A feature film with a duration of 192 minutes consists of four parts. It is known that the duration of any two parts differs by at least 6 minutes. What is the maximum duration the shortest part can have? Express the answer in minutes.


The general term of the sequence is given by \( a_{n} = b[\sqrt{n+c}] + d \). The sequence is calculated successively as follows:
\[ 1, 3, 3, 3, 5, 5, 5, 5, 5, \cdots \]

In this sequence, each positive odd number \( m \) appears exactly \( m \) times consecutively. Determine the value of \( b + c + d \), where \( b, c, \) and \( d \) are integers to be determined.


As shown in Figure 1, in rectangle \(ABCD\), \(AB = 3\), \(AD = 4\), and \(E\) is a point on \(AB\) with \(AE = 1\). Now fold \(\triangle BCE\) along \(CE\) so that the projection of point \(B\) onto plane \(AECD\) falls on \(AD\). Find the volume of the tetrahedron \(B-AECD\).


As shown in the figure, consider the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) with \(a > b > 0\). The left and right foci of the ellipse are \(F_{1}\) and \(F_{2}\), respectively. The line passing through \(F_{2}\) intersects the ellipse at the points \(A(x_{1}, y_{1})\) and \(B(x_{2}, y_{2})\). If the area of the inscribed circle in triangle \(A F_{1} B\) is \(\pi\) and \(\left| y_{1} - y_{2} \right| = 4\), what is the eccentricity of the ellipse?


Given the ellipse \(C: \frac{x^{2}}{2} + y^{2} = 1\) with foci \(F_{1}\) and \(F_{2}\), and a point \(P(x_{0}, y_{0})\) satisfying \(0 < \frac{x_{0}^{2}}{2} + y_{0}^{2} \leq 1\), find the range of the value \(\left|P F_{1}\right| + \left|P F_{2}\right|\).

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


In triangle $ABC$, $AB = c$, $AC = b > c$, and $AD$ is the angle bisector. A line through point $D$ perpendicular to $AD$ intersects $AC$ at point $E$. Find $AE$.


The real numbers \( x_1, x_4, y_1, y_2 \) are positive and the real numbers \( x_2, x_3, y_3, y_4 \) are negative. We have \( (x_i - a)^2 + (y_i - b)^2 \leq c^2 \) for \( i = 1, 2, 3, 4 \). Show that \( a^2 + b^2 \leq c^2 \). State the result in geometric language.

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


Find two integers A and B, located on a number line at a distance of 1 from each other, that satisfy the property: a root of the equation \(x^{3} + 3x^{2} - x + 1 = 0\) belongs to the interval \((A, B)\).

Solution: \boxed{A = -4, B = -3}


If we select $a_1, a_2, a_3$ from the numbers $1, 2, \cdots, 14$ in increasing order, such that $a_2 - a_1 \geq 3$ and $a_3 - $a_2 \geq 3$, how many different ways can this be done?


In an isosceles right triangle \( \triangle ABC \), \( CA = CB = 1 \). Let point \( P \) be any point on the boundary of \( \triangle ABC \). Find the maximum value of \( PA \cdot PB \cdot PC \).


Two trains depart simultaneously from two cities and travel towards each other. They meet after 3.3 hours. If train A departs 24 minutes earlier, then after train B has traveled for 3 hours, the two trains will still need to travel 14 kilometers to meet. If train B departs 36 minutes earlier, then after train A has traveled for 3 hours, the two trains will still need to travel 9 kilometers to meet. How far apart are the two cities? (in kilometers)

Solution: \boxed{660 \text{ km}}


Given that \( I \) is the incenter of \( \triangle ABC \), and
\[ 9 \overrightarrow{CI} = 4 \overrightarrow{CA} + 3 \overrightarrow{CB}. \]

Let \( R \) and \( r \) be the circumradius and inradius of \( \triangle ABC \), respectively. Find \(\frac{r}{R} = \).


What is the largest number of acute angles that can occur in a convex polygon?


For what smallest natural number \( n \) do there exist natural numbers \( a \) and \( b \) such that

\[ \gcd(a, b) = 999 \text{ and } \operatorname{lcm}(a, b) = n! \]

(where \( n! = 1 \cdot 2 \cdot \ldots \cdot n \))?


For which values of the parameter \(a\) does the equation \(x^{4} - 40 x^{2} + 144 = a(x^{2} + 4x - 12)\) have exactly three distinct solutions?


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?


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 \( x \).


There are 37 ones written on a board. Every minute, Karlson erases two arbitrary numbers and writes their sum on the board. Then, he eats a number of candies equal to the product of the two erased numbers. What is the maximum number of candies he could eat in 37 minutes?


Given a parallelogram with points $M$ on side $AB$ and $N$ on side $CD$ such that $A M: M B = C N: N D = m$. The diagonal $AC$ intersects lines $DM$ and $BN$ at points $R$ and $S$ respectively. What is the ratio $\frac{AC}{RS}$?

Solution: \boxed{2m + 1}


Arrange the numbers $1, 2, \cdots, 6$ in any order. If the number $k$ appears exactly in the $k$-th position, it is called a match. What is the expected number of matches?


Define \( f(n, k) \) as the number of ways to distribute \( k \) chocolates to \( n \) children such that each child receives 0, 1, or 2 chocolates. For example, \( f(3,4) = 6 \), \( f(3,6) = 1 \), and \( f(3,7) = 0 \).

(a) List all 6 ways to distribute 4 chocolates to 3 children, with each child receiving a maximum of two chocolates.

(b) Given 2015 children, verify that \( f(2015, k) = 0 \) for any \( k \) greater than or equal to an appropriate value.

(c) Show that the equation

\[
f(2016, k) = f(2015, k) + f(2015, k-1) + f(2015, k-2)
\]

is true for all positive integers \( k \geq 2 \).

(d) Compute the value of the expression

\[
f(2016, 1) + f(2016, 4) + f(2016, 7) + \ldots + f(2016, 4027) + f(2016, 4030)
\]


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


Find the smallest natural numbers \( a \) and \( b \) (where \( b > 1 \)), such that \( \sqrt{a \sqrt{a \sqrt{a}}} = b \).

Solution: \boxed{a=256, b=128}


How many positive five-digit integers exist such that the product of their five digits is 900?


We have polynomials \( p(x) = ax^2 + bx + c \) and \( P(x) = Ax^2 + Bx + C \). We know that for any \( x \), substituting \( p(x) \) into \( P \) gives \( P(p(x)) = x \). What is \( p(P(x)) \) equal to?

Solution: \boxed{p(P(x)) = x}


Given a linear function \( f(x) \), it is known that the distance between the points of intersection of the graphs \( y = x^2 - 1 \) and \( y = f(x) \) is \( \sqrt{34} \), and the distance between the points of intersection of the graphs \( y = x^2 + 1 \) and \( y = f(x) + 3 \) is \( \sqrt{42} \). Find the distance between the points of intersection of the graphs of the functions \( y = x^2 \) and \( y = f(x) - 1 \).


Poplar, willow, locust, birch, and phoenix trees are planted in a row, with a distance of 1 meter between any two adjacent trees. The distance between the poplar and both the willow and locust trees is equal. The distance between the birch and both the poplar and locust trees is equal. Find the distance between the phoenix tree and the birch tree in meters.


Let $n \in \mathbb{N}$. We define

$$
S=\left\{(x, y, z) \in\{0, \ldots, n\}^{3} \mid x+y+z>0\right\}
$$

a set of $(n+1)^{3}-1$ points in three-dimensional space. Determine the minimal number of planes whose union contains $S$ but not the point $(0,0,0)$.


Find the four-digit number that is a perfect square, where the thousands digit is the same as the tens digit, and the hundreds digit is 1 greater than the units digit.


Inside the circle $\omega$ are two intersecting circles $\omega_{1}$ and $\omega_{2}$, which intersect at points $K$ and $L$, and are tangent to the circle $\omega$ at points $M$ and $N$. It turns out that the points $K$, $M$, and $N$ are collinear. Find the radius of the circle $\omega$, given that the radii of the circles $\omega_{1}$ and $\omega_{2}$ are 3 and 5, respectively.


In the equation \( x5 \cdot \overline{3yz} = 7850 \), find \( x, y, z \).

Solution: \boxed{x=2, y=1, z=4}


Given the function \( f(x) = x^{3} + a x^{2} + x + 1 \) (with \( a \in \mathbf{R} \)) is decreasing in the interval \(\left(-\frac{2}{3}, -\frac{1}{3}\right)\) and increasing in the interval \(\left(-\frac{1}{3}, +\infty\right)\), determine the value of \( a \).


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


Given the function
$$
f(x) = 3 \lg (x+2) - \log_{2} x,
$$

and the set \( M = \{ n \in \mathbf{Z} \mid f(n^3 - 3n^2 - 10n + 32) \geq 0 \} \), find \( M \).

Solution: \boxed{\{-3, 2, 3, 4\}}


From the integers $1, 2, \cdots, 20$, three different numbers are randomly selected. What is the probability that these three numbers form an arithmetic sequence?


Let point \( P \) be inside triangle \( \triangle ABC \), and
$$
\overrightarrow{AP} = \frac{1}{3} \overrightarrow{PB} + \frac{1}{4} \overrightarrow{PC}.
$$
Then, find \( \frac{S_{\triangle PBC}}{S_{\triangle ABC}} \).


Given distinct natural numbers \(a, b, c, d\) that satisfy the following conditions: \(a > d\), \(ab = cd\), and \(a + b + c + d = ac\). Find the sum of all four numbers.


Arrange the fractions $\frac{1}{2}$, $\frac{1}{3}$, $\frac{1}{4}$, $\frac{1}{5}$, $\frac{1}{6}$, $\frac{1}{7}$, and the average of these 6 fractions in ascending order. Determine the position of this average value in the order.


For transportation between points located hundreds of kilometers apart on the Earth's surface, people of the future will likely dig straight tunnels through which capsules will travel frictionlessly under the influence of Earth's gravity. Let points \( A, B, \) and \( C \) lie on the same meridian, with the surface distance from \( A \) to \( B \) related to the surface distance from \( B \) to \( C \) in the ratio \( m : n \). A capsule travels through the tunnel \( AB \) in approximately 42 minutes. Estimate the travel time through tunnel \( AC \). Provide the answer in minutes.

Solution: \boxed{42 \ \text{minutes}}


In a class, the percentage of girls is more than 25% but less than 30%. What is the minimum number of students that can be in such a class?


In a trapezoid with bases respectively equal to \(a\) and \(b\) (\(a > b\)), a line parallel to the bases is drawn through the intersection point of the diagonals. Find the length of the segment of this line intercepted by the non-parallel sides of the trapezoid.

Given (Fig. 33): \(ABCD\) is a trapezoid, \(|AD| = a\), \(|BC| = b\), \([AC]\) and \([BD]\) are diagonals, \((PQ) \parallel (AD)\), \(M \in (PQ)\), \(M = [AC] \cap [BD]\), \(P \in [AB]\), \(Q \in [CD]\).

Find: \(|PQ|\).


How many paths are there on $\mathbb{Z}^{2}$ starting from $(0,0)$, taking steps of $+(1,0)$ or $+(0,1)$, and ending at $(m, n)$, where $m, n \geqslant 0$?

Solution: \boxed{\binom{m+n}{m}}


Lines parallel to the sides of a square form a small square whose center coincides with the center of the original square. It is known that the area of the cross, formed by the small square, is 17 times larger than the area of the small square. By how many times is the area of the original square larger than the area of the small square?


The difference between the highest common factor and the lowest common multiple of \( x \) and 18 is 120. Find the value of \( x \).


A copper wire with cross-sectional area $A$ carries a current $I$ during a short circuit. Initially, the wire's temperature was $T_{0}$. Estimate how long it will take for the wire to melt. Ignore radiation losses. The resistivity of the material depends on temperature: $\varrho=\varrho_{0}\left[1+\alpha\left(T-T_{0}\right)\right]$, where $\varrho_{0}$ is the resistivity at temperature $T_{0}$ and $\alpha$ is the temperature coefficient of the material. The temperature $T$ is given in Kelvin. Data: $\alpha=\frac{1}{235} \ {}^{\circ} \mathrm{C}^{-1}$, $\varrho_{0}=1.65 \cdot 10^{-8} \Omega \mathrm{m}$, $A=2 \mathrm{~mm}^{2}$, $I=120 \mathrm{~A}$, $T_{0}=313 \mathrm{~K}$.

Solution: \boxed{23 \text{ s}}


Calculate \(\operatorname{tg} \alpha\) if \(3 \operatorname{tg} \alpha - \sin \alpha + 4 \cos \alpha = 12\).


Xiao Wang and Xiao Li have to process the same number of identical parts and start working at the same time. It is known that Xiao Wang processes 15 parts per hour and must rest for 1 hour after every 2 hours of work. Xiao Li works continuously, processing 12 parts per hour. Both of them finish at the exact same time. How many parts did Xiao Wang process?


Calculate the value of \( (1 \div 2) \div (2 \div 3) \div (3 \div 4) \div \cdots \div (2019 \div 2020) \div (2020 \div 2021) \).


A thermally insulated vessel is divided into two parts by a heat-conducting partition. The first part contains helium at a temperature of \(127{ }^{\circ} \mathrm{C}\) with a quantity of \(v_{1}=0.2\) moles. The second part contains helium at a temperature of \(7{ }^{\circ} \mathrm{C}\) with a quantity of \(v_{2}=0.8\) moles.

1) What temperature (in degrees Celsius) will be established in the vessel after reaching thermal equilibrium?
2) Find the ratio of the final to the initial pressures in the first part of the vessel.


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

$$
A=\left(3\left(x_{1}+\ldots+x_{n}\right)-5\left(y_{1}+\ldots+y_{n}\right)\right) \cdot\left(5\left(x_{1}+\ldots{x_{n}}\right)+3\left(y_{1}+\ldots{y_{n}}\right)\right)
$$


There were 320 seats in the club's auditorium, arranged in equal rows. After increasing the number of seats in each row by 4 and adding one more row, the auditorium had 420 seats. How many rows are there now in the auditorium?


As shown in Figure 3, in \(\triangle ABC\), \(O\) is the midpoint of side \(BC\). A line through point \(O\) intersects lines \(AB\) and \(AC\) at different points \(M\) and \(N\) respectively. If
$$
\begin{array}{l}
\overrightarrow{AB}=m \overrightarrow{AM}, \\
\overrightarrow{AC}=n \overrightarrow{AN},
\end{array}
$$

then the value of \(m+n\) is


The diagonals of a convex quadrilateral \(ABCD\) intersect at point \(E\). It is known that the area of each of the triangles \(ABE\) and \(DCE\) is 1, and the area of the entire quadrilateral does not exceed 4. Given that \(AD = 3\), find the length of side \(BC\).


The histogram shows the distribution of passenger cars by the number of passengers (in $4\%$ of cars there are from 10 to 19 passengers, in $6\%$ of cars - from 20 to 29, etc.). If a car has 60 or more passengers, we will call such a car overcrowded.

a) Find the proportion of overcrowded cars.

b) Find the minimum possible proportion of passengers traveling in overcrowded cars.

c) Can the proportion of passengers traveling in overcrowded cars be less than the proportion of overcrowded cars?

Solution: \boxed{\text{No}}


Given that out of 8 teams, there are 3 weak teams, these 8 teams are divided into two groups $A$ and $B$ with 4 teams in each group by drawing lots.

1. The probability that one of the groups $A$ or $B$ has exactly two weak teams.
2. The probability that group $A$ has at least two weak teams.


Given positive real numbers \( a, b, c \) such that \( abc = 1 \), show that

\[
\left(a - 1 + \frac{1}{b}\right)\left(b - 1 + \frac{1}{c}\right)\left(c - 1 + \frac{1}{a}\right) \leq 1
\]


Calculate the limit of the function:
$$\lim _{x \rightarrow 0} \frac{e^{\sin 2x} - e^{\sin x}}{\tan x}$$


The circle is divided into 30 equal parts by 30 points on the circle. Randomly selecting 3 different points, what is the probability that these 3 points form an equilateral triangle?


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

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


Solve the system

$$
\left\{\begin{array}{l}
x^{3}+3 y^{3}=11 \\
x^{2} y+x y^{2}=6
\end{array}\right.
$$

Calculate the values of the expression $\frac{x_{k}}{y_{k}}$ for each solution $\left(x_{k}, y_{k}\right)$ of the system and find the smallest among them. If necessary, round your answer to two decimal places.


Solution: \boxed{y^2 x^3 + 7x = 0}


In our country, personal income tax on wages and salaries is calculated in segments. If the total income does not exceed 800 yuan, the personal income tax on wages and salaries is exempted; for the part exceeding 800 yuan, tax is levied. Let the taxable monthly amount be \( x \), \( x = \) total monthly income - 800 yuan, and see the table below for the tax rate:

1. If the taxable amount is \( f(x) \), express the calculation formula for the tax amount \( f(x) \) in levels 1-3 using a piecewise function.
2. For a person whose total income for March 2000 is 3000 yuan, calculate how much personal income tax this person should pay in March.

Solution: \boxed{205 \text{ yuan}}


Find the smallest natural number $m$ for which there exists a natural number $n$ such that the sets of the last 2014 digits in the decimal representation of the numbers $a=2015^{3 m+1}$ and $b=2015^{6 n+2}$ are identical, with the condition that $a<b$.


The sequence $\left\{a_{n}\right\}$ is: $1,1,2,1,1,2,3,1,1,2,1,1,2,3,4,\cdots$. The sequence is formed as follows: start with $a_{1}=1$ and copy this term to obtain $a_{2}=1$. Then add the number 2 as $a_{3}=2$. Next, copy all terms $1,1,2$ and place them as $a_{4}, a_{5}, a_{6}$, then add the number 3 as $a_{7}=3$, and so on. What is $a_{2021}$?


Given \( x, y \in\left[-\frac{\pi}{4}, \frac{\pi}{4}\right] \) and \( a \in \mathbf{R} \), satisfy the equations
\[
\begin{cases}
x^{3}+\sin x-2a=0, \\
4y^{3}+\sin y \cos y+a=0.
\end{cases}
\]
Find the value of \( \cos (x + 2y) \).


In how many ways can we choose two different integers between -100 and 100 inclusive, so that their sum is greater than their product?


The germination rate of seeds is $90\%$. What is the probability that 5 out of 7 sown seeds will germinate?


Determine, with proof, the smallest positive integer \( n \) with the following property: For every choice of \( n \) integers, there exist at least two whose sum or difference is divisible by 2009.


Calculate the limit of the function:

\[
\lim _{x \rightarrow 0} \frac{1-\cos 2x}{\cos 7x-\cos 3x}
\]


In trapezoid \(ABCD\), the smaller base \(BC\) is equal to 3, and the legs \(AB\) and \(CD\) are each equal to 3. The diagonals of the trapezoid form an angle of \(60^\circ\) with each other. Find the base \(AD\).


Find the area of triangle ABC, whose vertices have coordinates A(0,0), B(1424233,2848467), C(1424234,2848469). Round the answer to two decimal places.


Let the sequence \( a_i \) be defined as \( a_{i+1} = 2^{a_i} \). Find the number of integers \( 1 \leq n \leq 1000 \) such that if \( a_0 = n \), then 100 divides \( a_{1000} - a_1 \).


Winnie-the-Pooh, Owl, Rabbit, and Piglet ate 70 bananas, with each of them getting at least one banana. Winnie-the-Pooh ate more than each of the others. Owl and Rabbit together ate 45 bananas. How many bananas did Piglet eat?


The lengths of the diagonals of a rhombus are in the ratio $3:4$. By how many times is the area of the rhombus larger than the area of the circle inscribed in it?


Person A and Person B start running at the same time and from the same point on a circular track, but in opposite directions. Person A's speed is 180 meters per minute, and Person B's speed is 240 meters per minute. In 30 minutes, they meet 24 times. What is the maximum possible length of the track in meters?

Solution: \boxed{525 \text{ meters}}


Given a line passing through the origin with a positive slope intersecting the ellipse \(\frac{x^{2}}{4}+y^{2}=1\) at points \(E\) and \(F\), and points \(A(2,0)\) and \(B(0,1)\). Find the maximum area of the quadrilateral \(A E B F\).


A 4x4 chessboard is painted such that 8 out of the 16 squares are black, with each row and column containing exactly 2 black squares. How many different ways are there to paint the chessboard? (Answer with a number)


How many 3-digit numbers are there with the property that the 2-digit number obtained by deleting the middle digit is equal to one ninth of the original 3-digit number?


Let \( a_{1}, a_{2}, \cdots, a_{2014} \) be a permutation of the positive integers \( 1, 2, \cdots, 2014 \). Define
\[ S_{k} = a_{1} + a_{2} + \cdots + a_{k} \quad (k=1, 2, \cdots, 2014). \]

What is the maximum number of odd numbers among \( S_{1}, S_{2}, \cdots, S_{2014} \)?


Find the rightmost non-zero digit of the expansion of \( (20)(13!) \).


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

For each black point: There are exactly two green points that are at a distance of 2020 from this black point.

Determine the smallest possible number of green points.


Compute the triple integral

$$
\iiint_{\Omega} \frac{x^{2}}{x^{2}+y^{2}} \, dx \, dy \, dz
$$

where the region $\Omega$ is bounded by the surfaces

$$
z = \sqrt{36 - x^{2} - y^{2}}, \quad z = \sqrt{\frac{x^{2} + y^{2}}{3}}
$$


Calculate the limit of the function:

$$\lim _{x \rightarrow 0} \frac{6^{2 x}-7^{-2 x}}{\sin 3 x-2 x}$$

Solution: \boxed{2 \ln 42}


Given integer \( n \geq 3 \), define \(\mathcal{A} = \left\{\{\mathrm{z}_{1}, \mathrm{z}_{2}, \ldots, \mathrm{z}_{n}\} \mid \mathrm{z}_{\mathrm{i}} \in \mathbb{C}, |\mathrm{z}_{\mathrm{i}}| = 1, 1 \leq \mathrm{i} \leq n\right\}\). Determine the value of \(\min_{\mathrm{A} \in \mathcal{A}}\left\{\max_{\substack{\mathrm{u} \in \mathbb{C} \\|\mathrm{u}|=1}}\left\{\prod_{\mathrm{z} \in \mathrm{A}}|\mathrm{u}-\mathrm{z}|\right\}\right\}\) and find all \(\mathrm{A} \in \mathcal{A}\) that attain this minimum value in the maximum expression.


In the diagram, \(ABCD\) is a parallelogram. \(E\) is on side \(AB\), and \(F\) is on side \(DC\). \(G\) is the intersection point of \(AF\) and \(DE\), and \(H\) is the intersection point of \(CE\) and \(BF\). Given that the area of parallelogram \(ABCD\) is 1, \(\frac{\mathrm{AE}}{\mathrm{EB}}=\frac{1}{4}\), and the area of triangle \(BHC\) is \(\frac{1}{8}\), find the area of triangle \(ADG\).


Let $A$ be a finite set of integers and $n \geq 1$. Let $N$ be the number of $n$-tuples $\left(x_{1}, \ldots, x_{n}\right)$ in $A^{n}$ satisfying $x_{1}+\cdots+x_{n}=-1$. Let $M$ be the number of $n$-tuples in $A^{n}$ satisfying $x_{1}+\cdots+x_{n}=-1$ and for all integers $i$ satisfying $1 \leq i \leq n-1$, $x_{1}+\cdots+x_{i} \geq 0$.

Show that $M = \frac{N}{n}$.

Solution: \boxed{M = \frac{N}{n}}


The vertices of a regular hexagon are labeled \(\cos (\theta), \cos (2 \theta), \ldots, \cos (6 \theta)\). For every pair of vertices, Bob draws a blue line through the vertices if one of these functions can be expressed as a polynomial function of the other (that holds for all real \(\theta\)), and otherwise Roberta draws a red line through the vertices. In the resulting graph, how many triangles whose vertices lie on the hexagon have at least one red and at least one blue edge?


In a certain class, there are 28 boys and 22 girls. If 5 students are to be elected to different class committee positions, and it's desired that both boys and girls are represented among the 5 students, how many different election outcomes are possible?


A pair of natural numbers \(a > b\) is called good if the LCM of these numbers is divisible by their difference. Among all natural divisors of the number \( n \), exactly one good pair was found. What can \( n \) be equal to?

Solution: \boxed{n = 2}


Given the function
\[ f(x)=\begin{cases}
2^{2-x}, & x<2, \\
\log_{3}(x+1), & x \geq 2,
\end{cases} \]
if the equation \( f(x)=m \) has two distinct real roots, then the range of real number \( m \) is (express in interval notation).


Let \( f(x) \) be a function defined on \(\mathbf{R}\) such that \( f(1) = 1 \) and for any \( x \in \mathbf{R} \), it holds that \( f(x+5) \geq f(x) + 5 \) and \( f(x+1) \leq f(x) + 1 \). If \( g(x) = f(x) + 1 - x \), find \( g(2002) \).

(Note: This problem is from the 2002 National High School Mathematics Competition in China.)


Solution: \boxed{(-3, -2)}


We have an infinite grid, and we need to write a positive integer in each cell with the following conditions. The number $n$ should appear exactly $n$ times (for example, the number 1 appears once, the number 2 appears twice, etc.), and the difference between numbers in any two adjacent cells (cells sharing a common side) should be less than a given $k$. What is the smallest integer $k$ for which this arrangement is possible?


a) What is the maximum number of edges in a 30-vertex graph that contains no triangles?

b) What is the maximum number of edges in a 30-vertex graph that contains no complete subgraph of four vertices?


Let \( S = \{ 1, 2, 3, \ldots, 280 \} \). Find the smallest integer \( n \) such that each \( n \)-element subset of \( S \) contains five numbers which are pairwise relatively prime.


Compute the limit of the function:

$$
\lim _{x \rightarrow 0}\left(\frac{1+\operatorname{tg} x \cdot \cos 2 x}{1+\operatorname{tg} x \cdot \cos 5 x}\right)^{\frac{1}{x^{3}}}
$$


Clara leaves home by bike at 1:00 p.m. for a meeting scheduled with Quinn later that afternoon. If Clara travels at an average of $20 \mathrm{~km} / \mathrm{h}$, she would arrive half an hour before their scheduled meeting time. If Clara travels at an average of $12 \mathrm{~km} / \mathrm{h}$, she would arrive half an hour after their scheduled meeting time. At what average speed, in $\mathrm{km} / \mathrm{h}$, should Clara travel to meet Quinn at the scheduled time?


Bernie has 2020 marbles and 2020 bags labeled \( B_{1}, \ldots, B_{2020} \) in which he randomly distributes the marbles (each marble is placed in a random bag independently). If \( E \) is the expected number of integers \( 1 \leq i \leq 2020 \) such that \( B_{i} \) has at least \( i \) marbles, compute the closest integer to \( 1000 E \).


Given \(0 < a < b\), lines \(l\) and \(m\) pass through two fixed points \(A(a, 0)\) and \(B(b, 0)\) respectively, intersecting the parabola \(y^{2} = x\) at four distinct points. When these four points lie on a common circle, find the locus of the intersection point \(P\) of lines \(l\) and \(m\).

Solution: \boxed{x = \frac{a+b}{2}}


All cells of an \( n \times n \) square grid are numbered in some order with numbers from 1 to \( n^{2} \). Petya makes moves according to the following rules. On the first move, he places a rook on any cell. On each subsequent move, Petya can either place a new rook on some cell or move the rook from a cell numbered \( a \) horizontally or vertically to a cell with a number greater than \( a \). Each time a rook lands on a cell, that cell is immediately colored; placing a rook on a colored cell is prohibited. What is the minimum number of rooks Petya will need to color all cells of the grid regardless of the initial numbering?

Solution: \boxed{n}


Two classes are planting trees. Each student in the first class plants 3 trees, and each student in the second class plants 5 trees. A total of 115 trees are planted. What is the maximum possible total number of students in both classes combined?


For which values of \(a\) does the equation \(|x-3| = a x - 1\) have two solutions? Enter the midpoint of the interval of parameter \(a\) in the provided field. Round the answer to three significant digits according to rounding rules and enter it in the provided field.


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


When young fishermen were asked how many fish each of them caught, the first one replied, "I caught half the number of fish that my friend caught, plus 10 fish." The second one said, "And I caught as many as my friend, plus 20 fish." How many fish did the fishermen catch?


What is the maximum number of closed intervals that can be given on the number line such that among any three intervals, there are two that intersect, but the intersection of any four intervals is empty?


For each real number \( x \), \( f(x) \) is defined to be the minimum of the values of \( 2x + 3 \), \( 3x - 2 \), and \( 25 - x \). What is the maximum value of \( f(x) \)?


The sum of two sides of a rectangle is 7 cm, and the sum of three of its sides is 9.5 cm. Find the perimeter of the rectangle.

Solution: \boxed{14 \text{ cm}}


Sasha chose a natural number \( N > 1 \) and wrote down all of its natural divisors in ascending order: \( d_{1}<\ldots<d_{s} \) (such that \( d_{1}=1 \) and \( d_{s}=N \)). Then for each pair of adjacent numbers, he calculated their greatest common divisor; the sum of the \( s-1 \) resulting numbers turned out to be equal to \( N-2 \). What values could \( N \) take?

Solution: \boxed{N = 3}


A palindrome is a string that does not change when its characters are written in reverse order. Let \( S \) be a 40-digit string consisting only of 0s and 1s, chosen uniformly at random out of all such strings. Let \( E \) be the expected number of nonempty contiguous substrings of \( S \) which are palindromes. Compute the value of \( \lfloor E \rfloor \).


Find the maximum value of \( S = \sin^2 \theta_1 + \sin^2 \theta_2 + \cdots + \sin^2 \theta_n \), where \( 0 \leq \theta_i \leq \pi \) and \( \theta_1 + \theta_2 + \cdots + \theta_n = \pi \).


Charlie folds a \(\frac{17}{2}\)-inch by 11-inch piece of paper in half twice, each time along a straight line parallel to one of the paper's edges. What is the smallest possible perimeter of the piece after two such folds?


\( AB \) is a diameter; \( BC \) and \( AC \) are chords, with \( \angle B C = 60^\circ \). \( D \) is the point where the extension of the diameter \( AB \) meets the tangent \( CD \). Find the ratio of the areas of triangles \( DCB \) and \( DCA \).


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?


Calculate \(\int_{0}^{1} e^{-x^{2}} \, dx\) to an accuracy of 0.001.


The three-digit number \(\overline{abc}\) consists of three non-zero digits. The sum of the other five three-digit numbers formed by rearranging \(a, b, c\) is 2017. Find \(\overline{abc}\).


A hotel has 5 distinct rooms, each with single beds for up to 2 people. The hotel has no other guests, and 5 friends want to stay there for the night. In how many ways can the 5 friends choose their rooms?


Neznayka, Doctor Pilyulkin, Knopochka, Vintik, and Znayka participated in a math contest. Exactly four out of them solved each problem. Znayka solved more problems than each of the others - 10 problems, while Neznayka solved fewer problems than each of the others - 6 problems. How many problems in total were there in the math contest?


Let \( S=\{1,2, \cdots, 10\} \). \( A_{1}, A_{2}, \cdots, A_{k} \) are subsets of \( S \) that satisfy the following conditions:

1. \( \left|A_{i}\right|=5 \) for \( i=1,2, \cdots, k \).
2. \( \left|A_{i} \cap A_{j}\right| \leqslant 2 \) for \( 1 \leqslant i < j \leqslant k \).

Find the maximum value of \( k \).


As shown in the figure, \( M \) and \( N \) are points on the diagonals \( AC \) and \( CE \) of the regular hexagon \( ABCDEF \), respectively. The points divide the diagonals such that \( AM:AC = CN:CE = r \). If the points \( B \), \( M \), and \( N \) are collinear, find the value of \( r \).


Find the minimum value of the algebraic expression $|\sin x+\cos x+\tan x+\cot x+\sec x+\csc x|$ where $x$ is a real number.

Solution: \boxed{2\sqrt{2} - 1}


The arithmetic mean of three two-digit natural numbers \( x, y, z \) is 60. What is the maximum value that the expression \( \frac{x + y}{z} \) can take?


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


There are eight cards, each with a natural number from 1 to 8. If three cards are drawn so that the sum of their numbers is 9, how many different ways can this be done?


How many common terms (terms with the same value) are there between the arithmetic sequences \(2, 5, 8, \cdots, 2015\) and \(4, 9, 14, \cdots, 2014\)?


The sequence $\left\{x_{n}\right\}: 1,3,3,3,5,5,5,5,5, \cdots$ is formed by arranging all positive odd numbers in increasing order, and each odd number $k$ appears consecutively $k$ times, where $k=1,3,5, \cdots$. If the general term formula of this sequence is given by $x_{n}=a[\sqrt{b n+c}]+d$, find the value of $a+b+c+d$.


Determine the value of the following expression with as little computation as possible:

$$
\frac{49 \cdot 91^{3}+338 \cdot 343^{2}}{66^{3}-176 \cdot 121}: \frac{39^{3} \cdot 7^{5}}{1331000}
$$


In a football championship with 16 teams, each team played with every other team exactly once. A win was awarded 3 points, a draw 1 point, and a loss 0 points. A team is considered successful if it scored at least half of the maximum possible points. What is the maximum number of successful teams that could have participated in the tournament?


Let \( P(x) \) be a polynomial of degree \( n \) such that for \( k = 1, 2, 4, \ldots, 2^{n} \), we have \( P(k) = \frac{1}{k} \). Find \( P(0) \).

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


Let \( x \) and \( y \) be complex numbers such that \( x + y = \sqrt{20} \) and \( x^2 + y^2 = 15 \). Compute \( |x - y| \).


Two of the lightest kittens weigh a total of 80 grams. Four of the heaviest kittens weigh 200 grams. The total weight of all the kittens is \(X\) grams. What is the smallest value of \(X\) for which it is not possible to uniquely determine the number of kittens born?


You need to find out a five-digit phone number by asking questions that can be answered with "yes" or "no". What is the minimum number of questions required to guarantee finding the number (assuming all answers are correct)?


Given that \( \boldsymbol{m} \) is a non-zero vector, \( \boldsymbol{n} \) is a unit vector, \( \boldsymbol{m} \neq \boldsymbol{n} \), and the angle between \( \boldsymbol{m} \) and \( \boldsymbol{m} - \boldsymbol{n} \) is \( 60^{\circ} \), with \( |\boldsymbol{m}| \in (0, a] \), find the minimum value of \( a \).


As shown in the diagram, there are two cubes of different sizes. The edge length of the larger cube is 6 times the edge length of the smaller cube. All 6 faces of the larger cube are painted red, and all 6 faces of the smaller cube are painted yellow. Then, the two cubes are glued together. What is the ratio of the red surface area to the yellow surface area on the resulting combined figure?


Optimus Prime departs in robot form from point $A$ to point $B$ and arrives at $B$ on time. If he transforms into a car from the beginning, his speed increases by $\frac{1}{4}$, allowing him to arrive at $B$ 1 hour earlier. If he travels 150 km in robot form first, then transforms into a car, and his speed increases by $\frac{1}{5}$, he can arrive 40 minutes earlier. How far apart are points $A$ and $B$ in kilometers?


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


A train has 18 identical cars. In some of the cars, half of the seats are free, in others, one third of the seats are free, and in the remaining cars, all the seats are occupied. In the entire train, exactly one ninth of all seats are free. How many cars have all seats occupied?


Find the sum of the coefficients of the even-degree terms in the polynomial obtained from the expression \( f(x) = \left( x^3 - x + 1 \right)^{100} \) after expanding and combining like terms.


In a regular 1000-gon, all diagonals are drawn. What is the maximum number of diagonals that can be selected such that among any three of the chosen diagonals, at least two have the same length?


If the four-digit number \(5ab4\) is a perfect square, then \(a + b =\)


The entire area is divided into squares, denoted by two integer indices \( M \) and \( N \) such that, for example, a point with coordinates \( x=12.25, y=20.9 \) is in the square numbered \([12; 20]\), and a point with coordinates \( x=-12.34, y=0.1239 \) is in the square numbered \([-13; 0]\), and so on. A mysterious object moves in the plane \( Oxy \) along the trajectory \( y=\left(\left(\left(x^{5}-2013\right)^{5}-2013\right)^{5}-2013\right)^{5} \), and a radar beam on the ground is directed along the line \( y=x+2013 \). Specify the numbers of all squares in which the radar will detect the appearance of the mysterious object.

Solution: \boxed{[4; 2017]}


Observe the following operation:
If \(\overline{\mathrm{abc}}\) is a three-digit number, then since \(\overline{\mathrm{abc}} = 100a + 10b + c = 99a + 9b + (a + b + c)\),
Therefore, if \(a + b + c\) is divisible by 9, \(\overline{\mathrm{abc}}\) is divisible by 9.
This conclusion can be generalized to any number of digits.

Using this conclusion, solve the following questions:
1. \(N\) is a 2011-digit number, and each digit is 2. Find the remainder when \(N\) is divided by 9.
2. \(N\) is an \(n\)-digit number, and each digit is 7. \(n\) leaves a remainder of 3 when divided by 9. Find the remainder when \(N\) is divided by 9.


The side of a triangle is $\sqrt{2}$, and the angles adjacent to it are $75^{\circ}$ and $60^{\circ}$. Find the segment connecting the bases of the altitudes dropped from the vertices of these angles.


Let \(a, b, c\) be integers. Define \(f(x) = ax^2 + bx + c\). Suppose there exist pairwise distinct integers \(u, v, w\) such that \(f(u) = 0\), \(f(v) = 0\), and \(f(w) = 2\). Find the maximum possible value of the discriminant \(b^2 - 4ac\) of \(f\).


Person A and Person B start from point A to point B at the same time. If both travel at a constant speed, Person A takes 4 hours to complete the journey, and Person B takes 6 hours. When the remaining distance for Person B is 4 times the remaining distance for Person A, how many hours have they been traveling?


Dima's mother told him he needed to eat 13 spoons of porridge. Dima told his friend that he ate 26 spoons of porridge. Each subsequent child, when talking about Dima's feat, increased the number of spoons by 2 or 3 times. Eventually, one of the children told Dima's mother that Dima ate 33,696 spoons of porridge. How many times in total, including Dima, did the children talk about Dima's feat?


The judging panel for the Teen Singer Grand Prix consists of several judges. Each judge can award a maximum score of 10 points to a singer. After the first singer's performance, the following scoring situation was observed: The average score given by all judges was 9.64 points; if the highest score was removed, the average score given by the remaining judges was 9.60 points; if the lowest score was removed, the average score given by the remaining judges was 9.68 points. Therefore, the minimum possible lowest score given by the judges is $\qquad$ points, and the total number of judges in this Grand Prix is $\qquad$.


What is the smallest square number that, when divided by a cube number, results in a fraction in its simplest form where the numerator is a cube number (other than 1) and the denominator is a square number (other than 1)?


Determine the smallest positive value of \( a \) for which the equation

$$
\frac{\frac{x-a}{2}+\frac{x-2a}{3}}{\frac{x+4a}{5}-\frac{x+3a}{4}}=\frac{\frac{x-3a}{4}+\frac{x-4a}{5}}{\frac{x+2a}{3}-\frac{x+a}{2}}
$$

has an integer root.


An empty swimming pool was filled with water by two taps \(A\) and \(B\), both with constant flow rates. For 4 hours, both taps were open and filled 50% of the pool. Then, tap \(B\) was closed and for the next 2 hours, tap \(A\) filled 15% of the pool's volume. After this period, tap \(A\) was closed and tap \(B\) was opened. How long did tap \(B\) need to remain open to fill the pool by itself?


I place \( n \) pairs of socks (thus \( 2n \) socks) in a line such that the left sock is to the right of the right sock for each pair. How many different ways can I place my socks in this manner?


Find the general solution of the equation \( y'' - 5y' + 6y = 0 \) and identify the particular solution that satisfies the initial conditions \( y = 1 \) and \( y' = 2 \) at \( x = 0 \).


Find the sum of all four-digit numbers in which the digits $0, 4, 5, 9$ are absent.


Compute the lengths of the arcs of the curves given by the equations in the rectangular coordinate system.

$$
y=\arccos \sqrt{x}-\sqrt{x-x^{2}}+4, \quad 0 \leq x \leq \frac{1}{2}
$$


Given the set \( I = \{1, 2, \ldots, 2020\} \):
$$
\begin{array}{l}
W = \{w(a, b) = (a + b) + ab \mid a, b \in I\} \cap I, \\
Y = \{y(a, b) = (a + b) \cdot ab \mid a, b \in I\} \cap I, \\
X = W \cap Y,
\end{array}
$$

where \( W \) is the 'Wu' set, \( Y \) is the 'Yue' set, and \( X \) is the 'Xizi' set. Find the sum of the maximum and minimum numbers in the set \( X \).


If \( 0 < x < \frac{\pi}{2} \) and \( \sin x - \cos x = \frac{\pi}{4} \) and \( \tan x + \frac{1}{\tan x} = \frac{a}{b - \pi^c} \), where \( a, b \), and \( c \) are positive integers, find the value of \( a + b + c \).


On the board, the natural numbers \(a, b, c, d\) are written. It is known that among the six sums

\[
a+b, \quad b+c, \quad c+d, \quad d+a, \quad a+c, \quad b+d
\]

three are equal to 23, and the other three are equal to 34.

(a) (1 point) What is the value of \(a+b+c+d\)?

(b) (3 points) What is the smallest of the numbers \(a, b, c, d\)?


Let \( S \) be a set of 2017 distinct points in the plane. Let \( R \) be the radius of the smallest circle containing all points in \( S \) on either the interior or boundary. Also, let \( D \) be the longest distance between two of the points in \( S \). Let \( a, b \) be real numbers such that \( a \leq \frac{D}{R} \leq b \) for all possible sets \( S \), where \( a \) is as large as possible and \( b \) is as small as possible. Find the pair \( (a, b) \).

Solution: \boxed{(\sqrt{3}, 2)}


Assume integers \( u \) and \( v \) satisfy \( 0 < v < u \), and let \( A \) be \((u, v)\). Points are defined as follows: \( B \) is the reflection of \( A \) over the line \( y = x \), \( C \) is the reflection of \( B \) over the \( y \)-axis, \( D \) is the reflection of \( C \) over the \( x \)-axis, and \( E \) is the reflection of \( D \) over the \( y \)-axis. The area of pentagon \( ABCDE \) is 451. Find \( u+v \).


How many ordered pairs of integers \((x, y)\) satisfy \(x^{2} \leq y \leq x+6\)?


It is known that for real numbers \(a\) and \(b\), the following equalities hold:

\[
a^3 - 3ab^2 = 11, \quad b^3 - 3a^2b = 2
\]

What values can the expression \(a^2 + b^2\) take?


Find the minimum value of \(\sum_{k=1}^{100} |n - k|\), where \(n\) ranges over all positive integers.


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


Given the equation

\[ x^{2} + p x + q = 0 \]

with roots \( x_{1} \) and \( x_{2} \), determine the coefficients \( r \) and \( s \) as functions of \( p \) and \( q \) such that the equation

\[ y^{2} + r y + s = 0 \]

has roots \( y_{1} \) and \( y_{2} \) which are related to the roots \( x_{1} \) and \( x_{2} \) by

\[ y_{1} = \frac{x_{1}}{x_{1}-1} \quad \text{and} \quad y_{2} = \frac{x_{2}}{x_{2}-1} \]

What condition must hold between \( p \) and \( q \) for the equations (1) and (2) to be identical?

Solution: \boxed{p + q = 0}


The lateral edges of a triangular pyramid are pairwise perpendicular and have lengths $a, b,$ and $c$. Find the volume of the pyramid.

Solution: \boxed{\frac{a b c}{6}}


There are $N$ ($N \geqslant 9$) distinct non-negative real numbers less than 1 written on the blackboard. It is known that for any eight numbers on the blackboard, there exists another number on the blackboard such that the sum of these nine numbers is an integer. Find all possible values of $N$.


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?


Sarah is deciding whether to visit Russia or Washington, DC for the holidays. She makes her decision by rolling a regular 6-sided die. If she gets a 1 or 2, she goes to DC. If she rolls a 3, 4, or 5, she goes to Russia. If she rolls a 6, she rolls again. What is the probability that she goes to DC?


A surveillance service will be installed in a park in the form of a network of stations. The stations must be connected by telephone lines, so that any one of the stations can communicate with all the others, either through a direct connection or through at most one other station.

Each station can be directly connected by a cable to at most three other stations. What is the largest number of stations that can be connected in this way?


In the plane Cartesian coordinate system \( xOy \), the set \( S=\left\{(x, y) \mid (|x|+|y|-1)\left(x^{2}+y^{2}-1\right) \leqslant 0\right\} \) represents a plane region. Find the area of this region.

Solution: \boxed{\pi - 2}


Inside an equilateral triangle, a point \( M \) is taken, which is at distances \( b, c, \) and \( d \) from its sides. Find the height of the triangle.

Solution: \boxed{b + c + d}


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

(A. Yakubov)


Out of the four inequalities \(2x > 70\), \(x < 100\), \(4x > 25\), and \(x > 5\), two are true and two are false. Find the value of \(x\), given that it is an integer.

Solution: \boxed{x = 6}


What is the largest integer \( k \) whose square \( k^2 \) is a factor of \( 10! \)?


A cube with a side length of 20 is divided into 8000 unit cubes, and a number is written in each small cube. It is known that in every column of 20 cubes parallel to the edge of the cube, the sum of the numbers is 1 (columns in all three directions are considered). In a certain unit cube, the number 10 is written. Through this cube, there are three $1 \times 20 \times 20$ layers parallel to the faces of the large cube. Find the sum of all numbers outside these layers.


Calculate the limit of the numerical sequence:

\[
\lim _{n \rightarrow \infty} \frac{\sqrt{n-1}-\sqrt{n^{2}+1}}{\sqrt[3]{3 n^{3}+3}+\sqrt[3]{n^{5}+1}}
\]


Given the convex quadrilateral \(ABCD\) with an area of 1 unit. Reflect vertex \(A\) over \(B\), \(B\) over \(C\), \(C\) over \(D\), and \(D\) over \(A\). What is the area of the quadrilateral formed by the reflected points?


The vertices of a rhombus are located on the sides of a parallelogram, and the sides of the rhombus are parallel to the diagonals of the parallelogram. Find the ratio of the areas of the rhombus and the parallelogram if the ratio of the diagonals of the parallelogram is \( k \).


A two-digit number ' $a b$ ' is multiplied by its reverse ' $b a$ '. The ones (units) and tens digits of the four-digit answer are both 0.
What is the value of the smallest such two-digit number ' $a b$ '?


Given that \( |\overrightarrow{AB}| = 10 \), if a point \( P \) on the plane satisfies \( |\overrightarrow{AP} - t \overrightarrow{AB}| \geq 3 \) for any \( t \in \mathbf{R} \), find the minimum value of \( \overrightarrow{PA} \cdot \overrightarrow{PB} \) and the corresponding value of \( |\overrightarrow{PA} + \overrightarrow{PB}| \).


Find the least positive integral value of \( n \) for which the equation
\[ x_{1}^{3} + x_{2}^{3} + \cdots + x_{n}^{3} = 2002^{2002} \]
has integer solutions \(\left(x_{1}, x_{2}, x_{3}, \cdots, x_{n}\right)\).


Suppose \(x\) and \(y\) are integers such that
\[
(x-2004)(x-2006)=2^{y}.
\]
Find the largest possible value of \(x+y\).


Show that for all \(a, b, c \geq 0\) such that \(abc = 1\), \(\left(a^2 + 1\right)\left(b^3 + 2\right)\left(c^6 + 5\right) \geq 36\).


A rectangular table with dimensions $x$ cm $\times 80$ cm is covered with identical sheets of paper measuring 5 cm $\times 8$ cm. The first sheet is placed in the bottom-left corner, and each subsequent sheet is placed one centimeter higher and one centimeter to the right of the previous sheet. The last sheet is positioned in the top-right corner. What is the length $x$ in centimeters?


In how many ways can two disjoint subsets be selected from a set with $n$ elements?


Two circles touch each other internally at point K. Chord \( AB \) of the larger circle touches the smaller circle at point L, with \( A L = 10 \). Find \( B L \) given that \( A K : B K = 2 : 5 \).


Define a sequence of polynomials as follows: let \( a_{1}=3 x^{2}-x \), let \( a_{2}=3 x^{2}-7 x+3 \), and for \( n \geq 1 \), let \( a_{n+2}=\frac{5}{2} a_{n+1}-a_{n} \). As \( n \) tends to infinity, what is the limit of the sum of the roots of \( a_{n} \)?


Find the maximum value of the expression \((\sqrt{36-4 \sqrt{5}} \sin x-\sqrt{2(1+\cos 2 x)}-2) \cdot (3+2 \sqrt{10-\sqrt{5}} \cos y-\cos 2 y)\). If the answer is not an integer, round it to the nearest whole number.


How many integers \( b \) exist for which the equation \( x^2 + b x - 9600 = 0 \) has an integer solution that is divisible by both 10 and 12? Indicate the largest possible \( b \).


A 10,000-meter race involves three participants: A, B, and C. A starts last. During the race, A exchanges positions with B and C a total of 9 times. Determine the final position of A in the race.


Given that for any \( x \geqslant 1 \), the following inequality holds:
\[
\ln x - a\left(1 - \frac{1}{x}\right) \geqslant 0.
\]

Find the range of the real number \( a \).

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


On an infinite tape, numbers are written in a row. The first number is one, and each subsequent number is obtained by adding the smallest non-zero digit of its decimal representation to the previous number. How many digits are in the decimal representation of the number that is in the $9 \cdot 1000^{1000}$-th place in this sequence?


Given \(a\) and \(b\) are real numbers, satisfying:
\[
\sqrt[3]{a} - \sqrt[3]{b} = 12, \quad ab = \left( \frac{a + b + 8}{6} \right)^3.
\]

Find \(a - b\).


In a drawer, there are 5 distinct pairs of socks. Four socks are drawn at random. The probability of drawing two pairs is one in $n$. Determine the value of $n$.


Given the function \( f(x) = x + \frac{4}{x} - 1 \), if there exist \( x_{1}, x_{2}, \cdots, x_{n} \in \left[ \frac{1}{4}, 4 \right] \) such that
$$
\sum_{i=1}^{n-1} f(x_{i}) = f(x_{n}),
$$
then the maximum value of the positive integer \( n \) is...


a) In how many different ways can we paint the circles in the letter "O," not necessarily using all 3 colors?

b) In how many different ways can we paint the circles in the letter "E," necessarily using all 3 colors?

c) In how many different ways can we paint the letter "B" such that circles connected by a segment must have different colors? (The three circles aligned vertically have TWO segments separating them).


Let \( f: \mathbf{N}^{*} \rightarrow \mathbf{N}^{\top} \), and for all \( m, n \in \mathbf{N}^{\top} \), \( f(f(m) + f(n)) = m + n \). Find \( f(2005) \).


How many solutions does the equation \( x_1 + x_2 + x_3 = 1000 \) have
a) in natural numbers?
b) in non-negative integers?


In an integer triangle, two sides are equal to 10. Find the third side, given that the radius of the inscribed circle is an integer.


A cuckoo clock is on the wall. At the beginning of every hour, the cuckoo makes a number of "cuckoo" sounds equal to the hour displayed by the hour hand (for example, at 19:00 the cuckoo makes 7 sounds). One morning, Maxim approached the clock when it showed 9:05. He started turning the minute hand until he moved the time forward by 7 hours. How many times did the cuckoo make a sound during this time?


The area of the base of a right triangular prism is 4, and the areas of the lateral faces are 9, 10, and 17. Find the volume of the prism.


Find all functions \( f: \mathbf{Z}^{*} \rightarrow \mathbf{R} \) (where \(\mathbf{Z}^{*}\) is the set of positive integers) that satisfy the equation \( f(n+m)+f(n-m)=f(3n) \) for \( m, n \in \mathbf{Z}^{*} \) and \( n \geq m \).

Solution: \boxed{f(n) = 0}


The sum of the lengths of all of the edges of rectangular prism \( ABCDEFGH \) is 24. If the total surface area of the prism is 11, determine the length of the diagonal \( AH \).


Find all values of \( a \) for which the equation \( a^{2}(x-2) + a(39-20x) + 20 = 0 \) has at least two distinct roots.


Given that a parabola \( P \) has the center of the ellipse \( E \) as its focus, \( P \) passes through the two foci of \( E \), and \( P \) intersects \( E \) at exactly three points, determine the eccentricity of the ellipse.


Show that the fractional part of \( (2 + \sqrt{3})^n \) tends to 1 as \( n \) tends to infinity.


Find all pairs of integers \((x, y)\) that are solutions to the equation

$$
7xy - 13x + 15y - 37 = 0.
$$

Indicate the sum of all found values of \(x\).


A $12 \times 12$ grid is colored such that each cell is either black or white. Ensure that within any $3 \times 4$ or $4 \times 3$ subgrid, there is at least one black cell. Find the minimum number of black cells needed.


Let $A, B, C,$ and $D$ be four cocyclic points (in this order) on a circle with center $O$. Suppose that $(AC) \perp (BD)$. Let $I$ be the foot of the perpendicular from $O$ in triangle $AOD$. Show that $OI = \frac{BC}{2}$.

Solution: \boxed{\frac{B C}{2}}


If \( a \) is the smallest cubic number divisible by 810, find the value of \( a \).


Find a curve passing through the point $(0, -2)$ such that the tangent of the angle of inclination of the tangent at any of its points equals the ordinate of that point increased by three units.

Solution: \boxed{y = e^x - 3}


Given the complex numbers \( z = \cos \alpha + i \sin \alpha \) and \( u = \cos \beta + i \sin \beta \), and that \( z + u = \frac{4}{5} + \frac{3}{5} i \), find \( \tan(\alpha + \beta) \).


布尔图 rode a motorcycle from home to pick peaches and observe monkeys at Huaguo Mountain. He rode slowly and enjoyed the scenery along the way. Upon reaching the top, he found no peaches and was attacked by monkeys. He immediately rode back home at a speed of 42 kilometers per hour. If the average speed for the entire journey up and down the mountain was 21 kilometers per hour, what was his speed in kilometers per hour when going up the mountain?


Point \( K \) lies on edge \( AB \) of pyramid \( ABCD \). Construct the cross-section of the pyramid with a plane passing through point \( K \) parallel to lines \( BC \) and \( AD \).

Solution: \boxed{K L M N}


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?


During the draw before the math marathon, the team captains were asked to name the smallest possible sum of the digits in the decimal representation of the number \( n+1 \), given that the sum of the digits of the number \( n \) is 2017. What answer did the captain of the winning team give?


The robot vacuum cleaner is programmed to move on the floor according to the law:

$$\left\{\begin{array}{l} x = t(t-6)^2 \\ y = 0, \quad 0 \leq t \leq 7 \\ y = (t-7)^2, \quad t \geq 7 \end{array}\right.$$

where the axes are chosen parallel to the walls and the movement starts from the origin. Time $t$ is measured in minutes, and coordinates are measured in meters. Find the distance traveled by the robot in the first 7 minutes and the absolute change in the velocity vector during the eighth minute.


Find the maximum real number \( k \) such that for any positive numbers \( a \) and \( b \), the following inequality holds:
$$
(a+b)(ab+1)(b+1) \geqslant k \, ab^2.
$$


Calculate the limit of the function:

$$
\lim_{{x \rightarrow 0}} \frac{\sqrt{1+x \sin x}-1}{e^{x^{2}}-1}
$$


Given $\overrightarrow{OA}=(4,3)$, the graph of the function $y=x^{2}+bx+c$ is translated by the vector $\overrightarrow{OA}$ to obtain a new graph that is tangent to the line $4x + y - 8 = 0$ at the point $T(1, 4)$. Find the analytical expression of the original function.

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


What is the largest \( x \) such that \( x^2 \) divides \( 24 \cdot 35 \cdot 46 \cdot 57 \)?


There are fewer than 100 students performing a group dance, and there are two possible formations: one formation is with a group of 5 students in the middle, with the remaining students in groups of 8 around the outer circle; the other formation is with a group of 8 students in the middle, with the remaining students in groups of 5 around the outer circle. What is the maximum number of students?


Given that \( x + y + z = xy + yz + zx \), find the minimum value of \( \frac{x}{x^2 + 1} + \frac{y}{y^2 + 1} + \frac{z}{z^2 + 1} \).


Determine the measure of angle $\hat{A}$ of triangle $ABC$ if it is known that the angle bisector of this angle is perpendicular to the line passing through the orthocenter and the circumcenter of the triangle.


Sarah and Hagar play a game of darts. Let \(O_0\) be a circle of radius 1. On the \(n\)th turn, the player whose turn it is throws a dart and hits a point \(p_n\) randomly selected from the points of \(O_{n-1}\). The player then draws the largest circle that is centered at \(p_n\) and contained in \(O_{n-1}\), and calls this circle \(O_n\). The player then colors every point that is inside \(O_{n-1}\) but not inside \(O_n\) her color. Sarah goes first, and the two players alternate turns. Play continues indefinitely. If Sarah's color is red, and Hagar's color is blue, what is the expected value of the area of the set of points colored red?

Solution: \boxed{\frac{6 \pi}{7}}


A person's age in 1962 was one more than the sum of the digits of the year in which they were born. How old are they?


In quadrilateral \( \square ABCD \), \(\angle B < 90^\circ\) and \(AB < BC\). Tangents are drawn from point \( D \) to the circumcircle \( \Gamma \) of triangle \( \triangle ABC \), touching the circle at points \( E \) and \( F \). Given that \(\angle EDA = \angle FDC\), find \(\angle ABC\).


In a $15 \times 15$ grid, some centers of adjacent squares are connected by segments to form a closed line that does not intersect itself and is symmetric with respect to one of the main diagonals. Show that the length of the line is less than or equal to 200.


There are knights who always tell the truth and liars who always lie living on an island. The island's population is 1000 people, spread across 10 villages (each village has at least two people). One day, each islander claimed that all other villagers in their village are liars. How many liars live on the island? (Two villagers are considered to be from the same village if they live in the same village.)


The sequence $1001, 1004, 1009$ has a general term $a_{n} = n^{2} + 1000$, where $n \in \mathbf{N}_{+}$. For each $n$, let $d_{n}$ denote the greatest common divisor (GCD) of $a_{n}$ and $a_{n+1}$. Determine the maximum value of $d_{n}$ as $n$ ranges over all positive integers.


In a $6 \times 6$ grid of numbers:
(i) All the numbers in the top row and the leftmost column are the same;
(ii) Each other number is the sum of the number above it and the number to the left of it;
(iii) The number in the bottom right corner is 2016.
What are the possible numbers in the top left corner?


In triangle \(K L M\), side \(K L\) is equal to 24, angle bisector \(L N\) is equal to 24, and segment \(M N\) is equal to 9. Find the perimeter of triangle \(L M N\).


Find \(\operatorname{ctg} 2 \alpha\) if it is known that \(\sin \left(\alpha-90^{\circ}\right)=-\frac{2}{3}\) and \(270^{\circ}<\alpha<360^{\circ}\).


Consider a circle of radius \( R \) centered at the origin on the Cartesian plane. Specify at least one value of \( R \) for which there are exactly 32 integer points (a point is called an integer point if both its x-coordinate and y-coordinate are integers) lying on this circle.


In the quadrilateral \( ABCD \), angle \( B \) is \( 150^{\circ} \), angle \( C \) is a right angle, and the sides \( AB \) and \( CD \) are equal.

Find the angle between side \( BC \) and the line passing through the midpoints of sides \( BC \) and \( AD \).


What is the third smallest number that is both a triangular number and a square number? Obviously, the first two numbers that have this property are 1 and 36. What is the next number?


A pedestrian walked 5.5 kilometers in 1 hour but did not reach point \( B \) (short by \(2 \pi - 5.5\) km). Therefore, the third option is longer than the first and can be excluded.

In the first case, they need to cover a distance of 5.5 km along the alley. If they move towards each other, the required time is \(\frac{5.5}{20 + 5.5}\) hours.

In the second case, moving towards each other, after \(\frac{2 \pi - 5.5}{5.5}\) hours the pedestrian will reach point \( B \), while the cyclist will still be riding on the highway (since \(\frac{4}{15} > \frac{2 \pi - 5.5}{5.5}\)). Thus, the cyclist will always be on the highway and the closing speed of the pedestrian and the cyclist will be \(15 + 5.5 = 20.5\) km/h. They will meet in \(\frac{2 \pi - 1.5}{20.5}\) hours.

Compare the numbers obtained in cases 1 and 2:

$$
\frac{5.5}{25.5} = \frac{11}{51} < 0.22 < 0.23 < \frac{4 \cdot 3.14 - 3}{41} < \frac{2 \pi - 1.5}{20.5}
$$

Therefore, the answer is given by the first case.

**Note**: Some participants interpreted the condition differently, considering that the pedestrian and the cyclist met after an hour, and at that time, the pedestrian asked the cyclist to bring him the keys. Since the condition does not specify where exactly the neighbor was, and with a different interpretation of the condition, a similar problem arises (although with slightly more cumbersome calculations), the jury decided to accept both interpretations of the condition. The plan for solving the second interpretation is provided.


The terms of a certain sequence are sums of the corresponding terms of two geometric progressions, given that the first two terms are equal to 0. Determine the sequence.


Find all positive \( x \) and \( y \) for which the expression

\[
\frac{x y}{2}+\frac{18}{x y}
\]

takes the minimum value and among all such \( x \) and \( y \), the expression

\[
\frac{y}{2}+\frac{x}{3}
\]

is also minimized.

Solution: \boxed{x=3, y=2}


Solve the equation

$$
\cos ^{2} 8 x+\cos ^{2} x=2 \cos ^{2} x \cdot \cos ^{2} 8 x
$$

In the answer, indicate the number equal to the sum of the roots of the equation that belong to the interval $[3 \pi ; 6 \pi]$, rounding this number to two decimal places if necessary.


Find the probability that, when five different numbers are randomly chosen from the set $\{1, 2, \ldots, 20\}$, at least two of them are consecutive.


By how much is the sum of the squares of the first one hundred even numbers greater than the sum of the squares of the first one hundred odd numbers?


Write the first 10 prime numbers in a row. How can you remove 6 digits to get the largest possible number?


Compute the limit of the numerical sequence:

$$
\lim _{n \rightarrow \infty} n\left(\sqrt{n^{4}+3}-\sqrt{n^{4}-2}\right)
$$


Given the sequence \(\left\{a_{n}\right\}\), which satisfies
\[
a_{1}=0,\left|a_{n+1}\right|=\left|a_{n}-2\right|
\]

Let \(S\) be the sum of the first 2016 terms of the sequence \(\left\{a_{n}\right\}\). Determine the maximum value of \(S\).


A container is already filled with water. There are three lead balls: large, medium, and small. The first time, the small ball is submerged in the water; the second time, the small ball is removed, and the medium ball is submerged in the water; the third time, the medium ball is removed, and the large ball is submerged in the water. It is known that the water spilled the first time is 3 times the water spilled the second time, and the water spilled the third time is 3 times the water spilled the first time. Find the ratio of the volumes of the three balls.

Solution: \boxed{3 : 4 : 13}


Let \( p \) be a given odd prime number. If there exists a positive integer \( k \) such that \( \sqrt{k^2 - pk} \) is also a positive integer, find the value of \( k \).

Solution: \boxed{\frac{(p + 1)^2}{4}}


A Tim number is a five-digit positive integer with the following properties:
1. It is a multiple of 15.
2. Its hundreds digit is 3.
3. Its tens digit is equal to the sum of its first three (leftmost) digits.

How many Tim numbers are there?


Calculate the following expression:

$$
\lim _{n \rightarrow \infty} \sum_{k=0}^{n} \frac{k^{2}+3k+1}{(k+2)!}
$$


Given that $f(x)$ is a linear function, $f[f(1)] = -1$, and the graph of $f(x)$ is symmetric about the line $x - y = 0$ is $C$. If the point $\left(n, \frac{a_{n+1}}{a_{n}}\right)$ $(n \in \mathbf{N}^{\cdot})$ lies on the curve $C$, and given $a_{1} = 1$, $\frac{a_{n+1}}{a_{n}} - \frac{a_{n}}{a_{n-1}} = 1 (n \geqslant 2)$:
1. Find the analytic expression of $f(x)$ and the equation of the curve $C$.
2. Let $S_{n} = \frac{a_{1}}{3!} + \frac{a_{2}}{4!} + \frac{a_{3}}{5!} + \cdots + \frac{a_{n}}{(n+2)!}$. For all $n \in \mathbf{N}^{\cdot}$, if $S_{n} > m$ holds, find the maximum value of the natural number $m$.


Find all functions \( f: \mathbf{Z} \rightarrow \mathbf{Z} \) such that for all \( n \in \mathbf{Z} \), \( f[f(n)]+f(n)=2n+3 \), and \( f(0)=1 \).

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


Let \( x \) and \( y \) be positive real numbers, and \( x + y = 1 \). Find the minimum value of \( \frac{x^2}{x+2} + \frac{y^2}{y+1} \).


Let \((a_{1}, a_{2}, \ldots, a_{8})\) be a permutation of \((1,2, \ldots, 8)\). Find, with proof, the maximum possible number of elements of the set
\[ \left\{a_{1}, a_{1}+a_{2}, \ldots, a_{1}+a_{2}+\cdots+a_{8}\right\} \]
that can be perfect squares.


Determine the largest integer \( n \) such that \( 7^{2048} - 1 \) is divisible by \( 2^{n} \).


In how many ways can the number \( n \) be represented as the sum of positive odd addends? (Representations that differ only in the order of the addends are considered different.)


For each positive integer \( n \geq 1 \), we define the recursive relation given by
\[ a_{n+1} = \frac{1}{1 + a_{n}}. \]

Suppose that \( a_{1} = a_{2012} \). Find the sum of the squares of all possible values of \( a_{1} \).


In an isosceles triangle \(ABC\) with base \(AB\), the angle bisectors \(CL\) and \(AK\) are drawn. Find \(\angle ACB\) of triangle \(ABC\), given that \(AK = 2CL\).


Given a five-digit palindromic number that equals 45 times a four-digit palindromic number (i.e., $\overline{\mathrm{abcba}}=45 \times \overline{\text{deed}}$), what is the largest possible value of the five-digit palindromic number?


A function \( f \) defined on ordered pairs of positive integers satisfies the following properties:
1. \( f(x, x) = x \);
2. \( f(x, y) = f(y, x) \);
3. \( (x + y) f(x, y) = y f(x, x + y) \).

Calculate \( f(14,52) \).


Consider triangle \(ABC\) with \(\angle A = 2 \angle B\). The angle bisectors from \(A\) and \(C\) intersect at \(D\), and the angle bisector from \(C\) intersects \(\overline{AB}\) at \(E\). If \(\frac{DE}{DC} = \frac{1}{3}\), compute \(\frac{AB}{AC}\).


Find a natural number \( n \), knowing that it has two prime divisors and satisfies the conditions \(\tau(n) = 6\) and \(\sigma(n) = 28\).


Two cars leave points A and B simultaneously and meet at 12 PM. If the speed of the first car is doubled while keeping the speed of the second car the same, the meeting will occur 56 minutes earlier. If the speed of the second car is doubled while keeping the speed of the first car the same, they will meet 65 minutes earlier. Determine the meeting time if the speeds of both cars were doubled.

Solution: \boxed{10:29 \text{ AM}}


It is known that the polynomial \( f(x) = 8 + 32x - 12x^2 - 4x^3 + x^4 \) has 4 distinct real roots \(\{x_{1}, x_{2}, x_{3}, x_{4}\}\). The polynomial of the form \( g(x) = b_{0} + b_{1} x + b_{2} x^2 + b_{3} x^3 + x^4 \) has roots \(\{x_{1}^2, x_{2}^2, x_{3}^2, x_{4}^2\}\). Find the coefficient \( b_{1} \) of the polynomial \( g(x) \).


On the board, 26 ones are written. Every minute, Karlson erases two arbitrary numbers and writes their sum on the board, and then eats a number of candies equal to the product of the two erased numbers. What is the maximum number of candies he could eat in 26 minutes?


What is the greatest possible number of rays in space emanating from a single point and forming obtuse angles pairwise?


Two pedestrians set out simultaneously from $A$ to $B$ and from $B$ to $A$, respectively. When the first pedestrian has traveled halfway, the second pedestrian has 24 kilometers left to go. When the second pedestrian has traveled halfway, the first pedestrian has 15 kilometers left to go. How many kilometers will the second pedestrian have left to walk after the first pedestrian finishes their journey?

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


The warehouse has nails in boxes weighing 24, 23, 17, and 16 kg. Can the storekeeper release 100 kg of nails from the warehouse without opening the boxes?

Solution: \boxed{Yes}


In the right triangle \(ABC\), the altitude \(BH\) is drawn to the hypotenuse \(AC\). Points \(X\) and \(Y\) are the centers of the circles inscribed in triangles \(ABH\) and \(CBH\) respectively. The line \(XY\) intersects the legs \(AB\) and \(BC\) at points \(P\) and \(Q\). Find the area of triangle \(BPQ\), given that \(BH = h\).


In triangle \(ABC\) with sides \(BC=7\), \(AC=5\), and \(AB=3\), an angle bisector \(AD\) is drawn. A circle is circumscribed around triangle \(ABD\), and a circle is inscribed in triangle \(ACD\). Find the product of their radii.


A number \( \mathrm{X} \) is called "25-supporting" if for any 25 real numbers \( a_{1}, \ldots, a_{25} \) whose sum is an integer, there is at least one for which \( \left|a_{i} - \frac{1}{2}\right| \geq X \).

Provide the largest 25-supporting \( X \), rounded to the nearest hundredth according to standard mathematical rules.


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


Rearrange the four digits of 2016 to form a four-digit perfect square. What is this four-digit perfect square? $\qquad$ .


The line $c$ is defined by the equation $y = 2x$. Points $A$ and $B$ have coordinates $A(2, 2)$ and $B(6, 2)$. On line $c$, find the point $C$ from which the segment $AB$ is seen at the largest angle.

Solution: \boxed{(2, 4)}


Find a polynomial with integer coefficients for which $\cos 18^{\circ}$ is a root.

Solution: \boxed{16x^4 - 20x^2 + 5}


If \( A \) is a prime number and \( A-4 \), \( A-6 \), \( A-12 \), \( A-18 \) are also prime numbers, then \( A = \) ?


Gavrila found that the front tires of a car last for 21,000 km, and the rear tires last for 28,000 km. He decided to switch them at some point to maximize the possible distance the car can travel. Find this maximum distance (in km).

Solution: \boxed{24000 \text{ km}}


Let the random variables $\xi$ and $\eta$ denote the lifetimes of the blue and red light bulbs, respectively. The lifetime of the flashlight is equal to the minimum of these two values. Clearly, $\min (\xi, \eta) \leq \xi$. Let's move to the expected values: $\operatorname{E} \min (\xi, \eta) \leq \mathrm{E} \xi=2$. Therefore, the expected lifetime of the flashlight is no more than 2 years.


How many ordered triples of positive integers \((a, b, c)\) are there for which \(a^{4} b^{2} c = 54000\)?


The non-negative numbers \(a, b, c\) sum up to 1. Find the maximum possible value of the expression

$$
(a + 3b + 5c) \cdot \left(a + \frac{b}{3} + \frac{c}{5}\right)
$$


Calculate the limit of the function:

\[
\lim _{x \rightarrow 0} \frac{e^{3 x}-e^{-2 x}}{2 \arcsin x - \sin x}
\]


In January 1859, an eight-year-old boy dropped a newly-hatched eel into a well in Sweden. The eel, named Ale, finally died in August 2014. How many years old was Åle when it died?
A. 135
B. 145
C. 155
D. 165
E. 175


Let \( A = \sin \left(\sin \frac{3}{8} \pi\right) \), \( B = \sin \left(\cos \frac{3}{8} \pi\right) \), \( C = \cos \left(\sin \frac{3}{8} \pi\right) \), and \( D = \cos \left(\cos \frac{3}{8} \pi\right) \). Compare the magnitudes of \(A\), \(B\), \(C\), and \(D\).

Solution: \boxed{B<C<A<D}


The number written on the board can be multiplied by 5 or have its digits rearranged (a zero cannot be placed in the first digit). Is it possible to get the 100-digit number $5222 \ldots 2221$ starting from the number 1 in this way?

Solution: \boxed{\text{No}}


Given an ellipse $b^{2} x^{2} + a^{2} y^{2} = a^{2} b^{2}(a > b > 0)$, its two directrices intersect the $x$-axis at $E_{1}$ and $E_{2}$. Point $P$ is on the ellipse, and $\angle E_{1} P E_{2} = \alpha$. Let $e$ be the eccentricity and $c$ be the focal distance. Then $\alpha$ is an obtuse angle, and if $e^{2} > \frac{1}{2}(\sqrt{5}-1)$, we have $\cot \alpha \leqslant -e$, with equality if and only if $\left|y_{P}\right| = \frac{a b^{2}}{c^{2}}$.

Solution: \boxed{\cot \alpha \leq -e}


Along the street, there are 100 houses. A craftsman was asked to make house numbers for all the houses from 1 to 100. To fulfill the order, he needs to stock up on digits. Without using a pencil and paper, calculate mentally how many nines the craftsman will need.


The natural numbers \(a, b, c\) are chosen such that \(a < b < c\). It is also known that the system of equations \(2x + y = 2023\) and \(y = |x - a| + |x - b| + |x - c|\) has exactly one solution. Find the minimum possible value of \(c\).


On the sides $AB$ and $CD$ of a cyclic quadrilateral $ABCD$ that is not a trapezoid, points $P$ and $R$ are marked respectively, such that $AP:PB=CR:RD$. On the segment $PR$, a point $Q$ is chosen such that $PQ:QR=AB:CD$. Find the ratio of the areas of triangles $AQD$ and $BQC$, given that $AD=x$ and $BC=y$.

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


A circle has two chords \(AB\) and \(CD\). Construct a point \(X\) on the circumference such that \(XA^{2} + XB^{2} = XC^{2} + XD^{2}\).

Solution: \boxed{X}


In a deck of $n$ cards, some are face up and the rest are face down. In one move, it is allowed to take a number of cards from the top, flip the stack, and place it back on top of the deck. What is the minimum number of moves required to ensure that all cards are face down, regardless of the initial arrangement of the cards?

Solution: \boxed{n}


In a taxi, a passenger can sit in the front and three passengers can sit in the back. In how many ways can four passengers sit in a taxi if one of these passengers wants to sit by the window?


The sequence of functions is defined by the formulas:

$$
f_{0}(x)=2 \sqrt{x}, f_{n+1}(x)=\frac{4}{2-f_{n}(x)}, n=0,1,2 \ldots, x \in[4 ; 9]
$$

Find \( f_{2023}(9) \).


In the base of the pyramid \( S A B C D \), there is a trapezoid \( A B C D \) with bases \( B C \) and \( A D \), where \( B C = 2 A D \). Points \( K \) and \( L \) are taken on the edges \( S A \) and \( S B \) such that \( 2 S K = K A \) and \( 3 S L = L B \). In what ratio does the plane \( K L C \) divide the edge \( S D \)?


A building has seven rooms numbered 1 through 7 on one floor, with various doors connecting these rooms. The doors can be either one-way or two-way. Additionally, there is a two-way door between room 1 and the outside, and there is a treasure in room 7. Design the arrangement of rooms and doors such that:

(a) It is possible to enter room 1, reach the treasure in room 7, and return outside.

(b) The minimum number of steps required to achieve this (each step involving walking through a door) is as large as possible.


Can the number of diagonals of a polygon be exactly twice the number of its sides?


A polynomial \( G(x) \) with real coefficients takes the value 2022 at exactly five different points \( x_{1}<x_{2}<x_{3}<x_{4}<x_{5} \). It is known that the graph of the function \( y=G(x) \) is symmetric relative to the line \( x=-6 \).

(a) Find \( x_{1}+x_{3}+x_{5} \).

(b) What is the smallest possible degree of \( G(x) \)?


ABC is a triangle with ∠B = 90° and M the midpoint of AB. Show that sin(∠ACM) ≤ 1/3.


Let \(a, b, c\) be three positive numbers whose sum is one. Show that

\[ a^{a^{2}+2 c a} b^{b^{2}+2 a b} c^{c^{2}+2 b c} \geq \frac{1}{3} \]


Vasya cut a triangle out of cardboard and numbered its vertices with the digits 1, 2, and 3. It turned out that if Vasya rotates his triangle clockwise around its vertex numbered 1 by an angle equal to the angle at this vertex 15 times, the triangle returns to its original position. If Vasya rotates the triangle clockwise around its vertex numbered 2 by an angle equal to the angle at this vertex 6 times, the triangle again returns to its original position. Vasya claims that if he rotates his triangle around the vertex numbered 3 by an angle equal to the angle at this vertex \( n \) times, the triangle will return to its original position. What is the minimum \( n \) that Vasya might have named such that his statement is true for at least some cardboard triangle?


For letters \( a \sim z \), encode them as $(\mathbf{a}=1, \mathbf{b}=2, \ldots, \mathbf{z}=26)$. Each English word (assuming all letters in the word are lowercase) can be assigned a product \(\mathbf{p}\) calculated from these letter codes. For example, for the word "good," its corresponding \(\mathbf{p}\) value is $7 \times 15 \times 15 \times 4 = 6300$ (because \(g = 7, o = 15, d = 4\)). If a composite number cannot be expressed as the \(\mathbf{p}\) value of any word (regardless of whether the word has meaning), such a composite number is called a "middle ring number". What is the smallest three-digit "middle ring number"?


Five children and a monkey want to share a bag of walnuts. One of the children—unbeknownst to the others—divided the walnuts in the bag into five equal parts. He found one extra walnut, gave it to the monkey, and left with one-fifth of the walnuts. The second child then divided the remaining walnuts into five equal parts, also found one extra walnut, gave it to the monkey, and took one-fifth of the portion. The remaining three children did the same. Finally, the five children gathered the remaining walnuts, divided them into five equal parts, found one extra walnut, and gave it to the monkey.

What is the minimum number of walnuts that were originally in the bag?


Given the positive number sequence $\left\{a_{n}\right\}$ that satisfies $a_{n+1} \geqslant 2 a_{n}+1$ and $a_{n} < 2^{n+1}$ for all $n \in \mathbf{Z}_{+}$, what is the range of values for $a_{1}$?

Solution: \boxed{(0, 3]}


From a square with a side length of 5, four corner unit squares are cut out (see figure). What is the area of the largest square that can be cut out from the remaining part?


Given the set \( A \) which is an \( n \)-element subset of the set \(\{1,2, \cdots, 2017\}\), where any two numbers in \( A \) are neither coprime to each other nor is one a divisor of the other, find the maximum value of \( n \).


The graph of the function \( f(x) = \frac{1}{12} x^2 + ax + b \) intersects the x-axis at points \( A \) and \( C \), and the y-axis at point \( B \), as shown in the figure. It turns out that for the point \( T \) with coordinates \( (3, 3) \), the condition \( TA = TB = TC \) is satisfied. Find \( b \).


On a bookshelf, there are 30 volumes of an encyclopedia arranged in some order. In one operation, you are allowed to swap any two neighboring volumes. What is the minimum number of operations required to guarantee that all the volumes are arranged in the correct order (from the first to the thirtieth from left to right), regardless of the initial arrangement?


Ahmad walks up a flight of eight steps, going up either one or two steps at a time. There is a hole on the sixth step, so he cannot use this step. In how many different ways can Ahmad reach the top step?


What is the maximum number of lattice points (i.e. points with integer coordinates) in the plane that can be contained strictly inside a circle of radius 1?


Two multiplication examples of natural numbers were written on the board. In one of the examples, the numbers were the same. A troublemaker named Petya erased the original numbers and almost all the digits of the resulting products, leaving only the last three digits: ... 689 and ... 759. Can an exemplary student named Misha determine which of these three-digit numbers belongs to the product of the multiplication of identical numbers?


We create a random sequence using the digits $0, 1, 2$. For what length of sequences will the probability that all three digits appear in the sequence be at least $61\%$?


The sum \( b_{7} + b_{6} + \ldots + b_{2019} \) of the terms of a geometric progression \(\left\{b_{n}\right\}\), where \( b_{n} > 0 \), equals 27, and the sum of their reciprocals \(\frac{1}{b_{7}} + \frac{1}{b_{6}} + \ldots + \frac{1}{b_{2019}} \) equals 3. Find the product \( b_{7} \cdot b_{6} \cdot \ldots \cdot b_{2019} \).


 1
$A B C D$ is a parallelogram and $E$ is the midpoint of $C D$. If the ratio of the area of the triangle $A D E$ to the area of the parallelogram $A B C D$ is $1: a$, find $a$.

 2
$A B C D$ is a parallelogram and $E$ is the midpoint of $C D$. $A E$ and $B D$ meet at $M$. If $D M: M B = 1: k$, find $k$.

 3
If the square root of 5 is approximately 2.236, the square root of 80 with the same precision is $d$. Find $d$.

 4
A square is changed into a rectangle by increasing its length by $20\%$ and decreasing its width by $20\%$. If the ratio of the area of the rectangle to the area of the square is $1: r$, find $r$.


In a box, there are multicolored balls: 5 white, 12 red, and 20 black. What is the minimum number of balls that need to be drawn from the box, without looking inside, to ensure that:
a) There is at least one ball of each color.
b) There are 10 balls of one color.


There are three video game systems: the Paystation, the WHAT, and the ZBoz2 π, and none of these systems will play games for the other systems. Uncle Riemann has three nephews: Bernoulli, Galois, and Dirac. Bernoulli owns a Paystation and a WHAT, Galois owns a WHAT and a ZBoz2 π, and Dirac owns a ZBoz2 π and a Paystation. A store sells 4 different games for the Paystation, 6 different games for the WHAT, and 10 different games for the ZBoz2 π. Uncle Riemann does not understand the difference between the systems, so he walks into the store and buys 3 random games (not necessarily distinct) and randomly hands them to his nephews. What is the probability that each nephew receives a game he can play?


The area of triangle \(ABC\) is 1. Let \(A_1\), \(B_1\), and \(C_1\) be the midpoints of the sides \(BC\), \(CA\), and \(AB\) respectively. Points \(K\), \(L\), and \(M\) are taken on segments \(AB_1\), \(CA_1\), and \(BC_1\) respectively. What is the minimum area of the common part of triangles \(KLM\) and \(A_1B_1C_1\)?


The number 1047 leaves a remainder of 23 when divided by \( A \), and a remainder of 7 when divided by \( A+1 \). Find \( A \).


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

$$
\begin{aligned}
& \left\{\begin{array}{l}
x=6 \cos t \\
y=2 \sin t
\end{array}\right. \\
& y=\sqrt{3}(y \geq \sqrt{3})
\end{aligned}
$$

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


An integer is a perfect number if and only if it is equal to the sum of all of its divisors except itself. For example, 28 is a perfect number since \(28 = 1 + 2 + 4 + 7 + 14\).

Let \(n!\) denote the product \(1 \cdot 2 \cdot 3 \cdots n\), where \(n\) is a positive integer. An integer is a factorial if and only if it is equal to \(n!\) for some positive integer \(n\). For example, 24 is a factorial number since \(24=4!=1 \cdot 2 \cdot 3 \cdot 4\).

Find all perfect numbers greater than 1 that are also factorials.


Sixteen wooden Cs are placed in a 4-by-4 grid, all with the same orientation, and each is to be colored either red or blue. A quadrant operation on the grid consists of choosing one of the four 2-by-2 subgrids of Cs found at the corners of the grid and moving each C in the subgrid to the adjacent square in the subgrid that is 90 degrees away in the clockwise direction, without changing the orientation of the C. Given that two colorings are considered the same if and only if one can be obtained from the other by a series of quadrant operations, determine the number of distinct colorings of the Cs.


Given that Bag A contains two red balls and three white balls, and Bag B contains three red balls and three white balls. If one ball is randomly drawn from Bag A and placed into Bag B, and then one ball is randomly drawn from Bag B and placed into Bag A, let $\xi$ represent the number of white balls in Bag A at that time. Find $\mathrm{E} \xi$.


Find all polynomials \( P \in \mathbb{R}[X] \) such that for all real numbers \( a, b, c \) the following equation holds:
$$
P(a + b - 2c) + P(b + c - 2a) + P(c + a - 2b) = 3P(a - b) + 3P(b - c) + 3P(c - a)
$$

Solution: \boxed{P(x) = ax^2 + bx}


Two bags \( U_{1} \) and \( U_{2} \) contain identical red and white balls. \( U_{1} \) contains \( A \) red balls and 2 white balls. \( U_{2} \) contains 2 red balls and \( B \) white balls. Take two balls out of each bag. If the probability that all four balls are red is \(\frac{1}{60}\), find the value of \( B \).


Let \( A B C D \) be an isosceles trapezoid with \( [A B] \) as the larger base. It is given that the diagonals intersect at a point \( O \) such that \(\frac{O A}{O C}=2\). Given that the area of triangle \( B O C \) is 10, what is the area of the trapezoid \( A B C D \)?


A factory produced an original calculator that performs two operations:

(a) the usual addition, denoted by \( + \)

(b) an operation denoted by \( \circledast \).

We know that, for any natural number \( a \), the following hold:

\[
(i) \quad a \circledast a = a \quad \text{ and } \quad (ii) \quad a \circledast 0 = 2a
\]

and, for any four natural numbers \( a, b, c, \) and \( d \), the following holds:

\[
(iii) \quad (a \circledast b) + (c \circledast d) = (a+c) \circledast(b+d)
\]

What are the results of the operations \( (2+3) \circledast (0+3) \) and \( 1024 \circledast 48 \)?


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


If the plane vectors \(\vec{a}, \vec{b}, \vec{c}\) satisfy \(\vec{a} \cdot (\vec{a} + \vec{c}) = 0\) and \(|\vec{a} + \vec{b} - 2\vec{c}| = 2\), find the maximum value of \(\vec{a} \cdot \vec{b}\).


How many ways can the numbers from 1 to 100 be rearranged such that no two adjacent numbers differ by more than 1?


Let $\triangle ABC$ be a right triangle with right angle at $C$. Let $I$ be the incenter of $\triangle ABC$, and let $M$ lie on $AC$ and $N$ on $BC$ such that $M$, $I$, and $N$ are collinear and $\overline{MN}$ is parallel to $\overline{AB}$. If $AB = 36$ and the perimeter of $\triangle CMN$ is 48, find the area of $\triangle ABC$.


Let \( X \) be the set of non-negative integers and \( f : X \to X \) a map such that \( ( f(2n+1) )^2 - ( f(2n) )^2 = 6 f(n) + 1 \) and \( f(2n) \geq f(n) \) for all \( n \) in \( X \). How many numbers in \( f(X) \) are less than 2003?


Given \( x \in (-1,1) \), if the function
\[ f(x) = x^2 - a x + \frac{a}{2} \]
is always positive, determine the range of the real number \( a \).

Solution: \boxed{(0, 2]}


From 125 sugar cubes, a $5 \times 5 \times 5$ cube was made. Ponchik picked all the cubes that have an odd number of neighbors and ate them (neighbors are those cubes that share a face). How many cubes did Ponchik eat in total?


Solve the following system of equations in the domain of positive numbers:

$$
\begin{aligned}
x^{4}+y^{4}-x^{2} y^{2} & =13 \\
x^{2}-y^{2}+2 x y & =1
\end{aligned}
$$

Solution: \boxed{x = 1, y = 2}


Simplify the following expression as much as possible:

$$
\frac{\frac{1}{a-b}-\frac{2 a b}{a^{3}-a^{2} b+a b^{2}-b^{3}}}{\frac{a^{2}+a b}{a^{3}+a^{2} b+a b^{2}+b^{3}}+\frac{b}{a^{2}+b^{2}}}
$$

Solution: \boxed{\frac{a-b}{a+b}}


Determine the remainder when $1+2+\cdots+2014$ is divided by 2012.


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

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


The first three numbers of a sequence are \(1, 7, 8\). Every subsequent number is the remainder obtained when the sum of the previous three numbers is divided by 4. Find the sum of the first 2011 numbers in this sequence.


Let \( g \) be a natural number, and suppose \( g^{4} + g^{3} + g^{2} + g + 1 \) is a perfect square. Find the sum of all such \( g \).


If \( S \) is a non-empty set of positive integers, let \( p(S) \) be the reciprocal of the product of the members of \( S \). Show that \( \sum p(S) = n \), where the sum is taken over all non-empty subsets of \(\{1, 2, \ldots, n\}\).

Solution: \boxed{n}


In how many ways can one arrange all the natural numbers from 1 to \(2n\) in a circle so that each number is a divisor of the sum of its two adjacent numbers? (Arrangements that differ by rotation and symmetry are considered the same.)

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


The base $ABCD$ of a tetrahedron $P-ABCD$ is a convex quadrilateral with diagonals $AC$ and $BD$ intersecting at $O$. If the area of $\triangle AOB$ is 36, the area of $\triangle COD$ is 64, and the height of the tetrahedron is 9, what is the minimum volume of such a tetrahedron?


Given that \( a \geq b \geq c > 0 \) are real numbers such that \( abc = 1 \) and \( a + b + c > \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \), show that \( a > 1 > b \).

Solution: \boxed{a > 1 > b}


Find all non-negative integers \( x, y, z \) such that \( x^{3} + 4y^{3} = 16z^{3} + 4xyz \).

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


The 200-digit number \( M \) is composed of 200 ones. What is the sum of the digits of the product \( M \times 2013 \)?


The quantity \(\Delta=\max_{a \leqslant x \leqslant b}|f(x)-g(x)|\) is called the absolute difference of the functions \(f(x)\) and \(g(x)\) on the closed interval \([a, b]\). Find the absolute difference of the functions \(f(x)=x^{2}\) and \(g(x)=x^{3}\) on the closed interval \([0,1]\).


A master makes an integer number of parts in one hour, more than 5, and an apprentice makes 2 parts less. The master completes an order in an integer number of hours, and two apprentices together complete it one hour faster. How many parts are in the order?


Find the positive integer values of \( x \) that satisfy the inequality \(\frac{5x+1}{x-1} > 2x + 2\).


Through point \( A \) of a circle with a radius of 10 cm, two mutually perpendicular chords \( AB \) and \( AC \) are drawn. Calculate the radius of a circle that is tangent to the given circle and the constructed chords, given that \( AB = 16 \) cm.

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


The line segments connecting the midpoints of opposite sides of a convex quadrilateral are equal to each other. Find the area of the quadrilateral if its diagonals are 8 and 12.


The foci of the ellipse $\frac{x^{2}}{5^{2}}+\frac{y^{2}}{3^{2}}=1$ are $F_{1}$ and $F_{2}$. If a point $P$ on the ellipse makes $P F_{1} \perp P F_{2}$, find the area of the triangle $\triangle P F_{1} F_{2}$.


The height of a triangle divides an angle of the triangle in the ratio \(2: 1\), and the base into segments whose ratio (of the larger to the smaller) is equal to \(k\). Find the sine of the smaller angle at the base and the permissible values of \(k\).


At what frequency does the digit 7 appear in the decimal representation of natural numbers? That is, find the limit of the ratio of the number of occurrences of the digit 7 in the decimal representation of all numbers from 1 to $n$ to the total number of decimal digits in the representation of those same numbers as $n \rightarrow \infty$.


In the trapezoid \(ABCD\), the side \(AB\) is perpendicular to the bases \(AD\) and \(BC\), and the diagonals of the trapezoid intersect at point \(E\). Point \(F\) is the foot of the perpendicular dropped from point \(E\) to side \(AB\). Given that \(\angle DFE = \alpha\), find \(\angle CFE\).

Solution: \boxed{\alpha}


Given vectors \(\boldsymbol{a}\) and \(\boldsymbol{b}\) such that \(|\boldsymbol{a} - \boldsymbol{b}| = 3\), \(|\boldsymbol{a} + 2\boldsymbol{b}| = 6\), and \(\boldsymbol{a}^2 + \boldsymbol{a} \cdot \boldsymbol{b} - 2\boldsymbol{b}^2 = -9\), find \(|\boldsymbol{b}|\).


Let \(a, b, c\) be positive integers such that
\[
\frac{a}{77}+\frac{b}{91}+\frac{c}{143}=1.
\]
What is the smallest possible value of \(a+b+c\)?


Given the angles of triangle ABC and the radius R of the circumscribed circle passing through the vertices, connect the feet of the altitudes of the triangle. Calculate the sides, angles, and the circumradius of the triangle formed by these feet (the orthic triangle).


Given \(n (n>1)\) as a positive integer, let \(a_{k}=\left(\frac{n-k}{k}\right)^{k-\frac{n}{2}}\) for \(0 < k \leq n\), \(k \in \mathbf{N}\). Find the maximum value of the sequence \(\{a_{k}\}\).


Adia writes a list in increasing order of the integers between 1 and 100, inclusive, that cannot be written as the product of two consecutive positive integers. What is the 40th integer in her list?


What is the minimum number of weights necessary to be able to weigh any number of grams from 1 to 100 on a balance scale, if the weights can only be placed on one side of the scale?


Masha has seven different dolls that she places in six different dollhouses so that each dollhouse contains at least one doll. In how many ways can Masha do this? It is important which doll is in which dollhouse. How the dolls are arranged in the dollhouse that has two dolls does not matter.


Find the sum of every even positive integer less than 233 not divisible by 10.


In triangle \( ABC \), \( AB = BC = 6 \). On side \( AB \), a circle is constructed as the diameter, which intersects side \( BC \) at point \( D \) such that \( BD : DC = 2 : 1 \).

Find \( AC \).


Evaluate the limit of the function:

\[ \lim _{x \rightarrow -1} \frac{\left(x^{3}-2 x-1\right)(x+1)}{x^{4}+4 x^{2}-5} \]


Points \( A, B, C \) in the plane satisfy \( \overline{AB} = 2002 \) and \( \overline{AC} = 9999 \). The circles with diameters \( AB \) and \( AC \) intersect at \( A \) and \( D \). If \( \overline{AD} = 37 \), what is the shortest distance from point \( A \) to line \( BC \)?


Compute the number of ordered quintuples of nonnegative integers \((a_1, a_2, a_3, a_4, a_5)\) such that \(0 \leq a_1, a_2, a_3, a_4, a_5 \leq 7\) and 5 divides \(2^{a_1} + 2^{a_2} + 2^{a_3} + 2^{a_4} + 2^{a_5}\).


For a triangle with sides 26 cm, 28 cm, and 30 cm, find the product of the radii of the circumscribed and inscribed circles.

Solution: \boxed{130 \text{ cm}^2}


Given a triangle \(ABC\) such that the foot \(P\) of the perpendicular from point \(C\) to the line \(AB\) lies within the segment \(AB\). From point \(P\), perpendiculars \(p\) and \(q\) are drawn to the lines \(AC\) and \(BC\) respectively (in the given order). Let \(S\) be the intersection of the line \(BC\) and the line \(q\), and \(T\) be the intersection of the line \(AC\) and the line \(p\). Calculate the angle \(\angle ACB\), given that \(|\angle APT| + |\angle BPS| = 20^\circ\).


25 chess players are participating in a tournament. Each of them has different levels of strength, and in each match, the stronger player always wins.

What is the minimum number of matches required to determine the two strongest players?


Petya and his three classmates started a 100-meter race simultaneously, and Petya finished first. Twelve seconds after the race began, no one had finished yet, and all four participants had collectively run a total of 288 meters. When Petya finished the race, the other three participants had a combined distance of 40 meters left to the finish line. How many meters did Petya run in the first 12 seconds? Justify your answer. It is assumed that each participant ran with a constant speed.

Solution: \boxed{80 \text{ meters}}


Find the derivative.

$$
y=e^{a x}\left(\frac{1}{2 a}+\frac{a \cdot \cos 2 b x+2 b \cdot \sin 2 b x}{2\left(a^{2}+4 b^{2}\right)}\right)
$$

Solution: \boxed{e^{ax} \cos^2(bx)}


What is the average of five-digit palindromic numbers? (A whole number is a palindrome if it reads the same backward as forward.)


The intersection point of the medians of a right triangle is at distances of 3 and 4 from the legs. Find the distance from this point to the hypotenuse.


If the distinct numbers \( a, b, \) and \( c \) satisfy the following equations:

$$
\begin{aligned}
& a^{3} + p a + q = 0 \\
& b^{3} + p b + q = 0 \\
& c^{3} + p c + q = 0
\end{aligned}
$$

then

$$
a + b + c = 0
$$

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


A $4 \times 4 \times 4$ cube is cut into 64 smaller $1 \times 1 \times 1$ cubes. Then, 16 of the $1 \times 1 \times 1$ small cubes are painted red. The requirement is that, in any group of 4 small cubes parallel to any edge, exactly 1 cube is painted red. How many different ways are there to paint the cubes? (Painting methods that are the same after rotation are also considered different methods.)


In how many ways can the faces of a cube be painted with 6 different colors such that each color appears exactly once? Two painted cubes are considered identical if they can be made to look the same by rotating them, i.e., they can be confused by turning them in your hands.