides. We will call a card a "duplicate" if the numbers on both sides of the card are the same. The magician wants to draw two cards such that at least one of them is a duplicate and no number appears on both drawn cards at the same time. In how many ways can he do this?


Farmer John is inside an ellipse with reflective sides, described by the equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), with \(a > b > 0\). He is standing at the point \((3, 0)\) and shines a laser pointer in the \(y\)-direction. The light reflects off the ellipse and proceeds directly toward Farmer Brown, traveling a distance of 10 before reaching him. Farmer John then spins around in a circle; wherever he points the laser, the light reflects off the wall and hits Farmer Brown. What is the ordered pair \((a, b)\)?

Solution: \boxed{(5, 4)}


As shown in the diagram, two lines intersect with two circles at 9 points. Choose 4 points from these 9 points such that any 3 points among these 4 are neither collinear nor lie on the same circle. How many different selections can be made?


There are \( n \) different positive integers, each one not greater than 2013, with the property that the sum of any three of them is divisible by 39. Find the greatest value of \( n \).


We construct a channel from three boards of equal width. The side walls form an angle of $x$ with the base board. What value of the angle $x$ will make the cross-sectional area (and thus the volume) of the channel (which forms the shape of an isosceles trapezoid) the largest possible?


If the real number $\alpha$ satisfies $\cos \alpha = \tan \alpha$, then the value of $\frac{1}{\sin \alpha} + \cos^4 \alpha$ is $\qquad$


In a math competition, 5 problems were assigned. There were no two contestants who solved exactly the same problems. However, for any problem that is disregarded, for each contestant there is another contestant who solved the same set of the remaining 4 problems. How many contestants participated in the competition?


The radius of the circumscribed circle $R$ is known.

a) Find $AP^{2} + BP^{2} + CP^{2} + DP^{2}$.

b) Find the sum of the squares of the sides of the quadrilateral $ABCD$.


Suppose \( d \) is the number of positive factors of \( 500 + c \). Find the value of \( d \).


\[
\frac{x^{3}}{x^{3}+2 y^{2} z}+\frac{y^{3}}{y^{3}+2 z^{2} x}+\frac{z^{3}}{z^{3}+2 x^{2} y} \geqslant 1
\]

On the table, there is a pile of \( n \) stones. In one move, any existing pile can be split into two smaller piles. At the same time, the sizes of any two piles on the table must be "similar."


Mário completed 30 hours in an extra math course. On days when he had class, it lasted only 1 hour, and it took place exclusively in the morning or in the afternoon. Additionally, there were 20 afternoons and 18 mornings without classes during the course period.

a) On how many days were there no classes?

b) How many days did the course last?


In a convex hexagon, two diagonals are chosen independently at random. Find the probability that these diagonals intersect inside the hexagon (inside meaning not at a vertex).


Two bodies simultaneously started moving in a straight line from a certain point in the same direction with velocities \( v_{1} = 6t^2 + 4t \) m/s and \( v_{2} = 4t \) m/s. After how many seconds will the distance between them be 250 meters?

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


Determine all natural numbers \( n \) so that for any real numbers \( x_{1}, \ldots, x_{n} \), the following inequality holds:

$$
\left(\frac{x_{1}^{n}+\ldots+x_{n}^{n}}{n}-x_{1} \cdot \ldots \cdot x_{n}\right)\left(x_{1}+\ldots+x_{n}\right) \geq 0
$$

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


Given that the points \((2,1)\) and \((5,-1)\) are on opposite sides of the line \(3y - 2x - m = 0\), determine the range of values for \(m\).

A. \(\frac{4}{3} < m < \frac{17}{3}\)

B. \(-13 < m < -1\)

C. \(4 < m < 17\)

D. \(\frac{3}{2} < m < \frac{39}{2}\)

Solution: \boxed{B. -13 < m < -1}


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

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

Find \( g(5) \).


Determine two integers $u$ and $v$ such that $364u + 154v = 14$.

Solution: \boxed{u = 3, v = -7}


Point \( A \) has coordinates \((-13, -23)\) and point \( B \) has coordinates \((-33, -43)\). What is the slope of a line that is perpendicular to the line segment \( AB \)?


Let point \( P \) be on the ellipse \( \frac{x^{2}}{5}+y^{2}=1 \), and let \( F_{1} \) and \( F_{2} \) be the two foci of the ellipse. If the area of \( \triangle F_{1} P F_{2} \) is \( \frac{\sqrt{3}}{3} \), find \( \angle F_{1} P F_{2} \).


Let \( n \) be a fixed positive integer, and choose \( n \) positive integers \( a_{1}, \ldots, a_{n} \). Given a permutation \( \pi \) on the first \( n \) positive integers, let \( S_{\pi}=\left\{i \left\lvert\, \frac{a_{i}}{\pi(i)}\right.\right. \text{ is an integer} \} \). Let \( N \) denote the number of distinct sets \( S_{\pi} \) as \( \pi \) ranges over all such permutations. Determine, in terms of \( n \), the maximum value of \( N \) over all possible values of \( a_{1}, \ldots, a_{n} \).

Solution: \boxed{2^n - n}


Given that \( a, b, c, d \) are certain real values, the equation \( x^{4}+a x^{3}+b x^{2}+c x+d=0 \) has 4 non-real roots. Among these roots, the product of 2 of the roots is \( 13 + \mathrm{i} \) and the sum of the other 2 roots is \( 3 + 4\mathrm{i} \). Here, \( \mathrm{i} \) represents the imaginary unit. Find \( b \).


A company produces a lemon drink by diluting lemon juice with water. Initially, the company manufactured a drink containing 15% lemon juice. After some time, the CEO ordered to reduce the lemon juice content to 10%. By what percentage will the quantity of the produced lemon drink increase with the same supply volumes of lemons?


Points $M$ and $N$ are located on side $AC$ of triangle $ABC$, and points $K$ and $L$ are on side $AB$, with $AM : MN : NC = 1 : 3 : 1$ and $AK = KL = LB$. It is known that the area of triangle $ABC$ is 1. Find the area of quadrilateral $KLNM$.


In cube \(ABCD A_1 B_1 C_1 D_1\), where \(A A_1\), \(B B_1\), \(C C_1\), and \(D D_1\) are parallel edges, plane \(P\) passes through point \(D\) and the midpoints of edges \(A_1 D_1\) and \(C_1 D_1\). Find the distance from the midpoint of edge \(A A_1\) to plane \(P\), given that the edge length of the cube is 2.


In the quadrilateral \( MARE \) inscribed in a unit circle \(\omega\), \( AM \) is a diameter of \(\omega\), and \( E \) lies on the angle bisector of \(\angle RAM\). Given that triangles \( RAM \) and \( REM \) have the same area, find the area of quadrilateral \( MARE \).


Give the closed form (as a function of $k$ and $n$) for the sum of the following series:

$$
\binom{k}{1} \sum_{i=1}^{n} i+\binom{k}{2} \sum_{i=1}^{n} i^{2}+\ldots+\binom{k}{k-1} \sum_{i=1}^{n} i^{k-1}
$$

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


If \( A : B = 1 \frac{2}{3} : 4 \frac{5}{6} \) and \( C : A = 2 \frac{1}{5} : 3 \frac{2}{3} \), then express \( A : B : C \) in the simplest integer ratio.

Solution: \boxed{10: 29: 6}


Place the numbers $0, 1, 2, 3, 4, 5, 6, 7$ on the eight vertices of a cube (each vertex having one number, with each number used exactly once) in such a way that the sum of the two numbers at the ends of each edge is always a prime number. What is the maximum sum of the four numbers on one face of the cube?


Compute the definite integral:

$$
\int_{0}^{1} \frac{(4 \sqrt{1-x}-\sqrt{x+1}) d x}{(\sqrt{x+1}+4 \sqrt{1-x})(x+1)^{2}}
$$

Solution: \boxed{\frac{1}{8} \ln 5}


Let $O$ be the incenter of $\triangle ABC$, with $AB=5$, $AC=6$, and $BC=7$. If $OP = xOA + yOB + zOC$, where $0 \leq x, y, z \leq 1$, determine the area of the plane region covered by the locus of point $P$.


Determine the greatest common divisor (gcd) of all the numbers of the form

$$
(a-b)(b-c)(c-d)(d-a)(a-c)(b-d)
$$

where $a, b, c, d$ are integers.


Find the smallest natural number that is divisible by 28 and leaves a remainder of 4 when divided by 15.


Let \( S \) be a set of \( n \) distinct real numbers, and \( A_S \) be the set consisting of the averages of all pairs of distinct elements from \( S \). Given \( n \geq 2 \), what is the minimum number of elements that \( A_S \) can have?


Line \( l \) passes through point \( A(2,0) \) and intersects the parabolic curve \( y = x^{2} + 2 \) at points \( B \) and \( C \). Points \( B_{1} \) and \( C_{1} \) are the feet of the perpendiculars from \( B \) and \( C \) to the \( x \)-axis, respectively. Point \( P \) is on the line segment \( BC \) such that \(\overrightarrow{B P} = \frac{\left|B B_{1}\right|}{\left|C C_{1}\right|} \cdot \overrightarrow{P C}\). Find the equation of the locus of the centroid \( G \) of triangle \( \triangle POA \) (where \( O \) is the origin).

Solution: \boxed{12x - 3y - 4 = 0}


Solve the system of equations:
\[
\begin{cases}
\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 3, \\
\frac{1}{xy} + \frac{1}{yz} + \frac{1}{zx} = 3, \\
\frac{1}{xyz} = 1.
\end{cases}
\]

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


A car traveled from one city to another with a load at a speed of 80 km/h and returned empty at a speed of 120 km/h. Find the average speed of the car for the entire route. Give the answer in kilometers per hour, rounding to the nearest whole number if necessary.


Let \( n \) be the maximum number of bishops that can be placed on the squares of a \( 6 \times 6 \) chessboard such that no two bishops are attacking each other. Let \( k \) be the number of ways to put \( n \) bishops on a \( 6 \times 6 \) chessboard such that no two bishops are attacking each other. Find \( n + k \). (Two bishops are considered to be attacking each other if they lie on the same diagonal. Equivalently, if we label the squares with coordinates \((x, y)\), with \( 1 \leq x, y \leq 6 \), then the bishops on \((a, b)\) and \((c, d)\) are attacking each other if and only if \( |a - c| = |b - d| \).)


The sequence consists of 19 ones and 49 zeros arranged in random order. We define a group as the maximal subsequence of identical symbols. For example, in the sequence 110001001111 there are five groups: two ones, followed by three zeros, then one one, then two zeros, and finally four ones. Find the expected length of the first group.


Let \( x \) be a real number such that \( x^{2} - 15x + 1 = 0 \). Find the value of \( x^{4} + \frac{1}{x^{4}} \).


In the cyclic quadrilateral \(ABCD\), there is a point \(X\) on side \(AB\) such that the diagonal \(BD\) bisects \(CX\) and \(AC\) bisects \(DX\). Find the minimum value of \(\frac{AB}{CD}\).


The positive integer \( N \) has five digits. The six-digit integer \( P \) is formed by appending the digit 2 to the front of \( N \). The six-digit integer \( Q \) is formed by appending the digit 2 to the end of \( N \). Given that \( Q = 3P \), what values of \( N \) are possible?


There are 38 ones written on a board. Each minute, Carlson erases two random numbers and writes their sum on the board, then eats an amount of candies equal to the product of the two erased numbers. What is the maximum number of candies he could have eaten in 38 minutes?


I live in a five-story building with four entrances, with four apartments on each landing. What is the minimum number of questions you can ask to determine which apartment I live in, if I answer all your questions truthfully, but only with "yes" or "no"?


Let $a_{n}$ denote the number of numbers among $2^{1}, 2^{2}, \ldots, 2^{n}$ that begin with the digit 1. Calculate the limit $\lim _{n \rightarrow \infty} \frac{a_{n}}{n}$.


The function \( f(x) \) is such that for all natural \( n > 1 \), there exists a prime divisor \( p \) of the number \( n \) such that

\[
f(n) = f\left(\frac{n}{p}\right) - f(p)
\]

It is known that \( f(1001) = 1 \). What is \( f(1002) \)?


Given the function \( f(x) = ax^3 + bx^2 + cx + d \) (where \( a \neq 0 \)), and the condition that \( \left| f'(x) \right| \leq 1 \) for \( 0 \leq x \leq 1 \), find the maximum value of \( a \).


Suppose \( r_{1}, r_{2}, r_{3} \) are the three roots of the real-coefficient equation
\[ x^{3} - x^{2} + ax - b = 0 \]
where \( 0 < r_{i} < 1 \) for \( i = 1, 2, 3 \). Find the maximum value of \( 7a - 9b \).


Five identical balls roll on a smooth horizontal surface towards each other. The speeds of the first and second balls are \( v_{1} = v_{2} = 0.5 \mathrm{~m/s} \), and the speeds of the others are \( v_{3} = v_{4} = v_{5} = 0.3 \mathrm{~m/s} \). The initial distances between the balls are all equal to \( l = 1 \mathrm{~m} \). All collisions are perfectly elastic. After what time will the last collision occur in this system?

1

2

3

4

5

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


There are 15 cities in a country, some of which are connected by roads. Each city is assigned a number equal to the number of roads leading out of it. It turns out that there are no roads between cities with the same number of roads. What is the maximum number of roads that can be in the country?


A batch of feed can supply 10 ducks and 15 chickens for 6 days, or 12 ducks and 6 chickens for 7 days. How many ducks can this batch of feed supply for 21 days?


In rectangle \(ABCD\), a diagonal \(AC\) and a line \(OY\) are drawn. The line \(OY\) divides side \(AB\) in the ratio \(1:3\) starting from vertex \(B\) and side \(CD\) in the ratio \(1:3\) starting from vertex \(D\). Find what fraction of the area of the rectangle is the smallest one among the four resulting parts.


In the number \(2 * 0 * 1 * 6 * 0 * 2 *\), each of the 6 asterisks needs to be replaced with any of the digits \(1,2,3,4,5,6,7,8,9\) (digits may repeat) so that the resulting 12-digit number is divisible by 18. How many ways can this be done?


The hourly teaching fees of Teacher Huanhuan and Teacher Xixi are in the ratio of 5:4. The company decided to accelerate the training for these two assistant teachers, and increased their hourly teaching fees by 20 yuan each. After the increase, the ratio of their hourly teaching fees became 6:5. After the increase, the sum of their hourly teaching fees is $\qquad$ yuan.

Solution: \boxed{220 \, \text{yuan}}


In a circle with a radius of $10 \text{ cm}$, segment $AB$ is a diameter and segment $AC$ is a chord of $12 \text{ cm}$. Determine the distance between points $B$ and $C$.

Solution: \boxed{16 \, \text{cm}}


For \(0 \leq y \leq 2\), let \(D_y\) be the half-disk of diameter 2 with one vertex at \((0, y)\), the other vertex on the positive \(x\)-axis, and the curved boundary further from the origin than the straight boundary. Find the area of the union of \(D_y\) for all \(0 \leq y \leq 2\).

Solution: \boxed{\pi}


Find the GCD (Greatest Common Divisor) of all the numbers of the form \( n^{13} - n \).


Dasha calls a natural number special if four different digits are used to write it. For example, the number 3429 is special, while the number 3430 is not special.

What is the smallest special number greater than 3429?


The symbol \(\begin{tabular}{l|l}
3 & 4 \\
\hline 5 & 6 \\
\end{tabular}\) is evaluated as \(3 \times 6 + 4 \times 5 = 38\). If \(\begin{tabular}{l|l}
2 & 6 \\
\hline 1 & \, \\
\end{tabular}\) is evaluated as 16, then the number that should be placed in the empty space is

(A) 1
(B) 2
(C) 3
(D) 4
(E) 5


For which natural numbers \( n \), for any numbers \( \alpha, \beta, \gamma \) that are the angles of an acute-angled triangle, the inequality

$$
\sin n \alpha + \sin n \beta + \sin n \gamma < 0
$$

holds true?


Let \( AA' \) and \( BB' \) be two line segments which are perpendicular to \( A'B' \). The lengths of \( AA' \), \( BB' \), and \( A'B' \) are 680, 2000, and 2010 respectively. Find the minimal length of \( AX + XB \) where \( X \) is a point between \( A' \) and \( B' \).


In the tetrahedron \(ABCD\), \(\angle ADB = \angle BDC = \angle CDA = 60^{\circ}\), \(AD = BD = 3\), \(CD = 2\). What is the volume of the circumscribed sphere of the tetrahedron \(ABCD\)?

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


Find all odd positive integers \( n \) such that \( n \) divides \( 3^{n} + 1 \).


Find the probability that the groups will be able to contact each other. Consider pairs of tourists where the first tourist is from the first group, and the second tourist is from the second group. There are $6 \cdot 7 = 42$ such pairs. Therefore, the probability that no tourist has the phone number of anyone from the other group is $(1-p)^{42}$.

Hence, the desired probability is $1 - (1-p)^{42}$.

Solution: \boxed{1 - (1-p)^{42}}


Petya approaches the entrance door with a combination lock, which has buttons numbered from 0 to 9. To open the door, three correct buttons need to be pressed simultaneously. Petya does not remember the code and tries combinations one by one. Each attempt takes Petya 2 seconds.

a) How much time will Petya need to definitely get inside?

b) On average, how much time will Petya need?

c) What is the probability that Petya will get inside in less than a minute?


For natural numbers \(a\) and \(b\) we are given that \(2019 = a^2 - b^2\). It is known that \(a < 1000\). What is the value of \(a\)?


In Figure 1, \( P R S \) is a straight line, \( P Q = P R = Q S \) and \( \angle Q P R = 30^{\circ} \). If \( \angle R Q S = w^{\circ} \), find the value of \( w \).


Anya is arranging stones on the sand. First, she placed one stone, then added more stones to form a pentagon, then made a larger outer pentagon with the stones, and so on, as shown in the drawings. The number of stones she had arranged in the first four pictures: 1, 5, 12, and 22. If she continues creating such pictures, how many stones will be in the 10th picture?


There are 3 pieces of wood, each cut into 3 sections. It takes a total of 18 minutes. How many minutes does each cut take?

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


The numbers 1 to 8 are to be placed, one per circle, in the circles shown. The number next to each arrow shows what the product of the numbers in the circles on that straight line should be.

What will be the sum of the numbers in the three circles at the bottom of the diagram?
A. 11
B. 12
C. 15
D. 16
E. 17


Let the integer \( n \) have exactly 9 odd factors and 18 even factors. Find the smallest value of \( n \).


As shown in the picture, the knight can move to any of the indicated squares of the $8 \times 8$ chessboard in 1 move. If the knight starts from the position shown, find the number of possible landing positions after 20 consecutive moves.


How many four-digit numbers can be formed using the digits 1, 2, 3, 4, and 5 if:

a) No digit is repeated more than once;

b) Repetition of digits is allowed;

c) The numbers must be odd and no digit is repeated?


Kolya wants to multiply all the natural divisors of the number 1024 (including the number itself) on his calculator. Will he be able to see the result on a screen that has 16 decimal places?

Solution: \boxed{\text{No}}


Given \( x, y \in \mathbf{R} \) such that \( x^2 + y^2 = 2 \) and \( |x| \neq |y| \), find the minimum value of \( \frac{1}{(x+y)^2} + \frac{1}{(x-y)^2} \).


Find the area of a right triangle with one leg measuring 6 and the hypotenuse measuring 10.


On another island of knights, liars, and normal people, the king held opposite views and gave his daughter different paternal advice: "Dear, I don't want you to marry any knight or liar. I would like you to marry a solid, normal person with a good reputation. You should not marry a knight, because all knights are hypocrites. You should not marry a liar either, because all liars are treacherous. No, no matter what, a decent normal person would be just right for you!"

Assume you are an inhabitant of this island and a normal person. Your task is to convince the king that you are a normal person.
a) How many true statements will you need for this?

b) How many false statements will you need for the same purpose?

(In both cases, the task is to find the minimum number of statements.).


The sequence \( a_n \) is defined as follows:
\( a_1 = 2, a_{n+1} = a_n + \frac{2a_n}{n} \) for \( n \geq 1 \). Find \( a_{200} \).


A 90-degree angle is covered by some angles with the same vertex in two or three layers, and the sum of the angles is 290 degrees. Find the measure of the angle covered in three layers.

Solution: \boxed{x = 20^\circ}


A line passes through the point with coordinates $(10, 0)$ and intersects the parabola $y = x^2$ at points with abscissae $x_1$ and $x_2$. Find $\frac{1}{x_1} + \frac{1}{x_2}$.


Four people went to a bookstore to buy books. Each person bought 4 different books and every two people have exactly 2 books in common. How many different books were bought by these 4 people at least?


Given the set \( A = \{1, 2, \cdots, 6\} \), the mapping \( f: A \rightarrow A \) is a bijection,
$$
\begin{array}{l}
f^{(2)}(x) = f(f(x)), \\
f^{(n+1)}(x) = f\left(f^{(n)}(x)\right) \quad (n \geqslant 1).
\end{array}
$$

The number of mappings \( f \) that satisfy \( f^{(12)}(x) = x \) for any \( x \in A \) is .


The polynomial \( P(x) \) with integer coefficients satisfies \( P(19) = P(94) = 1994 \). Find the constant term of \( P(x) \) (given that its absolute value is less than 1000).


Let \( f(x) = x^2 - r_2 x + r_3 \) for all real numbers \( x \), where \( r_2 \) and \( r_3 \) are some real numbers. Define a sequence \( \{ g_n \} \) for all nonnegative integers \( n \) by \( g_0 = 0 \) and \( g_{n+1} = f(g_n) \). Assume that \( \{ g_n \} \) satisfies the following three conditions:
1. \( g_{2i} < g_{2i+1} \) and \( g_{2i+1} > g_{2i+2} \) for all \( 0 \leq i \leq 2011 \),
2. There exists a positive integer \( j \) such that \( g_{i+1} > g_i \) for all \( i > j \),
3. \( \{ g_n \} \) is unbounded.

If \( A \) is the greatest number such that \( A \leq |r_2| \) for any function \( f \) satisfying these properties, find \( A \).


A $5 \times 5$ grid consists of 25 $1 \times 1$ small squares. Each small square is divided into four identical isosceles triangles, three of which are colored black. If a side of a small square is on a black part, it is called a black edge; otherwise, it is a white edge. In the $5 \times 5$ grid, the common edge of adjacent (sharing a common edge) small squares must be the same color. What is the minimum number of black edges on the four sides of the $5 \times 5$ grid?


In square \(ABCD\), point \(P\) lies on \(AB\), and \(DP\) bisects \(BC\) at point \(N\). Point \(Q\) lies on \(BA\), and \(CQ\) bisects \(AD\) at point \(M\). Line \(DP\) intersects line \(CQ\) at point \(O\). Given that the area of square \(ABCD\) is 192, find the area of \(\triangle POQ\).


Find any solution to the rebus

$$
\overline{A B C A}=182 \cdot \overline{C D}
$$

where \( A, B, C, D \) are four distinct non-zero digits (the notation \(\overline{X Y \ldots Z}\) denotes the decimal representation of a number).

As an answer, write the four-digit number \(\overline{A B C D}\).


As shown in the figure, in $\triangle ABC$, $\cos \frac{C}{2} = \frac{2 \sqrt{5}}{5}$, $\overrightarrow{AH} \cdot \overrightarrow{BC} = 0$, $\overrightarrow{AB} \cdot (\overrightarrow{CA} + \overrightarrow{CB}) = 0$. Find the eccentricity of the hyperbola passing through point $C$ with foci at points $A$ and $H$.


A store purchased a batch of New Year cards at a price of 21 cents each and sold them for a total of 14.57 yuan. If each card is sold at the same price and the selling price does not exceed twice the purchase price, how many cents did the store earn in total?

Solution: \boxed{470 \text{ fen}}


C is a circle and L a line not meeting it. M and N are variable points on L such that the circle diameter MN touches C but does not contain it. Show that there is a fixed point P such that the ∠MPN is constant.

Solution: \boxed{P}


Let $ABC$ be an acute-angled triangle and $D, E,$ and $F$ be the feet of the altitudes from $A, B,$ and $C,$ respectively. Let $P$ (respectively $Q$, respectively $R$) be the foot of the perpendicular from $A$ (respectively $B$, respectively $C$) to $(EF)$ (respectively $(FD)$, respectively $(DE)$). Show that the lines $(AP),(BQ)$, and $(CR)$ are concurrent.

Solution: \boxed{O}


In a psychiatric hospital, there is a chief doctor and many madmen. During the week, each madman bit someone once a day (possibly themselves). At the end of the week, it was found that each patient has two bites, and the chief doctor has one hundred bites. How many madmen are there in the hospital?


Given that $[r, s]$ denotes the least common multiple (LCM) of positive integers $r$ and $s$, find the number of ordered triples of positive integers $(a, b, c)$ such that $[a, b]=1000$, $[b, c]=2000$, and $[c, a]=2000$.


A group with 4 boys and 4 girls was randomly divided into pairs. Find the probability that at least one pair consists of two girls. Round your answer to two decimal places.


Find all values of \( x \) such that one of the three given numbers \( \log _{x^{2}}\left(x^{2}-3 x+2\right) \), \( \log _{x^{2}} \frac{x^{2}}{x-2} \), and \( \log _{x^{2}} \frac{x^{2}}{x-1} \) is equal to the sum of the other two.

Solution: \boxed{x = 3}


At the World Meteorological Conference, each participant announced the average monthly temperature in their hometown in turn. All the other participants at that moment recorded the product of the temperatures in their own city and the announced city. A total of 36 positive and 36 negative numbers were recorded. What is the minimum number of times a positive temperature could have been announced?


A ship model with a mass of \(m = 0.5 \,\text{kg}\) is pushed with an initial velocity of \(v = 10 \,\text{m/s}\). The water's resistance force is proportional to the ship model's velocity (\(F = -kv, \, k = 0.5 \,\text{kg/s}\)). Calculate the distance traveled by the ship model:

a) During the time it takes for the velocity to decrease to half,

b) Until it comes to a complete stop!

Solution: \boxed{10 \, \text{m}}


An astronomer discovered that the intervals between the appearances of comet $2011 Y$ near the planet $12 I V 1961$ are successive terms of a decreasing geometric progression. The three latest intervals (in years) are the roots of the cubic equation \( t^{3}-c t^{2}+350 t-1000=0 \), where \( c \) is a constant. What will be the duration of the next interval before the comet's appearance?

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


Let \( x_{1}, x_{2}, x_{3} \) be the three roots of the equation \( x^{3}-17 x-18=0 \), with \( -4 < x_{1} < -3 \) and \( 4 < x_{3} < 5 \).
(1) Find the integer part of \( x_{2} \).
(2) Find the value of \( \arctan x_{1} + \arctan x_{2} + \arctan x_{3} \).


With what minimal force must you press on a cube with a volume of $10 \text{ cm}^3$, floating in water, so that it is completely submerged? The density of the cube's material is $500 \text{ kg/m}^3$, and the density of water is $1000 \text{ kg/m}^3$. Give the answer in SI units. Assume the acceleration due to gravity is $10 \text{ m/s}^2$.

Solution: \boxed{0.05 \, \text{N}}


For positive integers \( n \), define \( n! = 1 \times 2 \times \cdots \times n \). Find all positive integers \( x \) and \( y \) such that \( 3^x + x^4 = y! + 2019 \).

Solution: \boxed{\{x = 6, y = 3\}}


Find the value of \( n \) for which the following equation holds:

$$
\frac{1}{1+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{4}}+\ldots+\frac{1}{\sqrt{n}+\sqrt{n+1}}=2012
$$


From the 4040 integers ranging from -2020 to 2019, three numbers are randomly chosen and multiplied together. Let the smallest possible product be $m$ and the largest possible product be $n$. What is the value of $\frac{m}{n}$? Provide the answer in simplest fraction form.


Let \( A, E, H, L, T \), and \( V \) be chosen independently and at random from the set \(\left\{0, \frac{1}{2}, 1\right\}\). Compute the probability that \(\lfloor T \cdot H \cdot E \rfloor = L \cdot A \cdot V \cdot A\).


Given points \( A(-1, 5) \) and \( B(3, -7) \). Find the distance from the origin to the midpoint of segment \( AB \).


Find natural numbers \( n \) such that for all positive numbers \( a, b, c \) satisfying the inequality

\[ n(ab + bc + ca) > 5(a^2 + b^2 + c^2) \]

there exists a triangle with sides \( a, b, c \).


Evgeny has 3 roses, 2 tulips, 4 daisies, and 6 lilies. If he randomly chooses one of these flowers, what is the probability that it is a tulip?
(A) $\frac{3}{15}$
(B) $\frac{12}{15}$
(C) $\frac{6}{15}$
(D) $\frac{4}{15}$
(E) $\frac{2}{15}$


Determine the smallest possible value of the expression

$$
\frac{a b+1}{a+b}+\frac{b c+1}{b+c}+\frac{c a+1}{c+a}
$$

where \(a, b, c \in \mathbb{R}\) satisfy \(a+b+c=-1\) and \(a b c \leq -3\).


It is known that \( x = 2a^{5} = 5b^{2} > 0 \), where \( a \) and \( b \) are integers. What is the smallest possible value of \( x \)?


Let \(A\) and \(B\) be the foci of a hyperbola, and \(C\) be a point on the hyperbola. It is known that the side lengths of \(\triangle ABC\) form an arithmetic sequence and \(\angle ACB = 120^{\circ}\). Find the eccentricity of the hyperbola. 🟪


Given the function
$$
f(x)=\left(1-x^{2}\right)\left(x^{2}+b x+c\right) \text{ for } x \in [-1, 1].
$$

Let $\mid f(x) \mid$ have a maximum value of $M(b, c)$. As $b$ and $c$ vary, find the minimum value of $M(b, c)$.

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


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 \( y = f(x) \) is \( \sqrt{30} \), and the distance between the points of intersection of the graphs \( y = x^2 \) and \( y = f(x) + 3 \) is \( \sqrt{46} \). Find the distance between the points of intersection of the graphs \( y = x^2 - 1 \) and \( y = f(x) + 1 \).


Let \( x \) and \( y \) be non-zero real numbers. If \( x \) is 250% of \( y \) and \( 2y \) is \( c \% \) of \( x \), find the value of \( c \).


Through two vertices of an equilateral triangle \(ABC\) with an area of \(21 \sqrt{3} \ \text{cm}^2\), a circle is drawn such that two sides of the triangle are tangent to the circle. Find the radius of this circle.

Solution: \boxed{R = 2\sqrt{7}}


Find all triples of positive numbers \( a, b, c \) that satisfy the conditions \( a + b + c = 3 \), \( a^2 - a \geq 1 - bc \), \( b^2 - b \geq 1 - ac \), and \( c^2 - c \geq 1 - ab \).

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


There are 211 students and four different types of chocolates, with more than 633 pieces of each type of chocolate. Each student can take up to three pieces of chocolate, or none at all. Grouping students based on both the type and number of chocolates they take, the maximum number of students in any group is at least $\qquad$ students.


A cone is formed by rotating a right triangle with an area $S$ around one of its legs. Find the volume of the cone if the length of the circle described by the rotation of the triangle's centroid is $L$.

Solution: \boxed{SL}


Swap the digit in the hundreds place with the digit in the units place of a three-digit number while keeping the digit in the tens place unchanged. The new number obtained is equal to the original number. How many such numbers are there? How many of these numbers are divisible by 4?


The following figure shows a regular dodecagon.

Answer the following questions:

a) How many equilateral triangles can be formed such that all three of their vertices are vertices of the dodecagon?

b) How many scalene triangles can be formed such that all three of their vertices are vertices of the dodecagon?


A digit was removed from a five-digit number, and the resulting four-digit number was added to the original number. The sum turned out to be 54321. Find the original number.


Let the real number \( a < -1 \) and the variable \( x \) satisfy \( x^2 + ax \leq -x \). The minimum value of \( x^2 + ax \) is \( -\frac{1}{2} \). What is \( a \)?


Tiago writes all four-digit numbers using distinct non-zero digits that have the same parity. What is the probability that, if we randomly choose one of these numbers, it will be even?


Given \(\alpha, \beta \in \mathbf{R}\), the intersection point of the lines \(\frac{x}{\sin \alpha+\sin \beta} + \frac{y}{\sin \alpha+\cos \beta} = 1\) and \(\frac{x}{\cos \alpha+\sin \beta} + \frac{y}{\cos \alpha+\cos \beta} = 1\) lies on the line \(y = -x\). Calculate \(\sin \alpha + \cos \alpha + \sin \beta + \cos \beta\).


1. \(\lim _{x \rightarrow 1}(1-x) \operatorname{tg} \frac{\pi x}{2}\)
2. \(\lim _{x \rightarrow \frac{\pi}{4}}\left(\frac{\pi}{4}-x\right) \operatorname{cosec}\left(\frac{3}{4} \pi+x\right)\)
3. \(\lim _{x \rightarrow+\infty} x \operatorname{arcctg} x\)
4. \(\lim _{x \rightarrow-\infty} x\left(\frac{\pi}{2}+\operatorname{arctg} x\right)\)


More than three natural numbers are written in a circle, and their sum is 19. The sums of any three consecutive numbers are equal. What numbers were written in the circle?


Let \( a \star b = ab + a + b \) for all integers \( a \) and \( b \). Evaluate \( 1 \star (2 \star (3 \star (4 \star \ldots (99 \star 100) \ldots))) \).

Solution: \boxed{101! - 1}


Malysh and Carlson together ate a jar of jam. Carlson ate 40% fewer spoons of jam than Malysh, but each of Carlson's spoons held 150% more jam than each of Malysh's spoons. What fraction of the jar of jam did Carlson eat?


16 travelers, each of whom is either a liar or a knight (liars always lie, knights always tell the truth), settled into 3 hotel rooms. When everyone gathered in their rooms, Vasily, residing in the first room, said: "There are more liars in this room than knights. No, wait - there are more knights in this room than liars." After that, Vasily entered the second room and repeated the same two statements there. Then he entered the third room and also said the same two statements. What could be the number of knights among these 16 travelers?


The year 2009 has a unique property: by rearranging the digits of the number 2009, it is impossible to form a smaller four-digit number (numbers do not start with zero). In which future year will this property first repeat again?


As shown in the diagram, square $ABCD$ has a side length of 4, square $CEFG$ has a side length of 12, and points $D$, $C$, and $E$ lie on a straight line. Connect $DF$, and construct $GI \parallel DF$ intersecting the extension of $DA$ at point $I$. Draw $IH \perp DF$ intersecting $DF$ at point $H$. Find the length of $IH$.


A monkey becomes happy when it eats three different fruits. What is the maximum number of monkeys that can be made happy if there are 20 pears, 30 bananas, 40 peaches, and 50 mandarins? Justify your answer.


Find the differential equation whose general solution is the function \( y = C_1 x + C_2 \), where \( C_1 \) and \( C_2 \) are arbitrary constants.

Solution: \boxed{y'' = 0}


How much does the fish weigh? The Creachams planned to stop during their journey at a place with good fishing, as Uncle Jabez was an avid fisherman and they wanted to please him. They chose a charming spot and took the opportunity to have a picnic there. When Uncle brought a magnificent trout, a discussion arose about how much it might weigh. The colonel presented it as a puzzle, saying:

- Suppose the tail weighs 9 ounces, the head weighs as much as the tail plus half the body, and the body weighs as much as the head and tail.

Now tell me, if all this is true, how much does the fish weigh?

Solution: \boxed{72 \text{ ounces}}


Let \( N \) be the set of positive integers, and define a function \( f \) on \( N \) as follows:
(i) \( f(1) = 1, \quad f(3) = 3 \);

(ii) For \( n \in N \),
$$
\begin{array}{l}
f(2n) = f(n), \\
f(4n+1) = 2f(2n+1) - f(n), \\
f(4n+3) = 3f(2n+1) - 2f(n).
\end{array}
$$

Find all \( n \) such that \( n \leq 1988 \) and \( f(n) = n \).
(This problem was from the 29th International Mathematical Olympiad, 1988)


Find the minimum value of the expression \(\frac{5 x^{2}-8 x y+5 y^{2}-10 x+14 y+55}{\left(9-25 x^{2}+10 x y-y^{2}\right)^{5 / 2}}\). If necessary, round the answer to hundredths.


Given the ellipse \(\frac{x^{2}}{16}+\frac{y^{2}}{9}=1\) with three points \(P\), \(Q\), and \(R\) on it, where \(P\) and \(Q\) are symmetric with respect to the origin. Find the maximum value of \(|RP| + |RQ|\).


If all the positive factors of 28 are \( d_{1}, d_{2}, \ldots, d_{n} \) and \( a = \frac{1}{d_{1}} + \frac{1}{d_{2}} + \cdots + \frac{1}{d_{n}} \), find the value of \( a \).


The distance between cities A and B is an integer number of kilometers. Along the road between the cities, there are markers every kilometer: on one side, the distance to city A is written, and on the other side, the distance to city B is written. Slava was walking from city A to city B. During his journey, Slava calculated the GCD (Greatest Common Divisor) for the numbers written on each marker. It turned out that only the numbers 1, 3, and 13 appeared as GCDs. What is the distance between the cities?


The smallest variance. In a set of $n$ numbers, where one of the numbers is 0 and another is 1, what is the smallest possible variance for such a set of numbers? What should the set be to achieve this?


Solve the equation

$$
\left(x^{2} + x\right)^{2} + \sqrt{x^{2} - 1} = 0
$$


The fraction \(\frac{1}{2015}\) has a unique "(restricted) partial fraction decomposition" of the form
\[ \frac{1}{2015}=\frac{a}{5}+\frac{b}{13}+\frac{c}{31}, \]
where \(a, b, c\) are integers with \(0 \leq a < 5\) and \(0 \leq b < 13\). Find \(a + b\).


Random variables \( X \) and \( Y \) are independent. Find the variance of the random variable \( Z = 3X + 2Y \), given that \( D(X) = 5 \) and \( D(Y) = 6 \).


In an isosceles triangle \(ABC\), let \(F\) be the midpoint of the base \(AB\), and \(D\) be the foot of the perpendicular from \(F\) to the side \(BC\). Let \(G\) be the midpoint of the segment \(FD\). Show that \(AD\) is perpendicular to \(CG\).

Solution: \boxed{AD \perp CG}


There are 100 chairs arranged in a circle. If \( n \) people are sitting on these chairs, such that any new person sitting down will always sit on a chair adjacent to one of the \( n \) people, what is the minimum value of \( n \)?


Consider the multiples of the natural number 15 in sequence:
\[ 0, 15, 30, 45, 60, \ldots \]

This sequence is strictly increasing, and for any natural number, it either belongs to the sequence or lies between two consecutive terms. Determine between which terms in the sequence the number 2016 falls. Additionally, find the difference between 2016 and the larger of these two terms.


Find the sum of the coefficients of the polynomial
$$
\left(4 x^{2}-4 x+3\right)^{4}\left(4+3 x-3 x^{2}\right)^{2}.
$$


The average of \( x, y, z \) is 10. The average of \( x, y, z, t \) is 12. Find \( t \).


In a right-angled triangle \(ABC\) (with right angle at \(A\)), the bisectors of the acute angles intersect at point \(P\). The distance from \(P\) to the hypotenuse is \(\sqrt{80000}\). What is the distance from \(P\) to \(A\)?


Inside the parallelogram \(ABCD\), a point \(E\) is chosen such that \(AE = DE\) and \(\angle ABE = 90^\circ\). Point \(M\) is the midpoint of segment \(BC\). Find the angle \(DME\).


Solution: \boxed{\sqrt{pq}}


Point \( M \) is the midpoint of the hypotenuse \( AC \) of right triangle \( ABC \). Points \( P \) and \( Q \) on lines \( AB \) and \( BC \) respectively are such that \( AP = PM \) and \( CQ = QM \). Find the measure of angle \( \angle PQM \) if \( \angle BAC = 17^{\circ} \).


Columbus discovered America in the 15th century. In which year was this discovery made, given the following information:

a) The sum of the digits representing this year is 16.

b) If the tens digit is divided by the units digit, the quotient is 4 and the remainder is 1.


In triangle \(ABC\), point \(D\) lies on side \(AC\), such that \(AD = 2DC\). Point \(E\) lies on side \(BC\). The area of triangle \(ABD\) is 3, and the area of triangle \(AED\) is 1. Segments \(AE\) and \(BD\) intersect at point \(O\). Find the ratio of the areas of triangles \(ABO\) and \(OED\).


Yesterday, Sasha cooked soup and added too little salt, requiring additional seasoning. Today, he added twice as much salt as yesterday, but still had to season the soup additionally, though with half the amount of salt he used for additional seasoning yesterday. By what factor does Sasha need to increase today's portion of salt so that tomorrow he does not have to add any additional seasoning? (Each day Sasha cooks the same portion of soup.)


In the first hour of the shift, the master craftsman manufactured 35 parts. He then realized that at his current pace, he would need to stay an extra hour to meet the shift’s quota. By increasing his speed by 15 parts per hour, he completed the quota half an hour before the shift ended. How many parts must the master craftsman manufacture during the shift?


Let \( H \) be the orthocenter of an acute triangle \( ABC \). Given that \(\angle A=30^\circ\) and \(BC=3\), find \( AH \).

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


Can we solve every cubic equation using only basic operations and root extraction based on the previous discussion? (See problem 65 in section IV as well)

Solution: \boxed{Yes}


Can the rectangle shown below be cut into five squares? (The squares do not have to be identical, and no extra parts should remain)

Solution: \boxed{Yes}


In the following diagram, two lines that meet at a point \( A \) are tangent to a circle at points \( B \) and \( C \). The line parallel to \( AC \) passing through \( B \) meets the circle again at \( D \). Join the segments \( CD \) and \( AD \). Suppose \( AB = 49 \) and \( CD = 28 \). Determine the length of \( AD \).


Dad gives Xiaojun the same amount of pocket money every day. Xiaojun originally had some amount of money. If he spends 10 yuan every day, he can last for 6 days; if he spends 15 yuan every day, he can last for 3 days. How much money did Xiaojun originally have?


Given that \(\sin \left(\frac{\pi}{4}-x\right)=\frac{5}{13}\), and \(x \in \left(0, \frac{\pi}{4}\right)\), find the value of \(\frac{\cos 2x}{\cos \left(\frac{\pi}{4}+x\right)}\).


A circle $\omega$ is circumscribed around triangle $ABC$. Tangents to the circle at points $A$ and $B$ intersect at point $K$. Point $M$ is the midpoint of side $AC$. A line passing through point $K$ parallel to $AC$ intersects side $BC$ at point $L$. Find the angle $AML$.


As shown in Figure 1, in a cylindrical container with a height of 10 and a base radius of 1, a ping-pong ball is glued to the upper wall and another to the lower wall. The ping-pong ball on the lower wall is tangent to both the bottom and side surfaces, and the ping-pong ball on the upper wall is tangent to both the top and side surfaces (assume the thickness of both the cylinder and the ping-pong balls is negligible). A plane is tangent to both ping-pong balls. Given that the cross-section of the cylinder's edge by this plane is an ellipse, find the eccentricity of this ellipse.


Jonas divided a square plot into eight equal rectangular parts, as shown in the figure, giving one part to each of his eight children. To fence her part, Antônia found that its perimeter was \(120 \, \text{m}\).

a) What is the area of the plot that Jonas divided?

b) If we represent the perimeter of one of the smaller plots by \(P\), what is the original area of the plot in terms of \(P\)?


All plane angles of a trihedral angle are equal to $90^{\circ}$. Find the angles between the angle bisectors of the plane angles.


Calculate the limit of the function:

$$
\lim _{x \rightarrow 0} \frac{\sqrt{1-2 x+x^{2}}-(1+x)}{x}
$$


In the Cartesian coordinate system \(xOy\), the equation of the hyperbola \(C\) is \(x^2 - y^2 = 1\). Find all real numbers \(a > 1\) that satisfy the following condition: Through the point \((a, 0)\), draw any two mutually perpendicular lines \(l_1\) and \(l_2\). If \(l_1\) intersects the hyperbola \(C\) at points \(P\) and \(Q\), and \(l_2\) intersects \(C\) at points \(R\) and \(S\), then \(|PQ| = |RS|\) always holds.


There are 20 people - 10 boys and 10 girls. How many ways are there to form a group with an equal number of boys and girls?


There are 32 ones written on the board. Each minute, Carlsson erases any two numbers, writes their sum on the board, and then eats an amount of candy equal to the product of the two erased numbers. What is the maximum number of candies he could eat in 32 minutes?


A freight company needs to transport 89 tons of cargo. Each large truck can carry 7 tons and each small truck can carry 4 tons. If each large truck consumes 14 liters of fuel and each small truck consumes 9 liters, what is the minimum amount of fuel needed to transport all the cargo?

Solution: \boxed{181 \text{ liters}}


Each subsequent number is chosen independently of the previous ones. If it matches one of the already chosen numbers, you should move clockwise to the first unchosen number.

In the end, $k$ different numbers are obtained.

a) Is it true that the appearance of each specific number in such a selection is equally likely?

b) Is it true that the appearance of all selections is equally likely?

Solution: \boxed{\text{No}}


Let $AB, BC, CD$ be three segments that are not in the same plane. The midpoints of $AB$, $BC$, and $CD$ are $P$, $Q$, and $R$, respectively, with $PQ = 2$, $QR = \sqrt{5}$, and $PR = 3$. Determine the angle between the lines $AC$ and $BD$.

(A) $60^{\circ}$.

(B) $30^{\circ}$.

(C) $90^{\circ}$.

(D) $120^{\circ}$.

Solution: \boxed{(C)}


On the side \(AC\) of triangle \(ABC\), points \(E\) and \(K\) are taken such that \(E\) lies between \(A\) and \(K\), and \(AE:EK:KC = 3:5:4\). Median \(AD\) intersects segments \(BE\) and \(BK\) at points \(L\) and \(M\) respectively. Find the ratio of the areas of triangles \(BLM\) and \(ABC\). The answer is \(\frac{1}{5}\).


Each digit of the natural number \( N \) is strictly greater than the digit to its left. What is the sum of the digits of the number \( 9N \)?


There is a pile of red and white balls, with the total number of balls not exceeding 50. It is known that the number of red balls is three times the number of white balls. What is the maximum number of red balls?


Given the integer pair \((m, n)\) satisfies \(\frac{m^{2}+mn+n^{2}}{m+2n}=\frac{13}{3}\), find the value of \(m+2n\).


Given a sequence $\left\{a_{n}\right\}$ where $a_{1}=1$ and $n \cdot a_{n+1}=2\left(a_{1}+a_{2}+\cdots+a_{n}\right)$ for $n \in \mathbf{N}_{+}$, find the general term formula for the sequence.

Solution: \boxed{a_n = n}


The base of a pyramid is a parallelogram with sides measuring 10 cm and 18 cm, and an area of 90 cm². The height of the pyramid passes through the intersection point of the diagonals of the base and is 6 cm. Determine the lateral surface area of the pyramid.

Solution: \boxed{192 \text{ cm}^2}


Find all triples \((x, y, z)\) of real numbers that are solutions to the system of equations

$$
\left.\begin{array}{l}
3 \cdot 2^{y}-1=2^{x}+2^{-x} \\
3 \cdot 2^{z}-1=2^{y}+2^{-y} \\
3 \cdot 2^{x}-1=2^{z}+2^{-z}
\end{array}\right\}
$$

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


Point \( M \) lies on the side of a regular hexagon with side length 12. Find the sum of the distances from point \( M \) to the lines containing the remaining sides of the hexagon.


From a set of 32 chess pieces, two pieces are drawn randomly. What is the probability that:

a) 2 dark pieces or 2 pieces of different colors,

b) 1 bishop and 1 pawn or 2 pieces of different colors,

c) 2 different-colored rooks or 2 pieces of the same color but different sizes,

d) 1 king and one knight of the same color, or two pieces of the same color,

e) 2 pieces of the same size or 2 pieces of the same color

are drawn?


Let \(ABC\) be a triangle, and let \(P, Q, X, Y\) be points on the same side as \(A\) of line segment \(BC\) such that \(ABYX\) and \(ACQP\) are squares. Let \(M\) be the midpoint of \([BC]\). Show that \(PX = 2AM\).

Solution: \boxed{PX = 2AM}


A function \( f \), defined on the set of integers, satisfies the following conditions:

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

Find \( f(10) \).


The force with which the airflow acts on the sail can be calculated using the formula:

\[ F = \frac{B S \rho (v_{0} - v)^{2}}{2} \]

where \( B \) is the aerodynamic force coefficient,
\( S \) is the area of the sail \( S = 7 \, \text{m}^2 \),
\( \rho \) is the air density,
\( v_{0} \) is the wind speed \( v_{0} = 6.3 \, \text{m/s} \),
and \( v \) is the speed of the sailboat.

At a certain moment in time, the instantaneous power of the wind reaches its maximum value. What is the speed of the sailboat at this moment?

Given:
\[ v_{0} = 6.3 \, \text{m/s} \]

Find:
\[ v(t') - ? \]

Solution: \boxed{2.1 \, \text{m/s}}


In the set of natural numbers from 1 to 100, find the sum of all numbers that are not divisible by 6.


At exactly noon, a truck left a village for the city, and at the same time, a car left the city for the village. If the truck had left 45 minutes earlier, they would have met 18 kilometers closer to the city. If the car had left 20 minutes earlier, they would have met $k$ kilometers closer to the village. Find $k$.


In the direction from $A$ to $B$, a car traveled for some time at a constant speed of $v_{1}=60$ km/h. It covered the remaining part of the way in the same amount of time, but at a speed of $v_{2}=40$ km/h. In the opposite direction, the car traveled one half of the way at a speed of $v_{3}=80$ km/h and the other half at a speed of $v_{4}=45$ km/h. What is the average speed of the trip: a) from $A$ to $B$? b) from $B$ to $A$?

Solution: \boxed{57.6 \text{ km/h}}


Calculate the circulation of the vector field \( \mathbf{a} = y \mathbf{i} + z^{2} \mathbf{j} - z \mathbf{k} \) along the contour \( L: \left\{ \begin{array}{r} x^{2}+y^{2}=4, \\ z=3, \end{array} \right. \) 1) directly, 2) using Stokes' theorem.


Dominic randomly picks between two words MATHEMATICS and MEMES, each with an equal chance of being chosen. From his chosen word, he then randomly draws one letter, with the probability of each letter being drawn directly proportional to the frequency of its occurrence in the word. Given that Dominic drew an M, what is the probability that he picked MEMES?


The number of boys at Jiaotong Primary School is 7 times the number of girls. Additionally, there are 900 more boys than girls. How many boys and girls are there in total at Jiaotong Primary School?


A rectangular parallelepiped has three edges originating from vertex \( A \) with lengths of 1, 2, and 3 units. The endpoints of these edges, which are not \( A \), form a triangle. How far is point \( A \) from the plane of this triangle?


The Absent-Minded Scientist in his laboratory has created a unicellular organism that has a probability of 0.6 to split into two such organisms and a probability of 0.4 to die without leaving any offspring. Find the probability that after some time the Absent-Minded Scientist will have none of such organisms left.


ABC is a triangle with inradius \( r \). The circle through \( A \) and \( C \) orthogonal to the incircle meets the circle through \( A \) and \( B \) orthogonal to the incircle at \( A \) and \( A' \). The points \( B' \) and \( C' \) are defined similarly. Show that the circumradius of \( A'B'C' \) is \( \frac{r}{2} \).


Point $O$ is the center of the circle inscribed in the isosceles triangle $ABC (AB = BC)$. Line $AO$ intersects segment $BC$ at point $M$. Find the angles and the area of triangle $ABC$, given that $AO = 3$ and $OM = \frac{27}{11}$.

Solution: \boxed{20 \sqrt{5}}


In a joint-stock company, there are 2017 shareholders, and any group of 1500 of them has a controlling stake (at least 50% of the shares). What is the maximum share of the company that one shareholder can own?


In an isosceles triangle \( ABC \) with \(\angle B = \arctan \frac{8}{15}\), a circle with radius 1 is inscribed in angle \( C \) and touches side \( CB \) at point \( M \) and cuts off a segment \( KE \) from the base. It is known that \( MB = \frac{15}{8} \). Find the area of triangle \( KMB \), given that points \( A, K, E, B \) follow in the mentioned order along the base \( AB \).


Determine the smallest integer $k \geq 1$ such that the value of the expression $\sin(kx) \cdot \sin^k(x) + \cos(kx) \cdot \cos^k(x) - \cos^k(2x)$ does not depend on $x$.


The cells of a $100 \times 100$ table are painted white. In one move, you are allowed to choose any 99 cells from one row or one column and repaint each of them in the opposite color – from white to black or from black to white. What is the minimum number of moves needed to obtain a table with a checkerboard pattern of cells?


Given \( x \in \mathbf{R} \), find the minimum value of the function \( f(x) = |2x - 1| + |3x - 2| + |4x - 3| + |5x - 4| \).


There are 2015 lighted lamps with numbers 1, 2, 3, ..., 2015, each controlled by a pull switch. If the lamps numbered by multiples of 2, multiples of 3, and multiples of 5 are each pulled once, how many lamps remain lighted?

Answer with a calculation showing how many lamps remain lighted.


What is the maximum number of axes of symmetry that the union of \( k \) line segments on a plane can have?


Find the sum of integers that belong to the range of the function $f(x)=\log _{3}(10 \cos 2 x + 17)$ for $x \in \left[1.25(\operatorname{arctg} 0.25) \cos (\pi - \arcsin (-0.6)), \operatorname{arctg} 3\right]$.


Find the roots of the following equations:

$$
\begin{gathered}
\log _{x-1} x - \log _{x-1} 6 = 2 \\
x^{4} - 39x^{3} + 462x^{2} - 1576x + 1152 = 0
\end{gathered}
$$


Anya and Vanya are climbing the stairs of a multi-story building. Anya is going to the second floor, while Vanya is going to the sixth floor. How many times longer is Vanya's journey compared to Anya's?


If \( a \), \( b \), and \( y \) are real numbers and satisfy:
\[ \begin{cases}
a + b + y = 5 \\
ab + by + ay = 3
\end{cases} \]
determine the greatest possible value of \( y \).


Given that the real numbers \( x \) and \( y \) satisfy the equations \(\left\{\begin{array}{c} x^{3} + \cos x + x - 2 = 0 \\ 8 y^{3} - 2 \cos^{2} y + 2 y + 3 = 0 \end{array}\right.\), determine \(\cos (x + 2y)\).


Calculate the mass of the material segment \( A B \), given \( A(-2,1,0) \) and \( B(-1,3,5) \), where the density at each point \( M \) on the segment is proportional to the distance from \( M \) to \( A \) with a proportionality coefficient \( k \).

Solution: \boxed{15 k}


In space, the points \(A(-1, 2, 0), B(5, 2, -1), C(2, -1, 4)\), and \(D(-2, 2, -1)\) are given. Find:

a) the distance from the vertex \(D\) of the tetrahedron \(ABCD\) to the centroid of the base \(ABC\);

b) the equation of the plane \(ABC\);

c) the height of the tetrahedron from the vertex \(D\);

d) the angle between lines \(BD\) and \(AC\);

e) the angle between the faces \(ABC\) and \(ACD\);

f) the distance between lines \(BD\) and \(AC\).


Each face of a $6 \times 6 \times 6$ cube is divided into $1 \times 1$ cells. The cube is covered with $2 \times 2$ squares such that each square covers exactly four cells, no squares overlap, and each cell is covered by an equal number of squares. What is the maximum value of this equal number? (A square can be folded over an edge.)


Given that \( x \) is a negative real number that satisfies \(\frac{1}{x+\frac{1}{x+2}}=a\). If \( b=x+\frac{7}{2} \), find the value of \( b \).


Is it possible to cover the entire plane with squares, among which only two are identical?

Solution: \boxed{Yes}


Let \( a_{n} \) be the number written in base 10 with \( 2^{n} \) digits '9', for \( n \geq 0 \). Let \( b_{n} = a_{0} a_{1} \cdots a_{n} \) be the product of the first \( a_{k} \). What is the sum of the digits of \( b_{n} \)?

Solution: \boxed{9 \times 2^n}


Is it true that any number divisible by 6 and greater than 1000 can be represented as

$$
n(n+1)(n+2)(n+3)(n+4) - m(m+1)(m+2)
$$

where $m$ and $n$ are natural numbers?

Solution: \boxed{\text{No}}


What is the smallest natural number \( n \) for which there exist natural numbers \( x \) and \( y \) satisfying the equations:
a) \( x \cdot (x + n) = y^{2} \);
b) \( x \cdot (x + n) = y^{3} \)?


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


Given that \( f(x) \) and \( g(x) \) are functions defined on \(\mathbf{R}\), the graph of \( f(x) \) is symmetric about the line \( x=1 \), the graph of \( g(x) \) is symmetric about the point \( (1, -2) \), and \( f(x) + g(x) = 9^x + x^3 + 1 \), find the value of \( f(2)g(2) \).


The graphs of the functions \( y = kx + b \) and \( y = bx + k \) intersect. Find the x-coordinate of the intersection point.

Solution: \boxed{x = 1}


As shown in the figure, in $\triangle ABC$, $BC > AC$ and $\angle A = 60^\circ$. Points $D$ and $E$ are the midpoints of $AB$ and $AC$ respectively. If $PC$ bisects $\angle ACB$ and $PD$ bisects $\angle ADE$, then $\angle DPC$ equals $\qquad$.


From 60 right-angled triangles with legs of 2 and 3, a rectangle was formed. What is the maximum possible value of its perimeter?


\(\sin^6 \alpha + \cos^6 \alpha + 3 \sin^2 \alpha \cos^2 \alpha = 1\)


A volleyball net is in the shape of a rectangle with dimensions of $50 \times 600$ cells.

What is the maximum number of strings that can be cut so that the net does not fall apart into pieces?


Evgeny is laying tiles on the floor of his living room, which measures 12 by 16 meters. He plans to place square tiles of size 1 m × 1 m along the border of the room and to lay the rest of the floor with square tiles of size 2 m × 2 m. How many tiles will he need in total?

Solution: \boxed{87 \, \text{tiles}}


Ancient Greek mathematicians associated natural numbers with polygons in the following manner:

Triangular numbers: \(1, 3, 6, 10, 15\)

Square numbers: \(1, 4, 9, 16, 25\)

Pentagonal numbers: \(1, 5, 12, 22, 35\)

Hexagonal numbers: \(1, 6, 15, 28, 45\)

Following the above sequence, the 6th pentagonal number is \(\_\_\_\_\_\_ \).


In how many ways can 4 numbers be chosen from the first 1000 natural numbers \( 1,2, \ldots, 1000 \) to form an increasing arithmetic progression?


Verify the following properties of the symmetric difference operation $\triangle$:

$$
\begin{aligned}
(A \triangle B) \triangle C= & A \triangle(B \triangle C), \quad(A \triangle B) \triangle(B \triangle C)=A \triangle C \\
& A \triangle B=C \Leftrightarrow A=B \triangle C
\end{aligned}
$$

Solution: \boxed{A = B \triangle C}


A sequence of positive integers is defined by \(a_{0}=1\) and \(a_{n+1}=a_{n}^{2}+1\) for each \(n \geq 0\). Find \(\operatorname{gcd}\left(a_{999}, a_{2004}\right)\).


Jitka worked a job with a fixed daily wage. In three days, she earned enough money to buy a board game and still had 490 Kč left. If she worked for five days, she could buy two such board games and still have 540 Kč left.

How much did the board game cost?


Let $\alpha, \beta, \gamma$ be the three roots of $x^{3} - x - 1 = 0$. Calculate the value of $\frac{1-\alpha}{1+\alpha} + \frac{1-\beta}{1+\beta} + \frac{1-\gamma}{1+\gamma}$.


Let \( N = a_1a_2\ldots a_n \) in binary. Show that if \( a_1 - a_2 + a_3 - \ldots + (-1)^{n-1}a_n \equiv 0 \pmod{3} \), then \( N \equiv 0 \pmod{3} \).


Let \(AB\) and \(CD\) be chords of a circle that do not intersect, and let \(K\) be an internal point of the chord \(CD\). Construct a point \(P\) on the circumference of the circle such that the segment of the chord \(CD\) falling inside the triangle \(ABP\) is bisected by the point \(K\).

Solution: \boxed{P}


Find the smallest natural number \( n \) that satisfies the following conditions:
1. The units digit of \( n \) is 6.
2. If the units digit 6 is moved to the front of the number, the new number is 4 times \( n \).


Find the range of the function \( y = f^{[2019]}(x) \), where \( f(x) = \log_{0.5} \left(\frac{\sin x}{\sin x + 15}\right) \).

Solution: \boxed{[4, +\infty)}


In the complex plane, the complex numbers \(3-\mathrm{i}\), \(2-2 \mathrm{i}\), and \(1+5 \mathrm{i}\) correspond to the points \(A\), \(B\), and \(C\) respectively. Find the area of triangle \(ABC\). \(S_{\triangle ABC} =\) \(\qquad\)


An isosceles triangle can be divided into two isosceles triangles (not necessarily equal). Find the possible values for the smallest angle of such a triangle. Provide the smallest of these values in degrees, multiplied by 6006.


If any two adjacent digits of a three-digit number have a difference of at most 1, it is called a "steady number". How many steady numbers are there?


Triangle \( A B C \) is divided by line segments as shown in the picture. Line segments \( D E \) and \( A B \) are parallel. Triangles \( C D H, C H I, C I E, \) and \( F I H \) have the same area of \( 8 \, \mathrm{dm}^2 \).

Determine the area of quadrilateral \( A F H D \).

Solution: \boxed{24 \ \text{dm}^2}


The recruits stood in a row one after another, all facing the same direction. Among them are three brothers: Peter, Nikolai, and Denis. There were 50 people in front of Peter, 100 in front of Nikolai, and 170 in front of Denis. At the command "About face!", everyone turned to face the opposite direction. After turning, it was found that the number of people now standing in front of one of the brothers is four times the number of people standing in front of another. How many recruits, including the brothers, could there be? List all possible options.


Given the function
$$
f(x) = \left|8x^3 - 12x - a\right| + a
$$

The maximum value of this function on the interval \([0, 1]\) is 0. Find the maximum value of the real number \(a\).


Solve the system of equations

$$
\left\{\begin{array}{l}
2x - 3|y| = 1 \\
|x| + 2y = 4
\end{array}\right.
$$

Solution: \boxed{(2, 1)}


Which of the following numbers is closest to $\sqrt{65}-\sqrt{63}$?
(A) 0.12
(B) 0.13
(C) 0.14
(D) 0.15
(E) 0.16

Solution: \boxed{(B)}


Compute the limit of the numerical sequence:

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


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


Find the cosine of the angle between the vectors $\overrightarrow{A B}$ and $\overrightarrow{A C}$.

$A(3, 3, -1), B(1, 5, -2), C(4, 1, 1)$


In an isosceles triangle with a side length of 4, a median is drawn to one of the equal sides. Find the length of the base of the triangle if the length of the median is 3.


In triangle \( ABC \), side \( AC \) equals \( b \), side \( AB \) equals \( c \), and \( AD \) is the angle bisector such that \( DA = DB \). Find the length of side \( BC \).

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


A car left the city for the village, and simultaneously, a cyclist left the village for the city. When the car and the cyclist met, the car immediately turned around and went back to the city. As a result, the cyclist arrived in the city 35 minutes later than the car. How many minutes did the cyclist spend on the entire trip, given that his speed is 4.5 times less than the speed of the car?


Let \(a\), \(b\), and \(c\) be real numbers. If 1 is a root of \(x^{2} + ax + 2 = 0\) and \(a\) and \(b\) are roots of \(x^{2} + 5x + c = 0\), find the value of \(a + b + c\).


Ivan Petrovich wants to save money for his retirement in 12 years. He decided to deposit 750,000 rubles in a bank account with an 8 percent annual interest rate. What will be the total amount in the account by the time Ivan Petrovich retires, assuming the interest is compounded annually using the simple interest formula?


Vanya came up with a three-digit prime number in which all digits are different.

What digit can it end with if its last digit is equal to the sum of the first two digits?


Mark the number 1 at the endpoints of a diameter of a circle. Divide each resulting semicircular arc into equal parts and label the division points with the sum of the numbers at their endpoints. Repeat this process, dividing each arc into equal parts and labeling new points with the sum of the numbers at their endpoints, for $n$ steps. Find the sum of all the numbers marked on the circle after $n$ steps.

Solution: \boxed{2 \times 3^n}


How many five-digit numbers read the same from left to right and from right to left (for example, numbers like 54345, 17071)?


If we replace the two middle digits $\overline{00}$ in 2009 with the two-digit numbers $\overline{ab}$ and $c\bar{d}$ respectively, we obtain perfect squares: $\overline{2ab9}=n^{2}$ and $\overline{2cd9}=m^{2}$, where $m > n$ and $m, n \in \mathbb{N}$. Determine the array $(m+n, \overline{ab}+\overline{cd})$.


Find all three-digit numbers that are 12 times the sum of their digits.