The number $\overline{x y z t}$ is a perfect square such that the number $\overline{t z y x}$ is also a perfect square, and the quotient of the numbers $\overline{x y z t}$ and $\overline{t z y x}$ is also a perfect square. Determine the number $\overline{x y z t}$. (The overline indicates that the number is written in the decimal system.)


Given points $A(a-1, a)$ and $B(a, a+1)$, and a moving point $P$ such that its distance to the point $M(1,0)$ is greater by 1 than its distance to the $y$-axis. The locus of $P$ is a curve $C$. The line segment $AB$ intersects curve $C$ at exactly one point. Determine the range of the value of $a$.

Solution: \boxed{[-1, 0] \cup [1, 2]}


The complex numbers \( z_1, z_2, \cdots, z_{100} \) satisfy: \( z_1 = 3 + 2i \), \( z_{n+1} = \overline{z_n} \cdot i^n \) for \( n = 1, 2, \cdots, 99 \) (where \( i \) is the imaginary unit). Find the value of \( z_{99} + z_{100} \).

Solution: \boxed{-5 + 5i}


Is there a natural number \( n \) such that the number \( 6n^2 + 5n \) is a power of 2?

Solution: \boxed{\text{No}}


Compute the positive integer less than 1000 which has exactly 29 positive proper divisors. (Here we refer to positive integer divisors other than the number itself.)


Pedro wants to paint a cube-shaped box in such a way that faces sharing a common edge are painted in different colors. Calculate the minimum number of colors needed to paint the box in this manner.


As shown in the figure, $\angle 1$ is equal to 100 degrees, $\angle 2$ is equal to 60 degrees, and $\angle 3$ is equal to 90 degrees. Find $\angle 4$.


Find the sum of the squares of the distances from a point \( M \), taken on the diameter of some circle, to the ends of any chord parallel to this diameter, if the radius of the circle is \( R \) and the distance from point \( M \) to the center of the circle is \( a \).

Solution: \boxed{2(a^2 + R^2)}


In quadrilateral \( \square ABCD \), point \( M \) lies on diagonal \( BD \) with \( MD = 3BM \). Let \( AM \) intersect \( BC \) at point \( N \). Find the value of \( \frac{S_{\triangle MND}}{S_{\square ABCD}} \).


There are 1993 boxes containing balls arranged in a line from left to right. If the leftmost box contains 7 balls, and every four consecutive boxes contain a total of 30 balls, then how many balls are in the rightmost box?


Two circles with radii 2 and 3 touch each other externally at point $A$. Their common tangent passing through point $A$ intersects their other two common tangents at points $B$ and $C$. Find $BC$.

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


Three frogs on a swamp jumped in sequence. Each one landed exactly in the middle of the segment between the other two. The jump length of the second frog is 60 cm. Find the jump length of the third frog.

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


Given: Three vertices \( A, B, \) and \( C \) of a square are on the parabola \( y = x^2 \).
Find: The minimum possible area of such a square.


Calculate the double integrals over the regions bounded by the specified lines:

a) \(\iint x^{2} y \, dx \, dy\); \(y=0\); \(y=\sqrt{2ax - x^{2}}\);

b) \(\iint_{\Omega} \sin(x+y) \, dx \, dy\); \(y=0\); \(y=x\); \(x+y=\frac{\pi}{2}\);

c) \(\iint_{\Omega} x^{2}(y-x) \, dx \, dy\); \(x=y^{2}\); \(y=x^{2}\).


On the board, the natural number \( N \) was written nine times (one below the other). Petya added a non-zero digit to the left or right of each of the 9 numbers; all added digits are distinct. What is the largest possible number of prime numbers that could result from these 9 new numbers?

(I. Efremov)


As shown in the figure, in quadrilateral $ABCD$, the diagonals $AC$ and $BD$ are perpendicular. The lengths of the four sides are $AB = 6$, $BC = m$, $CD = 8$, and $DA = n$. Find the value of $m^{2} + n^{2}$.


In the coordinate plane, points with integer values for both coordinates are called lattice points. For a certain lattice point \( P \) and a positive number \( d \), if there are exactly \( k(>0) \) distinct lattice points at a distance \( d \) from \( P \), the range of values for \( k \) is denoted as \( \left\{k_1, k_2, \cdots\right\} \) where \( 0<k_1<k_2<\cdots \). What is \( k_2 \)?


In the vertices of a regular 300-gon, the numbers from 1 to 300 are arranged in some order, each number appearing exactly once. It turns out that for each number \(a\), there are as many numbers smaller than \(a\) among the 15 closest numbers to it clockwise as there are among the 15 closest numbers to it counterclockwise. A number that is larger than all 30 numbers closest to it is called "huge." What is the smallest possible number of huge numbers?


Quadrilateral \(ABCD\) is inscribed in a circle. At point \(C\), a tangent \(\ell\) to this circle is drawn. Circle \(\omega\) passes through points \(A\) and \(B\) and touches the line \(\ell\) at point \(P\). Line \(PB\) intersects segment \(CD\) at point \(Q\). Find the ratio \(\frac{BC}{CQ}\), given that \(B\) is tangent to circle \(\omega\).


Yiyi lives at Huashi and is planning to ride a shared bicycle to the park. There are 3 stations between Huashi station and the park, and each station has shared bicycles of four different colors: yellow, orange, red, and green. If Yiyi does not want to ride a bicycle of the same color at two consecutive stations, how many different riding strategies are there for Yiyi from home to the park?


In a city, there are 9 bus stops and several buses. Any two buses have at most one common stop. Each bus stops at exactly three stops. What is the maximum number of buses that can be in the city?


For a point \(P=\left(a, a^{2}\right)\) in the coordinate plane, let \(\ell(P)\) denote the line passing through \(P\) with slope \(2a\). Consider the set of triangles with vertices of the form \(P_{1}=\left(a_{1}, a_{1}^{2}\right)\), \(P_{2}=\left(a_{2}, a_{2}^{2}\right)\), \(P_{3}=\left(a_{3}, a_{3}^{2}\right)\), such that the intersection of the lines \(\ell\left(P_{1}\right)\), \(\ell\left(P_{2}\right)\), \(\ell\left(P_{3}\right)\) form an equilateral triangle \(\Delta\). Find the locus of the center of \(\Delta\) as \(P_{1} P_{2} P_{3}\) ranges over all such triangles.

Solution: \boxed{y = -\frac{1}{4}}


In triangle \(ABC\) with area 51, points \(D\) and \(E\) trisect \(AB\), and points \(F\) and \(G\) trisect \(BC\). Find the largest possible area of quadrilateral \(DEFG\).


In the sequence of positive integers \(1, 2, 3, 4, \cdots\), remove multiples of 3 and 4, but keep all multiples of 5 (for instance, 15 and 120 should not be removed). The remaining numbers form a new sequence: \(a_{1} = 1, a_{2} = 2, a_{3} = 5, a_{4} = 7, \cdots\). Find \(a_{1999}\).


In the triangle \( ABC \) the points \( M \) and \( N \) lie on the side \( AB \) such that \( AN = AC \) and \( BM = BC \). We know that \(\angle MCN = 43^\circ\). Find the size in degrees of \(\angle ACB\).


Given that \( F_1 \) and \( F_2 \) are the left and right foci of the ellipse \( C: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \) where \( a > b > 0 \), and \( P \) is a point on the ellipse \( C \). The incenter of triangle \( \triangle F_{1}PF_{2} \) is \( I \). If there exists a real number \( \lambda \) such that:
$$
(1+\lambda) \overrightarrow{PF_{1}} + (1-\lambda) \overrightarrow{PF_{2}} = 3 \overrightarrow{PI},
$$
then the eccentricity of the ellipse \( C \) is $\qquad$ .


The work was done together by team $C$ and team $D$. How many days did they take to complete it, given that team $C$ alone would have finished the work $c$ days later, and team $D$ alone would have required $d$ times more time? Apply the result to the cases $c=10, d=2$ and $c=8, d=3$.


How many different integral solutions \((x, y)\) does \(3|x| + 5|y| = 100\) have?


Small and Big islands have a rectangular shape and are divided into rectangular counties. Each county has a road along one of the diagonals. On each island, these roads form a closed path that does not pass through any point more than once. Here's how the Small Island, consisting of 6 counties, is arranged (see Figure 1). Draw how the Big Island, which has an odd number of counties, could be arranged. How many counties did you get?


Arsenius has 2018 buckets, each containing 1, 2, 3, ... 2017, 2018 liters of water respectively. Arsenius is allowed to take any two buckets and pour into the second bucket exactly as much water as is already in the second bucket from the first bucket. Can Arsenius collect all the water into one bucket? All buckets are large enough to hold all the water they get.

Solution: \boxed{\text{No}}


Four students, Andrey, Vanya, Dima, and Sasha, participated in a school chess tournament. Each student played two games against each of the other participants. Each win awarded 1 point, a draw awarded 0.5 points, and a loss awarded 0 points.

Given the following conditions at the end of the tournament:

- Each student scored a different number of points.
- Andrey secured first place, Dima secured second, Vanya secured third, and Sasha secured fourth.
- Andrey won the same number of games as Sasha.

How many points did each student score?

Solution: \boxed{4, 3.5, 2.5, 2}


A plane, perpendicular to an edge of a regular tetrahedron, passes through a point that divides this edge in the ratio 1:4. Find the ratio of the volumes of the resulting parts of the tetrahedron.


The height of a right triangle dropped to the hypotenuse is 1, one of the acute angles is $15^{\circ}$.

Find the hypotenuse.


A natural number, which does not end in zero, had one of its digits replaced with zero (if it was the leading digit, it was simply erased). As a result, the number became 9 times smaller. How many such numbers exist for which this is possible?


In front of the elevator doors, there are people with masses of 50, 51, 55, 57, 58, 59, 60, 63, 75, and 140 kg. The elevator's load capacity is 180 kg. What is the minimum number of trips needed to transport all the people?


Given that the equation \( |x| - \frac{4}{x} = \frac{3|x|}{x} \) has \( k \) distinct real root(s), find the value of \( k \).


In a convex quadrilateral \(ABCD\), \(\overrightarrow{BC} = 2 \overrightarrow{AD}\). Point \(P\) is a point in the plane of the quadrilateral such that \(\overrightarrow{PA} + 2020 \overrightarrow{PB} + \overrightarrow{PC} + 2020 \overrightarrow{PD} = \mathbf{0}\). Let \(s\) and \(t\) be the areas of quadrilateral \(ABCD\) and triangle \(PAB\), respectively. Then \(\frac{t}{s} =\) ______.


Find the smallest positive integer \( n \) such that in any two-coloring of the complete graph \( K_{n} \), there exist 3 monochromatic triangles that are pairwise edge-disjoint.


Find the largest positive number \( x \) such that
\[
\left(2 x^{3} - x^{2} - x + 1\right)^{1 + \frac{1}{2 x + 1}} = 1.
\]


Five runners ran a relay race. If the first runner had run twice as fast, they would have spent 5% less time. If the second runner had run twice as fast, they would have spent 10% less time. If the third runner had run twice as fast, they would have spent 12% less time. If the fourth runner had run twice as fast, they would have spent 15% less time. By what percentage less time would they have spent if the fifth runner had run twice as fast?


Find the value of \(a + b + c + d + e\) given the system of equations:
\[
\begin{array}{c}
3a + 2b + 4d = 10 \\
6a + 5b + 4c + 3d + 2e = 8 \\
a + b + 2c + 5e = 3 \\
2c + 3d + 3e = 4 \\
a + 2b + 3c + d = 7
\end{array}
\]


Given a cube \( A B C D A_1 B_1 C_1 D_1 \) with edge length \( a \), find the distance between the lines \( A A_1 \) and \( B D_1 \) and construct their common perpendicular.


It is known that the sum of the absolute values of the pairwise differences of five nonnegative numbers is equal to one. Find the smallest possible sum of these numbers.


In how many different ways can integers \( a, b, c \in [1, 100] \) be chosen such that the points with coordinates \( A(-1, a), B(0, b) \), and \( C(1, c) \) form a right triangle?


Let \( m \) and \( n \) be two integers such that \( n \) divides \( m(n+1) \). Show that \( n \) divides \( m \).

Solution: \boxed{ n \mid m }


Determine the largest positive integer \( n \) such that there exist positive integers \( x, y, z \) so that
\[ n^{2} = x^{2} + y^{2} + z^{2} + 2xy + 2yz + 2zx + 3x + 3y + 3z - 6 \]


In triangle $ABC$, the altitude from vertex $A$ is the harmonic mean of the two segments into which it divides side $BC$. What is $\operatorname{tg} \beta + \operatorname{tg} \gamma$ (where $\beta$ and $\gamma$ are the angles of the triangle)?


The parabola with the equation \( y = 25 - x^2 \) intersects the \( x \)-axis at points \( A \) and \( B \).

(a) Determine the length of \( AB \).

(b) Rectangle \( ABCD \) is formed such that \( C \) and \( D \) are below the \( x \)-axis and \( BD = 26 \). Determine the length of \( BC \).

(c) If \( CD \) is extended in both directions, it meets the parabola at points \( E \) and \( F \). Determine the length of \( EF \).


Suppose \( a \) and \( b \) are the roots of \( x^{2}+x \sin \alpha+1=0 \) while \( c \) and \( d \) are the roots of the equation \( x^{2}+x \cos \alpha-1=0 \). Find the value of \( \frac{1}{a^{2}}+\frac{1}{b^{2}}+\frac{1}{c^{2}}+\frac{1}{d^{2}} \).


Given two arbitrary positive integers \( n \) and \( k \), let \( f(n, k) \) denote the number of unit squares that one of the diagonals of an \( n \times k \) grid rectangle passes through. How many such pairs \( (n, k) \) are there where \( n \geq k \) and \( f(n, k) = 2018 \)?


Integrate the equation

$$
x \, dy = (x + y) \, dx
$$

and find the particular solution that satisfies the initial condition $y = 2$ when $x = -1$.

Solution: \boxed{y = x \ln |x| - 2x}


The hedgehogs collected 65 mushrooms and divided them so that each received at least one mushroom, but no two hedgehogs had the same number of mushrooms. What is the maximum number of hedgehogs that could be?


Dr. Aibolit gave 2006 miraculous pills to four sick animals. The rhinoceros received one more than the crocodile, the hippopotamus received one more than the rhinoceros, and the elephant received one more than the hippopotamus. How many pills did the elephant get?


The probability that a purchased light bulb will work is 0.95.

How many light bulbs need to be bought so that there is a probability of 0.99 that at least five of them will be working?

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


Is there a 6-digit number that becomes 6 times larger when its last three digits are moved to the beginning of the number, keeping the order of the digits?


Point \( P \) is inside triangle \( \triangle ABC \). Line segments \( APD \), \( BPE \), and \( CPF \) are drawn such that \( D \) is on \( BC \), \( E \) is on \( AC \), and \( F \) is on \( AB \). Given that \( AP = 6 \), \( BP = 9 \), \( PD = 6 \), \( PE = 3 \), and \( CF = 20 \), find the area of \( \triangle ABC \).


Let \( f(x) = x^2 + 3x + 2 \). Compute the expression

\[
\left(1 - \frac{2}{f(1)}\right) \cdot\left(1 - \frac{2}{f(2)}\right) \cdot\left(1 - \frac{2}{f(3)}\right) \cdots \left(1 - \frac{2}{f(2019)}\right).
\]


Given the parabola $y = x^2$ and a point $M(1,1)$ on it as the vertex of right-angled triangles inscribed in the parabola, find the coordinates of the intersection point $E$ of the line segments $AB$ and $CD$.


We have one $10 \mathrm{Ft}$ coin and several (at least four) $20 \mathrm{Ft}$ coins in our pocket. We successively draw one coin at a time at random until the sum of the drawn coins becomes a positive integer multiple of $30 \mathrm{Ft}$. On average, how many draws are needed to achieve this?


Given ten 5-element sets \( A_{1}, A_{2}, \cdots, A_{10} \), where the intersection of any two sets contains at least two elements. Let \( A=\bigcup_{i=1}^{10} A_{i}=\left\{x_{1}, x_{2}, \cdots, x_{n}\right\} \). For any \( x_{i} \in A \), let \( k_{i}(i=1,2, \cdots, n) \) be the number of sets among \( A_{1}, A_{2}, \cdots, A_{10} \) that contain the element \( x_{i} \). Define \( m = \max \left\{k_{1}, k_{2}, \cdots, k_{n}\right\} \). Determine the minimum value of \( m \).


The base of the rectangular parallelepiped $\boldsymbol{A}BCD A_{1}B_{1}C_{1}D_{1}$ is a square $ABCD$. Find the maximum possible value of the angle between the line $B D_{1}$ and the plane $BDC_{1}$.

Solution: \boxed{\arcsin \frac{1}{3}}


There are four weights of different masses. Katya weighs the weights in pairs. The results are 1800, 1970, 2110, 2330, and 2500 grams. How many grams does the sixth weighing result in?


Let \( P(n) \) be the number of permutations \(\left(a_{1}, \ldots, a_{n}\right)\) of the numbers \((1,2, \ldots, n)\) for which \( k a_{k} \) is a perfect square for all \( 1 \leq k \leq n \). Find with proof the smallest \( n \) such that \( P(n) \) is a multiple of 2010.


A circle inscribed in an angle with vertex \( O \) touches its sides at points \( A \) and \( B \). The ray \( OX \) intersects this circle at points \( C \) and \( D \), with \( OC = CD = 1 \). If \( M \) is the intersection point of the ray \( OX \) and the segment \( AB \), what is the length of the segment \( OM \)?


Given distinct real numbers \(a_{1}, a_{2}, a_{3}\) and \(b\). It turns out that the equation \((x-a_{1})(x-a_{2})(x-a_{3}) = b\) has three distinct real roots \(c_{1}, c_{2}, c_{3}\). Find the roots of the equation \((x+c_{1})(x+c_{2})(x+c_{3}) = b\).

Solution: \boxed{-a_1, -a_2, -a_3}


There are 29 students in a class: some are honor students who always tell the truth, and some are troublemakers who always lie.

All the students in this class sat at a round table.

- Several students said: "There is exactly one troublemaker next to me."
- All other students said: "There are exactly two troublemakers next to me."

What is the minimum number of troublemakers that can be in the class?


The numerator and denominator of a fraction are natural numbers that add up to 101. It is known that the fraction does not exceed \( \frac{1}{3} \).

Determine the greatest possible value of such a fraction.


Expand the function

$$
f(z)=\frac{z}{z^{2}-2 z-3}
$$

into a Taylor series in the vicinity of the point \( z_{0}=0 \), using expansion (12), and find the radius of convergence of the series.

Solution: \boxed{R = 1}


In triangle $ABC$, the angles $A$ and $C$ at the base are $20^{\circ}$ and $40^{\circ}$, respectively. It is given that $AC - AB = 5$ cm. Find the length of the angle bisector of angle $B$.

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


Let \( a, b, c \) be the side lengths of a right triangle, with \( a \leqslant b < c \). Determine the maximum constant \( k \) such that \( a^{2}(b+c) + b^{2}(c+a) + c^{2}(a+b) \geqslant k a b c \) holds for all right triangles, and identify when equality occurs.

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


There is a ten-digit number. From left to right:
- Its first digit indicates the number of zeros in the ten-digit number.
- Its second digit indicates the number of ones in the ten-digit number.
- Its third digit indicates the number of twos in the ten-digit number.
- ...
- Its tenth digit indicates the number of nines in the ten-digit number.

What is this ten-digit number?


The four-digit number $\overline{abcd}$ and $\overline{cdab}$ have a sum of 3333 and a difference of 693. Find the four-digit number $\overline{abcd}$.


How many integers from 1 to 2001 have a digit sum that is divisible by 5?


A cube of size \(1000 \times 1000 \times 1000\) is located in space with one vertex at the origin and faces parallel to the coordinate planes. Vectors are drawn from the origin to all integer points inside and on the boundary of this cube. Find the remainder when the sum of the squares of the lengths of these vectors is divided by 11.


Let \( p \) and \( q \) be distinct prime numbers. How many divisors does the number have:
a) \(pq\);

b) \(p^2 q\);

c) \(p^2 q^2\);

d) \(p^m q^n\)?


What is the maximum possible area of a triangle if the sides \(a, b, c\) satisfy the following inequalities:

$$
0 < a \leq 1 \leq b \leq 2 \leq c \leq 3
$$


In triangle \(ABC\), the angles are known: \(\angle A = 45^{\circ}\) and \(\angle B = 15^{\circ}\). On the extension of side \(AC\) beyond point \(C\), point \(M\) is taken such that \(CM = 2AC\). Find \(\angle AMB\).


Find the integer solutions of the equation

$$
x^{4} + y^{4} = 3 x^{3} y
$$


Find the smallest positive integer \( n \) that satisfies the following properties:
1. The units digit of \( n \) is 6.
2. If the units digit 6 is moved to the front of the remaining digits of \( n \), the resulting new number is 4 times \( n \).


A sandwich and a meal plate cost $\mathrm{R} \$ 5.00$ and $\mathrm{R} \$ 7.00$, respectively. In how many ways can one buy only sandwiches, only meal plates, or some combination of sandwiches and meal plates with $\mathrm{R} \$ 90.00$, without leaving any change?


Determine the coefficients of \(x^{17}\) and \(x^{18}\) after expanding and combining like terms in the expression

\[
\left(1 + x^5 + x^7\right)^{20}
\]


Compute the limit of the function:
$\lim _{x \rightarrow 0} \frac{2 x \sin x}{1-\cos x}$


Find the smallest prime number $p$ such that $p^{3}+2 p^{2}+p$ has exactly 42 divisors.


Let \(ABC\) be an equilateral triangle. Let \(\overrightarrow{AB}\) be extended to a point \(D\) such that \(B\) is the midpoint of \(\overline{AD}\). A variable point \(E\) is taken on the same plane such that \(DE = AB\). If the distance between \(C\) and \(E\) is as large as possible, what is \(\angle BED\)?


In how many ways can two distinct squares be chosen from an $8 \times 8$ chessboard such that the midpoint of the line segment connecting their centers is also the center of a square on the board?


Show that the expression

$$
3^{2 n+2}-2^{n+1}
$$

is divisible by 7 for any positive integer $n$.


In the complex plane, the points corresponding to the complex numbers \( z_1, z_2, z_3 \) are \( Z_1, Z_2, Z_3 \), respectively. Given that \(\left| z_1 \right| = \left| z_2 \right| = \sqrt{2}\), \(\overrightarrow{OZ_1} \cdot \overrightarrow{OZ_2} = 0\), and \(\left| z_1 + z_2 - z_3 \right| = 2\), find the range of the value of \(\left| z_3 \right|\).

Solution: \boxed{[0, 4]}


Originally, there were three equally sized grassy fields, and the grass grew at a fixed rate. To eat the grass from all three fields, 50 sheep take exactly 18 days. To eat the grass from two fields, 40 sheep take exactly 12 days. Eating the grass from one field, 70 sheep take $\qquad$ days exactly. How many days will it take?


With what minimum force must you press on a cube with a volume of 10 cm³, floating in water, to submerge it completely? The density of the cube's material is 400 kg/m³, and the density of water is 1000 kg/m³. Provide the answer in SI units. Assume the acceleration due to gravity is 10 m/s².

Solution: \boxed{0.06 \, \text{N}}


Given three composite numbers \( A, B, C \) that are pairwise coprime and \( A \times B \times C = 11011 \times 28 \). What is the maximum value of \( A + B + C \)?


Given that \( x \) is a real number and \( y = \sqrt{x^2 - 2x + 2} + \sqrt{x^2 - 10x + 34} \). Find the minimum value of \( y \).


The difference between the maximum and minimum values of the function \( f(x)=|\sin x|+\sin ^{+} 2x+|\cos x| \) is equal to ______.


The number 2 can be represented as \(2 = 2 = 1 + 1\). The number 3 can be represented as \(3 = 3 = 1 + 2 = 2 + 1 = 1 + 1 + 1\). The number 4 can be represented as \(4 = 4 = 1 + 3 = 3 + 1 = 2 + 2 = 2 + 1 + 1 = 1 + 2 + 1 = 1 + 1 + 2 = 1 + 1 + 1 + 1\), and so on. This kind of representation of positive integers is called ordered partitions. Find the number of ordered partitions of a natural number \(n\).


Solution: \boxed{-2 \sin^2 2 \alpha}


In $\triangle ABC$, $\angle A = 90^\circ$ and $\angle B = \angle C = 45^\circ$. $P$ is a point on $BC$, and $Q, R$ are the circumcenters of $\triangle APB$ and $\triangle APC$ respectively. If $BP = \sqrt{2}$ and $QR = 2$, find $PC$.


The proportion of top-quality products at this enterprise is $31 \%$. What is the most likely number of top-quality products in a randomly selected batch of 75 products?


Several consecutive natural numbers are written on the board. Exactly 52% of them are even. How many even numbers are written on the board?


How many boards, each 6 arshins long and 6 vershoks wide, are needed to cover the floor of a square room with a side of 12 arshins? The answer is 64 boards. Determine from this data how many vershoks are in an arshin.


In an ornithological park, there are birds of several species, with a total of 2021 individuals. The birds are seated in a row, and it turns out that between any two birds of the same species, there is an even number of birds. What is the smallest number of bird species that could be?


Petrov writes down the odd numbers: $1, 3, 5, \ldots, 2013$, and Vasechkin writes down the even numbers: $2, 4, \ldots, 2012$. Each of them calculates the sum of all digits of all their numbers and tells the result to their classmate Masha. Masha subtracts Vasechkin's result from Petrov's result. What is the final result?


Find all integers \( x, y \geq 1 \) such that \( x^3 - y^3 = xy + 61 \).


Let \( \triangle ABC \) be a triangle such that \( AB = 7 \), and let the angle bisector of \( \angle BAC \) intersect line \( BC \) at \( D \). If there exist points \( E \) and \( F \) on sides \( AC \) and \( BC \), respectively, such that lines \( AD \) and \( EF \) are parallel and divide triangle \( ABC \) into three parts of equal area, determine the number of possible integral values for \( BC \).


In the vertices of a convex 2020-gon, numbers are placed such that among any three consecutive vertices, there is both a vertex with the number 7 and a vertex with the number 6. On each segment connecting two vertices, the product of the numbers at these two vertices is written. Andrey calculated the sum of the numbers written on the sides of the polygon and obtained the sum \( A \), while Sasha calculated the sum of the numbers written on the diagonals connecting vertices one apart and obtained the sum \( C \). Find the largest possible value of the difference \( C - A \).


Given an acute triangle \( \triangle ABC \), construct isosceles triangles \( \triangle DAC, \triangle EAB, \triangle FBC \) outside of \( \triangle ABC \) such that \( DA = DC, EA = EB, FB = FC \), and the angles \( \angle ADC = 2 \angle BAC, \angle BEA = 2 \angle ABC, \angle CFB = 2 \angle ACB \). Define \( D' \) as the intersection of line \( DB \) with \( EF \), \( E' \) as the intersection of line \( EC \) with \( DF \), and \( F' \) as the intersection of line \( FA \) with \( DE \). Find the value of \( \frac{DB}{DD'} + \frac{EC}{EE'} + \frac{FA}{FF'} \).


Find the maximum value for \(a, b \geq 1\) of the expression

$$
\frac{|7a + 8b - ab| + |2a + 8b - 6ab|}{a \sqrt{1 + b^2}}
$$


A point \( D \) is marked on the altitude \( BH \) of triangle \( ABC \). Line \( AD \) intersects side \( BC \) at point \( E \), and line \( CD \) intersects side \( AB \) at point \( F \). It is known that \( BH \) divides segment \( FE \) in the ratio \( 1:3 \), starting from point \( F \). Find the ratio \( FH:HE \).


Find the first differential of the function \( y(x) = e^{3x} \ln(1 + x^2) \) and evaluate it at \( x = 0 \) with \( dx = \Delta x = 0.1 \).

Solution: \boxed{dy(0) = 0}


The product of three consecutive even numbers equals $87_{* * * * *}8$. Find these numbers and fill in the blanks in this product.


Solve the equation \(\frac{15}{x\left(\sqrt[3]{35-8 x^{3}}\right)}=2x+\sqrt[3]{35-8 x^{3}}\). Write the sum of all obtained solutions as the answer.


Color 101 cells blue in an $n \times n$ grid. It is known that there is a unique way to cut the grid along square lines into rectangles such that each rectangle contains exactly one blue cell. Find the smallest possible value of $n$.


Given that \(\alpha\) and \(\beta\) are acute angles, and the following equations hold:
$$
\left\{\begin{array}{l}
3 \sin ^{2} \alpha + 2 \sin ^{2} \beta = 1, \\
3 \sin 2 \alpha - 2 \sin 2 \beta = 0.
\end{array}\right.
$$

Determine \(\alpha + 2\beta\).


In the polar coordinate system, the distance from the pole to the center of the circle given by the equation \(\rho=3 \sqrt{2} \cos \left(\theta+\frac{\pi}{4}\right)+7 \sin \theta\) is what?


If the sets of real numbers
$$
A=\{2x, 3y\} \text{ and } B=\{6, xy\}
$$
have exactly one common element, then the product of all elements in $A \cup B$ is ____.


Suppose \( x_{1}, x_{2}, \ldots, x_{2011} \) are positive integers satisfying
\[ x_{1} + x_{2} + \cdots + x_{2011} = x_{1} x_{2} \cdots x_{2011} \]

Find the maximum value of \( x_{1} + x_{2} + \cdots + x_{2011} \).


Define a function \( f(x) \) on \(\mathbf{R}\), satisfying:

\[
\begin{cases}
f(x+2016) \leqslant f(x) + 2016, \\
f(x+2017) \geqslant f(x) + 2017,
\end{cases}
\]

and \( f(1) = 2 \). Let \( a_n = f(n) \) for \( n \in \mathbf{N}^* \). Then find \( a_{2018} \).


Let the positive integer \( n \) satisfy \( 31 \mid (5^n + n) \). Find the smallest value of \( n \).


Using 5 stars (*) and 4 circles (○) to populate a $3 \times 3$ grid, how many distinct configurations are there, considering rotations and reflections that result in identical patterns as the same?


The sequence \(a_n\) is defined by \(a_1 = 20\), \(a_2 = 30\), and \(a_{n+1} = 3a_n - a_{n-1}\). Find all \(n\) for which \(5a_{n+1} \cdot a_n + 1\) is a perfect square.

Solution: \boxed{n = 3}


In triangle \( \triangle ABC \), \( a, b, c \) are the sides opposite to angles \( \angle A, \angle B, \angle C \) respectively. If \( \frac{a}{b}+\frac{b}{a}=4 \cos C \), and \( \cos (A-B)=\frac{1}{6} \), then \( \cos C \) is?

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


Let $ABCD$ be a rectangle with an area of 4. Let $I$ be the midpoint of $[AD]$ and $J$ be the midpoint of $[BC]$. Let $X$ be the intersection point of the lines (AJ) and (BI), and let $Y$ be the intersection point of the lines (DJ) and (CI). What is the area of the quadrilateral $IXJY$?


For any positive integer \( n \), let \( f(n) \) represent the last digit of \( 1 + 2 + 3 + \cdots + n \). For example, \( f(1) = 1 \), \( f(2) = 3 \), \( f(5) = 5 \), and so on. Find the value of \( f(2) + f(4) + f(6) + \cdots + f(2012) \).


Arrange the numbers in the set \(\left\{2^{x}+2^{y} \mid x, y\ \text{are non-negative integers,}\ x < y\right\}\) in ascending order. What is the 60th number? (Answer in digits).


For the quadratic equation $a(1+\mathrm{i}) x^{2}+\left(1+a^{2} \mathrm{i}\right) x+a^{2}+\mathrm{i}=0$ to have real roots, find the value of the real number $a$.


Calculate the definite integral:

$$
\int_{1 / 8}^{1} \frac{15 \sqrt{x+3}}{(x+3)^{2} \sqrt{x}} \, dx
$$


A parallelogram has 3 of its vertices at \((1,2)\), \((3,8)\), and \((4,1)\). Compute the sum of the possible \(x\)-coordinates for the 4th vertex.


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


A random variable \( X \) is given by the distribution function:

$$
F(x)=\left\{\begin{array}{ccc}
0 & \text{if} & x \leq -c \\
\frac{1}{2}+\frac{1}{\pi} \arcsin \frac{x}{c} & \text{if} & -c < x \leq c \\
1 & \text{if} & x > c
\end{array}\right.
$$

(law of arcsine).

Find the mathematical expectation (expected value) of this variable.


The Pythagorean school considered numbers to be the origin of all things. They called numbers like $1, 3, 6, 10, \cdots$ triangular numbers. What is the sum of the first 100 triangular numbers when arranged in ascending order?


In the sequence $O \bullet \bullet \bigcirc \bigcirc \bullet \bullet \bigcirc \bigcirc \bullet \bullet \bullet \cdots$, what is the 16th item, and what is the 29th item?

Solution: \boxed{\bullet}


If \(2 a t^{2}+12 t+9=0\) has equal roots, find \(a\).

If \(a x+b y=1\) and \(4 x+18 y=3\) are parallel, find \(b\).

The \(b\text{-th}\) prime number is \(p\). Find \(p\).

If \(k=\frac{4 \sin \theta+3 \cos \theta}{2 \sin \theta-\cos \theta}\) and \(\tan \theta=3\), find \(k\).


In the plane, three lines are given. How many circles exist that are tangent to all three lines? Explore all possible cases.


What is the largest integer that cannot be expressed as the sum of 100 not necessarily distinct composite numbers? (A composite number is one that can be expressed as the product of two integers greater than 1.)


Let \(\alpha_{n}\) be a real root of the cubic equation \(n x^{3}+2 x-n=0\), where \(n\) is a positive integer. If \(\beta_{n}=\left\lfloor(n+1) \alpha_{n}\right\rfloor\) for \(n=234 \cdots\), find the value of \(\frac{1}{1006} \sum_{k=2}^{2013} \beta_{k}\).


How many divisors does 24 have? And 140? And $2012^{2013}$?


Vasya thought of a two-digit number and told Petya the product of the digits of this number, while Sasha was told the sum of these digits. The following dialogue took place between the boys:

Petya: "I can guess the conceived number in three attempts, but two may not be enough for me."

Sasha: "If that's the case, then I can guess it in four attempts, but three may not be enough for me."

What number was told to Sasha?


A regular 12-sided polygon is inscribed in a circle of radius 1. How many chords of the circle that join two of the vertices of the 12-gon have lengths whose squares are rational?


If a 31-day month is taken at random, find \( c \), the probability that there are 5 Sundays in the month.


A line perpendicular to a chord of a segment divides the chord in the ratio 1:4 and the arc in the ratio 1:2. Find the cosine of the central angle subtended by this arc.


How many ways are there to choose 2010 functions \( f_{1}, \ldots, f_{2010} \) from \(\{0,1\}\) to \(\{0,1\}\) such that \( f_{2010} \circ f_{2009} \circ \cdots \circ f_{1} \) is constant? Note: a function \( g \) is constant if \( g(a)=g(b) \) for all \( a, b \) in the domain of \( g \).

Solution: \boxed{4^{2010} - 2^{2010}}


Let $\{x\}$ denote the smallest integer not less than the real number \(x\). Find the value of the expression $\left\{\log _{2} 1\right\}+\left\{\log _{2} 2\right\}+\left\{\log _{2} 3\right\}+\cdots+\left\{\log _{2} 1991\right\}$.


Let \(ABC\) be a triangle such that the altitude from \(A\), the median from \(B\), and the internal angle bisector from \(C\) meet at a single point. If \(BC = 10\) and \(CA = 15\), find \(AB^2\).


In the convex quadrilateral \(ABCD\), \(AC \perp BD\), \(\angle BCA = 10^\circ\), \(\angle BDA = 20^\circ\), \(\angle BAC = 40^\circ\). Find \(\angle BDC\).


The number 2017 has 7 ones and 4 zeros in binary representation. When will the nearest year come where the binary representation of the year has no more ones than zeros? (Enter the year.)


Given \( a, b > 0 \) satisfying the equation \( \sqrt{|x|} + \sqrt{|x + a|} = b \) having exactly three distinct real solutions \( x_{1}, x_{2}, x_{3} \), with \( x_{1} < x_{2} < x_{3} = b \), find the value of \( a + b \).


Find the area enclosed by the closed curves:
1) The ellipse \( x = a \cos t, \ y = b \sin t \)
2) The loop of the Cartesian leaf \( x^{3} + y^{3} - 3 a x y = 0 \).

Figures 187 and 188.

Solution: \boxed{S = \frac{3 a^2}{2}}


What is the smallest number of weights needed to measure any arbitrary number of grams from 1 to 1000 using a balance scale (weights can be placed on either side of the balance)?


Let \( n \) be a positive integer. Determine the smallest number \( k \) of colors needed to color the edges of any directed simple graph with \( n \) vertices such that there is no monochromatic cycle.

Solution: \boxed{k = 2}


Calculate the limit of the numerical sequence:
$$\lim _{n \rightarrow \infty} \frac{\sqrt{n^{5}-8}-n \sqrt{n\left(n^{2}+5\right)}}{\sqrt{n}}$$


Let \( a \) and \( b \) be the roots of \( x^2 + 2000x + 1 = 0 \) and let \( c \) and \( d \) be the roots of \( x^2 - 2008x + 1 = 0 \). Find the value of \( (a+c)(b+c)(a-d)(b-d) \).


The year of the current mathematical celebration is divisible by its number: \( 2006 \div 17 = 118 \).

a) Name the first number of the mathematical celebration for which this was also true.

b) Name the last number of the mathematical celebration for which this will also be true.


There are 196 students numbered from 1 to 196 arranged in a line. Students at odd-numbered positions (1, 3, 5, ...) leave the line. The remaining students are renumbered starting from 1 in order. Then, again, students at odd-numbered positions leave the line. This process repeats until only one student remains. What was the initial number of this last remaining student?


Peter has a lock with a three-digit code. He knows that all the digits of his code are different and that if he divides the second digit by the third and then squares his answer, he will get the first digit. What is the difference between the largest and smallest possible codes?


A sphere is inscribed in a cone. What maximum fraction of the cone's volume does the sphere occupy?


Pedro uses $1 \mathrm{~mL}$ of gray paint to cover $100 \mathrm{~cm}^{2}$ of surface area.

a) The solid in the figure was made by joining one face of a cube with a side length of $10 \mathrm{~cm}$ to one face of a cube with a side length of $20 \mathrm{~cm}$. How many milliliters of paint does Pedro need to paint this solid?

b) Pedro used $54 \mathrm{~mL}$ of paint to paint a cube and then divided this painted cube into two equal rectangular blocks. How many milliliters of paint will he need to paint the remaining unpainted areas of these two blocks?

c) Pedro used $54 \mathrm{~mL}$ of paint to paint another cube. After painting, this cube was divided into smaller identical cubes, and Pedro used an additional $216 \mathrm{~mL}$ of paint to cover all the unpainted faces of these smaller cubes. Into how many smaller cubes was the original cube divided?


The sequences \(\{a_{n}\}\) and \(\{b_{n}\}\) satisfy \(a_{1}=1, b_{1}=7\), and the recursive relations \(\begin{cases} a_{n+1} = b_{n} - 2a_{n}, \\ b_{n+1} = 3b_{n} - 4a_{n} \end{cases}\). Find the value of \(\lim_{n \rightarrow \infty} \frac{a_{n}}{b_{n}}\).


Vasya loves picking mushrooms. He calculated that during the autumn, he collected a three-digit number of mushrooms with the sum of its digits equal to 14. Then Vasya calculated that 8% of the mushrooms were white, and 14% were boletus. How many mushrooms did Vasya collect?


On a chessboard, the centers of all 64 squares are marked. Is it possible to divide the board into parts with thirteen straight cuts such that each part contains at most one center of a square?

Solution: \boxed{\text{No}}


If \( a \) and \( b \) are given real numbers, and \( 1 < a < b \), then the absolute value of the difference between the average and the median of the four numbers \( 1, a+1, 2a+b, a+b+1 \) is ______.


A finite arithmetic sequence \(\{a_{n}\}\) has the first term \(a_1 = 1\) and the common difference of 2. The arithmetic mean of all its terms is 2015. If one term is removed and the arithmetic mean of the remaining terms is an integer, how many methods are there to remove the term?


What is the 18th digit after the decimal point of \(\frac{10000}{9899}\)?


A cuckoo clock strikes the number of times corresponding to the current hour (for example, at 19:00, it strikes 7 times). One morning, Max approached the clock when it showed 9:05. He started turning the minute hand until it moved forward by 7 hours. How many times did the cuckoo strike during this period?


Solution: \boxed{4 \mid n}


If the average of \( a, b \) and \( c \) is 12, and the average of \( 2a+1, 2b+2, 2c+3 \) and 2 is \( P \), find the value of \( P \).


Let \(P\) be a regular polygon with 2026 sides. A diagonal of \(P\) is said to be good if its endpoints divide the contour of \(P\) into two parts, each having an odd number of sides of \(P\). The sides of \(P\) are also said to be good. Suppose that \(P\) has been subdivided into triangles by 2023 diagonals, no two of which share a common point inside \(P\). What is the maximum number of such a subdivision that can contain isosceles triangles whose two sides are good?


The villages "Verkhnie Vasyuki" and "Nizhnie Vasyuki" are located on the riverbank. A steamboat travels the distance from Verkhnie Vasyuki to Nizhnie Vasyuki in one hour, while a motorboat covers the same distance in 45 minutes. It is known that the speed of the motorboat in still water is twice the speed of the steamboat (also in still water). Determine the time (in minutes) required for a raft to drift from Verkhnie Vasyuki to Nizhnie Vasyuki.

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


The last three digits of \( 1978^n \) and \( 1978^m \) are the same. Find the positive integers \( m \) and \( n \) such that \( m+n \) is minimized (here \( n > m \geq 1 \)).


There are three $a$s, four $b$s, and two $c$s. Using these nine letters to form a permutation, if the condition is that the same letters cannot all be adjacent, how many such permutations are there?


A function \( f(x) \) defined on the interval \([1,2017]\) satisfies \( f(1)=f(2017) \), and for any \( x, y \in [1,2017] \), \( |f(x) - f(y)| \leqslant 2|x - y| \). If the real number \( m \) satisfies \( |f(x) - f(y)| \leqslant m \) for any \( x, y \in [1,2017] \), find the minimum value of \( m \).


For which values of \(a\) are both roots of the equation \(x^{2}-(a+1)x+a+4=0\) negative?

Solution: \boxed{a \in (-4, -3]}


A certain lottery has tickets labeled with the numbers \(1, 2, 3, \ldots, 1000\). The lottery is run as follows: First, a ticket is drawn at random. If the number on the ticket is odd, the drawing ends; if it is even, another ticket is randomly drawn (without replacement). If this new ticket has an odd number, the drawing ends; if it is even, another ticket is randomly drawn (again without replacement), and so forth, until an odd number is drawn. Then, every person whose ticket number was drawn (at any point in the process) wins a prize.
You have ticket number 1000. What is the probability that you get a prize?


Vanya wrote the number 1347 on the board.

- Look! - Petya noticed. - In this number, each of the last two digits is equal to the sum of the two preceding digits.
- Exactly! - Vasya agreed.
- Can you write the largest four-digit number with this property?

Help Vasya solve Petya's task.


Point \( M \) is located on side \( CD \) of a square such that \( CM:MD = 1:3 \). Line \( AM \) intersects the circle circumscribed about the square at point \( E \). The area of triangle \( ACE \) is 14. Find the side length of the square.


Find the minimum value of the expression

\[
(3 \sqrt{2(1+\cos 2 x)}-\sqrt{8-4 \sqrt{3}} \sin x+2) \cdot(3+2 \sqrt{11-\sqrt{3}} \cos y-\cos 2 y)
\]

If the answer is not an integer, round it to the nearest whole number.


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

Solution: \boxed{5: 1}


How many five-digit numbers are there in which no two digits of the same parity are adjacent?


Find the maximum value of the function \( f(x) = 8 \sin x + 15 \cos x \).


For each value of \( n \in \mathbf{N} \), find the largest value of \( k \in \mathbf{Z}^{+} \) for which the number \(\left\lfloor (3+\sqrt{11})^{2n-1} \right\rfloor\) is divisible by \( 2^k \).

Solution: \boxed{n}


If the inequality
$$
\left|a x^{2}+b x+a\right| \leqslant x
$$

holds for $x \in [1,2]$, then the maximum value of $3a + b$ is ___.


Define the function \( y = f(x) \) on \(\mathbf{R}\), which has the following properties:
(1) For any \( x \in \mathbf{R} \), \( f\left(x^{3}\right) = f^{3}(x) \);
(2) For any \( x_{1}, x_{2} \in \mathbf{R}, x_{1} \neq x_{2} \), \( f\left(x_{1}\right) \neq f\left(x_{2}\right) \). Find the value of \( f(0) + f(1) + f(-1) \).


From twelve schoolchildren and three teachers, a school committee of nine people needs to be formed. The committee must include at least one teacher. How many ways can the committee be formed?


Find all prime numbers \( p \) such that \( 5^p + 12^p \) is a perfect square.


Three intersecting lines form 12 angles, with $n$ of them being equal. What can be the maximum value of $n$?


Compute the sum of all positive integers \( n \) for which
$$
9 \sqrt{n} + 4 \sqrt{n+2} - 3 \sqrt{n+16}
$$
is an integer.


Let \( C \) be a circle with radius 2006. Suppose \( n \) points are placed inside the circle and the distance between any two points exceeds 2006. What is the largest possible \( n \)?


Let \( k \) be a positive integer. How many non-negative integer solutions are there to the equation

\[ \left\lfloor \frac{x}{k} \right\rfloor = \left\lfloor \frac{x}{k+1} \right\rfloor? \]


The number \( N \) is of the form \( p^{\alpha} q^{\beta} r^{\gamma} \), where \( p, q, r \) are prime numbers, and \( p q - r = 3, p r - q = 9 \). Additionally, the numbers \( \frac{N}{p}, \frac{N}{q}, \frac{N}{r} \) respectively have 20, 12, and 15 fewer divisors than the number \( N \). Find the number \( N \).


There is a calculator with digits from 0 to 9 and two operation symbols. Initially, the display shows the number 0. You can press any buttons. The calculator performs actions in the sequence of button presses. If an operation symbol is pressed multiple times in a row, the calculator only remembers the last press. A scattered scientist pressed a lot of buttons in a random sequence. Estimate the probability that the result of the resulting chain of operations is an odd number.


Find the sum of all positive irreducible fractions less than 1 whose denominator is 1991.


Let \( A, B, C, D \) be four points on a circle in that order. Also, \( AB=3 \), \( BC=5 \), \( CD=6 \), and \( DA=4 \). Let diagonals \( AC \) and \( BD \) intersect at \( P \). Compute \( \frac{AP}{CP} \).


In an isosceles triangle \(ABC \) (\(AB = BC\)), a point \(D\) is taken on the side \(BC\) such that \(BD : DC = 1 : 4\).

In what ratio does the line \(AD\) divide the height \(BE\) of the triangle \(ABC\), counted from the vertex \(B\)?


Determine a value of \( n \) for which the number \( 2^8 + 2^{11} + 2^n \) is a perfect square.


From an external point \( P \), a tangent \( [P D] \) is drawn to a circle with center \( O \). The point of contact \( D \) is connected to the endpoints \( A \) and \( B \) of the diameter perpendicular to \( (P O) \). The lines \( (D A) \) and \( (D B) \) intersect \( (P O) \) at points \( M \) and \( N \). Show that \( P M = P N = P D \).

Solution: \boxed{PM = PN = PD}


An electronic watch displays the time as $09:15:12$ at 9:15:12 AM and 13:11:29 at 1:11:29 PM. How many times in a 24-hour day does the six digits of the time form a symmetric sequence (i.e., the time reads the same forwards and backwards, such as 01:33:10)?


$N_{0}$ is the set of all non-negative integers. The function $f(n)$ is such that $f: N_{0} \rightarrow N_{0}$, and for each $n \in N_{0}$, $f(f(n)) + f(n) = 2n + 3$. Find $f(1993)$.


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


Find an integer \( x \) such that \( x \equiv 2 \pmod{3} \), \( x \equiv 3 \pmod{4} \), and \( x \equiv 1 \pmod{5} \).


Let \( S = \{1, 2, 3, \cdots, 280\} \). Find the smallest natural number \( n \) such that every subset of \( S \) with \( n \) elements contains 5 pairwise relatively prime numbers.


Two people, Person A and Person B, start at the same time from point $A$ to point $B$: Person A is faster than Person B. After reaching point $B$, Person A doubles their speed and immediately returns to point $A$. They meet Person B at a point 240 meters from point $B$. After meeting, Person B also doubles their speed and turns back. When Person A returns to point $A$, Person B is still 120 meters away from point $A$. What is the distance between points $A$ and $B$ in meters?

Solution: \boxed{420 \text{ meters}}


Oleg drew an empty $50 \times 50$ table and wrote a non-zero number above each column and to the left of each row. It turned out that all 100 numbers written were different, with 50 of them being rational and the remaining 50 irrational. Then, in each cell of the table, he wrote the product of the numbers corresponding to its row and column ("multiplication table"). What is the maximum number of products in this table that could be rational numbers?


Six nests are placed in a circle and in each of the nests there is a bird. Every night each bird moves to a neighboring nest. Is it possible for all the birds to end up in the same nest after a finite number of nights?

Solution: \boxed{\text{No}}


How many triangles with different shapes are there, where the measures of the angles expressed in degrees are integers?


In a social event, it was originally planned that each pair of participants would shake hands exactly once. However, 4 individuals left the event after shaking hands twice each. As a result, there were a total of 60 handshakes. Determine the initial number of participants in the event.


André, Bianca, Carlos, and Dalva want to raffle a book among themselves. To do this, they place 3 white balls and 1 black ball in a box and agree that, in alphabetical order of their names, each of them will draw a ball, without returning it to the box. Whoever draws the black ball will win the book.

a) What is the probability that André wins the book?

b) What is the probability that Dalva wins the book?

To raffle another book among them, André suggested using 2 black balls and 6 white balls. As before, the first person to draw a black ball will win the book; if the first four balls drawn are white, they will continue to draw balls in the same order. In this new raffle:

c) What is the probability that André wins the book?

d) What is the probability that Dalva wins the book?


Nadya wants to cut a sheet of paper into 48 identical rectangles for a game. What is the minimum number of cuts she needs to make, given that the pieces of paper can be rearranged but not folded, and Nadya is able to cut any number of layers of paper simultaneously? (Each cut is a straight line from edge to edge of the piece.)


A circle, whose center lies on the line \( y = b \), intersects the parabola \( y = \frac{5}{12} x^{2} \) in at least three points; one of these points is the origin, and two of the remaining points lie on the line \( y = \frac{5}{12} x + b \). Find all values of \( b \) for which this configuration is possible.


Vasya is inventing a 4-digit password for a combination lock. He does not like the digit 2, so he does not use it. Moreover, he doesn't like when two identical digits stand next to each other. Additionally, he wants the first digit to match the last one. How many possible combinations need to be checked to guarantee guessing Vasya's password?


Given \(a > 0\), \(b > 0\), \(c > 1\), and \(a + b = 1\). Find the minimum value of \(\left(\frac{2a + b}{ab} - 3\right)c + \frac{\sqrt{2}}{c - 1}\).

Solution: \boxed{4 + 2\sqrt{2}}


What is the sum of the integers from 1 to $n$?


A permutation of $\{1, \ldots, n\}$ is randomly selected. How many fixed points does it have on average?


Formulate the Taylor series expansion for \( n=2 \) of the function \( f(x, y) = x^y \) near the point \( M_0(1,1) \) and approximately calculate \( 1.1^{1.02} \).


Calculate the following sum given \(xyz = 1\):

\[
\frac{1}{1+x+xy}+\frac{1}{1+y+yz}+\frac{1}{1+z+zx}
\]


How do the volumes of a regular tetrahedron and a regular octahedron with the same edge length compare?


What is the smallest positive integer \( n \) which cannot be written in any of the following forms?
- \( n = 1 + 2 + \cdots + k \) for a positive integer \( k \).
- \( n = p^k \) for a prime number \( p \) and integer \( k \).
- \( n = p + 1 \) for a prime number \( p \).
- \( n = pq \) for some distinct prime numbers \( p \) and \( q \).


It is known that for three consecutive natural values of the argument, the quadratic function \( f(x) \) takes on the values -9, -9, and -15, respectively. Find the maximum possible value of \( f(x) \).


The calculation result of the expression: \( 2016 \times \frac{1}{1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}} \) is __.


How many solutions in natural numbers \( x \) and \( y \) does the equation \( x + y + 2xy = 2023 \) have?


A four-digit number with digits in the thousands, hundreds, tens, and units places respectively denoted as \(a, b, c, d\) is formed by \(10 \cdot 23\). The sum of these digits is 26. The tens digit of the product of \(b\) and \(d\) equals \((a+c)\). Additionally, \(( b d - c^2 )\) is an integer power of 2. Find the four-digit number and explain the reasoning.


The height of an isosceles trapezoid is 14 cm, and the bases are 16 cm and 12 cm. Determine the area of the circumscribed circle.

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


Let \( f(n) \) be a function defined on the set of all positive integers and having its values in the same set. Suppose that \( f(f(m) + f(n)) = m + n \) for all positive integers \( n, m \). Find all possible values for \( f(1988) \).


For which values of the parameter \(a\) does the system of equations

$$
\left\{\begin{array}{l}
x^{2}+y^{2}+z^{2}+4 y=0 \\
x+a y+a z-a=0
\end{array}\right.
$$

have a unique solution?

Solution: \boxed{a = \pm 2}


The diagonals of the faces of a rectangular parallelepiped are $\sqrt{3}, \sqrt{5}$, and 2. Find its volume.


At an interview, ten people were given a test consisting of several questions. It is known that any group of five people together answered all the questions (i.e., for each question, at least one of the five gave the correct answer), but any group of four did not. What is the minimum number of questions this could have been?


In 1860, someone deposited 100,000 florins at 5% interest with the goal of building and maintaining an orphanage for 100 orphans from the accumulated amount. When can the orphanage be built and opened if the construction and furnishing costs are 100,000 florins, the yearly personnel cost is 3,960 florins, and the maintenance cost for one orphan is 200 florins per year?


This year, the son and daughter are of such ages that the product of their ages is 7 times less than the age of their father. In three years, the product of their ages will equal the age of their father. Find the age of the father.


Pass a line \( l \) through the point \( P(1,1) \) such that the midpoint of the chord intercepted by the line from the ellipse \( \frac{x^{2}}{9} + \frac{y^{2}}{4} = 1 \) is exactly \( P \). Determine the equation of the line \( l \).

Solution: \boxed{4x + 9y = 13}


The numbers from 1 to 600 are divided into several groups. It is known that if a group contains more than one number, then the sum of any two numbers in this group is divisible by 6. What is the minimum number of groups?


How many numbers, divisible by 4 and less than 1000, do not contain any of the digits 6, 7, 8, 9, or 0?


An automatic production line for processing casing parts included several identical machines. The line processed 38,880 parts daily. After modernization, all the machines were replaced with more productive but still identical machines, and their number increased by 3. The automatic line subsequently processed 44,800 parts per day. How many parts per day did each machine process initially?


Find the largest integer $x$ such that the number

$$
4^{27} + 4^{1000} + 4^{x}
$$

is a perfect square.

Solution: \boxed{x = 1972}


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

a) the expected number of matches;

b) the variance of the number of matches.


Given \( a_n = 1 + 2 + \cdots + n \) where \( n \in \mathbf{Z}_{+} \), and
\[ S_m = a_1 + a_2 + \cdots + a_m \text{ for } m = 1, 2, \cdots, \]
find the number of terms among \( S_1, S_2, \cdots, S_{2017} \) that are divisible by 2 but not by 4.


If \( c \) is a 2-digit positive integer such that the sum of its digits is 10 and the product of its digits is 25, find the value of \( c \).


In a coordinate system, a quadrilateral is given with its vertices: \( A(0, 0), B(5, 0), C(3, 2), D(0, 1) \). Show that the diagonals of the quadrilateral form a \( 45^\circ \) angle with each other.


In triangle \(ABC\), the angle bisectors \(CF\) and \(AD\) are drawn. Find the ratio \(\frac{S_{AFD}}{S_{ABC}}\), if \(AB: AC: BC = 21: 28: 20\).


Given four points on a plane that are not on the same circle and no three of them are collinear. Construct a circle that is equidistant from all four points. What is the maximum number of such circles?


In space, there is a cube with dimensions \(1000 \times 1000 \times 1000\) with one vertex at the origin and faces parallel to the coordinate planes. Vectors have been drawn from the origin to all integer points inside and on the boundary of this cube. Find the remainder when the sum of the squares of the lengths of these vectors is divided by 13.


Martin is standing in a queue. The number of people in the queue is a multiple of 3. He notices that he has as many people in front of him as behind him. He sees two friends, both standing behind him in the queue, one in 19th place and the other in 28th place. In which position in the queue is Martin?
A) 14
B) 15
C) 16
D) 17
E) 18


To enter Ali Baba's cave, it is necessary to reset 28 counters, each of which is set to a natural number from 1 to 2017. In one move, treasure seekers are allowed to decrease the values of some counters by the same number, which can be changed from move to move. Indicate the minimum number of moves in which the treasure seekers will definitely reset the counters (regardless of the initial values) and enter the cave.


Find the number of subsets \( B \) of the set \(\{1,2,\cdots, 2005\}\) such that the sum of the elements in \( B \) leaves a remainder of 2006 when divided by 2048.


Knot must travel from point \( K \) to point \( T \) on a figure where \( OK = 2 \) and \( OT = 4 \). Before reaching \( T \), Knot must first reach both solid lines depicted in the figure in order to collect the two pieces of the Lineforce. What is the minimal distance Knot must travel to accomplish this?


Find \( N = p^2 \) if \( \varphi(N) = 42 \) and \( p \) is a prime number.


The numbers \( x, y, z \) are such that \( x \in[-3,7], y \in[-2,5], z \in[-5,3] \).

(a) Find the smallest possible value of \( x^{2} + y^{2} \).

(b) Find the smallest possible value of \( x y z - z^{2} \).


A long wooden stick has three types of marked lines. The first type divides the stick into 10 equal parts; the second type divides the stick into 12 equal parts; and the third type divides the stick into 15 equal parts. If the stick is sawed at each marked line, into how many pieces will the stick be divided in total?


The store received 20 kg of cheese, and a queue formed for it. Each time the seller serves the next customer, she accurately calculates the average weight of cheese sold and informs how many people can still be served if everyone buys exactly this average weight. Could the seller, after each of the first 10 customers, inform that there will be exactly enough cheese for 10 more customers? If yes, how much cheese remains in the store after the first 10 customers?

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


Nine nonnegative numbers have an average of 10. What is the greatest possible value for their median?


Two girls knit at constant, but different speeds. The first girl takes a tea break every 5 minutes, and the second girl every 7 minutes. Each tea break lasts exactly 1 minute. When the girls went for a tea break together, it turned out that they had knitted the same amount. By what percentage is the first girl's productivity higher if they started knitting at the same time?


On a circle, 60 red points and one blue point are marked. Consider all possible polygons with vertices at the marked points. Which type of polygons is more common, those with the blue vertex or those without? And by how many?


A bug moves in the coordinate plane, starting at $(0,0)$. On the first turn, the bug moves one unit up, down, left, or right, each with equal probability. On subsequent turns the bug moves one unit up, down, left, or right, choosing with equal probability among the three directions other than that of its previous move. For example, if the first move was one unit up then the second move has to be either one unit down or one unit left or one unit right.

After four moves, what is the probability that the bug is at $(2,2)$?


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


Solution: \boxed{a_4 b_4 = -6}


Find all functions \( f: \mathbb{Q} \rightarrow \mathbb{Q} \) such that for all \( x, y \in \mathbb{Q} \):

\[ f(x + y) + f(x - y) = 2f(x) + 2f(y) \]

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


A three-digit number \( \mathrm{abc} \) divided by the sum of its digits leaves a remainder of 1. The three-digit number \( \mathrm{cba} \) divided by the sum of its digits also leaves a remainder of 1. If different letters represent different digits and \( a > c \), then \( \overline{\mathrm{abc}} = \) ____.


Find all triples of strictly positive integers \((m, n, p)\) with \(p\) being a prime number, such that \(2^{m} p^{2} + 1 = n^{5}\).


Find the smallest natural number that begins with the digit five, which is reduced by four times if this five is removed from the beginning of its decimal representation and appended to its end.


In the tetrahedron \(ABCD\), the ratios of the lengths are:
\[ BD : CD : AB : AC : AD : BC = \sqrt{3} : \sqrt{2} : 1 : 1 : 1 : 1 \]

Find the angle between \(AD\) and \(BC\).


In the sequence \(\{a_n\}\), let \(S_n = \sum_{i=1}^{n} a_i\) (with \(n \in \mathbb{Z}_+\)), and it is agreed that \(S_0=0\). It is known that
\[
a_k =
\begin{cases}
k, & \text{if } S_{k-1} < k; \\
-k, & \text{if } S_{k-1} \geq k
\end{cases}
\quad (1 \leq k \leq n, k, n \in \mathbb{Z}_+).
\]
Find the largest positive integer \(n\) not exceeding 2019 such that
\[
S_n = 0.
\]


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


When programming a computer to print the first 10,000 natural numbers greater than 0: $1,2,3, \cdots, 10000$, the printer unfortunately has a malfunction. Each time it prints the digits 7 or 9, it prints $x$ instead. How many numbers are printed incorrectly?


The Dorokhov family plans to purchase a vacation package to Crimea. The vacation will include the mother, father, and their eldest daughter Polina, who is 5 years old. They have chosen to stay at the "Bristol" hotel and have reached out to two travel agencies, "Globus" and "Around the World," to calculate the cost of the tour from July 10 to July 18, 2021.

The deals from each agency are as follows:
- At "Globus":
  - 11,200 rubles per person for those under 5 years old.
  - 25,400 rubles per person for those above 5 years old.
  - A 2% discount on the total cost as regular customers.

- At "Around the World":
  - 11,400 rubles per person for those under 6 years old.
  - 23,500 rubles per person for those above 6 years old.
  - A 1% commission fee is applied to the total cost.

Determine which travel agency offers the best deal for the Dorokhov family and identify the minimum cost for their vacation in Crimea. Provide only the number in your answer, without units of measurement.


The diagonal of an isosceles trapezoid is $a$, and the midline is $b$. Find the height of the trapezoid.

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


Five points are chosen on a sphere of radius 1. What is the maximum possible volume of their convex hull?


Given the quadratic polynomial \( f(x) = x^2 + ax + b \), it is known that for any real number \( x \), there exists a real number \( y \) such that \( f(y) = f(x) + y \). Find the maximum possible value of \( a \).


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


Students were assigned to write several different three-digit numbers that do not contain the digit 7. How many such numbers can be written in total?


In triangle \( \triangle ABC \), if \( \overrightarrow{AB} \cdot \overrightarrow{AC} = 7 \) and \( |\overrightarrow{AB} - \overrightarrow{AC}| = 6 \), find the maximum possible area of \( \triangle ABC \).


There are fewer than 30 students in a class. The probability that a randomly selected girl is an honor student is $\frac{3}{13}$, and the probability that a randomly selected boy is an honor student is $\frac{4}{11}$. How many honor students are there in the class?


40 red, 30 blue, and 20 green points are marked on a circle. A number is placed on each arc between neighboring red and blue points (1), red and green points (2), and blue and green points (3). (On arcs between points of the same color, 0 is placed.) Find the maximum possible sum of the placed numbers.


For a positive integer \( x \), let \( m = \left\lfloor \frac{x}{2} \right\rfloor + \left\lfloor \frac{x}{2^2} \right\rfloor + \left\lfloor \frac{x}{2^3} \right\rfloor + \cdots + \left\lfloor \frac{x}{2^k} \right\rfloor \), where \( k \) is the smallest integer such that \( 2^k \geq x \) and the symbol \(\left\lfloor x \right\rfloor\) denotes the greatest integer less than or equal to \( x \). The difference \( x - m \), referred to as the "loss number" of the positive integer \( x \), is given. For example, when \( x = 100 \),

\[ \begin{aligned}
m & = \left\lfloor \frac{100}{2} \right\rfloor + \left\lfloor \frac{100}{2^2} \right\rfloor + \left\lfloor \frac{100}{2^3} \right\rfloor + \left\lfloor \frac{100}{2^4} \right\rfloor + \left\lfloor \frac{100}{2^5} \right\rfloor + \left\lfloor \frac{100}{2^6} \right\rfloor + \left\lfloor \frac{100}{2^7} \right\rfloor \\
& = 50 + 25 + 12 + 6 + 3 + 1 + 0 = 97,
\end{aligned} \]

\[ x - m = 100 - 97 = 3, \]

so the "loss number" of \( 100 \) is \( 3 \). Find the smallest positive integer \( x \) for which the "loss number" is \( 9 \).


The participants of the olympiad left 9 pens in the room. Among any four pens, at least two belong to the same person. And among any five pens, no more than three belong to the same person. How many students forgot their pens, and how many pens does each student own?


101. Write the number 100 using six non-zero identical digits in such a way that the method of writing does not depend on the digits used.

102. Calculate:

\(\lg \operatorname{tg} 37^{\circ} \cdot \lg \operatorname{tg} 38^{\circ} \cdot \lg \operatorname{tg} 39^{\circ} \ldots \lg \operatorname{tg} 47^{\circ}\)


Given that sequences \(\{a_n\}\) and \(\{b_n\}\) are both arithmetic sequences, and the sequence \(\{c_n\}\) is defined by \(c_n = a_n b_n\) (for \(n \in \mathbb{N}^*\)). If \(c_1 = 1440\), \(c_2 = 1716\), and \(c_3 = 1848\), find the value of \(c_8\).


Show that the product of two sides of a triangle, divided by the length of the perpendicular dropped to the third side from the opposite vertex, is equal to the diameter of the circumscribed circle.


It is known that the function \( f(x) \) is odd, i.e., \( f(-x) = -f(x) \) for each real \( x \). Additionally, it is known that \( f(x+5) = f(x) \) for each \( x \), and that \( f\left( \frac{1}{3} \right) = 2022 \), and \( f\left( \frac{1}{2} \right) = 17 \). What is the value of

\[
f(-7) + f(12) + f\left( \frac{16}{3} \right) + f\left( \frac{9}{2} \right) ?
\]


A ballpoint pen costs 10 rubles, a gel pen costs 50 rubles, and a fountain pen costs 80 rubles. What is the maximum number of gel pens that can be bought under the conditions that exactly 20 pens in total must be purchased, all three types of pens must be included, and the total expenditure must be exactly 1000 rubles?


In a sequence of 99 consecutive natural numbers, the largest number is 25.5 times the smallest number. What is the average of these 99 natural numbers?


In the right triangle \(ABC\) with the right angle at \(B\), the angle bisector \(BL\) and the median \(CM\) intersect at point \(D\). Line \(AD\) intersects side \(BC\) at point \(E\). Find the area of triangle \(AEL\), given that \(EL = x\).


A perpendicular dropped from the vertex $C$ of parallelogram $A B C D$ to the line $C D$ intersects at point $F$ a perpendicular dropped from vertex $A$ to the diagonal $B D$. A perpendicular dropped from point $B$ to the line $A B$ intersects at point $E$ the perpendicular bisector of segment $A C$. In what ratio does segment $E F$ divide side $B C$?


Maximum number. Let there be a set of distinct complex numbers \( z_i, i=1, 2, \ldots, n \), that satisfy the inequality

\[
\min _{i \neq j}\left|z_{i}-z_{j}\right| \geqslant \max _{i}\left|z_{i}\right|
\]

Find the maximum possible \( n \) and, for this \( n \), all sets that satisfy the problem condition.

Solution: \boxed{n = 7}


The Absent-Minded Scientist boards a bus with \( n \) seats. All \( n \) tickets have been sold to \( n \) passengers. The Absent-Minded Scientist is the first to enter the bus and, without looking at his ticket, takes a random seat. Subsequently, each passenger enters one-by-one. If a passenger finds their assigned seat free, they take it. If their seat is occupied, they take the first available free seat they find. What is the probability that the last passenger who boards the bus will sit in their assigned seat?


A certain intelligence station has four different kinds of passwords $A$, $B$, $C$, and $D$. Each week, one of these passwords is used, and each week a password is chosen uniformly at random from the three passwords that were not used the previous week. Given that password $A$ is used in the first week, what is the probability that password $A$ is also used in the seventh week? (Express your answer in the simplest fractional form.)


The store sells 20 items, each priced uniquely in whole numbers from 1 to 20 rubles. The store has a promotion: when purchasing any 5 items, one of them is given for free, and the customer can choose which one. Vlad wants to buy all 20 items in this store, paying as little as possible. How many rubles does he need? (Each of the 20 items is sold in a single unit.)


Two sides of a triangle are 10 and 12, and the median drawn to the third side is 5. Find the area of the triangle.


The number \( N \) has the smallest positive divisor 1, the second largest positive divisor \( k \), and the third largest positive divisor \( m \). Moreover, \( k^k + m^m = N \). What is \( N \)?


A right triangle has legs of lengths 3 and 4. Find the volume of the solid formed by revolving the triangle about its hypotenuse.


The sequence \(\left\{a_{n}\right\}_{n \geq 1}\) is defined by \(a_{n+2}=7 a_{n+1}-a_{n}\) for positive integers \(n\) with initial values \(a_{1}=1\) and \(a_{2}=8\). Another sequence, \(\left\{b_{n}\right\}\), is defined by the rule \(b_{n+2}=3 b_{n+1}-b_{n}\) for positive integers \(n\) together with the values \(b_{1}=1\) and \(b_{2}=2\). Find \(\operatorname{gcd}\left(a_{5000}, b_{501}\right)\).


The sum of the non-negative numbers \(a\), \(b\), and \(c\) is equal to 3. Find the maximum value of the expression \(ab + bc + 2ca\).


Define a new operation: \( A \oplus B = A^2 + B^2 \), and \( A \otimes B \) is the remainder of \( A \) divided by \( B \). Calculate \( (2013 \oplus 2014) \otimes 10 \).


The integer sequence \(\left\{a_{n}\right\}\) is defined by \(a_{1}=1\), \(a_{2}=2\), and \(a_{n+2}=5a_{n+1}+a_{n}\). Determine the value of the expression \(\left[\frac{a_{2}}{a_{1}}\right]\left\{\left[\frac{a_{3}}{a_{2}}\right\}\left\{\frac{a_{4}}{a_{3}}\right\} \cdots \left\{\left\{\frac{a_{20225}}{a_{2024}}\right\}\left[\frac{a_{20224}}{a_{2}}\right]\right\}\right.\).


There are 30 people sitting around a round table - knights and liars (knights always tell the truth, and liars always lie). It is known that each of them has exactly one friend at the table, and if one is a knight, their friend is a liar, and vice versa (friendship is always mutual). In response to the question "Is your friend sitting next to you?" the individuals who are sitting in every other seat answered "Yes." How many of the remaining individuals could also answer "Yes"?


Given a natural number \( x = 9^n - 1 \), where \( n \) is a natural number. It is known that \( x \) has exactly three distinct prime divisors, one of which is 11. Find \( x \).


Initially, there were 20 balls of three colors in a box: white, blue, and red. If we double the number of blue balls, then the probability of drawing a white ball will decrease by $\frac{1}{25}$. If we remove all the white balls, the probability of drawing a blue ball will increase by $\frac{1}{16}$ compared to the initial probability of drawing a blue ball. How many white balls were in the box?


a) Given the quadratic equation \( x^{2} - 9x - 10 = 0 \). Let \( a \) be its smallest root. Find \( a^{4} - 909a \).

b) For the quadratic equation \( x^{2} - 9x + 10 = 0 \), let \( b \) be its smallest root. Find \( b^{4} - 549b \).


Given that there are $m$ distinct positive even numbers and $n$ distinct positive odd numbers such that their sum is 2015. Find the maximum value of $20m + 15n$.


A function \( f \) is defined on the positive integers by: \( f(1) = 1 \); \( f(3) = 3 \); \( f(2n) = f(n) \), \( f(4n + 1) = 2f(2n + 1) - f(n) \), and \( f(4n + 3) = 3f(2n + 1) - 2f(n) \) for all positive integers \( n \). Determine the number of positive integers \( n \) less than or equal to 1988 for which \( f(n) = n \).


The symmedian from point $A$ of a triangle $ABC$ intersects the circumcircle at point $D$. Let $P, Q, R$ be the orthogonal projections of $D$ onto sides $BC$, $CA$, and $AB$, respectively. Show that $PQ = PR$.

Solution: \boxed{PQ = PR}


Find the smallest natural number \( n \) such that the equation \(\left[\frac{10^{n}}{x}\right]=1989\) has an integer solution \( x \).
(The 23rd All-Soviet Union Math Olympiad, 1989)


If \( x, y, z \) are real numbers such that \( xy = 6 \), \( x - z = 2 \), and \( x + y + z = 9 \), compute \( \frac{x}{y} - \frac{z}{x} - \frac{z^2}{xy} \).


Find all positive integers \( k \) such that for any positive numbers \( a, b, c \) satisfying the inequality
\[ k(ab + bc + ca) > 5(a^2 + b^2 + c^2), \]
there must exist a triangle with side lengths \( a \), \( b \), and \( c \).


Given the polynomial

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

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


Suppose that \( f(x) = a(x - b)(x - c) \) is a quadratic function where \( a, b \) and \( c \) are distinct positive integers less than 10. For each choice of \( a, b \) and \( c \), the function \( f(x) \) has a minimum value. What is the minimum of these possible minimum values?


A number contains only the digits 3 or 4, and both digits appear at least once. This number is a multiple of both 3 and 4. What is the smallest such number?


The numbers \( a \) and \( b \) are positive integers and satisfy \( 96a^2 = b^3 \). What is the smallest value of \( a \)?


The sum of ten numbers is zero. The sum of all their pairwise products is also zero. Find the sum of their fourth powers.


The measurements of the sides of a rectangle are even numbers. How many such rectangles exist with an area equal to 96?


An infinite sequence of decimal digits is obtained by writing the positive integers in order: 123456789101112131415161718192021 ... . Define f(n) = m if the 10^n th digit forms part of an m-digit number. For example, f(1) = 2, because the 10th digit is part of 10, and f(2) = 2, because the 100th digit is part of 55. Find f(1987).


When dividing the numbers 312837 and 310650 by some three-digit natural number, the remainders are the same. Find this remainder.


Perpendiculars $BE$ and $DF$ dropped from vertices $B$ and $D$ of parallelogram $ABCD$ onto sides $AD$ and $BC$, respectively, divide the parallelogram into three parts of equal area. A segment $DG$, equal to segment $BD$, is laid out on the extension of diagonal $BD$ beyond vertex $D$. Line $BE$ intersects segment $AG$ at point $H$. Find the ratio $AH: HG$.


A fair coin is tossed 5 times. The probability of getting exactly one heads is not zero and is the same as the probability of getting exactly two heads.

Let the reduced fraction \(\frac{i}{j}\) represent the probability of getting exactly 3 heads when the coin is tossed 5 times. Find the value of \(i + j\).


The difference between the longest and shortest diagonals of the regular n-gon equals its side. Find all possible n.


Every day, Xiaoming goes to school along a flat road \(AB\), an uphill road \(BC\), and a downhill road \(CD\) (as shown in the diagram). Given that \(AB : BC : CD = 1 : 2 : 1\) and that Xiaoming's speeds on flat, uphill, and downhill roads are in the ratio 3 : 2 : 4, respectively, find the ratio of the time Xiaoming takes to go to school to the time he takes to come home.


The rules of a card arrangement game are as follows: Arrange nine cards marked with $1,2, \cdots, 9$ randomly in a row. If the number on the first card (from the left) is $k$, then reverse the order of the first $k$ cards, which is called an operation. The game stops when no operation can be performed (i.e., the number on the first card is "1"). If an arrangement cannot be operated on and can be obtained by performing one operation on a unique other arrangement, it is called a "second stop arrangement". Find the probability of a second stop arrangement occurring among all possible arrangements.


Calculate the limit of the function:
$$\lim_{x \rightarrow 0} \frac{\sqrt{1-2x+3x^{2}}-(1+x)}{\sqrt[3]{x}}$$


Three distinct numbers are drawn at random from the set \(\{1, 2, \cdots, 10\}\). Find the probability that the sample variance \(s^2 \leq 1\).


In a regular hexagon \( ABCDEF \), the diagonals \( AC \) and \( CE \) are divided by interior points \( M \) and \( N \) respectively in the ratio \(\frac{AM}{AC} = \frac{CM}{CE} = r\). If the points \( B \), \( M \), and \( N \) are collinear, find the value of \( r \).


We have 1000 solid cubes with edge lengths of 1 unit each. We want to use these small cubes to create a hollow cube with a wall thickness of 1 unit. The small cubes can be glued together but not cut. What is the maximum (external) volume of the cube we can thus create?


Find the sum of all four-digit numbers that can be formed using the digits \(1, 2, 3, 4, 5\) such that each digit appears at most once.


How many five-digit numbers are there, not divisible by 1000, in which the first, third, and last digits are even?


The weight of cows of the Red Gorbatov breed is a random variable \(X\) distributed according to the normal law with a mean \(a = 470 \, \text{kg}\) and a standard deviation \(\sigma = 30 \, \text{kg}\). What is the probability that two out of three randomly selected cows will have a weight more than 470 kg and less than 530 kg?


How to refute the statement: "If a number is divisible by 5, then it ends with the digit 5"?


Carlson and Baby have several jars of jam, each weighing an integer number of pounds.

The total weight of all Carlson's jars of jam is 13 times the total weight of all Baby's jars. Carlson gave Baby the jar with the smallest weight (among those he had), after which the total weight of his jars turned out to be 8 times the total weight of Baby's jars.

What is the maximum possible number of jars Carlson could have initially had?


$O$ is the center of a circle, $C$ is the intersection point of the chord $AB$ and the radius $OD$, which is perpendicular to the chord. Given $OC = 9$ and $CD = 32$. Find the length of the chord $AB$.


Given the sum \( \sum_{n=1}^{100} n=1+2+3+4+\cdots+100=5050 \), which represents the sum of 100 consecutive natural numbers starting from 1, find the units digit of \( \sum_{n=1}^{2021} n\left(n^{2}+n+1\right) \).


A cyclist rode 96 km 2 hours faster than expected. At the same time, he covered 1 km more per hour than he expected to cover in 1 hour 15 minutes. What was his speed?

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


Let T be the triangle with vertices (0, 0), (a, 0), and (0, a). Find \(\lim_{{a \to \infty}} a^4 \exp(-a^3) \int_{T} \exp(x^3 + y^3) \, dx \, dy\).


Find the number of integers from 1 to 250 that are divisible by any of the integers 2, 3, 5, or 7.


The points \(A\), \(B\), and \(C\) are the centers of three faces of a cuboid that meet at a vertex. The lengths of the sides of the triangle \(ABC\) are 4, 5, and 6. What is the volume of the cuboid?


A notebook sheet was colored in 23 colors, one color per cell. A pair of colors is called good if there are two adjacent cells colored in these colors. What is the minimum number of good pairs?


Let the function \( f^{\prime}(x) \) be the derivative of an even function \( f(x) \) with \( x \neq 0 \), and suppose \( f(-1)=0 \). Given that \( x f^{\prime}(x) - f(x) < 0 \) for \( x > 0 \), determine the range of \( x \) such that \( f(x) > 0 \).

Solution: \boxed{(-1, 0) \cup (0, 1)}


A line passing through the focus \( F \) of the parabola \( y^2 = 4x \) intersects the parabola at points \( M \) and \( N \). Let \( E(m, 0) \) be a point on the x-axis. The extensions of \( M E \) and \( N E \) intersect the parabola at points \( P \) and \( Q \). If the slopes \( k_1 \) of \( M N \) and \( k_2 \) of \( P Q \) satisfy \( k_1 = 3 k_2 \), then the value of the real number \( m \) is ______.


Find the real polynomial \( p(x) \) of degree 4 with the largest possible coefficient of \( x^4 \) such that \( p(x) \in [0, 1] \) for all \( x \in [-1, 1] \).

Solution: \boxed{4x^4 - 4x^2 + 1}


Find all real numbers \(k\) such that the inequality
\[
a^{3} + b^{3} + c^{3} + d^{3} + 1 \geq k(a + b + c + d)
\]
holds for any \(a, b, c, d \in [-1, +\infty)\).


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

a) the expected number of matches;

b) the variance of the number of matches.


Let \( A B C \) be a triangle with \(\widehat{C A B}=20^{\circ}\). Let \( D \) be the midpoint of the segment \([A B]\). It is given that \(\widehat{C D B}=40^{\circ}\). What is the value of the angle \(\widehat{A B C}\)?


In a group consisting of $2n$ members, the following conditions hold: there exists a number $d$ such that each member of the group knows at most $d$ other members, and there are more than $d$ people in the group such that none of them know each other. Show that, no matter what $d$ is, the total number of acquaintances is less than $n^{2}$.


A metal sphere with a radius of $\sqrt[z]{16}$ is melted and recast into a cone whose lateral surface area is three times the area of its base. Find the height of the cone.


The lengths of the sides of a triangle are $\sqrt{3}, \sqrt{4}(=2), \sqrt{5}$.

In what ratio does the altitude perpendicular to the middle side divide it?


(a) What is the smallest (positive) multiple of 9 that is written using only the digits 0 and 1?

(b) What is the smallest (positive) multiple of 9 that is written using only the digits 1 and 2?


For which natural numbers \( n \) is the inequality

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

true for any angles \( \alpha, \beta, \gamma \) that are the angles of an acute triangle?


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

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

for all real numbers \( x \) and \( y \).

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


Let \( f(x) \) be a differentiable function defined on \((- \infty, 0)\), with its derivative denoted by \( f^{\prime}(x) \). Given the inequality \( 2 f(x) + x f^{\prime}(x) > x^2 \), determine the solution set for the inequality \((x+2017)^{2} f(x+2017) - f(-1) > 0\).

Solution: \boxed{(-\infty, -2018)}


\(r\) is the inradius of the triangle \( \triangle ABC \) and \( r' \) is the exradius for the circle touching side \( AB \). Show that \( 4rr' \leq c^2 \), where \( c \) is the length of side \( AB \).

Solution: \boxed{4rr' \leq c^2}


Let \( p^{n} (p \text{ is a prime number}, n \geqslant 2) \) be a "good number". If the difference between 100 and 23 in base \( b \) is a good number, find all possible values of \( b \).


The segment $A B=2 d$ is bisected perpendicularly by the line $e$. On this line $e$, the segment $R S=d$ slides. The lines $A R$ and $B S$ intersect at point $P$. What is the locus of point $P$?

Solution: \boxed{x^2 + 2xy = d^2}


December 31, 2013 is a Tuesday. What day of the week is June 1, 2014? (Answer using numbers: Monday is 1, Tuesday is 2, Wednesday is 3, Thursday is 4, Friday is 5, Saturday is 6, Sunday is 7.)


If \( a \) is the sum of all the positive factors of 2002, find the value of \( a \).

It is given that \( x>0, y>0 \) and \( \sqrt{x}(\sqrt{x}+\sqrt{y})=3 \sqrt{y}(\sqrt{x}+5 \sqrt{y}) \). If \( b=\frac{2 x+\sqrt{x y}+3 y}{x+\sqrt{x y}-y} \), find the value of \( b \).

Given that the equation \(||x-2|-1|=c\) has only 3 integral solutions, find the value of \( c \).

If \( d \) is the positive real root of the equation \(\frac{1}{2}\left\{\frac{1}{2}\left[\frac{1}{2}\left(\frac{1}{2} x^{2}+2\right)+2\right]+2\right\}=2\), find the value of \( d \).


The base of the pyramid is an isosceles right triangle, where each leg measures 8. Each of the pyramid's lateral edges is 9. Find the volume of the pyramid.


Given a tetrahedron \( A B C D \) with side lengths \( A B = 41 \), \( A C = 7 \), \( A D = 18 \), \( B C = 36 \), \( B D = 27 \), and \( C D = 13 \), let \( d \) be the distance between the midpoints of edges \( A B \) and \( C D \). Find the value of \( d^{2} \).


Given a tetrahedron $A B C D$. All plane angles at vertex $D$ are right angles; $D A = 1, D B = 2, D C = 3$. Find the median of the tetrahedron drawn from vertex $D$.


On a board are written the numbers \(1, 2, 3, \ldots, 57\). What is the maximum number of these numbers that can be selected so that no two selected numbers differ by exactly 2.5 times?


Given the set \( M = \{ (a, b) \mid a \leq -1, b \leq m \} \). If for any \((a, b) \in M\), it always holds that \(a \cdot 2^b - b - 3a \geq 0\), then the maximum value of the real number \( m \) is _____.


In a class, some students study only English, some study only German, and some study both languages. What percentage of the students in the class study both languages if 80% of all students study English and 70% study German?


Each hotel room can accommodate no more than 3 people. The hotel manager knows that a group of 100 football fans, who support three different teams, will soon arrive. A room can only house either men or women; and fans of different teams cannot be housed together. How many rooms need to be booked to accommodate all the fans?


In triangle \(PQR\), the angle \(QRP\) is \(60^\circ\). Find the distance between the points of tangency on side \(QR\) of the inscribed circle with radius 2 and the circle with radius 3 that is tangent to the extensions of sides \(PQ\) and \(PR\).


In a tournament, each participant was supposed to play exactly one game with each of the remaining participants. However, two participants dropped out during the tournament after playing only 4 games each. As a result, the total number of games played was 62. How many participants were there in total?


Find the total number of sets of positive integers \((x, y, z)\), where \(x, y\) and \(z\) are positive integers, with \(x < y < z\) such that
$$
x + y + z = 203.
$$


Randomly select elements $m$ and $n$ (which can be the same) from the set $\{1, 2, \cdots, 100\}$. What is the probability that the unit digit of $2^{m} + 3^{n}$ is 3?


Let \( n \) be a three-digit number such that \( 100 \leqslant n \leqslant 999 \). Find all \( n \) for which the last three digits of \( n^2 \) are equal to \( n \).

Solution: \boxed{376 \text{, } 625}


Given \( x \cdot y \cdot z + y + z = 12 \), find the maximum value of \( \log_{4} x + \log_{2} y + \log_{2} z \).


Let \( f(x) \) be a quadratic function such that \( 2x^{2} - 4x + 3 \leqslant f(x) \leqslant 3x^{2} - 6x + 4 \) and \( f(3) = 11 \). Find \( f(5) \).


Find the maximum constant \( k \) such that for \( x, y, z \in \mathbb{R}_+ \), the following inequality holds:
$$
\sum \frac{x}{\sqrt{y+z}} \geqslant k \sqrt{\sum x},
$$

where " \( \sum \) " denotes a cyclic sum.


A math interest group in a school has 14 students. They form \( n \) different project teams. Each project team consists of 6 students, each student participates in at least 2 project teams, and any two project teams have at most 2 students in common. Find the maximum value of \( n \).


A rhombus, where the shorter diagonal is equal to its side of length 1, rotates around a line passing through the end of the longer diagonal and perpendicular to this diagonal. Find the volume of the resulting solid of revolution.

Solution: \boxed{\frac{3 \pi}{2}}


Find the smallest possible α such that if p(x) ≡ ax^2 + bx + c satisfies |p(x)| ≤ 1 on [0, 1], then |p'(0)| ≤ α.


The point \( P(-1,4) \) is reflected in the \( y \)-axis to become \( Q \). The point \( Q \) is reflected in the line \( y=x \) to become \( R \). The point \( R \) is reflected in the \( x \)-axis to become \( S \). What is the area of quadrilateral \( PQRS \)?


Inside an isosceles triangle \( ABC \), a point \( K \) is marked such that \( CK = AB = BC \) and \(\angle KAC = 30^\circ\). Find the angle \(\angle AKB\).


There are three simplest proper fractions with their numerators in the ratio $3: 2: 4$ and their denominators in the ratio $5: 9: 15$. When these three fractions are added together and then reduced to simplest form, the result is $\frac{28}{45}$. What is the sum of the denominators of these three fractions?


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


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


What is the greatest number of consecutive natural numbers, each of which has exactly four natural divisors (including 1 and the number itself)?

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


In an infinite sequence of natural numbers, the product of any fifteen consecutive terms is equal to one million, and the sum of any ten consecutive terms is equal to \(S\). Find the maximum possible value of \(S\).


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

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


The 97 numbers \( \frac{49}{1}, \frac{49}{2}, \frac{49}{3}, \ldots, \frac{49}{97} \) are written on a blackboard. We repeatedly pick two numbers \( a, b \) on the board and replace them by \( 2ab - a - b + 1 \) until only one number remains. What are the possible values of the final number?


Point \( D \) lies on side \( CB \) of right triangle \( ABC \left(\angle C = 90^{\circ} \right) \), such that \( AB = 5 \), \(\angle ADC = \arccos \frac{1}{\sqrt{10}}, DB = \frac{4 \sqrt{10}}{3} \). Find the area of triangle \( ABC \).


On the board, the number 27 is written. Every minute, the number is erased from the board and replaced with the product of its digits increased by 12. For example, after one minute, the number on the board will be $2 \cdot 7 + 12 = 26$. What number will be on the board after an hour?


Is it possible to select a number \( n \geq 3 \) and fill an \( n \times (n+3) \) table (with \( n \) rows and \( n+3 \) columns) with distinct natural numbers from 1 to \( n(n+3) \) such that in each row there are three numbers, one of which is equal to the product of the other two?

Solution: \boxed{\text{No}}


A school economics teacher believes that it's extremely important to know the basics of both microeconomics and macroeconomics. Therefore, the teacher has introduced the following system for the final grade in his subject. First, the student's understanding of the basics of microeconomics ($O_{\text {mic}}$) is assessed out of 5 points, and then the basics of macroeconomics ($O_{\text {mac}}$) are assessed out of 5 points. The final grade in economics is the minimum of the two values:
\[0.25 * O_{\text {mic}}+0.75 * O_{\text {mac}}\]
and
\[0.75 * O_{\text {mic}}+0.25 * O_{\text {mac}}\]
(The final value is rounded up to the nearest whole number according to the rules of mathematical rounding). Student Ivanov has planned his study time for each school subject and decided that he will spend no more than 4.6 time units on studying economics. However, studying one area of economics is more difficult for Ivanov than the other. Each time unit spent on studying macroeconomics increases his final result (in points) in this area by 1.5 points, while the same time unit spent on preparing for microeconomics increases his final result in this area by 2.5 points.

(a) What is the maximum grade that student Ivanov can achieve in economics?

(b) Explain why the final grade in economics reflects the teacher's opinion that it is extremely important to know both microeconomics and macroeconomics. Will it always turn out like this with such a grading scheme?


Solve the inequality \(\left(\sqrt{x^{3}-18 x-5}+2\right) \cdot\left|x^{3}-4 x^{2}-5 x+18\right| \leqslant 0\).

Solution: \boxed{x = 1 - \sqrt{10}}


Construct regular triangles outwardly on sides $AB$ and $BC$ of parallelogram $ABCD$. The third vertices of these triangles are $E$ and $F$, respectively. Show that the sum of angles $CED$ and $AFD$ is $60^{\circ}$.


In the sequence \(\{a_n\}\), \(a_4=1\), \(a_{11}=9\), and the sum of any three consecutive terms is always 15. Find \(a_{2016}\).


A standard deck of cards, excluding jokers, has 4 suits with 52 cards in total. Each suit has 13 cards, with face values from 1 to 13. If Feifei draws 2 hearts, 3 spades, 4 diamonds, and 5 clubs from this deck, and the sum of the face values of the spades is 11 times the sum of the face values of the hearts, while the sum of the face values of the clubs is 45 more than the sum of the face values of the diamonds, what is the sum of the face values of these 14 cards?


If \( S_{n} = 1 - 2 + 3 - 4 + \ldots + (-1)^{n-1} n \), where \( n \) is a positive integer, determine the value of \( S_{17} + S_{33} + S_{50} \).


How many pairs of natural numbers \(a\) and \(b\) exist such that \(a \geq b\) and the equation

\[ \frac{1}{a} + \frac{1}{b} = \frac{1}{6} \]

is satisfied?


Joe has written 5 questions of different difficulties for a test, numbered 1 through 5. He wants to make sure that problem \(i\) is harder than problem \(j\) whenever \(i - j \geq 3\). In how many ways can he order the problems for his test?


Given that the complex numbers \( z_{1} \) and \( z_{2} \) satisfy
\[ \left|z_{1} + z_{2}\right| = 20, \quad \left|z_{1}^{2} + z_{2}^{2}\right| = 16, \]
find the minimum value of \( \left|z_{1}^{3} + z_{2}^{3}\right| \).


Find the value of the expression \(\frac{a}{b} + \frac{b}{a}\), where \(a\) and \(b\) are the largest and smallest roots of the equation \(x^3 - 9x^2 + 9x = 1\), respectively.


Let \( P \) be an arbitrary point on the ellipse \(\frac{x^{2}}{16}+\frac{y^{2}}{9}=1\) other than the endpoints of the major axis. \( F_{1} \) and \( F_{2} \) are its left and right foci respectively, and \( O \) is the center. Find the value of \( \left|P F_{1}\right| \cdot \left|P F_{2}\right| + |O P|^{2} \).


Find the millionth digit after the decimal point in the decimal expansion of the fraction \(1 / 41\).


Let \( \triangle ABC \) be a triangle such that \( AB = 7 \), and let the angle bisector of \( \angle BAC \) intersect line \( BC \) at \( D \). If there exist points \( E \) and \( F \) on sides \( AC \) and \( BC \), respectively, such that lines \( AD \) and \( EF \) are parallel and divide triangle \( ABC \) into three parts of equal area, determine the number of possible integral values for \( BC \).


On the board, there are three quadratic equations written:

$$
\begin{gathered}
2020 x^{2}+b x+2021=0 \\
2019 x^{2}+b x+2020=0 \\
x^{2}+b x+2019=0
\end{gathered}
$$

Find the product of the roots of all the equations written on the board, given that each of them has two real roots.


Andrey, Boris, and Valentin participated in a 1 km race. (Assume each of them ran at a constant speed). Andrey finished 100 meters ahead of Boris. Boris finished 60 meters ahead of Valentin. What was the distance between Andrey and Valentin at the moment Andrey finished?

Solution: \boxed{154 \text{ meters}}


Given the sequence $\left\{a_{n}\right\}$ that satisfies $a_{1}=1$ and $a_{n}=2 a_{n-1}+n-2$ for $n \geqslant 2$, find the general term $a_{n}$.

Solution: \boxed{a_n = 2^n - n}


In Sally's sequence, every term after the second is equal to the sum of the previous two terms. Also, every term is a positive integer. Her eighth term is 400. Find the minimum value of the third term in Sally's sequence.


Solve the inequality:

$$
x \log _{1 / 10}\left(x^{2}+x+1\right)>0
$$

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


From May 1st to May 3rd, the provincial hospital plans to schedule 6 doctors to be on duty, with each person working 1 day and 2 people scheduled per day. Given that doctor A cannot work on the 2nd and doctor B cannot work on the 3rd, how many different scheduling arrangements are possible?


A sphere is circumscribed around a regular triangular prism, the height of which is twice the side of the base. How does its volume compare to the volume of the prism?

Solution: \boxed{\frac{64 \pi}{27}}


Let \( ABC \) be an isosceles triangle at \( A \) with \( \angle CAB = 20^\circ \). Let \( D \) be a point on the segment \( [AC] \) such that \( AD = BC \). Calculate the angle \( \angle BDC \).


The first term of an arithmetic sequence is 9, and the 8th term is 12. How many of the first 2015 terms of this sequence are multiples of 3?


A chess player has 5 points after the 7th round (from 7 games). In how many different ways could this result have been achieved? (Winning a game gives 1 point, drawing a game gives $1 / 2$ point, and losing a game gives 0 points.)


In the rectangular coordinate system \( xOy \), find the area of the graph formed by all points \( (x, y) \) that satisfy \( \lfloor x \rfloor \cdot \lfloor y \rfloor = 2013 \), where \( \lfloor x \rfloor \) represents the greatest integer less than or equal to the real number \( x \).