Several people participated in a race, including Andrei, Dima, and Lenya. No two participants finished simultaneously.

- The number of people who finished before Andrei is half the number of people who finished after him.
- The number of people who finished before Dima is a third of the number of people who finished after him.
- The number of people who finished before Lenya is a quarter of the number of people who finished after him.

What is the minimum number of people that could have participated in the race?


The total corn yield in centners, harvested from a certain field area, is expressed as a four-digit number composed of the digits 0, 2, 3, and 5. When the average yield per hectare was calculated, it was found to be the same number of centners as the number of hectares of the field area. Determine the total corn yield.


Calculate the value of

$$
A=\frac{1001 \cdot 1002 \cdot 1003 \cdot \ldots \cdot 2000}{1 \cdot 3 \cdot 5 \cdot \ldots \cdot 1999}
$$


As shown in the figure, $M$ and $N$ are points on the sides $AD$ and $BC$ of the quadrilateral $ABCD$. Given $AB = 1$, $\frac{AM}{AD} = \frac{BN}{BC} = \frac{1}{3}$, and the angle between $MN$ and $AB$ is equal to the angle between $MN$ and $CD$, find the length of $CD$.


Write the consecutive terms of the natural number sequence next to each other. Determine the 10000th digit in this sequence.


Given an ellipse \( C \) defined by \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) with \(a > b > 0\) and an eccentricity of \(\frac{2}{3}\). Points \(A_{1}\) and \(A_{2}\) are the left and right vertices of the ellipse \(C\), \(B\) is the top vertex, and \(F_{1}\) is the left focus of the ellipse \(C\). The area of triangle \(\triangle A_{1} F_{1} B\) is \(\frac{\sqrt{5}}{2}\).

(1) Find the equation of the ellipse \(C\).

(2) Suppose a moving line \(l\) passes through the point \(D(1,0)\) and intersects the ellipse \(C\) at points \(E\) and \(F\) (point \(E\) is above the x-axis). Points \(M\) and \(N\) are the intersections of lines \(A_{1} E\) and \(A_{2} F\) with the y-axis, respectively. Find the value of \(\frac{|OM|}{|ON|}\).


Given three different prime numbers \( p, q, r \) such that \( pqr = 1899\ldots962 \) (where the number contains \( n \) nines), define \( f(n) \) as the sum of the digits of the natural number \( n \). Determine the value of \( f(p) + f(q) + f(r) - f(pqr) \).


As is well known, balance scales reach equilibrium when both pans have the same weight. On one pan of the scales lie 9 identical diamonds, and on the other - 4 identical emeralds. If one more of the same emerald is added to the side with the diamonds, the scales will balance. How many diamonds will balance one emerald? Justify your answer.


Each of the 10 dwarfs either always tells the truth or always lies. It is known that each of them likes exactly one type of ice cream: vanilla, chocolate, or fruit. First, Snow White asked those who like vanilla ice cream to raise their hands, and everyone raised their hands; then she asked those who like chocolate ice cream, and half of the dwarfs raised their hands; finally, she asked those who like fruit ice cream, and only one dwarf raised their hand. How many among the dwarfs are truthful?


Given that \( r = 2006 \times \frac{\sqrt{8}-\sqrt{2}}{\sqrt{2}} \), find the value of \( r \).


From a given point, a tangent and a secant are drawn to a given circle. The length of the tangent is $18 \mathrm{~cm}$, and the part of the secant that lies inside the circle is $27 \mathrm{~cm}$. How long is the other part of the secant?

Solution: \boxed{9 \, \text{cm}}


In the image, there is a square \(A B C D\) with a side length of \(10 \, \text{cm}\) and a smaller square \(E F G H\), whose vertices \(E, F, G, H\) lie on the diagonals \(A C, B D\) of the square \(A B C D\) (see the image). The area that lies inside the square \(A B C D\) and outside the square \(E F G H\) is shaded. A line \(p\), which is parallel to \(A B\) at a distance of \(6.5 \, \text{cm}\) and passes through the square \(E F G H\), divides the shaded area into two parts. The area of one of these parts is \(13.8 \, \text{cm}^2\) larger than the area of the other. Calculate the side length of \(E F\).

Solution: \boxed{5.4 \ \text{cm}}


Find the sum of all real roots of the equation \(3 \tan ^{2} x + 8 \tan x + 3 = 0\) in the range \(0 < x < 2\pi\).


Let's call a number palindromic if it reads the same left to right and right to left. For example, the number 12321 is palindromic.

a) Write down a five-digit palindromic number that is divisible by 5.

b) How many five-digit palindromic numbers that are divisible by 5 exist?


Determine the positive real numbers \(a\) and \(b\) satisfying \(9a^{2} + 16b^{2} = 25\) such that \(a \cdot b\) is maximal. What is the maximum of \(a \cdot b\)?


Solve the system of equations \(\left\{\begin{array}{l}
x^{2} y - x y^{2} - 5 x + 5 y + 3 = 0, \\
x^{3} y - x y^{3} - 5 x^{2} + 5 y^{2} + 15 = 0.
\end{array}\right.\)

Solution: \boxed{(4, 1)}


The sequence \(\left(a_{n}\right)\) is an arithmetic progression with a common difference of 1. It is known that \( \mathrm{S}_{2022} \) is the smallest sum among all sums \( S_{n} \) (smaller than the sum of the first \( n \) terms for any other value of \( n \) ). What values can the first term of the progression take?

Solution: \boxed{(-2022, -2021)}


Given a trapezoid \(ABCD\) with \(BC \parallel AD\). Point \(H\) on side \(AB\) is such that \(\angle DHA = 90^\circ\). It is known that \(CH = CD = 13\) and \(AD = 19\). Find the length of segment \(BC\).


Calculate the result of the expression $\left(9 \frac{1}{2}+7 \frac{1}{6}+5 \frac{1}{12}+3 \frac{1}{20}+1 \frac{1}{30}\right) \times 12$.


Let the function \( f(x) = \frac{(x+1)^2 + \sin x}{x^2 + 1} \) have a maximum value of \( M \) and a minimum value of \( N \). Find \( M + N \).


Rachelle picks a positive integer \( a \) and writes it next to itself to obtain a new positive integer \( b \). For instance, if \( a=17 \), then \( b=1717 \). To her surprise, she finds that \( b \) is a multiple of \( a^{2} \). Find the product of all the possible values of \( \frac{b}{a^{2}} \).


As shown in the figure, a small square of the same size is cut from the corner and the edge of two identical large rectangular sheets of paper, respectively. If the perimeter of the shaded part in figure (2) is 17 centimeters more than the perimeter of the shaded part in figure (1), what is the perimeter of the cut-out small square in centimeters?


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


Calculate the limit of the function:

$$\lim _{x \rightarrow 0} \frac{1-\sqrt{\cos x}}{1-\cos \sqrt{x}}$$


Given \( x \neq y \) and two arithmetic sequences \( x, a_1, a_2, a_3, y \) and \( b_1, x, b_2, b_3, y, b_4 \), find the value of \( \frac{b_4 - b_3}{a_2 - a_1} \).


Jakob wrote down four consecutive positive integers. He then calculated the four possible totals made by taking three of the integers at a time. None of these totals was a prime. What is the smallest integer Jakob could have written?


Given \( f: A_{3} \rightarrow A_{3} \), find the number of functions \( f \) that satisfy \( f^{(3)} = f \), and list all such functions.


Let \( A=\{1,2,3\}, B=\{2x+y \mid x, y \in A, \ x < y\}, C=\{2x+y \mid x, y \in A, \ x > y\} \). Find the sum of all elements of \( B \bigcap C \).


A pool has two drain taps, A and B. If both taps are opened simultaneously, it takes 30 minutes to drain the full pool. If both taps are opened for 10 minutes and then tap A is closed while tap B continues, it also takes 30 more minutes to drain the full pool. How many minutes will it take to completely drain the full pool with only tap B open?


What are the last 8 digits of
$$
11 \times 101 \times 1001 \times 10001 \times 100001 \times 1000001 \times 111 ?
$$


A force of $60 \mathrm{H}$ stretches a spring by 2 cm. The initial length of the spring is $14 \mathrm{~cm}$. How much work is required to stretch it to 20 cm?

Solution: \boxed{5.4 \, \text{J}}


The force exerted by the airflow on a sail can be calculated using the formula:

\[ F = \frac{C S \rho (v_0 - v)^2}{2}, \]

where \(C\) is the coefficient of aerodynamic force, \(S\) is the area of the sail (\(S = 5 \, \text{m}^2\)), \(\rho\) is the air density, \(v_0\) is the wind speed (\(v_0 = 6 \, \text{m/s}\)), and \(v\) is the speed of the sailboat. At some point in time, the instantaneous power of the wind reaches its maximum value. What is the speed of the sailboat at this moment?

Given:

\[ F = \frac{C S \rho (v_0 - v)^2}{2} \]

\[ N(t') = N_{\max} \]

\[ v_0 = 6 \, \text{m/s} \]

\[ \overline{v(t') = ?} \]

Solution: \boxed{2 \, \text{m/s}}


The price of the "Voskhod" skating rink in February 2020 was 300 rubles per person. In March, the price was reduced, and the number of visitors increased by $50 \%$, while the revenue increased by $25 \%$. By how many rubles was the price reduced?


Calculate the limit of the numerical sequence:

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

Solution: \boxed{-\infty}


Find the number of ordered integer tuples \((k_1, k_2, k_3, k_4)\) that satisfy \(0 \leq k_i \leq 20\) for \(i = 1, 2, 3, 4\), and \(k_1 + k_3 = k_2 + k_4\).


Let \( P, Q \in \mathbb{R}[X] \) be two non-constant real polynomials having the same complex roots, such that \( P-1 \) and \( Q-1 \) also have the same complex roots. Show that \( P = Q \).

Solution: \boxed{P = Q}


Find the maximum value of the function \( f(x) = 9 \sin x + 12 \cos x \).


For a positive integer $k$, there exist positive integers $n$ and $m$ such that $\frac{1}{n^{2}} + \frac{1}{m^{2}} = \frac{k}{n^{2} + m^{2}}$. Find all such values of $k$.


James chooses five different positive integers, each at most eight, so that their mean is equal to their median. In how many different ways can he do this?


Given the sets \( U = \{1,2,3,4,5,6,7,8\} \), \( A = \{1,2,3,4,5\} \), and \( B = \{4,5,6,7,8\} \), determine the number of subsets of \( U \) that are not subsets of \( A \) and also not subsets of \( B \).


Let \( x \) be an acute angle. What is the maximum value of the function \( y = \sin x \sin 2x \)?


How many sides can a convex polygon have if all its diagonals are of the same length?


A quadrilateral pyramid \(SABCD\) is given, with a base that is a trapezoid \(ABCD\). The ratio of the bases \(AD\) and \(BC\) of this trapezoid is 2. Construct the cross-section of the pyramid with a plane passing through point \(D\) and the midpoints of the edges \(SA\) and \(SB\). In what ratio does this plane divide the edge \(SC\)?


Two circles touch externally. The line segment \(AB\) is the diameter of one circle and has a length of \(6 \ \text{cm}\), and the line segment \(CD\) is the diameter of the other circle and has a length of \(14 \ \text{cm}\). The quadrilateral \(ABCD\) is a trapezoid with bases \(AB\) and \(CD\).

Determine the maximum possible area of the trapezoid \(ABCD\), and explain why it cannot be larger.

Solution: \boxed{100 \text{ cm}^2}


The sum of the first and fifth terms of an arithmetic progression is 5/3, and the product of the third and fourth terms is 65/72. Find the sum of the first 17 terms of this progression.


In a rectangle, a segment \( AB \) of length \( l = 4 \), which is less than the lengths of its sides, is positioned such that the endpoints of the segment lie on the sides of the rectangle. Point \( A \), making a full revolution around the rectangle, travels a distance equal to its perimeter. During this motion, point \( C \), the midpoint of segment \( AB \), also traces out a certain path. By how much is this path shorter than the perimeter of the rectangle?

Solution: \boxed{16 - 4\pi}


Giuseppe has a sheet of plywood that measures $22 \times 15$. Giuseppe wants to cut out as many rectangular pieces of $3 \times 5$ as possible from it. How can he do this?


In a committee of 360 people, each member belongs to exactly three subcommittees, and each subcommittee has exactly 6 members. Determine the number of subcommittees.


The sum of the ages of Tom and John is 23, the sum of the ages of John and Alex is 24, and the sum of the ages of Tom and Alex is 25. What is the sum of the squares of their three ages?


A number was multiplied by the sum of its digits and the result was 2008. Find this number.


The quality control department found five defective books in a batch of 100 randomly selected books. Find the relative frequency of defective books.


A certain 4-digit number is a perfect square. If the first digit on the left is removed, it becomes a perfect cube, and if the first two digits are removed, it becomes the fourth power of an integer. Find this number.


Polina went to the cafeteria and saw that 2 puff pastries cost 3 times more than 2 pies. Polina didn't have enough money for 2 puff pastries, but she had enough for 1 pie and 1 puff pastry. She then wondered how many times more money she spent buying 1 puff pastry and 1 pie instead of 2 pies. Help Polina answer this question.


A group of schoolchildren arrived at a camp, among them Petya, Vasya, and Timofey, who were not acquainted with each other but each had acquaintances among the other children. Petya noticed that exactly \( \frac{1}{2} \) of his acquaintances are acquainted with Vasya, and exactly \( \frac{1}{7} \) are acquainted with Timofey. Vasya noticed that \( \frac{1}{3} \) of his acquaintances are acquainted with Petya, and \( \frac{1}{6} \) are acquainted with Timofey. Finally, Timofey noticed that exactly \( \frac{1}{5} \) of his acquaintances are acquainted with Petya. What fraction of Timofey's acquaintances are also acquainted with Vasya?


How many different ways are there to partition the number 2004 into natural summands that are approximately equal? There can be one or several summands. Numbers are called approximately equal if their difference is no more than 1. Ways that differ only by the order of summands are considered the same.


Let \( n \) be a natural number. A volleyball team consisting of \( n \) women and \( n \) men lines up for a game. Each team member occupies one of the positions \( 1, 2, \ldots, 2n \), whereby exactly positions \( 1 \) and \( n+1 \) are outside the court. During the game, all team members rotate, switching from position \( i \) to position \( i+1 \) (or from \( 2n \) to \( 1 \). How many possibilities are there for the starting line-up so that there are always at least \( n-1 \) women on the pitch, no matter how often the team rotates?

Note: Two starting line-ups are different if at least one team member occupies a different position.

Solution: \boxed{(n!)^2 \cdot 2^n}


The discrete random variable \(X\) is defined by the cumulative distribution function

$$
F(x)=\left\{\begin{array}{lll}
0 & \text { if } & x \leq 2 \\
0.5 & \text { if } & 2 < x \leq 4 \\
0.7 & \text { if } & 4 < x \leq 8 \\
1 & \text { if } & x > 8
\end{array}\right.
$$

Find the probability that \(X\) takes a value not less than 4 and less than 8.


Among the 100 numbers from 1 to 100, how many numbers have non-zero remainders when divided by 2, 3, and 5, and the remainders are all different from each other?


The sum of the terms of an infinite geometric series is 2 and the sum of the squares of the corresponding terms of this series is 6. Find the sum of the cubes of the corresponding terms.


A company's employees work 5 days a week and rest for 2 days. The company requires that at least 45 employees must be on duty every day from Monday to Sunday. What is the minimum number of employees needed by the company?


Let \(a\) and \(b\) be any positive real numbers. Determine the value of the expression

\[
\frac{\sqrt{\frac{a b}{2}}+\sqrt{8}}{\sqrt{\frac{a b+16}{8}+\sqrt{a b}}}
\]


Each member of the sequence \(a_1, a_2, \ldots, a_n\) belongs to the set \(\{1, 2, \ldots, n-1\}\) and \(a_1 + a_2 + \ldots + a_n < 2n\). Show that we can find a subsequence with sum \(n\).

Solution: \boxed{n}


Suppose \( a, b, c \), and \( d \) are pairwise distinct positive perfect squares such that \( a^{b} = c^{d} \). Compute the smallest possible value of \( a + b + c + d \).


What is the minimum width required for an infinite strip of paper from which any triangle with an area of 1 can be cut?


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


There are 7 safes and 7 codes for them, but it is unknown which code matches which safe. What is the minimum number of attempts needed to accurately match all the codes to the safes?


In the convex quadrilateral \(ABCD\), the diagonals \(AC\) and \(BD\) are equal. Moreover, \(\angle BAC = \angle ADB\) and \(\angle CAD + \angle ADC = \angle ABD\). Find the angle \(BAD\).


ABC is an acute-angled triangle. The tangent at A to the circumcircle meets the tangent at C at the point B'. BB' meets AC at E, and N is the midpoint of BE. Similarly, the tangent at B meets the tangent at C at the point A'. AA' meets BC at D, and M is the midpoint of AD. Show that ∠ABM = ∠BAN. If AB = 1, find BC and AC for the triangles which maximize ∠ABM.

Solution: \boxed{BC = AC = \sqrt{2}}


The sequence of numbers \( a_1, a_2, a_3, \ldots, a_n, \ldots \) satisfies the relation \( a_n = a_{n-1} \cdot a_{n-3} \) for \( n = 4, 5, 6, \ldots \). Find \( a_{2019} \) given that \( a_1 = 1, a_2 = 1 \), and \( a_3 = -1 \).


Let set $A=\left\{x \mid x^{2}-[x]=2\right\}$ and set $B=\{x \mid x<2\}$, where the symbol $[x]$ denotes the greatest integer less than or equal to $x$. Find $A \cap B$.

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


A music enthusiast, having labeled five newly listened to CDs with the numbers $1,2,3,4,5$, placed them in a cassette holder in a random order. What is the probability that CDs numbered 1 and 2 will be placed next to each other in the cassette holder and in ascending numerical order?


Two liters of hot oil at a temperature of \(100^{\circ} \mathrm{C}\) are mixed with one liter of cold oil at a temperature of \(20^{\circ} \mathrm{C}\). What will the volume of the mixture be when thermal equilibrium is established in the mixture? Heat losses to the external environment can be neglected. The coefficient of thermal expansion of the oil is \(2 \cdot 10^{-3} 1 /{ }^{\circ} \mathrm{C}\).


The integer \( n \) has a total of 10 divisors. These divisors are arranged in ascending order, and the 8th divisor is \( \frac{n}{3} \). Find the maximum value of the integer \( n \).


The length of the midsegment $E F$ of trapezoid $A B C D$ is 15 cm. Given that $\angle A B C = \angle A E F = 90^{\circ}$, and $G$ is a point on $E F$, if the area of triangle $A B G$ is $\frac{1}{5}$ of the area of trapezoid $A B C D$, what is the length of $E G$ in centimeters?

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


On some planet, there are \(2^{N}\) countries \((N \geq 4)\). Each country has a flag \(N\) units wide and one unit high composed of \(N\) fields of size \(1 \times 1\), each field being either yellow or blue. No two countries have the same flag.

We say that a set of \(N\) flags is diverse if these flags can be arranged into an \(N \times N\) square so that all \(N\) fields on its main diagonal will have the same color. Determine the smallest positive integer \(M\) such that among any \(M\) distinct flags, there exist \(N\) flags forming a diverse set.


Given that the function \( f(x)=\log _{\frac{1}{3}}\left(3^{x}+1\right)+\frac{1}{2} a b x \) is an even function, and \( g(x)=2^{x}+\frac{a+b}{2^{x}} \) is an odd function, where \( a \) and \( b \) are constants, find the value of \( \sum_{k=1}^{2008}\left(a^{k}+b^{k}\right) \).


Given real numbers \( x, y \in (1,+\infty) \) such that \( xy - 2x - y + 1 = 0 \), find the minimum value of \( \frac{3}{2} x^{2} + y^{2} \).


Let \( x \) and \( y \) be two strictly positive real numbers. Show that
\[
x^2 + \frac{8}{xy} + y^2 \geq 8
\]

Investigate the case of equality.

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


How many matches can be made from a wooden cube with an edge length of 1 meter if each match must be 5 cm long, 2 mm wide, and 2 mm high? (Assume that no material is lost in the cutting process.)


Given points \( A(5, 5) \), \( B(8, -3) \), and \( C(-4, 1) \). Find the coordinates of the centroid of triangle \( ABC \).

Solution: \boxed{(3, 1)}


In triangle \( \triangle ABC \), \(\sin A = \frac{3}{5}\) and \(\cos B = \frac{5}{13}\). What is the value of \(\cos C\)?


Knights, who always tell the truth, and liars, who always lie, live on an island. One day, 65 islanders gathered for a meeting. Each of them made the following statement in turn: "Among the statements made earlier, the true ones are exactly 20 less than the false ones." How many knights were present at this meeting?


If \( a, b, \) and \( c \) are positive real numbers such that
\[ ab + a + b = bc + b + c = ca + c + a = 35, \]
find the value of \( (a+1)(b+1)(c+1) \).


In the plane, a square \(ABCD\) is given. Find the minimum of the ratio \(\frac{OA + OC}{OB + OD}\), where \(O\) is an arbitrary point in the plane.


The five symbols $@, *, \#, \&$, and $\wedge$ used in the equations below represent different digits.

\[ @ + @ + @ = * \]
\[ \# + \# + \# = \wedge \]
\[ * + \wedge = \& \]

What is the value of $\&$?

A) 0
B) 2
C) 3
D) 6
E) 9


While one lion cub, who was 6 minutes away, went to the watering hole, the second one, having already quenched its thirst, headed back along the same path 1.5 times faster than the first. At the same time, a turtle, which was 32 minutes away from the watering hole, also headed towards it along the same path. After some time, the first lion cub stepped on the turtle, and 2 minutes and 24 seconds later, the second lion cub did as well. How many minutes after the second incident did the turtle reach the watering hole, knowing that all three moved at constant speeds?


Find the number of eight-digit numbers for which the product of the digits equals 7000. The answer must be given as an integer.


A smooth sphere with a radius of 1 cm is dipped in red paint and placed between two perfectly smooth concentric spheres with radii of 4 cm and 6 cm, respectively (the painted sphere is outside the smaller sphere but inside the larger one). As it moves, the sphere leaves a red mark when it touches either of the concentric spheres. After traveling through a closed route, a region enclosed by the red mark forms on the smaller sphere, with an area of 47 square centimeters. Find the area of the region enclosed by the red mark on the larger sphere. Provide the answer in square centimeters, rounded to the nearest hundredth if necessary.


Compose the equation of an equilateral hyperbola with foci on the $x$-axis that passes through the point $M(4, -2)$.

Solution: \boxed{x^2 - y^2 = 12}


The area of a quadrilateral formed by the midpoints of the bases and diagonals of a trapezoid is four times smaller than the area of the trapezoid itself. Find the ratio of the lengths of the bases of the trapezoid.


Find the area of a triangle if it is known that its medians \(CM\) and \(BN\) are 6 and 4.5 respectively, and \(\angle BKM = 45^\circ\), where \(K\) is the point of intersection of the medians.


A real number \( x \) satisfies \( 9^{x} + 3^{x} = 6 \). Compute the value of \( 16^{1/x} + 4^{1/x} \).


An engraver makes plates with letters. He engraves the same letters in the same amount of time, but different letters may take different times. On two plates, "ДОМ МОДЫ" (DOM MODY) and "ВХОД" (VKHOD), together he spent 50 minutes, and one plate "В ДЫМОХОД" (V DYMOHOD) took him 35 minutes. How much time will it take him to make the plate "ВЫХОД" (VYKHOD)?

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


Let \(\triangle ABC\) be a scalene triangle. Let \(h_{a}\) be the locus of points \(P\) such that \(|PB - PC| = |AB - AC|\). Let \(h_{b}\) be the locus of points \(P\) such that \(|PC - PA| = |BC - BA|\). Let \(h_{c}\) be the locus of points \(P\) such that \(|PA - PB| = |CA - CB|\). In how many points do all of \(h_{a}, h_{b}\), and \(h_{c}\) concur?


Four pumps of equal capacity, working together, filled the first tanker and one-third of a second tanker (of different volume) in 11 hours. If three pumps filled the first tanker, and then one of them filled one-quarter of the second tanker, the job would take 18 hours. How many hours would three pumps take to fill the second tanker?

Solution: \boxed{8 \text{ hours}}


Given 2018 real numbers \(a_1, a_2, \ldots, a_{2018}\) satisfying
$$
\sum_{i=1}^{2018} a_i = 0, \quad \sum_{i=1}^{2018} a_i^2 = 2018,
$$
find the maximum possible value of the minimum product of any two of these numbers.


Given a parallelogram \(ABCD\). Points \(M\) and \(N\) are the midpoints of the sides \(BC\) and \(CD\), respectively. Find the ratio of the area of quadrilateral \(AMND\) to the area of the parallelogram.


The lengths of the three sides $a, b, c$ of a triangle satisfy the equation $\frac{c^{2}}{a+b}+\frac{a^{2}}{b+c}=b$. What is the measure of the angle $B$ opposite the side of length $b$?


Determine the largest constant \( C \) such that for all real numbers \( x_1, x_2, \ldots, x_6 \), the inequality

\[
\left(x_1 + x_2 + \cdots + x_6\right)^2 \geq C \cdot \left(x_1 \left(x_2 + x_3\right) + x_2 \left(x_3 + x_4\right) + \cdots + x_6 \left(x_1 + x_2\right) \right)
\]

holds.

Also, find for this \( C \) all \( x_1, x_2, \ldots, x_6 \) for which equality holds.

Solution: \boxed{C = 3}


Find the number of points in the $xy$-plane with natural number coordinates $(x, y)$ that lie on the parabola $y = -\frac{x^2}{4} + 11x + 23$.


On a game show, Merble will be presented with a series of 2013 marbles, each of which is either red or blue. Each time he sees a marble, he can either keep it or pass, but cannot return to a previous marble; he receives 3 points for keeping a red marble, loses 2 points for keeping a blue marble, and gains 0 points for passing. All distributions of colors are equally likely, and Merble can only see the color of his current marble. If his goal is to end with exactly one point and he plays optimally, what is the probability that he fails?


Given an equilateral triangle of side 10, divide each side into three equal parts, construct an equilateral triangle on the middle part, and then delete the middle part. Repeat this step for each side of the resulting polygon. Find \( S^2 \), where \( S \) is the area of the region obtained by repeating this procedure infinitely many times.


Let \( a_{1}, a_{2}, a_{3}, a_{4}, a_{5} \) be real numbers whose sum is 20. Determine with proof the smallest possible value of
\[
\sum_{1 \leq i < j \leq 5} \left\lfloor a_{i} + a_{j} \right\rfloor.
\]


Two typists need to retype a manuscript consisting of three chapters, where the first chapter is twice as short as the second and three times longer than the third. Working together, the typists retyped the first chapter in 3 hours and 36 minutes. The second chapter was retyped in 8 hours, with the first typist working alone for 2 hours and the rest of the time they worked together. How much time will the second typist need to retype the third chapter alone?

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


Another professor enters the same room and says, "Each of you has to write down an integer between 0 and 200. I will then compute \( X \), the number that is 3 greater than half the average of all the numbers that you will have written down. Each student who writes down the number closest to \( X \) (either above or below \( X \)) will receive a prize." One student, who misunderstood the question, announces to the class that he will write the number 107. If among the other 99 students it is common knowledge that all 99 of them will write down the best response, and there is no further communication between students, what single integer should each of the 99 students write down?


Find an eight-digit palindrome that is a multiple of three, composed of the digits 0 and 1, given that all its prime divisors only use the digits 1, 3, and %. (Palindromes read the same forwards and backwards, for example, 11011).


The function \( f(x) \) is not defined at \( x = 0 \). Determine the value of \( f(0) \) so that \( f(x) \) is continuous at \( x = 0 \), given

\[
f(x) = \frac{\sqrt[3]{1+x} - 1}{\sqrt{4+x} - 2}
\]


In a $20 \times 19$ grid, use $3 \times 1$ dominoes to cover the grid without overlapping and without common points. Find the maximum possible number of dominoes needed.


Find the largest five-digit positive integer such that it is not a multiple of 11, and any number obtained by deleting some of its digits is also not divisible by 11.


Find the maximum value of the expression \(a + b + c + d - ab - bc - cd - da\), if each of the numbers \(a\), \(b\), \(c\), and \(d\) belongs to the interval \([0, 1]\).


In how many ways can 4 purple balls and 4 green balls be placed into a 4 × 4 grid such that every row and column contains one purple ball and one green ball? Only one ball may be placed in each box, and rotations and reflections of a single configuration are considered different.


$A, B, C, D$ attended a meeting, and each of them received the same positive integer. Each person made three statements about this integer, with at least one statement being true and at least one being false. Their statements are as follows:

$A:\left(A_{1}\right)$ The number is less than 12;
$\left(A_{2}\right)$ 7 cannot divide the number exactly;
$\left(A_{3}\right)$ 5 times the number is less than 70.

$B:\left(B_{1}\right)$ 12 times the number is greater than 1000;
$\left(B_{2}\right)$ 10 can divide the number exactly;
$\left(B_{3}\right)$ The number is greater than 100.

$C:\left(C_{1}\right)$ 4 can divide the number exactly;
$\left(C_{2}\right)$ 11 times the number is less than 1000;
$\left(C_{3}\right)$ 9 can divide the number exactly.

$D:\left(D_{1}\right)$ The number is less than 20;
$\left(D_{2}\right)$ The number is a prime number;
$\left(D_{3}\right)$ 7 can divide the number exactly.

What is the number?


In a regular hexagon divided into six regions, each region is planted with ornamental plants under the constraint that the same plant is planted within a single region and different plants are planted in adjacent regions. Given 4 different types of plants to choose from, determine the number of different planting schemes.


In how many different ways can 25 identical coins be distributed among four schoolchildren? (Two ways are considered different if at least one of the schoolchildren receives a different amount of money in each way.)


Enter any three-digit number into a calculator. Then, type the same number again to form a six-digit number in the format \(abcabc\). Divide this number by 7, then divide the result by 11, and finally, divide the obtained number by 13. What happened? Why did you get this result?

Solution: \boxed{abc}


On a certain segment, the endpoints and three internal points were marked. It turned out that all pairwise distances between the five marked points are different and expressed in whole numbers of centimeters. What is the smallest possible length of the segment? Justify your answer.

Solution: \boxed{11 \, \text{cm}}


In the given diagram, \( A, B, C, D \) are the midpoints of the sides of a regular hexagon, and the area of the hexagon is 16. Find the area of the shaded region.


Ten tennis players have arrived at a competition, including 4 from Russia. According to the rules for the first round, players are paired randomly. Find the probability that in the first round, all Russian players will play only with other Russian players.


Calculate the limit of the function:

$\lim_{x \rightarrow 0} \sqrt[x^{2}]{2-\cos x}$

Solution: \boxed{\sqrt{e}}


There are 6 visually identical coins, 4 of which are genuine and of the same mass, and 2 are counterfeit, which are lighter and also weigh the same. What is the minimum number of weighings required on a balance scale without weights to identify both counterfeit coins?


Find all prime numbers whose decimal representation has the form 101010 ... 101 (units and zeros alternate).


Given the sequence: $1 \times 9 + 2 = 11, 12 \times 9 + 3 = 111, 123 \times 9 + 4 = 1111, \cdots, \triangle \times 9 + O = 111111$, find the value of $\triangle + O$.


One of the three cards had the number 18, another had the number 75, and the third had some two-digit number. If you sum all the distinct six-digit numbers that can be obtained by arranging these cards in a row, you get the number 2606058. What number is written on the third card?


Let \( A \) be a set with 225 elements, and \( A_{1}, A_{2}, \cdots, A_{11} \) be 11 subsets of \( A \) each containing 45 elements, such that for any \( 1 \leq i < j \leq 11 \), \(|A_{i} \cap A_{j}| = 9\). Find the minimum value of \(|A_{1} \cup A_{2} \cup \cdots \cup A_{11}|\).


If the roots of \( x^{2} - 2x - R = 0 \) differ by 12, find the value of \( R \).


In a trapezoid \(ABCD\) with bases \(AD=12\) and \(BC=8\), circles constructed on the sides \(AB\), \(BC\), and \(CD\) as diameters intersect at one point. The length of diagonal \(AC\) is 12. Find the length of \(BD\).


Simplify the expression \(\frac{2 a \sqrt{1+x^{2}}}{x+\sqrt{1+x^{2}}}\) where \(x=\frac{1}{2} \cdot\left(\sqrt{\frac{a}{b}} - \sqrt{\frac{b}{a}}\right)\) and \(a>0, b>0\).

Solution: \boxed{a+b}


Find the value of \(10 \cdot \operatorname{ctg}(\operatorname{arcctg} 3 + \operatorname{arcctg} 7 + \operatorname{arcctg} 13 + \operatorname{arcctg} 21)\).


Petya is thinking of a four-digit number of the form \( \overline{20 * *} \).

Vasya consecutively checks whether the number chosen by Petya is divisible by 1, 3, 5, 7, 9, 11. If the number is divisible, Vasya pays Petya 1, 3, 5, 7, 9, or 11 rubles respectively. For example, for the number 2000, Vasya would pay Petya \(1+5=6\) rubles.

What is the maximum number of rubles Petya can receive?


For any finite set \( S \), let \( f(S) \) be the sum of the elements of \( S \) (if \( S \) is empty then \( f(S) = 0 \) ). Find the sum over all subsets \( E \) of \( S \) of \(\frac{f(E)}{f(S)}\) for \( S = \{1, 2, \ldots, 1999\} \).


Let \( a_{n} \) denote the number of \( n \)-digit decimal numbers in the interval \([0,1)\) that do not contain the digit 9, and let \( S_{n} \) denote the sum of these numbers. Then,
$$
\lim _{n \rightarrow \infty} \frac{S_{n}}{a_{n}}=
$$


In the figure, the triangle $\triangle ABC$ is isosceles, with $\angle BAC = 20^\circ$.

Given that $BC = BD = BE$, determine the measure of the angle $\angle BDE$.


A man was building a house for himself. Only the roof remains to be completed. The builder wanted the raindrops falling on the roof to roll off as quickly as possible. Determine the angle of inclination of the roof required to achieve this goal. Ignore friction between the raindrops and the roof.


As shown in the figure, in triangle \(ABC\), \(AF : FB = BD : DC = CE : AE = 3 : 2\). Given that the area of triangle \(ABC\) is 1, find the area of triangle \(GHI\).


For \( x, y \in (0,1] \), find the maximum value of the expression

\[
A = \frac{\left(x^{2} - y\right) \sqrt{y + x^{3} - x y} + \left(y^{2} - x\right) \sqrt{x + y^{3} - x y} + 1}{(x - y)^{2} + 1}
\]


If a square with a side length of 8 cm is placed over a triangle, the overlapping area occupies half of the triangle's area and three-quarters of the square's area. What is the area of this triangle in square centimeters?


Eight dwarfs are seated around a round table, each with three diamonds. Every minute, the dwarfs simultaneously perform the following action: they divide all their diamonds into two piles (it is possible for one or both piles to be empty), then give one pile to their left neighbor and the other pile to their right neighbor. At some point, all the diamonds end up with three dwarfs. One of them has seven diamonds. How many diamonds do the other two dwarfs have?


1. Determine the remainder of the Euclidean division of $9^{100}$ by 8.
2. Find the last digit of $2012^{2012}$.


We are given the values of the differentiable real functions \( f, g, h \), as well as the derivatives of their pairwise products, at \( x=0 \):
\[
f(0) = 1, \quad g(0) = 2, \quad h(0) = 3, \quad (gh)^{\prime}(0) = 4, \quad (hf)^{\prime}(0) = 5, \quad (fg)^{\prime}(0) = 6.
\]
Find the value of \( (fgh)^{\prime}(0) \).


For \( n \in \mathbf{N}^{*} \), if \( n \cdot 2^{n} + 1 \) is divisible by 3, the set of remainders when \( n \) is divided by 6 is \(\quad\).

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


Let \( f(x) \) be a real-valued function defined on the set of real numbers, and suppose it satisfies \( 2 f(x-1) - 3 f(1-x) = 5x \). Find \( f(x) \).

Solution: \boxed{f(x) = x - 5}


The eccentricity of the conic section $6 x^{2} + 4 x y + 9 y^{2} = 20$ is $\qquad$.


Points \(A, B, C, D,\) and \(E\) are vertices of an irregular five-pointed star, see the illustration. Determine the sum of the marked angles.

Note: The illustration is for reference only.

Hint: What is the sum of the interior angles of a triangle?


A firework was thrown vertically upward with a speed of $20 \, \mathrm{m/s}$. One second after the start of the flight, it exploded into two unequal parts, with their masses in the ratio of $1:2$. The smaller fragment immediately after the explosion flew horizontally at a speed of $16 \, \mathrm{m/s}$. Find the magnitude of the speed of the second fragment (in $\mathrm{m/s}$) immediately after the explosion. Assume the acceleration due to gravity is $10 \, \mathrm{m/s}^2$.

Solution: \boxed{17 \, \text{m/s}}


On the planet Lemniscate, the people use an advanced version of the periodic table. They know that $\mathrm{Kr}$ deteriorates into $\mathrm{Pl}$ and then into a stable element As. Starting with a block of $\mathrm{Kr}$ of size 10, at time $t$, let:
- $x(t)$ be the amount of $\mathrm{Kr}$,
- $y(t)$ be the amount of $\mathrm{Pl}$,
- $z(t)$ be the amount of As.

Given $x^{\prime}(t)=-x$, and in the absence of $\mathrm{Kr}$, $y^{\prime}(t)=-2y$, find the time $t$ at which the quantity of $\mathrm{Pl}$ will be the largest. Assume the entire amount of deteriorating $\mathrm{Kr}$ turns into $\mathrm{Pl}$.

Solution: \boxed{t = \ln 2}


Let $n$ be the smallest positive integer such that the last three digits of $n^3$ are 888. Find $n$.


Express the following as an irreducible fraction:

$$
6 \frac{7}{2015} \times 4 \frac{5}{2016} - 7 \frac{2008}{2015} \times 2 \frac{2011}{2016} - 7 \times \frac{7}{2015}
$$


In each cell of a square grid of size \( \left(2^{n}-1\right) \times \left(2^{n}-1\right) \) one of the numbers 1 or -1 is placed. The arrangement is called successful if each number is equal to the product of all its neighboring numbers (neighboring numbers are considered those that are in cells with a common side). Find the number of successful arrangements.


Given \( n \in \mathbb{N} \), show that there exists \( r \in \mathbb{N} \) such that none of the numbers \( r+1, r+2, r+3, \ldots, r+n \) are prime.

Solution: \boxed{(n+1)! + 1}


In a grid square of size \( n \times n \), where each cell is painted in one of two colors: white or black, what is the smallest \( n \) for which a rectangle always exists (i.e., for any coloring) whose vertices coincide with the centers of four cells of the same color?


The diagram below shows a pentagon (made up of region \( A \) and region \( B \)) and a rectangle (made up of region \( B \) and region \( C \)) that overlaps. The overlapped region \( B \) is \(\frac{3}{16}\) of the pentagon and \(\frac{2}{9}\) of the rectangle. If the ratio of region \( A \) of the pentagon to region \( C \) of the rectangle is \(\frac{m}{n}\) in its lowest term, find the value of \( m+n \).


A square and a regular hexagon are drawn with the same side length. If the area of the square is \(\sqrt{3}\), what is the area of the hexagon?


An amount needs to be divided among $A, B, C,$ and $D$ such that the ratios of their shares are $3:4:5:6$. However, the amount was divided in the ratio $3:4:6:7$, resulting in $C$ and $D$ together receiving 1400 crowns more than they should have. What was the total amount to be divided?


Find the maximum value of the function

$$
f(x)=\frac{x}{x^{2}+9}+\frac{1}{x^{2}-6 x+21}+\cos 2 \pi x
$$


Calculate the limit of the numerical sequence:

\[
\lim _{n \rightarrow \infty} \frac{n \sqrt[4]{11 n}+\sqrt{25 n^{4}-81}}{(n-7 \sqrt{n}) \sqrt{n^{2}-n+1}}
\]


Which is greater: $\log _{3} 4$ or $\log _{4} 5$?

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


In \(\triangle ABC\), \(\angle A = 80^\circ\), \(\angle B = 30^\circ\), and \(\angle C = 70^\circ\). Let \(BH\) be an altitude of the triangle. Extend \(BH\) to a point \(D\) on the other side of \(AC\) so that \(BD = BC\). Find \(\angle BDA\).


Given that \( p \) and \( q \) are both integers greater than 1, and that \( 2p - 1 \) is a multiple of \( q \), and \( 2q - 1 \) is a multiple of \( p \), find the value of \( p + q \).


For natural numbers \( m \) and \( n \), it is known that \( 3n^3 = 5m^2 \). Find the smallest possible value of \( m + n \).


Vovochka approached a game machine, on the screen of which the number 0 was displayed. The rules of the game stated: "The screen shows the number of points. If you insert a 1-ruble coin, the number of points increases by 1. If you insert a 2-ruble coin, the number of points doubles. If you score 50 points, the machine gives a prize. If the resulting number is greater than 50, then all the points scored are lost." What is the minimum amount of rubles Vovochka needs to get the prize? Answer: 11 rubles.

Solution: \boxed{11 \text{ rubles}}


In a scalene triangle \( ABC \), one of the angles is equal to the difference of the other two angles, and one of the angles is twice as large as another. The angle bisectors of angles \( A \), \( B \), and \( C \) intersect the circumcircle of the triangle at points \( L \), \( O \), and \( M \) respectively. Find the area of triangle \( LOM \) if the area of triangle \( ABC \) is 2. If the answer is not an integer, round it to the nearest whole number.


Given that the union of sets \( A \) and \( B \) is \( A \cup B = \left\{a_{1}, a_{2}, a_{3}\right\} \), and \( A \neq B \), where the pairs \((A, B)\) and \((B, A)\) are considered different, how many such pairs \( (A, B) \) are there?


Solve the equation $\frac{n!}{2} = k! + l!$ in natural numbers, where $n! = 1 \cdot 2 \cdot \ldots n$. In your answer, indicate 0 if there are no solutions, $n$ if there is one solution, and the sum of values of $n$ for all solutions if there are several solutions. Note that a solution is a triple $(n, k, l)$; if solutions differ in at least one component, they are considered different.


Given that \( x + y = 12 \) and \( x y = 50 \), find the exact value of \( x^2 + y^2 \).


Find the limit $\lim _{x \rightarrow 1} \frac{3 x^{2}-x-2}{4 x^{2}-5 x+1}$.


Find the sum $\frac{1}{2!}+\frac{2}{3!}+\frac{3}{4!}+\ldots+\frac{n}{(n+1)!}$ and compute its limit as $n \rightarrow \infty$.


In triangle \( KLM \), with all sides distinct, the bisector of angle \( KLM \) intersects side \( KM \) at point \( N \). A line through point \( N \) intersects side \( LM \) at point \( A \), such that \( MN = AM \). Given that \( LN = a \) and \( KL + KN = b \), find \( AL \).


Joe already has a son, Tom, and a daughter, Anna, and he wants to have twins. If the newborns are both boys, then the number of Tom's brothers is exactly $k$ times the number of his sisters. If the newborns are both girls, then the number of Anna's brothers is exactly $t$ times the number of her sisters. If the newborns are one boy and one girl, find:
(1) The ratio of the number of brothers to sisters for the boys.
(2) The ratio of the number of brothers to sisters for the girls.

Solution: \boxed{k}


Determine the positive integer based on the following conditions: if you write the number down and append the digit 4 to the right, the resulting number will be divisible by a number that is 4 more than the original number, and the quotient will be 27 less than the divisor.


How many squares are there on an infinitely large chessboard that can be reached by a king in exactly \( k \) moves from a designated square, but not in fewer moves? (In chess, the king can move to any of the 8 squares that share either a side or a corner with the square on which the king stands.)


There are candies in five bags. The first bag contains 2 candies, the second contains 12, the third contains 12, the fourth contains 12, and the fifth contains 12. You can transfer any number of candies from any bag to any other bag. What is the minimum number of transfers required to make the number of candies in all bags equal?


In a mathematics olympiad consisting of two rounds, a total of 14 problems need to be solved. For each correctly solved problem, a contestant receives 7 points, while for each incorrectly solved problem, 12 points are deducted. How many problems did the student solve correctly if they scored 60 points?


What is the probability of randomly drawing three different numbers from the set {1, 2, ..., 10} such that their sample variance \( s^2 \leqslant 1 \)?


\(ABCD\) is a square and \(X\) is a point on the side \(DA\) such that the semicircle with diameter \(CX\) touches the side \(AB\). Find the ratio \(AX: XD\).

Solution: \boxed{1 : 3}


Determine the coefficients $a, b, c$ given that the following system of equations has no solution, but if $|x-1|$ is written instead of $x-1$ in the second equation, the pair $x = \frac{3}{4}, y = \frac{5}{8}$ satisfies the new system of equations.

\[
\begin{aligned}
& a(x-1) + 2y = 1 \\
& b(x-1) + cy = 3
\end{aligned}
\]

Solution: \boxed{a=1, b=2, c=4}


Petya and Vasya competed in a 60-meter race. When Petya finished, Vasya was 9 meters behind him. In the second race, Petya started exactly 9 meters behind Vasya. Who finished first in the second race, and by how many meters did he outpace his opponent? (Assuming each of the boys ran at their own constant speed both times).

Solution: \boxed{1.35 \text{ meters}}


Suppose \( x \) and \( y \) are real numbers satisfying \( x + y = 5 \). What is the largest possible value of \( x^2 + 2xy \)?


In figure 1, \(MN\) is a straight line, \(\angle QON = a^{\circ}\), \(\angle POQ = b^{\circ}\) and \(\angle POM = c^{\circ}\). If \(b : a = 2 : 1\) and \(c : b = 3 : 1\), find the value of \(b\).

Solution: \boxed{b = 40^\circ}


Find the integer whose cube is equal to the sum of the cubes of the three preceding consecutive numbers.


Given that \( m \) and \( n \) are known positive integers, and the number of digits of \( m \) in decimal notation is \( d \), where \( d \leq n \). Find the sum of all digits in the decimal representation of \((10^n - 1)m\).


Cars A and B start from city $A$ heading towards city $B$, which is 300 kilometers away. It is known that car A leaves 1 hour later than car B but arrives at city $B$ 1 hour earlier. How many kilometers away from city $B$ does car A catch up with car B?

Solution: \boxed{150 \text{ km}}


Find the smallest natural number such that when multiplied by 9, the resulting number consists of the same digits but in some different order.


A common year has 365 days and a leap year has 366 days. A leap year, when February has 29 days, occurs every four years.

(a) It is often said, "A common year has 52 weeks." Is this statement correct? And what about a leap year? Justify your answers.

(b) If a common year starts on a Tuesday, then on which day of the week will the following year start?
(c) Answer the previous question for a leap year.

Solution: \boxed{\text{Thursday}}


In how many ways can 8 people be arranged in a line if Alice and Bob must be next to each other, and Carol must be somewhere behind Dan?


Find a four-digit number that is a perfect square if its first two digits are the same as its last two digits.


Given that \( m \) is a constant, and the function
\[
f(x) = \sin^{4} x + m \sin x \cdot \cos x + \cos^{4} x
\]
has the range \(\left[0, \frac{9}{8}\right]\). Find the value of \( m \).

Solution: \boxed{\pm 1}


A pool is being filled by three pipes. The first and second pipes together can fill the pool in 70 minutes, the first and third pipes together can fill it in 84 minutes, and the second and third pipes together can fill it in 140 minutes. How many minutes will it take to fill the pool using each pipe individually, and how many minutes will it take if all three pipes are open at the same time?

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


Let the solution sets of the inequality $\left|x-\frac{(a+1)^{2}}{2}\right| \leqslant \frac{(a-1)^{2}}{2}$ and $x^{2}-3(a+1)x+2(3a+1) \leq 0$ (where $a \in \mathbf{R}$) be $A$ and $B$ respectively. If $A \subseteq B$, find the range of $a$.

Solution: \boxed{[1, 3] \cup \{-1\}}


A ball rolled into a pool and floated on the water. Its highest point was $2 \mathrm{~cm}$ above the water surface. The diameter of the circle marked by the water surface on the ball was $8 \mathrm{~cm}$.

Determine the diameter of the ball.

(L. Hozová)

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


Given the sets:
$$
\begin{array}{l}
A=\left\{x \mid x^{2}+x-6>0\right\}, \\
B=\left\{x \mid x^{2}-2 a x+3 \leqslant 0\right\}.
\end{array}
$$

If \(a>0\) and there are exactly two integers in the intersection \(A \cap B\), then the range of values for \(a\) is .

Solution: \boxed{[2.375, 2.8)}


A computer can apply three operations to a number: "increase by 2," "increase by 3," "multiply by 2." The computer starts with the number 1 and is made to go through all possible combinations of 6 operations (each combination is applied to the initial number 1). After how many of these combinations will the computer end up with an even number?


Lesha's summer cottage has the shape of a nonagon with three pairs of equal and parallel sides. Lesha knows that the area of the triangle with vertices at the midpoints of the remaining sides of the nonagon is 12 sotkas. Help him find the area of the entire summer cottage.


The factorial of a number \( n \) is defined as the product of all integers from 1 to \( n \) inclusive. Find all three-digit numbers that are equal to the sum of the factorials of their digits.


How many integers \( n \) are there with the property that the product of the digits of \( n \) is 0, where \( 5000 \leq n \leq 6000 \)?


As shown in the figure, fill each cell of a $5 \times 5$ grid with a natural number (natural numbers include 0) such that the average of the four numbers in every $2 \times 2$ sub-square is 100. What is the minimum average of the 25 natural numbers in the entire $5 \times 5$ grid?


In triangle \( \triangle ABC \), the angles are \( \angle B = 30^\circ \) and \( \angle A = 90^\circ \). 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 \( MC = 3 \).


What is the value of \( 6 \times (5 - 2) + 4 \)?


If the number of sides in a polygon is doubled, the difference between the diagonals and sides increases by 99. How many sides does the polygon have?


In the plane, there are \( n \) lines such that every pair of lines intersects, but no four lines pass through a single point. There are a total of 16 intersection points, and 6 of these points are intersections of three lines each. Find the value of \( n \).


What is the sum of the exterior angles at the three vertices of a triangle?


Galya thought of a number, multiplied it by \( \mathrm{N} \), then added \( \mathrm{N} \) to the result, divided the resulting number by \( \mathrm{N} \), and subtracted \( \mathrm{N} \). As a result, she obtained a number that is 7729 less than the original thought number. What is \( \mathrm{N} \)?


Let \(A B C D E F\) be a regular hexagon of area 1. Let \(M\) be the midpoint of \(D E\). Let \(X\) be the intersection of \(A C\) and \(B M\), let \(Y\) be the intersection of \(B F\) and \(A M\), and let \(Z\) be the intersection of \(A C\) and \(B F\). If \([P]\) denotes the area of polygon \(P\) for any polygon \(P\) in the plane, evaluate \([B X C]+[A Y F]+[A B Z]-[M X Z Y]\).


A positive integer is thought of. The digit 7 is appended to the end of this number, and from the resulting new number, the square of the original number is subtracted. The remainder is then reduced by 75% of this remainder and the original number is subtracted again. The final result is zero. What number was thought of?


An acute-angled isosceles triangle and a trapezoid are inscribed in a circle. One base of the trapezoid is the diameter of the circle, and the lateral sides of the trapezoid are parallel to the lateral sides of the triangle. Find the ratio of the areas of the trapezoid and the triangle.


Let \( a \in \mathbb{N}^{*}, n, m \in \mathbb{N} \). Calculate the GCD of \( a^{n}-1 \) and \( a^{m}-1 \).

Solution: \boxed{a^{\gcd(n, m)} - 1}


The first tourist travels for 1.5 hours on a bicycle at a speed of 16 km/h, then takes a break for 1.5 hours, and then continues at the initial speed. Four hours after the first tourist starts, a second tourist starts chasing the first tourist on a motorcycle at a speed of 56 km/h. What distance will they each have traveled by the time the second tourist catches up to the first tourist?

Solution: \boxed{56 \text{ km}}


For arbitrary real numbers \(a\) and \(b\) (\(a \neq 0\)), find the minimum value of the expression \(\frac{1}{a^{2}} + 2a^{2} + 3b^{2} + 4ab\).


Given a triangle $ABC$ where $AC=\sqrt{2}$, $BC=1$, and $\angle B=45^\circ$. Find the angle $A$.


Solve the equation \(x^{3}-3 y^{3}-9 z^{3}=0\) in integers.

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


There are $N$ points marked on a plane. Any three of them form a triangle whose angles are expressible in degrees as natural numbers. What is the maximum possible $N$ for which this is possible?


A smooth ball with a radius of 1 cm was dipped in red paint and set between two absolutely smooth concentric spheres with radii of 4 cm and 6 cm, respectively (the ball is outside the smaller sphere but inside the larger one). Upon contact with both spheres, the ball leaves a red mark. During its movement, the ball traveled along a closed path, resulting in a region on the smaller sphere outlined in red with an area of 37 square centimeters. Find the area of the region outlined in red on the larger sphere. Give your answer in square centimeters, rounding to the nearest hundredth if necessary.


Consider an alphabet of 2 letters. A word is any finite combination of letters. We will call a word unpronounceable if it contains more than two identical letters in a row. How many unpronounceable words of 7 letters exist? Points for the problem: 8.


Given non-negative real numbers \( x_1, x_2, \ldots, x_n \) such that \( x_1 + x_2 + \cdots + x_n = s \), show that:
\[ x_1 x_2 + x_2 x_3 + \cdots + x_{n-1} x_n \leq \frac{s^2}{4}. \]


In the ellipse, \( O \) is the center, \( P \) is the endpoint of the minor axis, and \( F_{1} \) and \( F_{2} \) are the two foci. Given that \(\left|F_{1} F_{2}\right| = 3\) and the distance from point \( O \) to the line \( F_{1} P \) is 1. Find the eccentricity of the ellipse.


For packing books during the school library move, you can buy small boxes that hold 12 books each, or large boxes that are designed for 25 books each. If all the books are placed in small boxes, 7 books remain. If all the books are packed in large boxes, there is space left for 5 more books. The library collection contains between 500 and 650 books. How many books are in the library?


Find a number \( N \) with five digits, all different and none zero, which equals the sum of all distinct three-digit numbers whose digits are all different and are all digits of \( N \).


Let $\alpha(n)$ be the measure of the interior angles of a regular $n$-sided polygon. What is the value of $n$ if $\alpha(n+3)-\alpha(n)=\alpha(n)-\alpha(n-2)$?


There are 10 different points \( P_{1}(x_{1}, y_{1}), P_{2}(x_{2}, y_{2}), \cdots, P_{10}(x_{10}, y_{10}) \) in the Cartesian plane. If \( x_{i}=x_{j} \) or \( y_{i}=y_{j} \), then \( P_{i} \) and \( P_{j} \) are called a "same-coordinate pair" (the order of \( P_{i} \) and \( P_{j} \) does not matter). It is known that among these 10 different points, the number of "same-coordinate pairs" that each point forms does not exceed \( m \); moreover, it is always possible to partition these points exactly into 5 pairs, where each pair is not a "same-coordinate pair." Find the maximum value of \( m \).


On the International Space Station, there was an electronic clock displaying time in the HH:MM format. Due to an electromagnetic storm, the device started malfunctioning, causing each digit on the display to either increase by 1 or decrease by 1. What was the actual time when the storm occurred, if immediately after the storm the clock showed 09:09?


A person starts from a certain point, moves forward 20 meters, then turns 30 degrees to the right, moves forward another 20 meters, turns 30 degrees to the right again, and continues this pattern. How many meters has the person walked in total by the time they return to the starting point?

Solution: \boxed{240 \text{ meters}}


The average of a list of three consecutive odd integers is 7. When a fourth positive integer, \( m \) , different from the first three, is included in the list, the average of the list is an integer. What is the sum of the three smallest possible values of \( m \)?


Knowing the edges of a tetrahedron, its volume can be calculated by determining the height of any vertex. Perform the calculation for the following example: the edge lengths of tetrahedron $ABCD$ are as follows: $AB=6$, $AC=7$, $AD=8$, $CD=9$, $BD=10$, $BC=11$ units. (We can choose the base, for example, the triangle $ABC$.)

Solution: \boxed{ V = 48 }


In the final round of a giraffe beauty contest, two giraffes named Tall and Spotted have made it to this stage. There are 105 voters divided into 5 districts, each district divided into 7 sections, with each section having 3 voters. Voters select the winner in their section by majority vote; in a district, the giraffe winning the majority of sections wins the district; finally, the giraffe winning the majority of districts is declared the contest winner. The giraffe named Tall won. What is the minimum number of voters who could have voted for him?


Let \(ABCD\) be a convex quadrilateral with \(\angle ABC = 90^\circ\), \(\angle BAD = \angle ADC = 80^\circ\). Let \(M\) and \(N\) be points on \([AD]\) and \([BC]\) such that \(\angle CDN = \angle ABM = 20^\circ\). Finally, assume \(MD = AB\). What is the measure of \(\angle MNB\)?


Calculate the sum

$$
S=\frac{2013}{2 \cdot 6}+\frac{2013}{6 \cdot 10}+\frac{2013}{10 \cdot 14}+\ldots+\frac{2013}{2010 \cdot 2014}
$$

Provide the remainder when the even number nearest to the value of $S$ is divided by 5.


Given 95 numbers \( a_{1}, a_{2}, \cdots, a_{95} \), each of which can only take one of the two values +1 or -1, find the smallest positive value of the sum of their pairwise products:
\[ a_{1} a_{2} + a_{1} a_{3} + \cdots + a_{94} a_{95} \]


How many three-digit numbers have at least two identical digits?


Find the four-digit number that, when divided by 131, leaves a remainder of 112, and when divided by 132, leaves a remainder of 98.


Given that \(a\) is a positive integer, the system of equations
\[
\begin{cases}
ax + y = -4 \\
2x + y = -2
\end{cases}
\]
has a solution such that \(x < 0\) and \(y > 0\). Find the value of \(a\).


Consider the following two sets of numbers:

$$
A=\left\{\frac{3 n-4}{5 n-3}: n \in \mathbb{Z}\right\} \quad \text{and} \quad B=\left\{\frac{4 k-3}{7 k-6}: k \in \mathbb{Z}\right\} \text{.}
$$

How many elements are in the set $A \cap B$?


From \( m \) boys and \( n \) girls \((10 \geqslant m > n \geqslant 4)\), 2 people are randomly chosen to be class representatives. Suppose event \( A \) denotes that the 2 chosen people are of the same gender, and event \( B \) denotes that the 2 chosen people are of different genders. If the probabilities of events \( A \) and \( B \) are the same, find the possible values of \((m, n)\).


Given a regular tetrahedron $ABCD$ with edge length of 1 cm, an ant starts at point $A$ and follows these rules: at each vertex, it randomly chooses one of the three edges connected to that vertex and crawls along this edge to the next vertex. What is the probability that, after crawling a total of 6 cm, the ant returns to vertex $A$?


A train passenger knows that the speed of their train is 40 km/h. As soon as a passing train started to go by the window, the passenger started a stopwatch and noted that the passing train took 3 seconds to pass completely. Determine the speed of the passing train, given that its length is 75 meters.

Solution: \boxed{50 \, \text{km/h}}


An \((l, a)\)-design of a set is a collection of subsets of that set such that each subset contains exactly \(l\) elements and that no two of the subsets share more than \(a\) elements. How many \((2,1)\)-designs are there of a set containing 8 elements?


In some cells of a $1 \times 2100$ strip, a chip is placed in each cell. A number equal to the absolute difference between the number of chips to the left and to the right of this cell is written in each empty cell. It is known that all recorded numbers are different and non-zero. What is the smallest number of chips that can be placed in the cells?


Let \( p \) be a prime number such that the next larger number is a perfect square. Find the sum of all such prime numbers.


In triangle $ABC$, the angles $\angle B=30^{\circ}$ and $\angle A=90^{\circ}$ are given. 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 the segment $BM$).

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


Let \( a_{1}, a_{2}, \cdots, a_{n} \) be distinct positive integers such that \( a_{1} + a_{2} + \cdots + a_{n} = 2014 \), where \( n \) is some integer greater than 1. Let \( d \) be the greatest common divisor of \( a_{1}, a_{2}, \cdots, a_{n} \). For all values of \( n \) and \( a_{1}, a_{2}, \cdots, a_{n} \) that satisfy the above conditions, find the maximum value of \( n \cdot d \).


Given the complex number \( z = \sqrt{\frac{\pi}{\mathrm{e} + \pi}}(1 + \mathrm{i}) + \sqrt{\frac{\mathrm{e}}{\pi + \mathrm{e}}}(1 - \mathrm{i}) \) (where \( \mathrm{e} \) is the base of the natural logarithm and \( \mathrm{i} \) is the imaginary unit), find the modulus of \( z \).

Solution: \boxed{|z| = \sqrt{2}}


Find all integer triplets \((x, y, z)\) such that \(x^{2} + y^{2} + z^{2} = 2xyz\).

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


Given a square \(ABCD\). On the extension of the diagonal \(AC\) beyond point \(C\), a point \(K\) is marked such that \(BK = AC\). Find the angle \(BKC\).


There are 62 boxes in total, with small boxes being 5 units per ton and large boxes being 3 units per ton. A truck needs to transport these boxes. If the large boxes are loaded first, after loading all the large boxes, exactly 15 small boxes can still be loaded. If the small boxes are loaded first, after loading all the small boxes, exactly 15 large boxes can still be loaded. How many large boxes are there among these boxes in total?


On a long strip of paper, the numbers $1, 2, 3, \ldots, n$ are written sequentially. The long strip is cut into five segments, each containing some consecutive natural numbers (the digits of a single number are not split across different segments). The average values of these five segments are given as $1234, 345, 128, 19,$ and $9.5$ (the order of these numbers is shuffled). What is the value of $\mathbf{n}$?


Given that \( 111111222222 = c \times (c+1) \), find the value of \( c \).

Solution: \boxed{c = 333333}


In the vertices of a regular 2019-gon, numbers are arranged such that the sum of the numbers in any nine consecutive vertices is equal to 300. It is known that the 19th vertex has the number 19, and the 20th vertex has the number 20. What number is at the 2019th vertex?


Baron Munchausen told a story. "There were a whole crowd of us. We reached a crossroads. Then half of our group turned left, a third turned right, and a fifth went straight." "But wait, the Duke remarked, the sum of half, a third, and a fifth isn't equal to one, so you are lying!" The Baron replied, "I'm not lying, I'm rounding. For example, there are 17 people. I say that a third turned. Should one person split in your opinion? No, with rounding, six people turned. From whole numbers, the closest to the fraction $17 / 3$ is 6. And if I say that half of the 17 people turned, it means 8 or 9 people." It is known that Baron Munchausen never lies. What is the largest number of people that could have been in the crowd?


Pankrác paid 60% of the price of the boat, Servác paid 40% of the remaining price, and Bonifác covered the missing amount, which was 30 zlatek.

How much did the boat cost?

Solution: \boxed{125 \text{ zlateks}}


Given a triangle \(ABC\). The tangent to the circumcircle of this triangle at point \(B\) intersects line \(AC\) at point \(M\). Find the ratio \(|AM| : |MC|\), if \(|AB| : |BC| = k\).


From the numbers $1, 2, \cdots, 2004$, select $k$ numbers such that among the selected $k$ numbers, there are three numbers that can form the side lengths of a triangle (with the condition that the three side lengths are pairwise distinct). Find the smallest value of $k$ that satisfies this condition.


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

$$
r=\frac{1}{2}+\sin \phi
$$


The germination rate of seeds of this plant variety is $80\%$. What is the probability that at least 4 out of 5 sown seeds will germinate?


In 12 days it will be the year 2016. Hao Hao remarked: "Up to now, I have only experienced 2 leap years, and my birth year is a multiple of 9. How old will Hao Hao be in 2016?"


From vertex \( B \) of parallelogram \( ABCD \), the perpendicular heights \( BK \) and \( BH \) are drawn. The segments \( KH = a \) and \( BD = b \) are known. Find the distance from point \( B \) to the orthocenter of triangle \( BKH \).

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


Given a large \(4 \times 4 \times 4\) cube composed of 64 unit cubes, 16 of these unit cubes are to be painted red such that each \(1 \times 1 \times 4\) column within the large cube contains exactly 1 red unit cube. How many different ways can the 16 red unit cubes be selected? Explain the reasoning.


Mrs. Kučerová was on a seven-day vacation, and Káta walked her dog and fed her rabbits during this time. Káta received a large cake and 700 CZK as compensation. After another vacation, this time lasting four days, Káta received the same cake and 340 CZK for the same tasks.

What was the cost of the cake?

Solution: \boxed{140 \, \text{KČ}}


Let the real numbers \(a_1, a_2, \cdots, a_{2016}\) satisfy
\[
9 a_i > 11 a_{i+1}^2 \quad (i = 1, 2, \cdots, 2015).
\]

Find the maximum value of
\[
\left(a_1 - a_2^2\right)\left(a_2 - a_3^2\right) \cdots \left(a_{2015} - a_{2016}^2\right)\left(a_{2016} - a_1^2\right).
\]


Let $I$ be the incenter of triangle $\triangle ABC$, and suppose $3 \overrightarrow{IA} + 4 \overrightarrow{IB} + 5 \overrightarrow{IC} = \mathbf{0}$. Determine the measure of angle $C$.


In the triangle $ABC$, $\angle ACB = 90^{\circ}$ and $AC > BC$. The midpoint of the arc $AB$ not containing $C$ of the circumcircle of the triangle is $X$. The perpendicular to $CX$ at $X$ intersects the line $CA$ at point $P$. Show that $AP = BC$.

Solution: \boxed{AP = BC}


Solve the equation \( 2 \sqrt{2} \sin ^{3}\left(\frac{\pi x}{4}\right) = \sin \left(\frac{\pi}{4}(1+x)\right) \).

How many solutions of this equation satisfy the condition: \( 2000 \leq x \leq 3000 \)?


What is the size of the largest rectangle that can be drawn inside a 3-4-5 right triangle with one of the rectangle's sides along one of the legs of the triangle?


A sequence \( U_{1}, U_{2}, U_{3}, \ldots \) is defined as follows:
- \( U_{1} = 2 \);
- if \( U_{n} \) is prime then \( U_{n+1} \) is the smallest positive integer not yet in the sequence;
- if \( U_{n} \) is not prime then \( U_{n+1} \) is the smallest prime not yet in the sequence.
The integer \( k \) is the smallest such that \( U_{k+1} - U_{k} > 10 \).
What is the value of \( k \times U_{k} \)?


Five cards labeled 1, 3, 5, 7, 9 are laid in a row in that order, forming the five-digit number 13579 when read from left to right. A swap consists of picking two distinct cards and then swapping them. After three swaps, the cards form a new five-digit number \( n \) when read from left to right. Compute the expected value of \( n \).


João and Maria each have a large jar with one liter of water. On the first day, João puts $1 \mathrm{ml}$ of water from his jar into Maria's jar. On the second day, Maria puts $2 \mathrm{ml}$ of water from her jar into João's jar. On the third day, João puts $3 \mathrm{ml}$ of water from his jar into Maria's jar, and so on. After 200 days, how many milliliters of water are in Maria's jar?


Let \( S = \{1, 2, \cdots, 2005\} \). If in any set of \( n \) pairwise coprime numbers in \( S \) there is at least one prime number, find the minimum value of \( n \).


On the board, there are 28 ones written. Each minute, Karlsson erases two arbitrary numbers and writes their sum on the board, then eats a number of candies equal to the product of the two erased numbers. What is the maximum number of candies he could eat in 28 minutes?


The segment \( AD \) is the bisector of the right triangle \( ABC \) (\(\angle C=90^{\circ}\)). A circle with radius \(\sqrt{15}\) passes through points \( A, C, D \) and intersects side \( AB \) at point \( E \) such that \( AE: AB = 3: 5 \). Find the area of triangle \( ABC \).


The triangle \( \triangle ABC \) has side \( AC \) with length \( 24 \text{ cm} \) and a height from vertex \( B \) with length \( 25 \text{ cm} \). Side \( AB \) is divided into five equal parts, with division points labeled \( K, L, M, N \) from \( A \) to \( B \). Each of these points has a parallel line drawn to side \( AC \). The intersections of these parallels with side \( BC \) are labeled \( O, P, Q, R \) from \( B \) to \( C \).

Calculate the sum of the areas of the trapezoids \( KLQR \) and \( MNOP \).

Solution: \boxed{120 \, \text{cm}^2}


For which strictly positive integers $n$ do we have $5^{n-1} + 3^{n-1} \mid 5^{n} + 3^{n}$?

Solution: \boxed{n = 1}


Pleasant Goat is learning addition and subtraction. He has three types of number cards: "3", "4", and "5", with a total of 10 cards. The sum of the numbers on these 10 cards is 39, and there are exactly 2 cards with the number "5". How many cards with the number "3" are there?


A lattice point is a point whose coordinates are both integers. Suppose Johann walks in a line from the point \((0,2004)\) to a random lattice point in the interior (not on the boundary) of the square with vertices \((0,0)\), \((0,99)\), \((99,99)\), \((99,0)\). What is the probability that his path, including the endpoints, contains an even number of lattice points?


Determine all positive integers \( x \) such that \( 2x + 1 \) is a perfect square, but among the numbers \( 2x + 2, 2x + 3, \cdots, 3x + 2 \), there are no perfect squares.


Suppose you have 9 evenly spaced dots in a circle on a piece of paper. You want to draw a 9-pointed star by connecting dots around the circle without lifting your pencil, skipping the same number of dots each time.

Determine the number of different stars that can be drawn, if the regular nonagon does not count as a star.


As shown in the diagram, a cube with a side length of 12 cm is cut once. The cut is made along \( IJ \) and exits through \( LK \), such that \( AI = DL = 4 \) cm, \( JF = KG = 3 \) cm, and the section \( IJKL \) is a rectangle. The total surface area of the two resulting parts of the cube after the cut is \( \quad \) square centimeters.


Fox Alice thought of a two-digit number and told Pinocchio that this number is divisible by $2, 3, 4, 5,$ and $6$. However, Pinocchio found out that exactly two of these five statements are actually false. What numbers could Fox Alice have thought of? Indicate the number of possible options in the answer.


In a trapezoid with bases $x$ and 5, express the distance between the midpoints of the diagonals as a function of $x$.

Solution: \boxed{\frac{|x - 5|}{2}}


A motorcyclist left city A for city B. If he travels at a speed of 50 km/h, he will arrive in city B one hour earlier than the scheduled time. If he travels at a speed of 35 km/h, he will be 2 hours late. Find the distance between cities A and B.


Compute the limit of the function:

$\lim _{x \rightarrow \frac{\pi}{4}}(\sin x+\cos x)^{\frac{1}{\operatorname{tg} x}}$


How many positive integers \(N\) possess the property that exactly one of the numbers \(N\) and \((N+20)\) is a 4-digit number?


For how many integer values of $x$ will the following polynomial be a perfect square?

$$
4 x^{4}-12 x^{3}+17 x^{2}-6 x-14
$$


In a class of 42 girls, three students are favorites: an actor ( $S z$ ), a fencing champion $(V)$, and a skating champion $(K)$. The girls collect autographed photos of their favorites. Six girls already have such photos from all three favorites. The class collectively has $21 S z$ photos, $20 V$ photos, and $18 K$ photos. In response to the question of who has both an $S z$ photo and a $V$ photo, 7 girls raised their hands. Similarly, 10 girls raised their hands for having $S z$ and $K$ photos, and 11 girls for having $V$ and $K$ photos. Is there any girl who does not have an autographed photo from any of her favorites, and if so, how many such girls are there?


For all \( k \leq n \in \mathbb{N} \), define the binomial coefficient \( \binom{n}{k} \) recursively:

$$
\begin{gathered}
\binom{n}{0}=\binom{n}{n}=1 \\
\binom{n+1}{k+1}=\binom{n}{k}+\binom{n}{k+1} \text { for all } k<n
\end{gathered}
$$

1. Show that

$$
\binom{n}{k}=\frac{n!}{k!(n-k)!}
$$

2. Show that

$$
(a+b)^{n}=\sum_{k=0}^{n}\binom{n}{k} a^{k} b^{n-k}
$$

3. Compute

$$
\sum_{k=0}^{n}\binom{2 n}{2 k}
$$


Find the expected value (mathematical expectation) of the random variable \( X \) if the distribution function of this variable is known:

$$
F(x)= \begin{cases}0 & \text { if } x \leq 0 \\ x^{2} & \text { if } 0 < x \leq 1 \\ 1 & \text { if } x > 1\end{cases}
$$


Determine all prime numbers \(p, q,\) and \(r\) such that \(p+q^{2}=r^{4}\).

Solution: \boxed{p=7, q=3, r=2}


A string of 31 beads has one most expensive bead in the middle. Starting from one end, each subsequent bead is 3 yuan more expensive than the previous one until reaching the middle bead. Starting from the other end, each subsequent bead is 4 yuan more expensive than the previous one until reaching the middle bead. The total value of the string of beads is 2012 yuan. How much is the middle bead worth?


Pilyulkin, Guslya, and Toropyzhka are decorating a Christmas tree. It is known that Guslya hung twice as many ornaments on the tree as Pilyulkin did, Pilyulkin hung 15 fewer ornaments than Toropyzhka, and Toropyzhka hung twice as many ornaments as Guslya and Pilyulkin together. How many ornaments are decorating the tree?


In the expression \(1 \square 2 \square 3 \square 4\), each \(\square\) is to be replaced by either \(+\) or \(\times\). What is the largest value of all the expressions that can be obtained in this way?

A. 10
B. 14
C. 15
D. 24
E. 25


The desired three-digit number begins with the digit 1. If this digit is erased and then written as the last digit of the number, the new three-digit number obtained will be greater than the original number by $9 a^{1 / \lg a}$. Find this number.


Determine the functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) such that:

\[
f(f(x+1) + y - 1) = f(x) + y
\]

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


During New Year's Day, the class committee spent 730 yuan to buy each classmate a souvenir worth 17 yuan. After the purchase, there were 16 yuan left. How many students are there in the class?


An ancient Greek was born on January 7, 40 B.C., and died on January 7, 40 A.D. How many years did he live?


Given that $a$ and $b$ are non-zero vectors, and $a+3b$ is perpendicular to $7a-5b$, and $a-4b$ is perpendicular to $7a-2b$, find the angle between $a$ and $b$.


Calculate the arc lengths of the curves given by the equations in polar coordinates.

$$
\rho=6 \sin \varphi, 0 \leq \varphi \leq \frac{\pi}{3}
$$


Andrey and Grandpa Grisha went mushroom picking sometime between six and seven o'clock in the morning, at the moment when the clock hands were aligned. They returned home between twelve and one o'clock in the afternoon, at the moment when the clock hands were pointing in exactly opposite directions. How long did their mushroom "hunt" last?


By writing successive natural numbers, we get the sequence

$$
12345678910111213141516171819202122 \ldots
$$

What is the digit that is in the $2009^{th}$ position of this sequence?


Integrate the equation
\[ 2xy \, dx + x^2 \, dy = 0 \]

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


The solution of the inequality \(x^{2}+5x-2c \leq 0\) is \(d \leq x \leq 1\). Find \(d\).


Let \( O \) be the origin, \( A \) be a point moving on the parabola \( x = \frac{1}{4} y^2 + 1 \), and \( B \) be a point moving on the parabola \( y = x^2 + 4 \). Find the minimum area of the triangle \( \triangle OAB \).


Let \( n \) be a positive integer and \( d \) be the greatest common divisor of \( n^2 + 1 \) and \( (n+1)^2 + 1 \). Find all the possible values of \( d \). Justify your answer.

Solution: \boxed{1 \text{ and } 5}


Given the sequence \(\{a_n\}\) which satisfies the relation: \(a_1 = 1\) and \(a_{n+1} = a_n + a_n^2\) for \(n \in \mathbb{N}^*\). Let
\[
S_n = \frac{1}{(1+a_1)(1+a_2)\cdots(1+a_n)}, \quad T_n = \sum_{k=1}^n \frac{1}{1+a_k}
\]
Find the value of \(S_n + T_n\).


Three clients are at the hairdresser, each paying their bill at the cash register.

- The first client pays the same amount that is in the register and takes 10 reais as change.
- The second client performs the same operation as the first.
- The third client performs the same operation as the first two.

Find the initial amount of money in the cash register, knowing that at the end of the three operations, the cash register is empty.


Given a convex pentagon, Petya wrote down the sine values of all its angles in his notebook, and Vasya wrote down the cosine values of all its angles. It was found that among the numbers written by Petya, there are not four different values. Can all the numbers written by Vasya be different?

Solution: \boxed{\text{No}}


There is a fraction where its numerator plus 2 can be simplified to $\frac{4}{7}$, and its denominator minus 2 can be simplified to $\frac{14}{25}$. What is this fraction in its simplest form?


In the isosceles triangle \(ABC\) with equal sides \(AB\) and \(CB\), the sides \(AB\) and \(CB\) are extended beyond point \(B\) to points \(D\) and \(E\) respectively. The segments \(AE\), \(ED\), and \(DC\) are equal to each other, and \(\angle BED \neq \angle BDE\). Find the angle \(ABE\).


Vovochka approached a game machine, on the screen of which the number 0 was displayed. The rules of the game stated: "The screen shows the number of points. If you insert a 1-ruble coin, the number of points increases by 1. If you insert a 2-ruble coin, the number of points doubles. If you score 50 points, the machine gives a prize. If the resulting number is greater than 50, then all the points scored are lost." What is the minimum amount of rubles Vovochka needs to get the prize? Answer: 11 rubles.

Solution: \boxed{11 \text{ rubles}}


Teacher Li brought a stack of craft paper, just enough to evenly distribute among 24 students. Later, 8 more students arrived, and each student received 2 fewer sheets than originally planned. How many sheets of craft paper did Teacher Li bring in total?


Let \( P_1 \) and \( P_2 \) be two points on a plane. Define \( P_{2k+1} \) as the point symmetric to \( P_{2k} \) with respect to \( P_1 \), and \( P_{2k+2} \) as the point symmetric to \( P_{2k+1} \) with respect to \( P_2 \), where \( k \in \mathbf{Z}_{+} \). If the distance \( \left| P_1 P_2 \right| = 1 \), find the distance \( \left| P_{2017} P_{2018} \right| \).


A piece of an alloy of copper and zinc weighing 36 kg contains 45% copper. How much copper must be added to this piece to ensure that the new alloy contains 60% copper?

Solution: \boxed{13.5 \text{ kg}}


The sum of the first four terms of an arithmetic progression, as well as the sum of the first seven terms, are natural numbers. Furthermore, its first term \(a_1\) satisfies the inequality \(a_1 \leq \frac{2}{3}\). What is the greatest value that \(a_1\) can take?


Linlin fills a glass with pure milk. The first time she drinks $\frac{1}{3}$ of the milk, then fills the glass with soy milk and stirs well. The second time, she drinks $\frac{1}{3}$ of the mixture, refills the glass with soy milk, and stirs well again. She repeats this process. What fraction of the total amount of pure milk will Linlin have drunk after the fourth time?


A math competition problem has probabilities of being solved independently by person \( A \), \( B \), and \( C \) as \( \frac{1}{a} \), \( \frac{1}{b} \), and \( \frac{1}{c} \) respectively, where \( a \), \( b \), and \( c \) are positive integers less than 10. When \( A \), \( B \), and \( C \) work on the problem simultaneously and independently, the probability that exactly one of them solves the problem is \( \frac{7}{15} \). Determine the probability that none of the three persons solve the problem.


Masha and Olya bought many identical pens for the new school year. It is known that one pen costs an integer number of rubles greater than 10. Masha bought pens for exactly 357 rubles, and Olya bought pens for exactly 441 rubles. How many pens did they buy in total?


Let \( M \) be the point of intersection of the diagonals of a convex quadrilateral \( A B C D \), in which the sides \( A B \), \( A D \), and \( B C \) are equal to each other.

Find the angle \( \angle C M D \), given that \( D M = M C \) and \( \angle C A B \neq \angle D B A \).


Determine the smallest natural number \( k \) such that for any \( a \in [0,1] \) and any \( n \in \mathbf{N}^{*} \), it always holds that \( a^{k} (1-a)^{n} < \frac{1}{(n+1)^{3}} \).


The function \( f(x) \) is defined on the positive half-axis and takes only positive values. It is known that \( f(1) + f(2) = 10 \) and \( f(a+b) = f(a) + f(b) + 2 \sqrt{f(a) f(b)} \) for any \( a \) and \( b \). Find \( f\left(2^{2011}\right) \).


Given triangle \(ABC\) with \(\angle A = \alpha\) and \(\angle B = \beta\). The lines \(O_1O_2\), \(O_2O_3\), and \(O_1O_3\) are the bisectors of the external angles of triangle \(ABC\), as shown in the figure. Point \(O\) is the center of the inscribed circle of triangle \(ABC\). Find the angle between the lines \(O_1O_2\) and \(OO_3\).


How can you measure 8 liters of water while being near a river, with two buckets having capacities of 10 liters and 6 liters? (You need to have 8 liters of water in one bucket).


Initially, a natural number was displayed on the calculator screen. Each time, Tanya added to the current number \( n \) a natural number by which \( n \) was not divisible. For example, if the number on the screen was 10, Tanya could add 7 to get 17.

Tanya performed such an operation five times, and the number 100 appeared on the screen. What is the largest possible initial number for which this could happen?


Find a natural number \( A \) such that if \( A \) is appended to itself on the right, the resulting number is a perfect square.


Given the function \( f(x)=\frac{a x}{2 x+3} \), if \( f(f(x))=x \) is always true, then the value of the real number \( a \) is ____ .


There is a sequence of numbers: \(1, 3, 8, 22, 60, 164, 448, \cdots\). The first number is 1, the second number is 3, and from the third number onwards, each number is exactly twice the sum of the previous two numbers. What is the remainder when the 2000th number in this sequence is divided by 9?


Determine the pressure force of water on a vertical parabolic segment, whose base is $4 \mathrm{m}$ and is located at the water surface, and its vertex is at a depth of $4 \mathrm{м}$.

Solution: \boxed{167424 \, \text{N}}


My age is a two-digit number that is a power of 5. My cousin's age is a two-digit number that is a power of 2. The sum of the digits of our ages is an odd number. What is the product of the digits of our ages?


If the number \(100^{10}\) is written as the sum of tens \((10+10+10+\ldots)\), how many terms will there be?


An unfair coin comes up heads with probability \(\frac{4}{7}\) and tails with probability \(\frac{3}{7}\). Aerith and Bob take turns flipping the coin until one of them flips tails, with Aerith going first. What is the probability that Aerith wins the game?


Find all such triple natural numbers \( m, n, \) and \( l \) that satisfy the following conditions:
\[
m + n = (\gcd(m, n))^2, \\
m + l = (\gcd(m, l))^2, \\
n + l = (\gcd(n, l))^2.
\]

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


a) A championship will have 7 competitors. Each of them will play exactly one game against every other competitor. What is the total number of games in the championship?

b) A championship will have \( n \) competitors, and each of them will play exactly one game against every other competitor. Verify that the total number of games is \( \frac{n \cdot (n-1)}{2} \).

c) A competitor \( A \) is called the master of the championship if every other player either lost to \( A \) or lost to someone who lost to \( A \). Verify that every championship where each player plays exactly once against every other and there are no ties always admits a master.

d) João and Maria were in a tournament where each athlete played against every other exactly once. However, Maria had a health issue and left the tournament after playing 10 games, and João had to leave after his first game. At the end of the day, a total of 55 games were counted. Did João play his game against Maria?

Solution: \boxed{Yes}


A class of 10 students took a math test. Each problem was solved by exactly 7 of the students. If the first nine students each solved 4 problems, how many problems did the tenth student solve?


What is the value of \(\frac{\left(1^{2}+1\right)\left(2^{2}+1\right)\left(3^{2}+1\right)}{\left(2^{2}-1\right)\left(3^{2}-1\right)\left(4^{2}-1\right)}\)? Give your answer in its simplest form.


In a reading room, there are six textbooks on probability theory, three of which are hardcover. A librarian randomly takes two textbooks. Find the probability that both textbooks are hardcover.


Given an integer \( n \geq 2 \), find the smallest positive real number \( c \) such that for any complex numbers \( z_1, z_2, \ldots, z_n \), the following inequality holds:
$$
\left|\sum_{i=1}^{n} z_{i}\right| + c \sum_{1 \leq i < j \leq n} \left|z_{i} - z_{j}\right| \geq \sum_{i=1}^{n} \left|z_{i}\right|.
$$


Each face of a tetrahedron is a triangle with sides \(a\), \(b\), \(c\) and the tetrahedron has circumradius 1. Find \(a^2 + b^2 + c^2\).


On the sides \( AB \) and \( CD \) of an inscribed quadrilateral \( ABCD \), which is not a trapezoid, points \( P \) and \( R \) are marked, respectively, such that \( AP:PB = CR:RD \). On the segment \( PR \), a point \( Q \) is chosen such that \( PQ:QR = AB:CD \). Find the ratio of the areas of triangles \( AQD \) and \( BQC \) if it is known that \( AD = x \) and \( BC = y \).

Solution: \boxed{\frac{x}{y}}


Enter any 3-digit number into a calculator. Then, enter the same number again, thus obtaining a 6-digit number in the form $abcabc$. Divide this number by 7, then divide the result by 11, and finally, divide the obtained number by 13. What happened? Why did you get this result?

Solution: \boxed{abc}


Given segments \(a\) and \(b\), construct a segment \(\sqrt{a b}\) using a compass and straightedge.

Solution: \boxed{\sqrt{ab}}


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


Before the geometry lesson, the teacher wrote on the board the values of all the angles (in degrees) of a certain convex polygon. However, the classroom helpers erased one of the numbers written on the board. When the lesson began, it turned out that the sum of the remaining numbers was 1703. What number did the helpers erase?


There are chickens and rabbits in the same cage, with a total of 40 heads. The number of rabbit feet is 8 less than 10 times the number of chicken feet. How many rabbits are there?


Find all the functions \( f: \mathbb{Q} \rightarrow \mathbb{Q} \) (where \( \mathbb{Q} \) is the set of rational numbers) that satisfy \( f(1)=2 \) and
\[ f(x y) \equiv f(x) f(y) - f(x + y) + 1, \quad \text{for all} \ x, y \in \mathbb{Q}. \]

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


Find the number which, when multiplied by 3, then divided by 5, increased by 6, after which the square root is taken, one is subtracted, and the result is squared, gives 4.


A farmer sold domestic rabbits. By the end of the market, he sold exactly one-tenth as many rabbits as the price per rabbit in forints. He then distributed the revenue between his two sons. Starting with the older son, the boys alternately received one-hundred forint bills, but at the end, the younger son received only a few ten-forint bills. The father then gave him his pocket knife and said that this made their shares equal in value. How much was the pocket knife worth?

Solution: \boxed{40 \text{ forints}}


The length of a crocodile from head to tail is three times smaller than ten ken, and from tail to head it is three ken and two shaku. It is known that one shaku is equal to 30 cm. Find the length of the crocodile in meters. (Ken and shaku are Japanese units of length.)

Solution: \boxed{6 \, \text{m}}


The area of a rectangle remains unchanged when either its length is increased by 6 units and width decreased by 2 units, or its length decreased by 12 units and its width increased by 6 units. If the perimeter of the original rectangle is \( x \) units, find the value of \( x \).


In triangle \( A B C \), angle \( B \) equals \( 45^\circ \) and angle \( C \) equals \( 30^\circ \). Circles are constructed on the medians \( B M \) and \( C N \) as diameters, intersecting at points \( P \) and \( Q \). The chord \( P Q \) intersects side \( B C \) at point \( D \). Find the ratio of segments \( B D \) to \( D C \).


Let \( m, n > 2 \) be integers. One of the angles of a regular \( n \)-gon is dissected into \( m \) angles of equal size by \( (m-1) \) rays. If each of these rays intersects the polygon again at one of its vertices, we say \( n \) is \( m \)-cut. Compute the smallest positive integer \( n \) that is both 3-cut and 4-cut.


A circle is circumscribed around quadrilateral \(A B C D\) with center at point \(O\). The diagonals of the quadrilateral are perpendicular. Find the length of side \(B C\) if the distance from point \(O\) to side \(A D\) is equal to 1.

Solution: \boxed{BC = 2}


Find the differentials of the functions:
1) \( y = x^3 - 3^x \);
2) \( F(\varphi) = \cos \frac{\varphi}{3} + \sin \frac{3}{\varphi} \);
3) \( z = \ln \left(1 + e^{10x}\right) + \operatorname{arcctg} e^{5x} \); calculate \(\left.dz\right|_{x=0; dx=0.1}\).


Find all triples \((x, y, z)\) of positive integers satisfying the system of equations:

\[
\left\{
\begin{array}{l}
x^{2}=2(y+z) \\
x^{6}=y^{6}+z^{6}+31\left(y^{2}+z^{2}\right)
\end{array}
\right.
\]

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


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

\[
\begin{cases}
x^{2}+xy+y^{2}=27 \\
y^{2}+yz+z^{2}=16 \\
z^{2}+xz+x^{2}=43
\end{cases}
\]

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


What is the maximum number of $1 \times 1 \times 4$ parallelepipeds that can be placed inside a $6 \times 6 \times 6$ cube so that their faces are parallel to the corresponding faces of the cube?


Calculate the volumes of the solids bounded by the following surfaces.

$$
\frac{x^{2}}{3}+\frac{y^{2}}{16}=1,
z=y \sqrt{3},
z=0\ (y \geq 0)
$$


Find the smallest natural number \( m \) such that the expression \( 148^{n} + m \cdot 141^{n} \) is divisible by 2023 for any odd natural \( n \).


The sides of the base of a brick are 28 cm and 9 cm, and its height is 6 cm. A snail crawls rectilinearly along the faces of the brick from one vertex of the lower base to the opposite vertex of the upper base. The horizontal and vertical components of its speed $v_{x}$ and $v_{y}$ are related by the equation $v_{x}^{2}+4 v_{y}^{2}=1$ (for example, on the upper face, $v_{y}=0$ cm/min, hence $v_{x}=v=1$ cm/min). What is the minimum time the snail can spend on its journey?

Solution: \boxed{35 \text{ min}}


A robot moves along straight segments, making a turn every minute by 90 degrees either to the right or left (the time taken to turn is negligible). The robot travels 10 meters in one minute. What is the minimum distance from the initial position the robot can be after 9 minutes of movement, given that the robot did not turn during the first minute?

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


Determine the increment of the argument and the function \( y = x^2 \) if the argument \( x \) changes from 2 to 2.5.


For which integers \( n \geq 3 \) does there exist a regular \( n \)-gon in the plane such that all its vertices have integer coordinates in a rectangular coordinate system?


A fisherman cast a net into a pond and caught 30 fish. Marking each fish with a tag, he released the catch back into the pond. The next day, the fisherman cast the net again and caught 40 fish, among which two were tagged. Using this information, approximately estimate the number of fish in the pond.


Two cars, Car A and Car B, are traveling from city A to city B at a speed of 55 km/hour. At 10 AM, the distance traveled by Car A is 5 times the distance traveled by Car B. At 12 noon, the distance traveled by Car A is 3 times the distance traveled by Car B. How many hours later did Car B start compared to Car A?

Solution: \boxed{8 \text{ hours}}


Each side of the triangle $ABC$ is divided into $p$ equal parts, where $p$ is a prime number. Then, from each side, one division point is connected to the opposite vertex of the triangle such that these three lines intersect at a single point. Determine the possible values of $p$.


In a country with 100 cities, 30 are in the mountainous region, and 70 are in the plains. Over three years, air routes were established between the cities. Each year, 50 new routes were inaugurated: all the cities were randomly divided into 50 pairs, and a route was opened between the cities in each pair. After three years, it was found that out of the 150 routes, exactly 21 connect pairs of mountainous cities. How many routes connect pairs of plains cities?


Let \( R \) be the region in the Cartesian plane of points \((x, y)\) satisfying \( x \geq 0, y \geq 0 \), and \( x + y + \lfloor x \rfloor + \lfloor y \rfloor \leq 5 \). Determine the area of \( R \).


Find the smallest natural number ending with the digit 2, which doubles if this digit is moved to the beginning.


A parallelogram is circumscribed around a circle with radius \( R \). The area of the quadrilateral formed by the points of tangency of the circle with the parallelogram is \( S \). Find the sides of the parallelogram.


A diagonal of a 20-gon divides it into a 14-gon and an 8-gon (see the figure). How many of the remaining diagonals of the 20-gon intersect the given diagonal? The vertex of the 14-gon is not considered an intersection.


Calculate

$$
\frac{1 \times 2 \times 4 + 2 \times 4 \times 8 + 3 \times 6 \times 12 + 4 \times 8 \times 16}{1 \times 3 \times 9 + 2 \times 6 \times 18 + 3 \times 9 \times 27 + 4 \times 12 \times 36}
$$

Only a numerical response is expected here. The answer must be given as a fraction in its irreducible form (i.e., in the form $\frac{a}{b}$ where $a$ and $b$ are integers with no common divisors).


Find the limit of the function

$$
\lim _{x \rightarrow \infty} \frac{3 x^{4}-2 x^{3}+6 x^{2}+x-2}{9 x^{4}+5 x^{3}+7 x^{2}+x+1}
$$


Given \( p(x) = a x^{3} + b x^{2} + c x + d \) is a cubic polynomial, and it satisfies
\[ p\left(\frac{1}{2}\right) + p\left(-\frac{1}{2}\right) = 1000 p(0). \]

Let \( x_{1}, x_{2}, x_{3} \) be the three roots of \( p(x) = 0 \). Find the value of \( \frac{1}{x_{1} x_{2}} + \frac{1}{x_{2} x_{3}} + \frac{1}{x_{1} x_{3}} \).


In a convex quadrilateral \( EFGH \), the vertices \( E, F, G, H \) lie on the sides \( AB, BC, CD, DA \) of another convex quadrilateral \( ABCD \), respectively, and satisfy the condition:
\[
\frac{AE}{EB} \cdot \frac{BF}{FC} \cdot \frac{CG}{GD} \cdot \frac{DH}{HA}=1
\]
Moreover, the points \( A, B, C, D \) lie on the sides \( H_1E_1, E_1F_1, F_1G_1, G_1H_1 \) of a third convex quadrilateral \( E_1F_1G_1H_1 \), respectively, such that \( E_1F_1 \parallel EF \), \( F_1G_1 \parallel FG \), \( G_1H_1 \parallel GH \), and \( H_1E_1 \parallel HE \). Given that:
\[
\frac{E_1A}{AH_1}=\lambda
\]
find the value of \( \frac{F_1C}{CG_1} \).

Solution: \boxed{\lambda}


Katya gave Vasya a series of books to read. He promised to return them in a month but did not. Katya asked her brother Kolya to talk to Vasya to get the books back. Kolya agreed, but in return, Katya must give him one of the books. In this example, Kolya is:
1) a financial pyramid
2) a collection agency
3) a bank
4) an insurance company


Petya bought himself football shorts in the store.

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

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


Given \( a+3=3a, b+4=4b, c+5=5c, d+6=6d \), find \( a \times b \times c \times d \).


Find the minimum value of \((s+5-3|\cos t|)^{2}+(s-2|\sin t|)^{2}\) as \(s\) and \(t\) range over all real numbers.


Two players play a game where they are each given 10 indistinguishable units that must be distributed across three locations. (Units cannot be split.) At each location, a player wins at that location if the number of units they placed there is at least 2 more than the units of the other player. If both players distribute their units randomly (i.e., there is an equal probability of them distributing their units for any attainable distribution across the 3 locations), the probability that at least one location is won by one of the players can be expressed as \(\frac{a}{b}\), where \(a, b\) are relatively prime positive integers. Compute \(100a + b\).


Given \( x \in \mathbf{R} \) and the inequality \( 2x^2 - a\sqrt{x^2+1} + 3 \geqslant 0 \) always holds, find the maximum value of the real number \( a \).