Let \( ABCD \) be a square, and let \( \ell \) be a line passing through the midpoint of segment \( \overline{AB} \) that intersects segment \( \overline{BC} \). Given that the distances from \( A \) and \( C \) to \( \ell \) are 4 and 7, respectively, compute the area of \( ABCD \).


Given the family of curves \(2(2 \sin \theta-\cos \theta+3) x^{2}-(8 \sin \theta+\cos \theta+1) y=0\), where \(\theta\) is a parameter. Find the maximum length of the chord intercepted by these curves on the line \(y = 2x\).

Solution: \boxed{8 \sqrt{5}}


Postman Pechkin is riding a bicycle down the road. He noticed that every 4.5 kilometers, a suburban bus overtakes him, and every 9 minutes, a suburban bus traveling in the opposite direction passes him. The interval between bus movements in both directions is 12 minutes. At what speed is Pechkin traveling?

Solution: \boxed{15 \text{ km/h}}


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


It is known that

$$
\begin{array}{r}
a_{1}-4 a_{2}+3 a_{3} \geqslant 0 \\
a_{2}-4 a_{3}+3 a_{4} \geqslant 0 \\
\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \\
\cdot \cdot \cdot \cdot \cdot \cdot \cdot \\
a_{98}-4 a_{99}+3 a_{100} \geqslant 0 \\
a_{99}-4 a_{100}+3 a_{1} \geqslant 0 \\
a_{100}-4 a_{1}+3 a_{2} \geqslant 0
\end{array}
$$

Let $a_{1}=1$; what are the values of $a_{2}, a_{3}, \ldots, a_{100}$?


In the trapezoid $ABCD$, a circle with radius 2 is inscribed, touching the base $CD$ at point $N$. Find the area of the trapezoid if $DN = 1$ and $AB = 12$.


A number is divided by \(7, 11, 13\). The sum of the quotients is 21, and the sum of the remainders is 21. What is the number?


A numerical sequence is defined by the conditions: \( a_{1} = 1 \), \( a_{n+1} = a_{n} + \left\lfloor \sqrt{a_{n}} \right\rfloor \).

How many perfect squares are there among the first terms of this sequence that do not exceed \( 1{,}000{,}000 \)?


As shown in the figure, circle $\odot O$ is tangent to sides $AB$ and $AD$ of the square $ABCD$ at points $L$ and $K$, respectively. The circle intersects side $BC$ at points $M$ and $P$. Given that $BM = 8$ cm and $MC = 17$ cm, what is the area of circle $\odot O$ in square centimeters?

Solution: \boxed{169 \pi}


The villages of "Upper Vasyuki" and "Lower Vasyuki" are located on the riverbank. The steamboat travels the distance from Upper Vasyuki to Lower Vasyuki in one hour, while the boat takes 45 minutes. It is known that the speed of the boat in still water is twice the speed of the steamboat (also in still water). Determine how much time (in minutes) it will take a raft to float downstream from Upper Vasyuki to Lower Vasyuki.

Solution: \boxed{90 \text{ minutes}}


In the rectangle \(ABCD\), \(BE = 5\), \(EC = 4\), \(CF = 4\), and \(FD = 1\) as shown in the figure. What is the area of \(\triangle AEF\)?


An equilateral triangle is divided into smaller equilateral triangles with side length 1 by lines parallel to its sides, dividing each side into $n$ parts (in the given figure, $n=5$).

What is the maximum number of segments of length 1 with endpoints at the vertices of these triangles that can be marked in such a way that no triangle has all its sides consisting of marked segments?


Several energy-saving devices with a total weight of 120 kg were delivered to the factory. It is known that the total weight of the three lightest devices is 31 kg, and the total weight of the three heaviest devices is 41 kg. How many energy-saving devices were delivered to the factory if the weights of any two devices are different?


Given the points $A(-6,-1)$, $B(2,3)$, and $C(-1,4)$ in a right-angled coordinate system, find the point $D$ such that the quadrilateral $ABCD$ is an isosceles trapezoid with $AB \parallel CD$.

Solution: \boxed{(-5, 2)}


In the plane Cartesian coordinate system $xOy$, a line $l$ passes through the origin, and $\vec{n}=(3,1)$ is a normal vector to $l$. Given that the sequence $\left\{a_{n}\right\}$ satisfies: for any positive integer $n$, the point $\left(a_{n+1}, a_{n}\right)$ lies on $l$. If $a_{2}=6$, then the value of $a_{1} a_{2} a_{3} a_{4} a_{5}$ is $\qquad$.


Let \( n \) be an integer. Determine the largest possible constant \( C \) such that for all \( a_1, \ldots, a_n \geq 0 \), we have \(\sum a_i^2 \geq C \sum_{i < j} a_i a_j\).


Points \(M\) and \(N\) lie on edges \(BC\) and \(AA_1\) of the parallelepiped \(ABCD A_1 B_1 C_1 D_1\). Construct the intersection point of line \(MN\) with the base plane \(A_1 B_1 C_1 D_1\).

Solution: \boxed{P}


In five pots standing in a row, Rabbit poured three kilograms of honey (not necessarily into each pot and not necessarily in equal amounts). Winnie the Pooh can take any two pots standing next to each other. What is the maximum amount of honey Winnie the Pooh will be able to eat for sure?

Solution: \boxed{1 \text{ kg}}


Determine the coordinates of the focus of the parabola $y=-\frac{1}{6} x^{2}$ and write the equation of its directrix.

Solution: \boxed{y = 1.5}


Find the sum of the first 10 elements that appear both in the arithmetic progression $\{4, 7, 10, 13, \ldots\}$ and the geometric progression $\{10, 20, 40, 80, \ldots\}$. (10 points)


At the namesake festival, 45 Alexanders, 122 Borises, 27 Vasily, and several Gennady attended. At the beginning of the festival, all of them lined up so that no two people with the same name stood next to each other. What is the minimum number of Gennadys that could have attended the festival?


Given that \( m \) and \( n \) are positive integers, with \( n \) being an odd number, find the greatest common divisor of \( 2^m + 1 \) and \( 2^n - 1 \).


Find the area of the figure defined on the coordinate plane by the inequalities \( |x| - 1 \leq y \leq \sqrt{1 - x^2} \).

Solution: \boxed{\frac{\pi}{2} + 1}


It is known that for three consecutive natural values of the argument, the quadratic function \( f(x) \) takes the values 13, 13, and 35 respectively. Find the smallest possible value of \( f(x) \).


The legs of a right triangle are 3 and 4. Find the distance between the centers of the circumscribed and inscribed circles.


Solution: \boxed{x = y}


a) How many zeros are at the end of the number \( A = 2^{5} \times 3^{7} \times 5^{7} \times 11^{3} \) ?

b) How many zeros are at the end of the number \( B = 1 \times 2 \times 3 \times 4 \times \cdots \times 137 \) ?


Find the smallest possible value of \(x+y\) where \(x, y \geq 1\) and \(x\) and \(y\) are integers that satisfy \(x^{2}-29y^{2}=1\).


The diagram shows a sector \( OAB \) of a circle, center \( O \) and radius \( 8 \text{ cm} \), in which \( \angle AOB = 120^\circ \). Another circle of radius \( r \text{ cm} \) is to be drawn through the points \( O \), \( A \), and \( B \). Find the value of \( r \).


Solution: \boxed{\text{No}}


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


In triangle \( \triangle ABC \), if \( \tan A \tan B = \tan A \tan C + \tan C \tan B \), then \( \frac{a^2 + b^2}{c^2} = \) ?


Let \(\alpha, \beta\) be the roots of \(x^{2}+bx-2=0\). If \(\alpha>1\) and \(\beta<-1\), and \(b\) is an integer, find the value of \(b\).


In the village where Glafira lives, there is a small pond that is filled by springs at the bottom. Glafira discovered that a herd of 17 cows completely drank this pond in 3 days. After some time, the springs refilled the pond, and then 2 cows drank it in 30 days. How many days will it take for one cow to drink this pond?


Find all values of \( x \) for which the minimum of the numbers \( 8 - x^{2} \) and \( \operatorname{ctg} x \) is not less than -1. In the answer, record the total length of the found intervals on the number line, rounding it to the nearest hundredth if necessary.


Let \( q(n) \) be the sum of the digits of the natural number \( n \). Determine the value of

$$
q\left(q\left(q\left(2000^{2000}\right)\right)\right)
$$


In a $5 \times 5$ grid, place an "L" shape composed of 4 small squares, which can be rotated and flipped. No two "L" shapes can overlap. What is the maximum number of "L" shapes that can be placed?


What is the minimum area of a triangle that contains a unit square?


There are 200 matches. How many ways are there to form, using all the matches, a square and (separately) an equilateral triangle? (Different ways are distinguished by the sizes of the square and the triangle).


Let \( x < y \) be positive real numbers such that
\[ \sqrt{x} + \sqrt{y} = 4 \]
and
\[ \sqrt{x+2} + \sqrt{y+2} = 5. \]
Compute \( x \).


In a circle with radius \( R \), two chords \( AB \) and \( AC \) are drawn. A point \( M \) is taken on \( AB \) or its extension such that the distance from \( M \) to the line \( AC \) is equal to \( |AC| \). Similarly, a point \( N \) is taken on \( AC \) or its extension such that the distance from \( N \) to the line \( AB \) is equal to \( |AB| \). Find the distance \( |MN| \).


Given \( a \in \mathbf{Z} \), and \( x^{6} - 33x + 20 \) is divisible by \( x^{2} - x + a \), determine the value of \( a \).


In an isosceles triangle, the side is divided by the point of tangency of the inscribed circle in the ratio 7:5 (starting from the vertex). Find the ratio of the side to the base.


There are 15 stones placed in a line. In how many ways can you mark 5 of these stones so that there are an odd number of stones between any two of the stones you marked?


Given that $A$, $B$, and $C$ are points on the circle $\odot O$, and
$$
\overrightarrow{A O}=\frac{1}{2}(\overrightarrow{A B}+\overrightarrow{A C}),
$$
find the dot product $\overrightarrow{A B} \cdot \overrightarrow{A C}$.


Find the sum of the digits in the decimal representation of the integer part of the number $\sqrt{\underbrace{11 \ldots 11}_{2018} \underbrace{55 \ldots 55}_{2017} 6}$.


From the vertex of the obtuse angle \( A \) of triangle \( ABC \), a perpendicular \( AD \) is drawn. From point \( D \), a circle is drawn with radius equal to \( AD \), intersecting the sides \( AB \) and \( AC \) at points \( M \) and \( N \), respectively. Find the side \( AC \), given that \( AB = c \), \( AM = m \), and \( AN = n \).

Solution: \boxed{\frac{mc}{n}}


A circle with a radius of 15 is tangent to two adjacent sides \( AB \) and \( AD \) of square \( ABCD \). On the other two sides, the circle intercepts segments of 6 and 3 cm from the vertices, respectively. Find the length of the segment that the circle intercepts from vertex \( B \) to the point of tangency.


Given a line \( l \) tangent to the unit circle \( S \) at point \( P \), point \( A \) lies on the same side of \( l \) as circle \( S \), and the distance from \( A \) to \( l \) is \( h \) where \( h > 2 \). From point \( A \), two tangents to \( S \) are drawn, intersecting \( l \) at points \( B \) and \( C \). Find the product of the lengths of line segments \( PB \) and \( PC \).


Under one of the squares of an $8 \times 8$ board, there is a treasure buried. Under each of the remaining squares, there is a tag indicating the minimum number of steps it takes to reach the treasure from that square (one step consists of moving to an adjacent square along one of its sides). What is the minimum number of squares that need to be dug to ensure retrieval of the treasure?


They multiplied several natural numbers and obtained 224, with the smallest number being exactly half of the largest number. How many numbers were multiplied?


Fill 2 $a$'s and 2 $b$'s into the 16 squares as shown in the diagram, with each square containing at most one letter. If the same letters cannot be in the same row or column, how many different ways can this be done? (Answer with a number.)


A certain store sells a brand of socks for 4.86 yuan per pair. Now they are running a promotion: "buy five, get one free". What is the actual price per pair of socks now?


Given a cyclic quadrilateral \(ABCD\) with side lengths \(AB = 2\), \(BC = 6\), and \(CD = DA = 4\), find the area of the quadrilateral \(ABCD\).


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


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


A set consisting of a finite number of points in the plane has the following property: for any two points A and B in this set, there exists a point C in this set such that triangle ABC is equilateral. How many points can such a set contain?


Given the quadratic function $y=f(x)=x^{2}+bx+c$ whose graph passes through the point $(1,13)$, and the function $y=f\left(x-\frac{1}{2}\right)$ is an even function:
(1) Find the explicit expression for $f(x)$;
(2) Determine whether there exists a point on the graph of $y=f(x)$ whose x-coordinate is a positive integer and y-coordinate is a perfect square. If such a point exists, find its coordinates; if not, provide an explanation.

Solution: \boxed{(10, 121)}


Let \( A \) be the sum of the digits of the decimal number \( 4444^{4444} \), and let \( B \) be the sum of the digits of \( A \). Find the sum of the digits of \( B \), where all numbers mentioned are in decimal form.


Misha wrote on the board 2004 pluses and 2005 minuses in some order. From time to time, Yura comes to the board, erases any two signs, and writes one in their place. If he erases two identical signs, he writes a plus; if the signs are different, he writes a minus. After several such actions, only one sign remains on the board. What is the final sign?

Solution: \boxed{-}


Triangle \(A B C\) has a right angle at \(C\), and \(D\) is the foot of the altitude from \(C\) to \(A B\). Points \(L\), \(M\), and \(N\) are the midpoints of segments \(A D, D C\), and \(C A\), respectively. If \(C L=7\) and \(B M=12\), compute \(B N^{2}\).


An n-digit number has the property that if you cyclically permute its digits, the result is always divisible by 1989. What is the smallest possible value of n? What is the smallest such number?


Gena went to the shooting range with his dad. They agreed that Gena makes 5 shots and for each hit, he gets the right to make 2 more shots. In total, Gena made 17 shots. How many times did he hit the target?


Determine the largest square number that is not divisible by 100 and, when its last two digits are removed, is also a square number.


Given the functions \( f(x) \) and \( g(x) \) as defined in Theorem 1, if \( T_1 = 1 \) and \( T_2 = \frac{1}{m} \) (with \( m \in \mathbb{N} \) and \( m > 1 \)), then the smallest positive period of the function \( h(x) = f(x) + g(x) \) is \( \frac{1}{k} \), where \( k = 1 \) or \( k \in \mathbb{N} \), \( k \) is not a multiple of \( m \), and \( m \) is not a multiple of \( k \).


27 If the sum of all positive divisors of a positive integer equals twice the number itself, the number is called a perfect number. Find all positive integers \( n \) such that \( n-1 \) and \( \frac{n(n+1)}{2} \) are both perfect numbers.


The base of a right prism is a right triangle with a hypotenuse of length \( c \) and an acute angle of \( 30^\circ \). A plane is drawn through the hypotenuse of the lower base and the vertex of the right angle of the upper base, forming an angle of \( 45^\circ \) with the base's plane. Determine the volume of the triangular pyramid cut off from the prism by this plane.


If a positive integer \( n \) makes the equation \( x^{3} + y^{3} = z^{n} \) have positive integer solutions \((x, y, z)\), then \( n \) is called a "good number". Find the number of "good numbers" not exceeding 2019.


Two small and large cubes are glued together to form a three-dimensional shape. The four vertices of the smaller cube's glued face are at the quarter points (not the midpoints) of the edges of the larger cube's glued face. If the edge length of the larger cube is 4, what is the surface area of this three-dimensional shape?


Let $a$ and $b$ be natural numbers, $a > b$, and the numbers $a+b$ and $a-b$ are relatively prime. Find all values of the greatest common divisor of the numbers $a$ and $b$.


Let \( \triangle ABC \) have centroid \( S \), midpoint of segment \( AS \) be \( H \), and midpoint of side \( AB \) be \( Q \). Let the line parallel to \( BC \) through \( H \) intersect \( AB \) at \( P \) and line \( CQ \) at \( R \). What is the ratio of the areas of triangles \( PQR \) and \( APH \)?


Stringing 6 red balls, 1 green ball, and 8 yellow balls into a necklace, how many possible different types can appear (balls of the same color are not distinguished)?


There are 6 different numbers from 6 to 11 on the faces of a cube. The cube was rolled twice. The sum of the numbers on the four lateral faces was 36 in the first roll and 33 in the second roll. What number is on the face opposite to the face with the number 10?


Bethany, Chun, Dominic, and Emily go to the movies. They choose a row with four consecutive empty seats. If Dominic and Emily must sit beside each other, in how many different ways can the four friends sit?


Take a clay sphere of radius 13, and drill a circular hole of radius 5 through its center. Take the remaining "bead" and mold it into a new sphere. What is this sphere's radius?


Show that the equation

$$
a x^{2} + b x + c = 0
$$

can only be satisfied by three distinct numbers $\left( x_{1}, x_{2}, x_{3} \right)$ if $a = b = c = 0$.

Solution: \boxed{a = b = c = 0}


Given a positive integer \(N\) (written in base 10), define its integer substrings to be integers that are equal to strings of one or more consecutive digits from \(N\), including \(N\) itself. For example, the integer substrings of 3208 are \(3, 2, 0, 8, 32, 20, 320, 208\), and 3208. (The substring 08 is omitted from this list because it is the same integer as the substring 8, which is already listed.)

What is the greatest integer \(N\) such that no integer substring of \(N\) is a multiple of 9? (Note: 0 is a multiple of 9.)


Represent the expression \(2x^{2} + 2y^{2}\) as a sum of two squares.

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


Find the smallest positive integer \( n \) such that for any positive integer \( k \geqslant n \), in the set \( M = \{1, 2, \cdots, k\} \), for any \( x \in M \), there always exists another number \( y \in M \) (with \( y \neq x \)) such that \( x + y \) is a perfect square.


Given the sequences \( \left\{a_{n}\right\} \) and \( \left\{b_{n}\right\} \) with general terms \( a_{n}=2^{n} \) and \( b_{n}=3n+2 \), respectively, arrange the common terms of \( \left\{a_{n}\right\} \) and \( \left\{b_{n}\right\} \) in ascending order to form a new sequence \( \left\{c_{n}\right\} \). Determine the general term of \( \left\{c_{n}\right\} \).

Solution: \boxed{c_n = 2^{2n+1}}


A three-digit number \( A \) is such that the difference between the largest three-digit number and the smallest three-digit number that can be formed using the digits of \( A \) is still \( A \). What is the value of this number \( A \)?


Find the value of $\cos \frac{2 \pi}{7} + \cos \frac{4 \pi}{7} + \cos \frac{6 \pi}{7}$.


Find all prime numbers \( p \) such that the numbers \( p + 4 \) and \( p + 8 \) are also prime.


The Fibonacci numbers are defined recursively by \( F_{0}=0 \), \( F_{1}=1 \), and \( F_{i}=F_{i-1}+F_{i-2} \) for \( i \geq 2 \). Given 15 wooden blocks of weights \( F_{2}, F_{3}, \ldots, F_{16} \), compute the number of ways to paint each block either red or blue such that the total weight of the red blocks equals the total weight of the blue blocks.


Towers are placed on an $n \times n$ chessboard such that if the square $(i, j)$ is empty, there are at least $n$ towers on the $i$-th row and the $j$-th column. Show that there are at least $n^{2} / 2$ towers on the chessboard.


Given nonzero real numbers \(a, b, c, d\) and the function \(f(x)=\frac{ax+b}{cx+d}\) for \(x \in \mathbb{R}\) such that \(f(19) = 19\) and \(f(97) = 97\). If for any real number \(x \neq -\frac{d}{c}\), it holds that \(f[f(x)] = x\), find the unique number that is outside the range of \(f(x)\).


For a homework assignment, Tanya was asked to come up with 20 examples of the form \( * + * = * \), where different natural numbers need to be inserted in place of \( * \) (i.e., a total of 60 different numbers should be used). Tanya loves prime numbers very much, so she decided to use as many of them as possible while still getting correct examples. What is the maximum number of prime numbers Tanya can use?


Find the minimum value of the expression

$$
\frac{|a-3b-2| + |3a-b|}{\sqrt{a^2 + (b+1)^2}}
$$

for \(a, b \geq 0\).


On the sides $AB$, $BC$, and $AC$ of triangle $ABC$, whose area is 75, points $M$, $N$, and $K$ are respectively located. It is known that $M$ is the midpoint of $AB$, the area of triangle $BMN$ is 15, and the area of triangle $AMK$ is 25. Find the area of triangle $CNK$.


0 < k < 1 is a real number. Define \( f: [0, 1] \to [0, 1] \) by
\[ f(x) =
\begin{cases}
0 & \text{for } x \leq k, \\
1 - \left( \sqrt{kx} + \sqrt{(1-k)(1-x)} \right)^2 & \text{for } x > k.
\end{cases}
\]
Show that the sequence \( 1, f(1), f(f(1)), f(f(f(1))), \ldots \) eventually becomes zero.


How many terms of the sum

$$
1+2+3+\ldots
$$

are needed for the result to be a three-digit number in which all digits are the same?


In an arithmetic progression with 12 terms, the sum of the terms is 354. The ratio of the sum of the terms with even indices to the sum of the terms with odd indices is 32:27. Determine the first term and the common difference of the progression.

Solution: \boxed{a = 2, \, d = 5}


Find all functions \( f: \mathbf{Z}_{+} \rightarrow \mathbf{Z}_{+} \) such that for any positive integers \( m \) and \( n \), \( f(m) + f(n) - mn \) is non-zero and divides \( mf(m) + nf(n) \).

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


Let points \( A_{1}, A_{2}, A_{3}, A_{4}, A_{5} \) be located on the unit sphere. Find the maximum value of \( \min \left\{A_{i} A_{j} \mid 1 \leq i < j \leq 5 \right\} \) and determine all cases where this maximum value is achieved.


Find all natural numbers \( x \) such that the product of all digits in the decimal representation of \( x \) is equal to \( x^{2} - 10x - 22 \).


Given a right triangle \(ABC\) with legs \(BC = 30\) and \(AC = 40\). Points \(C_1\), \(A_1\), and \(B_1\) are chosen on the sides \(AB\), \(BC\), and \(CA\), respectively, such that \(AC_1 = BA_1 = CB_1 = 1\). Find the area of triangle \(A_1 B_1 C_1\).


In the expression below, each letter has replaced a certain digit in the base-6 numbering system:

$$
F A R E S = (F E E)^{2}
$$

(identical letters represent identical digits). Restore the original digits.


A state issues car license plates consisting of 6 digits (each digit ranging from $0$ to $9$), with the condition that any two license plate numbers must differ in at least two places. (For example, license numbers 027592 and 020592 cannot both be used). Determine the maximum number of license plate numbers possible under this condition. Provide a proof.


Seven fishermen are standing in a circle. Each fisherman has a professional habit of exaggerating numbers, with a distinct measure of exaggeration (an integer) indicating by how many times the number mentioned by the fisherman exceeds the true value. For example, if a fisherman with an exaggeration measure of 3 catches two fish, he will claim to have caught six fish. When asked: "How many fish did your left neighbor catch?" the answers were (not necessarily in the seated order) $12, 12, 20, 24, 32, 42,$ and $56$. When asked: "How many fish did your right neighbor catch?" six of the fishermen answered $12, 14, 18, 32,$ $48,$ and $70$. What did the seventh fisherman answer?


Find all prime numbers for which \( p^{3} + p^{2} + 11p + 2 \) is also prime.


For how many integers \( x \) is the expression \(\frac{\sqrt{75-x}}{\sqrt{x-25}}\) equal to an integer?


In the city of Omsk, a metro line was built in a straight line. On this same line is the house where Nikita and Yegor live. Every morning, they simultaneously leave the house for their lessons, after which Yegor runs to the nearest metro station at a speed of 12 km/h, while Nikita walks along the metro line to another station at a speed of 6 km/h. Despite this, Nikita always manages to reach his lesson on time, whereas Yegor does not, even though he doesn't delay anywhere. Find the maximum possible speed of the metro trains, given that it is constant and an integer. (Assume the school is located directly at a certain metro station, distinct from the given ones).


(a) Find the distance from the point \((1,0)\) to the line connecting the origin and the point \((0,1)\).

(b) Find the distance from the point \((1,0)\) to the line connecting the origin and the point \((1,1)\).

(c) Find the distance from the point \((1,0,0)\) to the line connecting the origin and the point \((1,1,1)\).


Given the sequence \( b_{k} \) defined as:
$$
b_{k}=\left\{\begin{array}{cc}
0 & k<l \\
a_{k-l} & k \geqslant l,
\end{array} \text{ then } B(x)=x^{l} A(x)
\right.
$$

Solution: \boxed{B(x) = x^l A(x)}


The sum of positive numbers \(a, b, c,\) and \(d\) does not exceed 4. Find the maximum value of the expression

\[
\sqrt[4]{a^{2}(a+b)}+\sqrt[4]{b^{2}(b+c)}+\sqrt[4]{c^{2}(c+d)}+\sqrt[4]{d^{2}(d+a)}
\]

Solution: \boxed{4 \sqrt[4]{2}}


Which values of \( x \) satisfy the following system of inequalities?

\[
\frac{x}{6} + \frac{7}{2} > \frac{3x + 29}{5}, \quad x + \frac{9}{2} > \frac{x}{8}, \quad \frac{11}{3} - \frac{x}{6} < \frac{34 - 3x}{5}.
\]

Solution: \boxed{\text{No solution}}


Find the largest natural number that cannot be represented as the sum of two composite numbers.


Find the sum of all natural numbers that have exactly four natural divisors, three of which are less than 15, and the fourth is not less than 15.


Let $A$ and $B$ be two opposite vertices of a cube with side length 1. What is the radius of the sphere centered inside the cube, tangent to the three faces that meet at $A$ and to the three edges that meet at $B$?

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


Two squares \(A B C D\) and \(B E F G\) share vertex \(B\), with \(E\) on the side \(B C\) and \(G\) on the side \(A B\). The length of \(C G\) is \(9 \text{ cm}\) and the area of the shaded region is \(47 \text{ cm}^2\). Calculate the perimeter of the shaded region.

Solution: \boxed{32 \text{ cm}}


A circle with a radius of \(1 + \sqrt{2}\) is circumscribed around an isosceles right triangle. Find the radius of a circle that is tangent to the legs of this triangle and internally tangent to the circumscribed circle.


Calculate the value of the product \(\left(1-\frac{1}{4}\right) \cdot\left(1-\frac{1}{9}\right) \cdot\left(1-\frac{1}{16}\right) \cdot \ldots \cdot\left(1-\frac{1}{2020^{2}}\right)\).


Given that one root of the quadratic trinomial \(a x^{2}+b x+b\) and one root of the quadratic trinomial \(a x^{2}+a x+b\) have a product equal to 1, determine the sum of the squares of these two roots.


Is it possible to find two lucky tickets among ten consecutive tickets? A ticket is considered lucky if the sum of the first three digits is equal to the sum of the last three digits.

Solution: \boxed{\text{Yes}}


What is the ratio of the volumes of an equilateral cone, an equilateral cylinder, and a sphere, if their surface areas are equal?

Solution: \boxed{2 : \sqrt{6} : 3}


Find all values of the parameters \(a, b, c\) for which the system of equations
\[
\left\{
\begin{array}{l}
a x + b y = c \\
b x + c y = a \\
c x + a y = b
\end{array}
\right\}
\]
has at least one negative solution (where \(x, y < 0\)).

Solution: \boxed{a + b + c = 0}


There are 10 sticks with lengths of 1 cm, 2 cm, $2^2$ cm, ..., $2^9$ cm. Is it possible to form an isosceles triangle using some or all of these sticks?

Solution: \boxed{\text{No}}


Each face and each vertex of a regular tetrahedron is colored red or blue. How many different ways of coloring are there? (Two tetrahedrons are said to have the same coloring if we can rotate them suitably so that corresponding faces and vertices are of the same color.)


Let \( k \) be a circle with center \( O \). Let \( A, B, C \), and \( D \) be four different points on the circle \( k \) in this order, such that \( AB \) is a diameter of \( k \). The circumcircle of the triangle \( COD \) intersects \( AC \) for the second time in \( P \). Show that \( OP \) and \( BD \) are parallel.

Solution: \boxed{OP \parallel BD}


There are more than 5000 toothpicks, which can be divided into small packs of six different specifications. If packed 10 toothpicks per pack, 9 toothpicks will remain in the end. If packed 9 toothpicks per pack, 8 toothpicks will remain. For the third, fourth, fifth, and sixth specifications, if packed 8, 7, 6, and 5 toothpicks per pack, respectively, 7, 6, 5, and 4 toothpicks will remain. How many toothpicks were there originally?


A point is located at a distance $a$ from the straight line $MN$. A circle with radius $r$ is described such that it passes through point $A$ and is tangent to the straight line $MN$. Find the distance between the point of tangency and the given point $A$.


21. $A$ and $B$ are two different non-zero natural numbers less than 1000. Find the maximum value of $\frac{A-B}{A+B}$.


The circle \(C\) has radius 1 and touches the line \(L\) at point \(P\). The point \(X\) lies on the circle \(C\) and \(Y\) is the foot of the perpendicular from \(X\) to the line \(L\). Find the maximum possible value of the area of triangle \(PXY\) as \(X\) varies.


Given an isosceles triangle with a vertex angle of $20^\circ$, the length of the base is $a$, and the length of the legs is $b$. Find the value of $\frac{a^3 + b^3}{ab^2}$.


As shown in the figure, the squares $\mathrm{ABCD}$ and $\mathbf{EFGH}$ have their sides parallel to each other. Connect $\mathbf{CG}$ and extend it to intersect BD at point $I_{0}$. Given $BD = 10$, the area of $\triangle BFC = 3$, and the area of $\triangle CHD = 5$, find the length of $\mathbf{BI}$.


A standard deck of 52 cards has the usual 4 suits and 13 denominations. What is the probability that two cards selected at random, and without replacement, from this deck will have the same denomination or have the same suit?


There are 5 integers written on a board. By summing these numbers in pairs, the following set of 10 numbers is obtained: $-1, 2, 6, 7, 8, 11, 13, 14, 16, 20$. Determine which numbers are written on the board. Write their product as the answer.


The school plans to arrange 6 leaders to be on duty from May 1st to May 3rd. Each leader must be on duty for 1 day, with 2 leaders assigned each day. If leader A cannot be on duty on the 2nd, and leader B cannot be on duty on the 3rd, how many different methods are there to arrange the duty schedule?


Two individuals, A and B, are undergoing special training on the same elliptical track. They start from the same point simultaneously and run in opposite directions. Each person, upon completing the first lap and reaching the starting point, immediately turns around and accelerates for the second lap. During the first lap, B's speed is $\frac{2}{3}$ of A's speed. During the second lap, A's speed increases by $\frac{1}{3}$ compared to the first lap, and B's speed increases by $\frac{1}{5}$. Given that the second meeting point between A and B is 190 meters from the first meeting point, what is the length of the elliptical track in meters?


Solve the following equations for real numbers:
(a) \(\sqrt{x+\sqrt{2x-1}} + \sqrt{x-\sqrt{2x-1}} = \sqrt{2}\);
(b) \(\sqrt{x+\sqrt{2x-1}} + \sqrt{x-\sqrt{2x-1}} = 1\);
(c) \(\sqrt{x+\sqrt{2x-1}} + \sqrt{x-\sqrt{2x-1}} = 2\).

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


An arithmetic sequence consists of two-digit even terms, where the sum of all odd terms is 100. Starting from the first term, each odd term is combined with the following adjacent even term to form a four-digit number without changing the order. Determine the difference between the sum of the new sequence and the sum of the original sequence.


We have drawn the circumcircle of a right triangle with legs of 3 and 4 units. What is the radius of the circle that is tangent to both legs of the triangle and the circumcircle from the inside?


A set consisting of a finite number of points in the plane has the following property: for any two points $A$ and $B$ in this set, there exists a point $C$ in this set such that the triangle $A B C$ is equilateral. How many points can such a set contain?


Calculate the limit of the function:
\[ \lim _{x \rightarrow \frac{\pi}{3}} \frac{1-2 \cos x}{\sin (\pi-3 x)} \]


Let the sequence \( a_1, a_2, \cdots, a_n, \cdots \) be defined such that \( a_1 = a_2 = 1, a_3 = 2 \), and for any positive integer \( n \), \( a_n \cdot a_{n+1} \cdot a_{n+2} \neq 1 \). Also, given that \( a_n \cdot a_{n+1} \cdot a_{n+2} \cdot a_{n+3} = a_n + a_{n+1} + a_{n+2} + a_{n+3} \), find the value of \( a_1 + a_2 + \cdots + a_{100} \).


Let $k$ be a real number. In the Cartesian coordinate plane $xOy$, there are two sets of points:
\[ A = \left\{ (x, y) \mid x^2 + y^2 = 2(x + y) \right\} \]
\[ B = \{ (x, y) \mid kx - y + k + 3 \geq 0 \} \]
If the intersection of $A$ and $B$ is a single-element set, what is the value of $k$?

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


Based on the circle graph, what percentage of Mr. Auckland's Grade 9 students do at least one hour of homework per day?
(A) $25 \%$
(B) $33 \%$
(C) $50 \%$
(D) $67 \%$
(E) $75 \%$


When the gym teacher whistles, all 10 boys and 7 girls line up in a random order. Find the expected value of the number of girls standing to the left of all the boys.


Find the greatest integer less than or equal to \((2+\sqrt{3})^{3}\).


Given a function \( f(x) \) that satisfies for any real numbers \( x \) and \( y \):
\[ f(x+y) = f(x) + f(y) + 6xy \]
and \( f(-1)f(1) \geq 9 \),
find \( f\left(\frac{2}{3}\right) \).


Vasya, Petya, and Kolya are in the same class. Vasya always lies in response to any question, Petya alternates between lying and telling the truth, and Kolya lies in response to every third question but tells the truth otherwise. One day, each of them was asked six consecutive times how many students are in their class. The responses were "Twenty-five" five times, "Twenty-six" six times, and "Twenty-seven" seven times. Can we determine the actual number of students in their class based on their answers?


An ideal gas is used as the working substance of a heat engine operating cyclically. The cycle consists of three stages: isochoric pressure reduction from $3 P_{0}$ to $P_{0}$, isobaric density increase from $\rho_{0}$ to $3 \rho_{0}$, and a return to the initial state, represented as a quarter circle in the $P / P_{0}, \rho / \rho_{0}$ coordinates with the center at point $(1,1)$. Determine the efficiency of this cycle, knowing that it is 8 times less than the maximum possible efficiency for the same minimum and maximum gas temperatures as in the given cycle.


In city $\mathrm{N}$, there are exactly three monuments. One day, a group of 42 tourists arrived in this city. Each tourist took no more than one photograph of each of the three monuments. It turned out that any two tourists together had photographs of all three monuments. What is the minimum number of photographs that all the tourists together could have taken?


Suppose that \(a\) and \(b\) are positive integers with \(2^a \times 3^b = 324\). Evaluate \(2^b \times 3^a\).


For what values of \( k \) does the equation
\[ |x-2007| + |x+2007| = k \]
have \((-\infty, -2007) \cup (2007, +\infty)\) as its solution set?

Solution: \boxed{k > 4014}


A positive integer \( \overline{ABC} \), where \( A, B, C \) are digits, satisfies
\[
\overline{ABC} = B^{C} - A
\]

Find \( \overline{ABC} \).


Given \( n \) points in a plane, where the distance between any two points is \(\geq 1\), the number of pairs of points at a distance of 1 is at most \( 3n \). For \( n \) points in space, where the distance between any two points is \(\geq 1\), the number of pairs of points at a distance of 1 is at most \( 7n \).


Let \( a, b, c, d \) such that \( a b c d = 1 \). Show that

\[ a^{2} + b^{2} + c^{2} + d^{2} + a b + a c + a d + b c + b d + c d \geq 10 \]


A, B, and C are guessing a two-digit number.
A says: Its number of factors is even, and it is greater than 50.
B says: It is an odd number, and it is greater than 60.
C says: It is an even number, and it is greater than 70.

If each of them only spoke half the truth, what is this number?


Points \(A, A_1, B, B_1, C,\) and \(C_1\) are located on a sphere of radius 11. Lines \(AA_1, BB_1,\) and \(CC_1\) are pairwise perpendicular and intersect at point \(M\), which is at a distance of \(\sqrt{59}\) from the center of the sphere. Find the length of \(AA_1\), given that \(BB_1 = 18\) and point \(M\) divides segment \(CC_1\) in the ratio \((8 + \sqrt{2}) : (8 - \sqrt{2})\).


A sequence of positive integers \(a_{n}\) begins with \(a_{1}=a\) and \(a_{2}=b\) for positive integers \(a\) and \(b\). Subsequent terms in the sequence satisfy the following two rules for all positive integers \(n\):
\[a_{2 n+1}=a_{2 n} a_{2 n-1}, \quad a_{2 n+2}=a_{2 n+1}+4 .\]

Exactly \(m\) of the numbers \(a_{1}, a_{2}, a_{3}, \ldots, a_{2022}\) are square numbers. What is the maximum possible value of \(m\)? Note that \(m\) depends on \(a\) and \(b\), so the maximum is over all possible choices of \(a\) and \(b\).


Given the operation defined as \(a \odot b \odot c = a \times b \times c + (a \times b + b \times c + c \times a) - (a + b + c)\), calculate \(1 \odot 43 \odot 47\).


Solve the inequality \(\left(\sqrt{x^{3}+2 x-58}+5\right)\left|x^{3}-7 x^{2}+13 x-3\right| \leqslant 0\).

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


For a real number \( x \), let \( [x] \) be \( x \) rounded to the nearest integer and \( \langle x \rangle \) be \( x \) rounded to the nearest tenth. Real numbers \( a \) and \( b \) satisfy \( \langle a \rangle + [b] = 98.6 \) and \( [a] + \langle b \rangle = 99.3 \). Compute the minimum possible value of \( [10(a+b)] \).

(Here, any number equally between two integers or tenths of integers, respectively, is rounded up. For example, \( [-4.5] = -4 \) and \( \langle 4.35 \rangle = 4.4 \).)


Positive integer \(a\) and integers \(b\) and \(c\), in the three-dimensional coordinate system \(O-xyz\), point \(O(0,0,0)\), \(A(a, b, c)\), and \(B\left(x^{2}, x, 1\right)\) satisfy that the angle between \(\overrightarrow{OA}\) and \(\overrightarrow{OB}\) is \(\frac{\pi}{2}\). Given that the real number \(x\) has exactly two distinct solutions \( x_{1} \) and \( x_{2} \) in the interval \((0,1)\), find the minimum value of \(a\).


Calculate the limit of the function:

\[
\lim _{x \rightarrow 0} \frac{1-\sqrt{\cos x}}{x \cdot \sin x}
\]


The diagram shows a square \(PQRS\) with sides of length 2. The point \(T\) is the midpoint of \(RS\), and \(U\) lies on \(QR\) so that \(\angle SPT = \angle TPU\). What is the length of \(UR\)?


This year is 2014. The number 2014 is not a perfect square, but its digits can be rearranged to form a new four-digit number that is a perfect square. For example, \(1024 = 32^2\). Given that using the digits \(2, 0, 1, 4\) exactly once can form another four-digit perfect square, find this new four-digit perfect square.


An arithmetic sequence \(\{a_{n}\}\) with \(a_1 > 0\) has a sum of the first \(n\) terms denoted by \(S_n\). Given that \(S_9 > 0\) and \(S_{10} < 0\), for which value of \(n\) is \(S_n\) maximized?

Solution: \boxed{n = 5}


Find all natural numbers \(a\) and \(b\) such that for all natural numbers \(n\), the number \((a n + 1)^{6} + b\) is divisible by \(n^{2} + n + 1\).

Solution: \boxed{a = 2, b = 27}


In a warehouse, there are 8 cabinets, each containing 4 boxes, and each box contains 10 mobile phones. The warehouse, each cabinet, and each box are locked with a key. The manager is tasked with retrieving 52 mobile phones. What is the minimum number of keys the manager must take with him?


In the country of Wonderland, an election campaign is being held for the title of the best tea lover, with the candidates being the Mad Hatter, March Hare, and Dormouse. According to a poll, $20\%$ of the residents plan to vote for the Mad Hatter, $25\%$ for the March Hare, and $30\%$ for the Dormouse. The remaining residents are undecided. Determine the smallest percentage of the undecided voters that the Mad Hatter needs to secure to guarantee not losing to both the March Hare and the Dormouse (regardless of how the undecided votes are allocated), given that each undecided voter will vote for one of the candidates. The winner is determined by a simple majority. Justify your answer.


A sequence of numbers is arranged in a line, and its pattern is as follows: the first two numbers are both 1. From the third number onward, each number is the sum of the previous two numbers: $1, 1, 2, 3, 5, 8, 13, 21, 34, 55$. How many even numbers are there among the first 100 numbers in this sequence (including the 100th number)?


Find the residue of the function

$$
f(z)=e^{1 / z^{2}} \cos z
$$

at the point $z=0$.


Given positive real numbers \(a\) and \(b\) that satisfy \(ab(a+b) = 4\), find the minimum value of \(2a + b\).


Graph the set of points on the $(x; y)$ plane whose coordinates satisfy the system $\left\{\begin{array}{l}(|x|+x)^{2}+(|y|-y)^{2} \leq 16, \\ y-3 x \leq 0\end{array}\right.$, and find the area of the resulting figure.

Solution: \boxed{\frac{20}{3} + \pi}


Given a triangle \(T\),

a) Place the centrally symmetric polygon \(m\) with the largest possible area inside \(T\).

What is the area of \(m\) if the area of \(T\) is 1?

b) Enclose \(T\) in the centrally symmetric convex polygon \(M\) with the smallest possible area.

What is the area of \(M\) if the area of \(T\) is 1?


Let the function \( y = f(x) \) have the domain \( \mathbf{R} \). When \( x < 0 \), \( f(x) > 1 \). Moreover, for any real numbers \( x, y \in \mathbf{R} \), the equation \( f(x+y) = f(x) f(y) \) holds. A sequence \( \left\{a_n\right\} \) is defined such that \( a_1 = f(0) \) and \( f\left(a_{n+1}\right) = \frac{1}{f\left(-2 - a_n\right)} \) for \( n \in \mathbf{N} \).
1. Find the value of \( a_{2003} \).
2. If the inequality \( \left(1+\frac{1}{a_1}\right)\left(1+\frac{1}{a_2}\right) \cdots\left(1+\frac{1}{a_n}\right) \geqslant k \cdot \sqrt{2n+1} \) holds for all \( n \in \mathbf{N} \), determine the maximum value of \( k \).


The sum of the absolute values of the terms of a finite arithmetic progression is equal to 100. If all its terms are increased by 1 or all its terms are increased by 2, in both cases the sum of the absolute values of the terms of the resulting progression will also be equal to 100. What values can the quantity \( n^{2} d \) take under these conditions, where \( d \) is the common difference of the progression, and \( n \) is the number of its terms?


Teacher Shi distributed cards with the numbers 1, 2, 3, and 4 written on them to four people: Jia, Yi, Bing, and Ding. Then the following conversation occurred:

Jia said to Yi: "The number on your card is 4."
Yi said to Bing: "The number on your card is 3."
Bing said to Ding: "The number on your card is 2."
Ding said to Jia: "The number on your card is 1."

Teacher Shi found that statements between people with cards of the same parity (odd or even) are true, and statements between people with cards of different parity are false. Additionally, the sum of the numbers on Jia's and Ding's cards is less than the sum of the numbers on Yi's and Bing's cards.

What is the four-digit number formed by the numbers on the cards of Jia, Yi, Bing, and Ding, in that order?


Given one hundred numbers: \(1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \ldots, \frac{1}{100}\). Calculate 98 differences: \(a_{1}=1-\frac{1}{3}, a_{2}=\frac{1}{2}-\frac{1}{4}, \ldots, a_{98}=\frac{1}{98}-\frac{1}{100}\). What is the sum of all these differences?


Barry has three sisters. The average age of the three sisters is 27. The average age of Barry and his three sisters is 28. What is Barry's age?
(A) 1
(B) 30
(C) 4
(D) 29
(E) 31

Solution: \boxed{31 \text{ (E)}}


Let \( ABC \) be any triangle. Let \( D \) and \( E \) be points on \( AB \) and \( BC \) respectively such that \( AD = 7DB \) and \( BE = 10EC \). Assume that \( AE \) and \( CD \) meet at a point \( F \). Determine \( \lfloor k \rfloor \), where \( k \) is the real number such that \( AF = k \times FE \).


In triangle \(ABC\), an incircle \(\omega\) with radius \(r\) is inscribed, touching side \(AB\) at point \(X\). A point \(Y\), diametrically opposite to \(X\), is marked on the circle. Line \(CY\) intersects side \(AB\) at point \(Z\). Find the area of triangle \(ABC\), given that \(CA + AZ = 1\).

Solution: \boxed{r}


Calculate the area of the figure bounded by the graphs of the functions:

$$
y = 4 - x^2, \quad y = x^2 - 2x
$$


Calculate the area of the set of points on the coordinate plane that satisfy the inequality \((y + \sqrt{x})(y - x^2) \sqrt{1 - x} \leq 0\).


The secant passing through the intersection points of two circles, one with center \( O_1 \) and radius \( 4 \text{ cm} \), and the other with center \( O_2 \) and radius \( 6 \text{ cm} \), intersects the segment \( O_1O_2 \) at point \( T \). It is known that the length of \( O_1O_2 \) is not less than \( 6 \text{ cm} \). The larger circle intersects the segment \( O_1O_2 \) at point \( A \), while the smaller circle intersects it at point \( B \), with the ratio \( AT : BT = 1 : 2 \). Calculate the length of the segment \( O_1O_2 \).

Solution: \boxed{6 \text{ cm}}


Let an ellipse have center \(O\) and foci \(A\) and \(B\). For a point \(P\) on the ellipse, let \(d\) be the distance from \(O\) to the tangent at \(P\). Show that \(PA \cdot PB \cdot d^2\) is independent of the position of \(P\).

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


Find the millionth digit after the decimal point in the decimal representation of the fraction \( \frac{3}{41} \).


What is the smallest natural number by which 720 must be multiplied to obtain the cube of a natural number?


If \(\sqrt{9-8 \sin 50^{\circ}}=a+b \csc 50^{\circ}\) where \(a, b\) are integers, find \(ab\).


At the first site, higher-class equipment was used, and at the second site, first-class equipment was used. There was less higher-class equipment than first-class equipment. First, 40% of the equipment from the first site was transferred to the second site. Then, 20% of the equipment at the second site was transferred back to the first site, with half of the transferred equipment being first-class. After this, the amount of higher-class equipment at the first site was 26 units more than at the second site, and the total amount of equipment at the second site increased by more than 5% compared to the original amount. Find the total amount of first-class equipment.


In triangle \( \triangle ABC \), \( AB + AC = 7 \), and the area of the triangle is 4. What is the minimum value of \( \sin A \)?


Arrange positive integers that are neither perfect squares nor perfect cubes (excluding 0) in ascending order as 2, 3, 5, 6, 7, 10, ..., and determine the 1000th number in this sequence.


Given a unit cube $A B C D-A_{1} B_{1} C_{1} D_{1}$, a black and a white ant start crawling from point $A$ along the edges. Each time an ant traverses one edge, it is said to have completed one segment. The white ant's path is $A A_{1} \rightarrow A_{1} D_{1} \rightarrow \cdots$, while the black ant's path is $A B \rightarrow B B_{1} \rightarrow \cdots$. Both ants follow the rule that the $(n+2)$-th segment they crawl must be on a line that is skew to the line of the $n$-th segment. Assume that after each of the black and white ants has crawled 2008 segments, each stops at a vertex of the cube. Determine the distance between the black and white ants at this moment.


Dan is holding one end of a 26-inch long piece of light string that has a heavy bead on it with each hand (so that the string lies along two straight lines). If he starts with his hands together and keeps his hands at the same height, how far does he need to pull his hands apart so that the bead moves upward by 8 inches?

Solution: \boxed{24 \ \text{inches}}


For what value of $\lambda$ does the equation

$$
\lambda x^{2}+4 x y+y^{2}-4 x-2 y-3=0
$$

represent a pair of lines?


Given that $\boldsymbol{a}$ and $\boldsymbol{b}$ are unit vectors and $|3 \boldsymbol{a} + 4 \boldsymbol{b}| = |4 \boldsymbol{a} - 3 \boldsymbol{b}|$, and that $|\boldsymbol{c}| = 2$, find the maximum value of $|\boldsymbol{a} + \boldsymbol{b} - \boldsymbol{c}|$.

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


Equilateral triangles \(ABC\) and \(A_1B_1C_1\) with side length 10 are inscribed in the same circle such that point \(A_1\) lies on arc \(BC\) and point \(B_1\) lies on arc \(AC\). Find \(AA_1^2 + BC_1^2 + CB_1^2\).


Let $F_{n}$ be the $n$-th term of the Fibonacci sequence (with the usual convention $F_{0}=0$ and $F_{1}=1$), and let $f_{n}$ be the number of ways to tile a $1 \times n$ rectangle with $1 \times 1$ squares and $1 \times 2$ dominos. Show that $F_{n}=f_{n-1}$.

Solution: \boxed{F_n = f_{n-1}}


In triangle \(ABC\), the angles \(\angle B = 30^\circ\) and \(\angle A = 90^\circ\) are known. Point \(K\) is marked on side \(AC\), and points \(L\) and \(M\) are marked on side \(BC\) such that \(KL = KM\) (point \(L\) lies on segment \(BM\)).

Find the length of segment \(LM\), given that \(AK = 4\), \(BL = 31\), and \(MC = 3\).


Given a sequence \(\{a_n\}\) with the sum of the first \(n\) terms \(S_n\), where
\[ a_1 = 3, \quad S_n = 2a_n + \frac{3}{2}((-1)^n - 1). \]

If there exist three terms \(a_1, a_p, a_q\) (\(p, q \in \mathbb{Z}_+\), \(1 < p < q\)) that form an arithmetic sequence, find \( q - p \).


Every day from Monday to Friday, an old man went to the blue sea and cast his net. Each day, the number of fish caught in the net was not greater than the number caught the previous day. Over the five days, the old man caught exactly 100 fish. What is the minimum total number of fish he could have caught over three days - Monday, Wednesday, and Friday?


In a plane, there are 2020 points, some of which are black and the rest are green.

For every black point, there are exactly two green points that are at a distance of 2020 from this black point.

Determine the minimum possible number of green points.


Let \( a_{1}, a_{2}, a_{3}, \ldots \) be an arithmetic sequence with common difference 1 and \( a_{1} + a_{2} + a_{3} + \ldots + a_{100} = 2012 \). If \( P = a_{2} + a_{4} + a_{6} + \ldots + a_{100} \), find the value of \( P \).


How many numbers between 100 and 999 (inclusive) have digits that form an arithmetic progression when read from left to right?

A sequence of three numbers \( a, b, c \) is said to form an arithmetic progression if \( a + c = 2b \).

A correct numerical answer without justification will earn 4 points. For full points, a detailed reasoning is expected.


Given that \( x \) and \( y \) are acute angles and \( x + y = 60^\circ \), find the maximum value of \( \sin x + \sin y \).


Sasha chose five numbers from 1, 2, 3, 4, 5, 6, and 7 and informed Anya of their product. Based on this information, Anya realized that she could not uniquely determine the parity of the sum of the numbers chosen by Sasha. What number did Sasha inform Anya of?


Each letter in the table represents a different digit, and different letters represent different digits. The leading digit of each number cannot be zero. The three numbers in each row from left to right form an arithmetic sequence, and the three numbers in each column from top to bottom also form an arithmetic sequence. What is the five-digit number $\overline{\mathrm{CDEFG}}$?

\begin{tabular}{|c|c|c|}
\hline
$A$ & $BA$ & $AA$ \\
\hline
$AB$ & $CA$ & $EF$ \\
\hline
$CD$ & $GA$ & $BDC$ \\
\hline
\end{tabular}


Each of 11 positive numbers is equal to the sum of the squares of the other 10 numbers. Determine the numbers.


Given that the hyperbola with asymptotes $y= \pm 2 x$ passes through the intersection point of the lines $x+y-3=0$ and $2 x-y+6=0$, find the length of the hyperbola's real axis.


Eighty students stand in a line facing the teacher, and count off from left to right: $1, 2, 3, \cdots, 80$. After the count, the teacher instructs the students whose numbers are multiples of 2 to turn around. Next, the teacher instructs the students whose numbers are multiples of 4 to turn around. Then the multiples of 8 turn around, followed by multiples of 16, 32, and finally 64. How many students are now facing away from the teacher?


An isosceles triangle has a square with a unit area inscribed in it, with one side of the square lying on the base of the triangle. Find the area of the triangle, given that the centers of gravity of the triangle and the square coincide.


A domino has a left end and a right end, each of a certain color. Alice has four dominos, colored red-red, red-blue, blue-red, and blue-blue. Find the number of ways to arrange the dominos in a row end-to-end such that adjacent ends have the same color. The dominos cannot be rotated.


A three-digit number is called a "cool number" if there is a digit in one of its places that is half the product of the other two digits. A three-digit number is called a "super cool number" if such digits appear in two or three of its places. How many different "super cool numbers" exist? (Zero is not allowed in the digits of "cool" and "super cool" numbers).


The inhabitants of a village of druids do not get along very well. It is even impossible to place 4 or more druids in a circle such that each druid is willing to shake hands with their two neighbors. Seeing that such a state of affairs tarnishes the reputation of the tribe, the chief tries to undertake a pacifying action to calm the situation. He collects 3 gold coins from each druid, then gives one gold coin to each person in any pair willing to shake hands. Show that the chief can keep at least 3 gold coins in his pocket at the end of the action.


$N$ friends simultaneously learned $N$ news items, with each friend learning one piece of news. They started calling each other and sharing the news.

Each call lasts 1 hour, and during one call, any number of news items can be shared.

What is the minimum number of hours required for everyone to know all the news items? Consider the following cases:
a) $N=64$,

b) $N=55$

c) $N=100$.

Solution: \boxed{7 \text{ hours}}


Solve the following equation where \( n \geq 2 \) is a given natural number and \( x \) is the unknown.

$$
\sum_{i=0}^{n-2} \frac{1}{(x+i)(x+i+1)}=x(x+1)(x+2) \cdot \ldots \cdot (x+n)+\frac{n-1}{x(x+n-1)}
$$

Solution: \boxed{x = -n}


The real numbers \( x, y, z, w \) satisfy
\[
\begin{array}{l}
2x + y + z + w = 1 \\
x + 3y + z + w = 2 \\
x + y + 4z + w = 3 \\
x + y + z + 5w = 25
\end{array}
\]

Find the value of \( w \).


795. Calculate the double integral \(\iint_{D} x y \, dx \, dy\), where region \(D\) is:

1) A rectangle bounded by the lines \(x=0, x=a\), \(y=0, y=b\);
2) An ellipse \(4x^2 + y^2 \leq 4\);
3) Bounded by the line \(y=x-4\) and the parabola \(y^2=2x\).


1. Given the equation \( x + y + z = 15 \), find the number of solutions in natural numbers.
2. How many non-negative integer solutions are there for the equation \( 2x_{1} + x_{2} + x_{3} + x_{4} + x_{5} + x_{6} + x_{7} + x_{8} + x_{9} + x_{10} = 3 \)?


Let the function
$$
f(x) = A \sin(\omega x + \varphi) \quad (A>0, \omega>0).
$$

If \( f(x) \) is monotonic on the interval \(\left[\frac{\pi}{6}, \frac{\pi}{2}\right]\) and
$$
f\left(\frac{\pi}{2}\right) = f\left(\frac{2\pi}{3}\right) = -f\left(\frac{\pi}{6}\right),
$$

then the smallest positive period of \( f(x) \) is ______.

Solution: \boxed{\pi}


Using 9 sheets of $2 \times 1$ rectangular paper to cover a $2 \times 9$ chessboard, there are $\qquad$ different methods.


How many ways can you mark 8 squares of an \(8 \times 8\) chessboard so that no two marked squares are in the same row or column, and none of the four corner squares is marked? (Rotations and reflections are considered different.)


Let \(ABCD\) be a rectangle with \(AB = 6\) and \(BC = 4\). Let \(E\) be the point on \(BC\) with \(BE = 3\), and let \(F\) be the point on segment \(AE\) such that \(F\) lies halfway between the segments \(AB\) and \(CD\). If \(G\) is the point of intersection of \(DF\) and \(BC\), find \(BG\).


A great-grandfather banker left an inheritance to his newborn great-grandson. According to the bank agreement, the amount in the great-grandson's account increases. Each year, on the day after his birthday, the current amount increases by one million rubles more than the previous year. Thus, if the initial amount was zero rubles, after one year it will be +1 million rubles; after 2 years, $1+2$ million rubles; after 3 years, $1+2+3$; and so on. According to the agreement, the process will stop, and the great-grandson will receive the money when the amount in the account is a three-digit number consisting of three identical digits.

How old will the great-grandson be when the conditions of the agreement are fulfilled?


In the sum \(1+3+9+27+81+243+729 \), one can strike out any terms and change some signs in front of the remaining numbers from "+" to "-". Masha wants to get an expression equal to 1 in this way, then (starting from scratch) get an expression equal to 2, then (starting again from scratch) get 3, and so on. Up to what maximum integer will she be able to do this without skipping any numbers?


Calculate the area of the figure bounded by the curves given by the following equations:

$$
\begin{aligned}
& \left\{\begin{array}{l}
x=16 \cos ^{3} t \\
y=2 \sin ^{3} t
\end{array}\right. \\
& x=2 \quad (x \geq 2)
\end{aligned}
$$


Given the sequences \( \left\{a_{n}\right\} \) and \( \left\{b_{n}\right\} \) such that
\[
\begin{array}{l}
a_{1} = -1, \quad b_{1} = 2, \\
a_{n+1} = -b_{n}, \quad b_{n+1} = 2a_{n} - 3b_{n} \quad (n \in \mathbb{Z}_{+}).
\end{array}
\]
Find the value of \( b_{2015} + b_{2016} \).

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


In an $8 \times 8$ table, 23 cells are black, and the rest are white. In each white cell, the sum of the black cells located in the same row and the black cells located in the same column is written. Nothing is written in the black cells. What is the maximum value that the sum of the numbers in the entire table can take?


Find the value of the constant \( c \) so that the function \( f(x) = \arctan \frac{2-2x}{1+4x} + c \) is an odd function on the interval \(\left(-\frac{1}{4}, \frac{1}{4}\right) \).

Solution: \boxed{-\arctan 2}


Select 3 numbers from the range 1 to 300 such that their sum is exactly divisible by 3. How many such combinations are possible?


Given that \( 0<a<b<c<d<300 \) and the equations:
\[ a + d = b + c \]
\[ bc - ad = 91 \]

Find the number of ordered quadruples of positive integers \((a, b, c, d)\) that satisfy the above conditions.


Let \( A_{10} \) denote the answer to problem 10. Two circles lie in the plane; denote the lengths of the internal and external tangents between these two circles by \( x \) and \( y \), respectively. Given that the product of the radii of these two circles is \( \frac{15}{2} \), and that the distance between their centers is \( A_{10} \), determine \( y^{2} - x^{2} \).


The vertex of a parabola is \( O \) and its focus is \( F \). When a point \( P \) moves along the parabola, find the maximum value of the ratio \( \left|\frac{P O}{P F}\right| \).


A square-based truncated pyramid is inscribed around a given sphere. What can be the ratio of the truncated pyramid's volume to its surface area?


Xiaopang and Xiaoya both have their birthdays in May, and both fall on a Wednesday. Xiaopang's birthday is later, and the sum of their birth dates is 38. What is Xiaopang's birthday in May?


Find the number of ordered triples of nonnegative integers \((a, b, c)\) that satisfy
\[
(a b+1)(b c+1)(c a+1)=84.
\]


The function \( f \) is defined on the set of natural numbers and satisfies the following conditions:
(1) \( f(1)=1 \);
(2) \( f(2n)=f(n) \), \( f(2n+1)=f(2n)+1 \) for \( n \geq 1 \).

Find the maximum value \( u \) of \( f(n) \) when \( 1 \leq n \leq 1989 \) and determine how many values \( n \) (within the same range) satisfy \( f(n)=u \).


Among any \( m \) consecutive natural numbers, if there is always a number whose sum of the digits is a multiple of 6, what is the smallest value of \( m \)?


In the plane, sequentially draw \( n \) segments end-to-end such that the endpoint of the \( n \)-th segment coincides with the starting point of the 1st segment. Each segment is called a "segment". If the starting point of one segment is exactly the endpoint of another segment, these two segments are called adjacent. We stipulate that:

1. Adjacent segments cannot be drawn on the same straight line.
2. Any two non-adjacent segments do not intersect.

A figure satisfying these criteria is called a "simple polyline loop". For instance, a simple polyline loop with ten segments might exactly lie on five straight lines.

If a simple polyline loop’s \( n \) segments exactly lie on six straight lines, find the maximum value of \( n \) and provide an explanation.


If \(a, b, c,\) and \(d\) are consecutive integers, then the sum

\[ ab + ac + ad + bc + bd + cd + 1 \]

is divisible by 12.


The symbol \(\diamond\) is defined so that \(a \diamond b = \frac{a+b}{a \times b}\). For example, \(2 \diamond 5 = \frac{2+5}{2 \times 5} = \frac{7}{10}\). What is the value of \(3 \diamond 6\)?
(A) 9
(B) \(\frac{1}{18}\)
(C) \(\frac{1}{6}\)
(D) 2
(E) \(\frac{1}{2}\)


As shown in the figure, each small square has a side length of $10 \mathrm{~km}$. There are 2 gas stations in the figure. A car starts from point $A$ and travels along the edges of the small squares to point $B$. If the car needs to refuel every $30 \mathrm{~km}$, how many shortest routes are there for the car to reach point $B$?


The sequence \(\left(x_{n}\right)\) is defined by the recurrence relations

$$
x_{1}=1, \, x_{2}=2, \, x_{n}=\left|x_{n-1}-x_{n-2}\right| \, \text{for} \, n > 2.
$$

Find \(x_{1994}\).


If the function
$$
f(x) = 3 \cos \left(\omega x + \frac{\pi}{6}\right) - \sin \left(\omega x - \frac{\pi}{3}\right) \quad (\omega > 0)
$$
has a minimum positive period of \(\pi\), then the maximum value of \(f(x)\) on the interval \(\left[0, \frac{\pi}{2}\right]\) is \(\qquad\)


Find all strictly increasing functions \( f: \mathbb{N} \rightarrow \mathbb{N} \) such that \( f(2) = 2 \) and for all \( m, n \) that are coprime, \( f(mn) = f(m) f(n) \).

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


200 people stand in a circle. Each of them is either a liar or a conformist. Liars always lie. A conformist standing next to two conformists always tells the truth. A conformist standing next to at least one liar can either tell the truth or lie. 100 of the standing people said: "I am a liar," the other 100 said: "I am a conformist." Find the maximum possible number of conformists among these 200 people.


Let the integer sequence \(a_1, a_2, \ldots, a_{10}\) satisfy \(a_{10} = 3a_1\) and \(a_2 + a_8 = 2a_5\), with \(a_{i+1} \in \{1 + a_i, 2 + a_i\}\) for \(i = 1, 2, \ldots, 9\). How many such sequences exist?


During a long voyage of a passenger ship, it was observed that at each dock, a quarter of the passenger composition is renewed, that among the passengers leaving the ship, only one out of ten boarded at the previous dock, and finally, that the ship is always fully loaded.

Determine the proportion of passengers at any time, while the ship is en route, who did not board at either of the two previous docks?


Let \( P \) be the intersection point of the directrix \( l \) of an ellipse and its axis of symmetry, and \( F \) be the corresponding focus. \( AB \) is a chord passing through \( F \). Find the maximum value of \( \angle APB \) which equals \( 2 \arctan e \), where \( e \) is the eccentricity.

Solution: \boxed{2 \arctan e}


Find the minimum value of the expression \(\frac{1}{1-x^{2}} + \frac{4}{4-y^{2}}\) under the conditions \(|x| < 1\), \(|y| < 2\), and \(xy = 1\).


The quadratic function \( f(x) = a + bx - x^2 \) satisfies \( f(1+x) = f(1-x) \) for all real numbers \( x \) and \( f(x+m) \) is an increasing function on \( (-\infty, 4] \). Find the range of the real number \( m \).

Solution: \boxed{(-\infty, -3]}


Let $\mathbb{R}$ denote the set of real numbers. Determine all functions $f: \mathbb{R} \rightarrow $\mathbb{R}$ such that

$$
f(x^{2}+f(y))=y+(f(x))^{2}
$$

holds for all $x, y$ in $\mathbb{R}$.

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


ABCD is a parallelogram with an area of 120. K is the midpoint of side AD, and L is the midpoint of side CD. Find the area of triangle BKL.


Write the product of the digits of each natural number from 1 to 2018 (for example, the product of the digits of the number 5 is 5; the product of the digits of the number 72 is \(7 \times 2=14\); the product of the digits of the number 607 is \(6 \times 0 \times 7=0\), etc.). Then find the sum of these 2018 products.


In a certain country, there are 21 cities, and the government intends to build $n$ roads (all two-way), with each road connecting exactly 2 of the country’s cities. What is the minimum value of $n$ such that, regardless of how the roads are built, it is possible to travel between any 2 cities (possibly passing through intermediate cities)?


(1) In the sequence $\left\{a_{n}\right\}$,

\[a_{1}=1, \quad a_{n+1}=\frac{10+4 a_{n}}{1+a_{n}} \quad (n \in \mathbf{Z}_{+})\]

find the general term formula of the sequence $\left\{a_{n}\right\}$.

(2) Given the sequence $\left\{a_{n}\right\}$ satisfies:

\[a_{1}=2, \quad a_{n+1}=\frac{a_{n}-1}{a_{n}+1} \quad (n \in \mathbf{Z}_{+})\]

find the value of $a_{2019}$.


Let \(\alpha\) and \(\beta\) be a pair of conjugate complex numbers. Given that \(|\alpha - \beta| = 2 \sqrt{3}\), and \(\frac{\alpha}{\beta^{2}}\) is a real number, what is \(|\alpha|\)?


Seven students in a class receive one failing grade every two days of school, while nine other students receive one failing grade every three days. The remaining students in the class never receive failing grades. From Monday to Friday, 30 new failing grades appeared in the class register. How many new failing grades will appear in the class register on Saturday?


In Mezhdugrad, houses stand along one side of the street, with each house having between $1$ and $9$ floors. According to an ancient law of Mezhdugrad, if two houses on the same side of the street have the same number of floors, then, no matter how far apart they are, there must be a house with more floors between them. What is the maximum possible number of houses that can stand on one side of the street in Mezhdugrad?


Point \( O \), lying inside a convex quadrilateral with area \( S \), is reflected symmetrically with respect to the midpoints of its sides.

Find the area of the quadrilateral with vertices at the resulting points.


In a class, there are \( n \) boys and \( n \) girls (\( n \geq 3 \)). They sat around a round table such that no two boys and no two girls sit next to each other. The teacher has \( 2n \) cards with the numbers \( 1, 2, 3, \ldots, 2n \), each appearing exactly once. He distributes one card to each student in such a way that the number on any girl's card is greater than the number on any boy's card. Then each boy writes down the sum of the numbers on three cards: his own and those of the girls sitting next to him. For which values of \( n \) could all the \( n \) sums obtained be equal?

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


Currently, the exchange rates for the dollar and euro are as follows: $D = 6$ yuan and $E = 7$ yuan. The People's Bank of China determines these exchange rates regardless of market conditions and follows a strategy of approximate currency parity. One bank employee proposed the following scheme for changing the exchange rates. Each year, the exchange rates can be adjusted according to the following four rules: Either change $D$ and $E$ to the pair $(D + E, 2D \pm 1)$, or to the pair $(D + E, 2E \pm 1)$. Moreover, it is prohibited for the dollar and euro rates to be equal at the same time.

For example: From the pair $(6, 7)$, after one year the following pairs are possible: $(13, 11)$, $(11, 13)$, $(13, 15)$, or $(15, 13)$. What is the smallest possible value of the difference between the higher and lower of the simultaneously resulting exchange rates after 101 years?


Let \( a \) and \( b \) be positive whole numbers such that \(\frac{4.5}{11} < \frac{a}{b} < \frac{5}{11} \). Find the fraction \(\frac{a}{b}\) for which the sum \( a+b \) is as small as possible. Justify your answer.


Let \( a, b, c, d \) be odd numbers with \( 0 < a < b < c < d \), and \( ad = bc \), \( a+d = 2^k \), \( b+c = 2^m \), where \( k \) and \( m \) are integers. Find the value of \( a \).


Given that $n$ and $k$ are positive integers with $n > k$, and real numbers $a_{1}, a_{2}, \cdots, a_{n} \in (k-1, k)$. Let positive real numbers $x_{1}, x_{2}, \cdots, x_{n}$ satisfy the condition that for any $k$-element subset $I$ of $\{1,2,\cdots, n\}$, the inequality $\sum_{i \in I} x_{i} \leqslant \sum_{i \in I} a_{i}$ holds. Find the maximum value of $x_{1} x_{2} \cdots x_{n}$.

Solution: \boxed{a_1 a_2 \cdots a_n}


Solve the following system of equations:

$$
\begin{aligned}
x^2 + 7y + 2 &= 2z + 4\sqrt{7x - 3} \\
y^2 + 7z + 2 &= 2x + 4\sqrt{7y - 3} \\
z^2 + 7x + 2 &= 2y + 4\sqrt{7z - 3}
\end{aligned}
$$

Solution: \boxed{x = y = z = 1}


In a family, there are six children. Five of them are respectively 2, 6, 8, 12, and 14 years older than the youngest, and the age of each child is a prime number. How old is the youngest?


Two cyclists set off simultaneously towards each other from points $A$ and $B$ and met 70 km from $A$. Continuing with the same speeds, they reached points $A$ and $B$ and turned back. The second time they met 90 km from $B$. Find the distance from $A$ to $B$.

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


In a $5 \times 18$ rectangle, the numbers from 1 to 90 are placed. This results in five rows and eighteen columns. In each column, the median value is chosen, and among the medians, the largest one is selected. What is the minimum possible value that this largest median can take?

Recall that among 99 numbers, the median is such a number that is greater than 49 others and less than 49 others.


Points A and B are 999 km apart, with 1000 milestones along the way. Each milestone indicates the distance to points A and B in the format (distance to A, distance to B), such as (0, 999), (1, 998), (2, 997), ..., (997, 2), (998, 1), (999, 0). How many of these milestones display exactly two different digits on the distances to A and B?


In an $8 \times 8$ chessboard, color the squares using 8 colors such that each square is colored in one color, neighboring squares sharing an edge have different colors, and each row contains all 8 colors. How many different coloring methods are there?

Solution: \boxed{8! \times 14833^7}


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


The center of one sphere is on the surface of another sphere with an equal radius. How does the volume of the intersection of the two spheres compare to the volume of one of the spheres?


For how many integers \( n \) between 1 and 2005, inclusive, is \( 2 \cdot 6 \cdot 10 \cdots(4n - 2) \) divisible by \( n! \)?


Given \( f(x) = \max \left| x^3 - a x^2 - b x - c \right| \) for \( 1 \leq x \leq 3 \), find the minimum value of \( f(x) \) as \( a, b, \) and \( c \) range over all real numbers.


Let \( n \) be a fixed positive integer. Let \( S \) be any finite collection of at least \( n \) positive reals (not necessarily all distinct). Let \( f(S) = \left( \sum_{a \in S} a \right)^n \), and let \( g(S) \) be the sum of all \( n \)-fold products of the elements of \( S \) (in other words, the \( n \)-th symmetric function). Find \( \sup_S \frac{g(S)}{f(S)} \).


Dodson, Williams, and their horse Bolivar want to reach City B from City A as quickly as possible. Along the road, there are 27 telegraph poles, dividing the whole path into 28 equal intervals. Dodson walks an interval between poles in 9 minutes, Williams in 11 minutes, and either can ride Bolivar to cover the same distance in 3 minutes (Bolivar cannot carry both simultaneously). They depart from City A at the same time, and the journey is considered complete when all three are in City B.

The friends have agreed that Dodson will ride part of the way on Bolivar, then tie Bolivar to one of the telegraph poles, and continue on foot, while Williams will initially walk and then ride Bolivar. At which telegraph pole should Dodson tie Bolivar so that they all reach City B as quickly as possible?

Answer: At the 12th pole, counting from City A.


Let \( n \) be a positive integer such that \(\sqrt{3}\) lies between \(\frac{n+3}{n}\) and \(\frac{n+4}{n+1}\). Determine the value of \( n \).


Among the complex numbers that satisfy the condition \( |z - 5i| \leq 4 \), find the one whose argument has the smallest positive value.

Solution: \boxed{2.4 + 1.8 i}


We start with three piles of coins, initially containing 51, 49, and 5 coins respectively. At each step, we are allowed either to combine two piles into one or to split a pile containing an even number of coins into two equal halves. Is it possible to create three new piles of 52, 48, and 5 coins using these operations?

Solution: \boxed{\text{No}}


Calculate the coefficient of $x^{80}$ in the polynomial $(1+x+x^2+\cdots+x^{80})^3$ after combining like terms.


João managed to paint the squares of an \( n \times n \) board in black and white so that the intersections of any two rows and any two columns did not consist of squares with the same color. What is the maximum value of \( n \)?


Let $A$ be the set of all non-empty subsets of the set $\{1,2, \cdots, n\}$. For $a \in A$, let $\delta(a)$ represent the sum of the elements in $a$. Find $\sum_{a \in A} \delta(a)$.

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


Let \( n \) be a positive integer. Consider the set
\[
S = \left\{\left(x_{1}, x_{2}, \cdots, x_{k}\right) \mid x_{1}, x_{2}, \cdots, x_{k} \in \{0, 1, \cdots, n\}, x_{1} + x_{2} + \cdots + x_{k} > 0 \right\}
\]
which consists of \((n+1)^{k} - 1\) points in \( k \)-dimensional space. Find the minimum number of hyperplanes whose union contains \( S \) but does not include the point \((0, 0, \cdots, 0)\).

Solution: \boxed{kn}


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


Let \(ABCD\) be a parallelogram with an area of 1, and let \(M\) be a point on the segment \([BD]\) such that \(MD = 3MB\). Let \(N\) be the point of intersection of the lines \( (AM) \) and \( (CB) \). Calculate the area of the triangle \(MND\).


Solve the system in positive numbers:

$$
\begin{cases}x^{y} & =z \\ y^{z} & =x \\ z^{x} & =y\end{cases}
$$

Solution: \boxed{x = y = z = 1}


Let \( n \) be a positive integer and define \(\mathrm{S}_{\mathrm{n}} = \left\{\left(\mathrm{a}_{1}, \mathrm{a}_{2}, \ldots, \mathrm{a}_{2^{n}}\right) \mid \mathrm{a}_{\mathrm{i}} \in\{0,1\}, 1 \leq \mathrm{i} \leq 2^{\mathrm{n}}\right\}\). For \( \mathrm{a}, \mathrm{b} \in \mathrm{S}_{\mathrm{n}} \) where \( a = \left(a_{1}, a_{2}, \ldots, a_{2^{n}}\right) \) and \( b = \left(b_{1}, b_{2}, \ldots, b_{2^{n}}\right) \), define \( d(a, b) = \sum_{i=1}^{2^{n}} \left|a_{i} - b_{i}\right| \). If for any \(a, b \in A\), \( d(a, b) \geq 2^{n-1} \), then \( A \subseteq \mathrm{S}_{\mathrm{n}} \) is called a "good subset". Find the maximum possible value of \( |A| \).


For the smallest value of $a$, the inequality \(\frac{\sqrt[3]{\operatorname{ctg}^{2} x}-\sqrt[3]{\operatorname{tg}^{2} x}}{\sqrt[3]{\sin ^{2} x}-\sqrt[3]{\cos ^{2} x}}<a\) holds for all permissible \( x \in\left(-\frac{3 \pi}{2} ;-\pi\right) \)? If necessary, round your answer to two decimal places.


Let \( A \) be the set of all two-digit positive integers \( n \) for which the number obtained by erasing its last digit is a divisor of \( n \). How many elements does \( A \) have?


In a clock workshop, there are several digital clocks (more than one), displaying time in a 12-hour format (the number of hours on the clock screen ranges from 1 to 12). All clocks run at the same speed but show completely different times: the number of hours on the screen of any two different clocks is different, and the number of minutes as well.

One day, the master added up the number of hours on the screens of all available clocks, then added up the number of minutes on the screens of all available clocks, and remembered the two resulting numbers. After some time, he did the same thing again and found that both the total number of hours and the total number of minutes had decreased by 1. What is the maximum number of digital clocks that could be in the workshop?


Vasya wrote consecutive natural numbers \(N\), \(N+1\), \(N+2\), and \(N+3\) in rectangular boxes. Below each rectangle, he wrote the sum of the digits of the corresponding number in a circle.

The sum of the numbers in the first and second circles equals 200, and the sum of the numbers in the third and fourth circles equals 105. What is the sum of the numbers in the second and third circles?


What are the integers $k$ such that for all real numbers $a, b, c$,
$$
(a+b+c)(ab + bc + ca) + kabc = (a+b)(b+c)(c+a)
$$


Consider a 99-sided polygon where its edges are initially colored in the sequence red, blue, red, blue, ..., red, blue, yellow, with one color per edge. Given that no two adjacent edges can have the same color, is it possible, through a series of operations where each operation consists of changing the color of exactly one edge, to transform the coloring into the sequence red, blue, red, blue, ..., red, yellow, blue?

Solution: \boxed{\text{No}}


Let \(\mathcal{C}\) be the hyperbola \(y^{2}-x^{2}=1\). Given a point \(P_{0}\) on the \(x\)-axis, we construct a sequence of points \((P_{n})\) on the \(x\)-axis in the following manner: let \(\ell_{n}\) be the line with slope 1 passing through \(P_{n}\), then \(P_{n+1}\) is the orthogonal projection of the point of intersection of \(\ell_{n}\) and \(\mathcal{C}\) onto the \(x\)-axis. (If \(P_{n}=0\), then the sequence simply terminates.)

Let \(N\) be the number of starting positions \(P_{0}\) on the \(x\)-axis such that \(P_{0}=P_{2008}\). Determine the remainder of \(N\) when divided by 2008.


Given $0<a<b<c<d<500$, how many ordered quadruples of integers $(a, b, c, d)$ satisfy $a + d = b + c$ and $bc - ad = 93$?


The factorial of an integer $m$ is denoted as $m!$ and is the product of all positive integers up to $m$. For example, $2!=1 \times 2=2$, and $(3!)!=1 \times 2 \times 3 \times 4 \times 5 \times 6=720$. Given that $((n!)!)!$ is a factor of $(2021!)!$, what is the maximum value of $n$?


Find all values of the parameters \(a, b, c\) for which the system of equations
\[
\left\{
\begin{array}{l}
a x + b y = c \\
b x + c y = a \\
c x + a y = b
\end{array}
\right\}
\]
has at least one negative solution (where \(x, y < 0\)).

Solution: \boxed{a + b + c = 0}


A circle with a radius of 10 cm has rays drawn from a point \(A\) that are tangent to the circle at points \(B\) and \(C\) such that triangle \(ABC\) is equilateral. Find its area.

Solution: \boxed{75 \sqrt{3}}


In triangle \(ABC\), points \(M\) and \(N\) are the midpoints of sides \(AC\) and \(BC\) respectively. It is known that the intersection of the medians of triangle \(AMN\) is the orthocenter of triangle \(ABC\). Find the angle \(ABC\).


Timur and Alexander are counting the trees growing around the house. They move in the same direction but start counting from different trees. How many trees are growing around the house if the tree that Timur counted as the 12th, Alexander counted as the 33rd, and the tree that Timur counted as the 105th, Alexander counted as the 8th?


It is known that the equation \(x^4 - 8x^3 + ax^2 + bx + 16 = 0\) has (with multiplicity) four positive roots. Find \(a - b\).


In a basketball shooting test, each participant needs to make 3 successful shots to pass and can stop once they achieve this, but each participant is allowed a maximum of 5 attempts. Given that a player's probability of making a shot is $\frac{2}{3}$, what is the probability that the player will pass the test?


Find a function \( f: \mathbf{Z}_{+} \rightarrow \mathbf{Z}_{+} \) such that for all \( m, n \in \mathbf{Z}_{+} \), the following condition holds:
$$
(n! + f(m)!) \mid (f(n)! + f(m!)).
$$

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


The sequence \( u_0, u_1, u_2, \ldots \) is defined as follows: \( u_0 = 0 \), \( u_1 = 1 \), and \( u_{n+1} \) is the smallest integer \( > u_n \) such that there is no arithmetic progression \( u_i, u_j, u_{n+1} \) with \( i < j < n+1 \). Find \( u_{100} \).


In the diagram, $\triangle PQR$ is right-angled at $P$ and $\angle PRQ=\theta$. A circle with center $P$ is drawn passing through $Q$. The circle intersects $PR$ at $S$ and $QR$ at $T$. If $QT=8$ and $TR=10$, determine the value of $\cos \theta$.


On side \( AB \) of triangle \( ABC \), a point \( K \) is marked, and on the side \( AC \), a point \( M \) is marked. The segments \( BM \) and \( CK \) intersect at point \( P \). It turned out that the angles \( \angle APB \), \( \angle BPC \), and \( \angle CPA \) are each \( 120^\circ \), and the area of the quadrilateral \( AKPM \) is equal to the area of triangle \( BPC \). Find the angle \( \angle BAC \).


The number of solutions to the equation $\sin |x| = |\cos x|$ in the closed interval $[-10\pi, 10\pi]$ is __.


The image shows a 3x3 grid where each cell contains one of the following characters: 华, 罗, 庚, 杯, 数, 学, 精, 英, and 赛. Each character represents a different number from 1 to 9, and these numbers satisfy the following conditions:
1. The sum of the four numbers in each "田" (four cells in a square) is equal.
2. 华 $\times$ 华 $=$ 英 $\times$ 英 + 赛 $\times$ 赛.
3. 数 > 学

According to the above conditions, find the product of the numbers represented by 华, 杯, and 赛.


There is one three-digit number and two two-digit numbers written on the board. The sum of the numbers containing the digit seven is 208. The sum of the numbers containing the digit three is 76. Find the sum of all three numbers.


Given that \( P \) is a point on a sphere \( O \) with radius \( r \), three mutually perpendicular chords \( PA \), \( PB \), and \( PC \) are drawn through \( P \). If the maximum distance from point \( P \) to the plane \( ABC \) is 1, then \( r \) equals \(\quad\)


Find the maximum value of the expression \( x^{2} + y^{2} \) if \( |x-y| \leq 2 \) and \( |3x + y| \leq 6 \).


Find the limit of the function:

$$
\lim _{x \rightarrow 0} \frac{(1+x)^{3}-(1+3 x)}{x+x^{5}}
$$


A $5 \times 5$ square contains a light bulb in each of its cells. Initially, only one light bulb is on. You can change the state of the light bulbs in any $k \times k$ square, with $k > 1$. What are the initial positions of the light bulb that is on that allow turning off all the light bulbs?


Darya Dmitrievna is preparing a test on number theory. She promised each student to give as many problems as the number of terms they create in the numerical example

$$
a_{1} + a_{2} + \ldots + a_{n} = 2021
$$

where all numbers \( a_{i} \) are natural, greater than 10, and are palindromes (they do not change if their digits are written in reverse order). If a student does not find any such example, they will receive 2021 problems on the test. What is the smallest number of problems a student can receive? (20 points)


If \( f(x) = \frac{25^x}{25^x + P} \) and \( Q = f\left(\frac{1}{25}\right) + f\left(\frac{2}{25}\right) + \cdots + f\left(\frac{24}{25}\right) \), find the value of \( Q \).


Given a positive integer \( n \), find the smallest positive integer \( u_n \) such that for any odd integer \( d \), the number of integers in any \( u_n \) consecutive odd integers that are divisible by \( d \) is at least as many as the number of integers among the odd integers \( 1, 3, 5, \ldots, 2n-1 \) that are divisible by \( d \).

Solution: \boxed{u_n = 2n - 1}


There are two sets of numbers from 1 to 20. All possible sums of two numbers (with each addend taken from a different set) are formed. How many of these sums are divisible by 3?


Ivan Semenovich leaves for work at the same time every day, travels at the same speed, and arrives exactly at 9:00 AM. One day, he overslept and left 40 minutes later than usual. To avoid being late, Ivan Semenovich increased his speed by 60% and arrived at 8:35 AM. By what percentage should he have increased his usual speed to arrive exactly at 9:00 AM?

Solution: \boxed{30 \%}


Let \( M = \{1, 2, \cdots, 2n + 1\} \). \( A \) is a subset of \( M \) with no two elements summing to \( 2n + 2 \). Find the maximum value of \( |A| \).


A square with a side length of 100 was cut into two equal rectangles. These rectangles were then placed next to each other as shown in the picture. Find the perimeter of the resulting figure.


Thirty-nine students from seven classes invented 60 problems, with the students from each class inventing the same number of problems (which is not zero), and the students from different classes inventing different numbers of problems. How many students invented one problem each?


Draw the height BH. \(\angle BCH = \angle CBH = 45^\circ\), \(\angle ABH = 60^\circ\). Let BH = HC = \(x\), then BC = \(x \sqrt{2}\), AB = \(2 x\).

\(\frac{AB}{BC} = \frac{BC}{MB} = \sqrt{2}\), \(\angle ABC\) is common. Therefore, \(\triangle MBC \sim \triangle CBA\) by the second criterion of similarity. From this similarity, it follows that \(\angle BMC = 45^\circ\). Then \(\angle AMC = 135^\circ\).


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


The coordinates of the vertices of a pentagon are $A(0, 0)$, $B(11, 0)$, $C(11, 2)$, $D(6, 2)$, and $E(0, 8)$. Write the equation of the line that is parallel to the $y$-axis and bisects the area of the pentagon.


Find the largest positive integer \( n \) such that \( n^3 + 100 \) is divisible by \( n + 10 \).


Two schools, Xiaoxin and Xiaoxiao, have the same number of students participating in the Golden Cup Competition. The schools use cars to transport the students to the competition site. Xiaoxin's cars each carry 15 people, while Xiaoxiao's cars each carry 13 people. As a result, Xiaoxiao needs one more car than Xiaoxin. Later, each school adds one more participant, which makes both schools require the same number of cars. Finally, it is decided that each school will add one more participant, causing Xiaoxiao to need one more car than Xiaoxin again. How many students in total participate in the competition from both schools in the end?


Simplify $\frac{\sin x+\sin 3 x+\sin 5 x+\cdots+\sin (2 n-1) x}{\cos x+\cos 3 x+\cos 5 x+\cdots+\cos (2 n-1) x}$.

Solution: \boxed{\tan(nx)}


A pedestrian and a cyclist are moving in one direction along a highway, while a cart and a car are moving in the opposite direction. All participants move at constant speeds (each with their own speed). The cyclist first overtakes the pedestrian, then after some time meets the cart, and then after the same amount of time meets the car. The car first meets the cyclist, then after some time meets the pedestrian, and then after the same amount of time overtakes the cart. The cyclist overtook the pedestrian at 10 o'clock, and the pedestrian met the car at 11 o'clock. When did the pedestrian meet the cart?

Solution: \boxed{10:40 \text{ am}}


Construct the center \(O\) of a rotational homothety with a given coefficient \(k \neq 1\), which maps the line \(l_{1}\) to the line \(l_{2}\), and the point \(A_{1}\) lying on \(l_{1}\) to the point \(A_{2}\).

Solution: \boxed{O}


The sides of triangle $ABC$ touch the incircle at points $K$, $P$, and $M$, with point $M$ located on side $BC$. Find the angle $KMP$ if $\angle A = 2\alpha$.

Solution: \boxed{90^\circ - \alpha}


If \( n \) is a positive integer, \( a(n) \) is the smallest positive integer such that \( (a(n))! \) is divisible by \( n \). Find all positive integers \( n \) such that \( \frac{a(n)}{n} = \frac{2}{3} \). (2003 German Mathematical Olympiad)


Inside a cylinder with a base radius of 6, there are two spheres each with a radius of 6, and the distance between their centers is 13. If a plane is made tangent to these two spheres and intersects with the cylinder to form an ellipse, then the sum of the lengths of the major and minor axes of this ellipse is $\qquad$


The legs of a compass are positioned at the nodes of an infinite sheet of ruled paper, where the cells are squares with side length 1. It is allowed, without changing the compass opening, to move one leg to another node on the sheet by rotating the compass around the other leg. Is it possible, in a few such steps, to swap the legs of the compass?

Solution: \boxed{\text{No}}


A four-digit (integer) number, when divided by 7 and 29, leaves a remainder of 0, and when its 19-times multiple is divided by 37, it leaves a remainder of 3. What is this number?


In how many ways can the letters in the word "ИНТЕГРАЛ" be rearranged such that the third, fifth, and seventh positions are occupied by consonant letters?


In an isosceles triangle, the center of the inscribed circle divides the altitude in a ratio of $12: 5$, and the lateral side is equal to 60. Find the base.


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

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


Let \( a_{1}, a_{2}, \cdots, a_{k}\left(k \in \mathbf{Z}_{+}\right) \) be integers greater than 1, and they satisfy
\[
\left(a_{1}!\right)\left(a_{2}!\right) \cdots\left(a_{k}!\right) \mid 2017!
\]

Determine the maximum value of \( \sum_{i=1}^{k} a_{i} \) as \( k \) varies.


For all positive real numbers \( x \) and \( y \), let

\[ f(x, y) = \min \left(x, \frac{y}{x^2 + y^2} \right). \]

Show that there exists \( x_0 \) and \( y_0 \) such that \( f(x, y) \leq f(x_0, y_0) \) for all positive \( x \) and \( y \), and find \( f(x_0, y_0) \).


Malvina and Pierro have several coins, each with a value of 1 soldi. Their total amount is less than 40 soldi. If Pierro manages to increase his amount of money by 8 times, he will have more money than Malvina. However, if Pierro only quadruples his capital and Buratino adds another 15 soldi to it, he still won't surpass Malvina. How much money does Malvina have?


In triangle \( ABC \), \( AB = 6 \), \( BC = 10 \), and \( CA = 14 \). If \( D \), \( E \), and \( F \) are the midpoints of sides \( BC \), \( CA \), and \( AB \), respectively, find \( AD^{2} + BE^{2} + CF^{2} \).


Given that \(\sin (x + \sin x) = \cos (x - \cos x)\), where \(x \in [0, \pi]\). Find \(x\).


Let \( x \) and \( y \) satisfy the constraints:
\[
\begin{cases}
x - y - 2 \geq 0, \\
x - 5 \leq 0, \\
y + 2 \geq 0.
\end{cases}
\]
Then, the sum of the minimum and maximum values of \( z = x^2 + y^2 \) is \(\quad\)


Consider the sequence $\left(a_{n}\right)$ with the first term $a_{1} = 1$ and defined by the following relation: For any positive integer $n$,

$$
a_{n} = a_{\left\lfloor \frac{n}{2} \right\rfloor} + a_{\left\lceil \frac{n}{2} \right\rceil}
$$

Express $a_{n}$ as a function of $n$.

Solution: \boxed{a_n = n}


The circles with centers \( O_1 \) and \( O_2 \) have a common chord \( AB \), and the angle \( \angle A O_1 B = 120^{\circ} \). The ratio of the length of the second circle to the length of the first circle is \( \sqrt{3} \). Find the angle \( \angle A O_2 B \).


Dani wrote the integers from 1 to \( N \). She used the digit 1 fifteen times. She used the digit 2 fourteen times.
What is \( N \) ?


There is a round table with 9 chairs, and 4 people are seated randomly. What is the probability that no two people are sitting next to each other?


Two parabolas with different vertices are the graphs of quadratic trinomials with leading coefficients $p$ and $q$. It is known that the vertex of each parabola lies on the other parabola. What can the value of $p+q$ be?


Given that \(a, b, c\) are positive integers satisfying
$$
a+b+c=\operatorname{gcd}(a, b)+\operatorname{gcd}(b, c)+\operatorname{gcd}(c, a)+120 \text {, }
$$
determine the maximum possible value of \(a\).


Given the set \( M = \{1, 2, \cdots, 2017\} \) which consists of the first 2017 positive integers, if one element is removed from \( M \) such that the sum of the remaining elements is a perfect square, what is the removed element?


For how many integers \( n \), where \( 1 \leq n \leq 1000 \), is the number \(\frac{1}{2}\binom{2n}{n}\) even?


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

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


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

\[
\left\{
\begin{array}{l}
x^{2}+x y+y^{2}=75 \\
y^{2}+y z+z^{2}=64 \\
z^{2}+x z+x^{2}=139
\end{array}
\right.
\]

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


Find all 4-digit numbers that are 8802 less than the number formed by reversing its digits.


If a four-digit number $\overline{a b c d}$ meets the condition $a + b = c + d$, it is called a "good number." For instance, 2011 is a "good number." How many "good numbers" are there?


As shown in the figure, the ratio of the height of a cylinder to that of a cone is 4:5, and the ratio of the circumference of their bases is 3:5. Given that the volume of the cone is 250 cubic centimeters, find the volume of the cylinder.


On the sides \( AB \) and \( BC \) of an equilateral triangle \( ABC \), points \( L \) and \( K \) are marked, respectively. \( M \) is the intersection point of segments \( AK \) and \( CL \). It is known that the area of triangle \( AMC \) is equal to the area of quadrilateral \( LBKM \). Find the angle \( \angle AMC \).


In a classroom, 34 students are seated in 5 rows of 7 chairs. The place at the center of the room is unoccupied. A teacher decides to reassign the seats such that each student will occupy a chair adjacent to his/her present one (i.e. move one desk forward, back, left or right). In how many ways can this reassignment be made?


A company gathered at a meeting. Let's call a person sociable if, in this company, they have at least 20 acquaintances, with at least two of those acquaintances knowing each other. Let's call a person shy if, in this company, they have at least 20 non-acquaintances, with at least two of those non-acquaintances not knowing each other. It turned out that in the gathered company, there are neither sociable nor shy people. What is the maximum number of people that can be in this company?


What are the last digits of the decimal representations of the following numbers:
1. \( 135^x + 31^y + 56^{x+y} \), where \( x \in \mathbb{N}, y \in \mathbb{N} \)?
2. \( 142 + 142^2 + 142^3 + \ldots + 142^{20} \)?
3. \( 34^x + 34^{x+1} + 34^{2x} \), where \( x \in \mathbb{N} \)?


P is a point inside the triangle ABC. Angle PAC = angle PBC. M is the midpoint of AB and L and N are the feet of the perpendiculars from P to BC and CA respectively. Show that ML = MN.

Solution: \boxed{ML = MN}


An isosceles trapezoid is circumscribed around a circle with a side length of $l$. One of the bases of the trapezoid is $a$. Find the area of the trapezoid.

Solution: \boxed{l \sqrt{a(2l - a)}}


Let the complex number \( z \) satisfy \( z + \frac{1}{z} \in [1, 2] \). Then, the minimum value of the real part of \( z \) is ______.


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


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


Let \( k_{1} \) be the smallest natural number that is a root of the equation

\[
\sin k^\circ = \sin 334 k^\circ
\]

(a) Find \( k_{1} \).

(b) Find the smallest natural number root of the same equation that is greater than \( k_{1} \).


In Figure 5, the area of triangle \(ABC\) is 1. \(BD : CD = 2 : 1\), \(E\) is the midpoint of \(AC\), and \(AD\) intersects \(BE\) at point \(P\). Find the area of quadrilateral \(PDCE\).


A student passed 31 exams over 5 years. Each subsequent year, he passed more exams than the previous year, and in his fifth year, he passed three times as many exams as in his first year. How many exams did he pass in his fourth year?


Find the smallest integer value of \( a \) for which the system of equations

$$
\left\{\begin{array}{l}
\frac{y}{a-\sqrt{x}-1}=4 \\
y=\frac{\sqrt{x}+5}{\sqrt{x}+1}
\end{array}\right.
$$

has a unique solution.


Find the maximum value of the expression $\sin x \sin y \sin z + \cos x \cos y \cos z$.


Let \(a\) and \(b\) be positive natural numbers such that \(a^{\mathfrak{n}} \mid b^{n+1}\) for every \(n\) in \(\mathbb{N}\).

Show that \(a \mid b\).

Solution: \boxed{a \mid b}


As shown in the figure, points $C$ and $D$ divide $AB$ into three equal parts. At 8:00, Person A starts walking uniformly from $A$ to $B$. At 8:12, Person B starts walking uniformly from $B$ to $A$. After a few more minutes, Person C starts walking uniformly from $B$ to $A$. When Person A and Person B meet at point $C$, Person C just reaches point $D$. When Person A and Person C meet at 8:30, Person B just reaches $A$. At what time does Person C start?