A sequence of numbers is defined using the relation
$$
a_{n} = -a_{n-1} + 6a_{n-2}
$$
where \( a_1 = 2 \) and \( a_2 = 1 \). Find \( a_{100} + 3a_{99} \).

Solution: \boxed{7 \cdot 2^{98}}


Calculate:
a) \(\sin \left(-60^{\circ}\right)\);
b) \(\cos \left(-45^{\circ}\right)\);
c) \(\operatorname{tg}\left(-945^{\circ}\right)\).


Let \( n \) be a positive integer greater than 1. Find the number of permutations \((a_{1}, a_{2}, \cdots, a_{n})\) of \(1, 2, \cdots, n\) such that there is exactly one \( i \in \{1, 2, \cdots, n-1\} \) where \( a_{i} > a_{i+1} \).

Solution: \boxed{p_n = 2^n - n - 1}


Find the number of permutations of \( n \) distinct elements \( a_{1}, a_{2}, \cdots, a_{n} \) such that \( a_{1} \) and \( a_{2} \) are not adjacent.


In an unfortunate accident, a maternity hospital lost all the tags for 8 babies born on the same day. Assuming there are no other markings, what is the probability that exactly 4 families will correctly identify their baby? $\qquad$


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

\[
\frac{(2 a + 2 a b - b(b + 1))^{2} + \left(b - 4 a^{2} + 2 a (b + 1)\right)^{2}}{4 a^{2} + b^{2}}
\]


Let \( P_{i}(x) = x^2 + b_{i} x + c_{i} \) be real coefficient polynomials (\( b_{i}, c_{i} \in \mathbb{R}, i = 1, 2, \ldots, n \)) that are distinct from each other, and for any \( 1 \leq i < j \leq n \), the polynomial \( P_{i}(x) + P_{j}(x) \) has exactly one real root. Find the maximal value of \( n \).


For a positive integer \( n \), let \( \tau(n) \) be the number of positive integer divisors of \( n \). How many integers \( 1 \leq n \leq 50 \) are there such that \( \tau(\tau(n)) \) is odd?


Inside a convex 13-sided polygon, there are 200 points such that no three of these 213 points (including the vertices of the polygon) lie on the same line. The polygon is divided into triangles, each vertex of which is any three of the given 213 points. What is the maximum number of triangles that could result?


Find the largest perfect square such that when the last two digits are removed, it is still a perfect square. (It is assumed that one of the digits removed is not zero.)

Solution: \boxed{41^2 = 1681}


Calculate the volume of the solid obtained by rotating around the x-axis the region bounded by the parabolas $y=x^{2}$ and $x=y^{2}$.


A two-meter gas pipe has rusted in two places. Determine the probability that all three resulting parts can be used as connections to gas stoves, if according to regulations, the stove should not be located at a distance closer than 50 cm from the main gas pipeline.


The task is given a finite increasing sequence \( a_{1}, a_{2}, \ldots, a_{n} \) (\(n \geq 3\)) of natural numbers, and for all \( k \leq n-2 \), the equality \( a_{k+2}=3 a_{k+1}-2 a_{k}-1 \) holds. The sequence must necessarily contain the term \( a_{k}=2021 \). Determine the maximum number of three-digit numbers divisible by 25 that this sequence can contain.


Counting losses. An English officer recounted that he was part of a unit initially consisting of 1000 men. During one operation, the unit suffered heavy losses, and the survivors were captured and sent to a prisoner of war camp.

On the first day of the march, $\frac{1}{6}$ of the remaining members of the unit escaped; on the second day, $\frac{1}{8}$ of the remaining members escaped and one person died; on the third day, $\frac{1}{4}$ of the remaining members escaped. The remaining prisoners were divided into 4 equal groups upon arrival at the camp.

How many people died during the operation?


Find all natural numbers \( n \) for which the sum of the digits of \( 5^{n} \) equals \( 2^{n} \).


There are 23 students in a class. During the year, each student celebrated their birthday once, attended by some (at least one, but not all) of their classmates. Could it be possible that every pair of students in this class met the same number of times at such celebrations? (It is assumed that any two guests met at each celebration, and the birthday student met with all guests.)

(I. Bogdanov)

Solution: \boxed{Yes}


Given a positive integer \( n \), taking the sum of any two of its positive divisors results in unique sums. The sums, in ascending order, are \( 4, 6, 8, 10, 12, 14, 16, 18, 20, 24, 46, 48, 50, 54, 60 \). Find the value of \( n \).


What is the last digit of the difference

$$
1 \cdot 2 \cdot 3 \cdot 4 \ldots 13 - 1 \cdot 3 \cdot 5 \cdot 7 \ldots 13 ?
$$


Katya correctly solves a problem with a probability of $4 / 5$, and the magic pen solves a problem correctly without Katya's help with a probability of $1 / 2$. In a test containing 20 problems, solving 13 correctly is enough to get a "good" grade. How many problems does Katya need to solve on her own and how many should she leave to the magic pen to ensure that the expected number of correct answers is at least 13?


On the sides \( BC \) and \( CD \) of the square \( ABCD \), points \( M \) and \( K \) are marked respectively so that \(\angle BAM = \angle CKM = 30^\circ\). Find \(\angle AKD\).


A prime number of the form \( 2^{p} - 1 \) is called a Mersenne prime, where \( p \) is also a prime number. To date, the largest known Mersenne prime is \( 2^{82589933} - 1 \). Find the last two digits of this number.


There were several whole heads of cheese stored in a warehouse. During the night, some rats came and ate 10 heads, with each rat eating an equal amount. Some of the rats got sick from overeating. The remaining seven rats came the next night and finished the leftover cheese, but each rat could only eat half as much cheese as they did the night before. How much cheese was originally stored in the warehouse?


If \(\frac{x-a-b}{c}+\frac{x-b-c}{a}+\frac{x-c-a}{b}=3\), where \(a, b, c\) are positive constants, find \(x\) in terms of \(a, b\) and \(c\).

Solution: \boxed{a+b+c}


In all the figures of this question, we see a triangle \(ABC\) divided into four parts; in these triangles, \(D\) is the midpoint of \(AB\), \(E\) is the midpoint of \(AC\), and \(FG\) measures \(\frac{1}{2} BC\).

a) The quadrilaterals \(DJMA\) and \(ELNA\) are obtained by rotating the quadrilaterals \(DHFB\) and \(EIGC\) by \(180^{\circ}\) around \(D\) and \(E\), respectively. Explain why the points \(M\), \(A\), and \(N\) are collinear, that is, why the measure of angle \(MAN\) is equal to \(180^{\circ}\).

b) In the figure below, point \(K\) is the intersection of lines \(JM\) and \(LN\). Explain why the triangles \(FGI\) and \(MNK\) are congruent.

The above items show that \(HJKL\) is a rectangle formed with the four parts into which triangle \(ABC\) was divided.

c) Show that \(LH = EF\).

d) In the figure below, triangle \(ABC\) has area 9 and \(HJKL\) is a square. Calculate the length of \(EF\).


Omar marks a sequence of 12 points on a straight line beginning with a point \(O\), followed by a point \(P\) with \(OP = 1\). He chooses the points so that each point is the midpoint of the two immediately following points. For example, \(O\) is the midpoint of \(PQ\), where \(Q\) is the third point he marks. What is the distance between the first point \(O\) and the 12th point \(Z\)?


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 \( x \).


Adil was born in 2015. His younger sister Bav was born in 2018. What is the minimum number of days by which Adil is older than Bav?


We place labeled points on a circle as follows. At step 1, take two points at opposite ends of a diameter and label them both 1. At step \(n > 1\), place a point at the midpoint of each arc created at step \(n-1\) and label it with the sum of the labels at the two adjacent points. What is the total sum of the labels after step \(n\)? For example, after step 4 we have: 1, 4, 3, 5, 2, 5, 3, 4, 1, 4, 3, 5, 2, 5, 3, 4.

Solution: \boxed{2 \times 3^{n-1}}


A tree grows in a rather peculiar manner. Lateral cross-sections of the trunk, leaves, branches, twigs, and so forth are circles. The trunk is 1 meter in diameter to a height of 1 meter, at which point it splits into two sections, each with diameter 0.5 meter. These sections are each one meter long, at which point they each split into two sections, each with diameter 0.25 meter. This continues indefinitely: every section of tree is 1 meter long and splits into two smaller sections, each with half the diameter of the previous. What is the total volume of the tree?


The line \( c \) is given by the equation \( y = x + 1 \). Points \( A \) and \( B \) have coordinates \( A(1, 0) \) and \( B(3, 0) \). Find the point \( C \) on the line \( c \) from which the segment \( AB \) is seen at the largest angle.

Solution: \boxed{(1, 2)}


Given a square \( ABCD \). Point \( N \) lies on side \( AD \) such that \( AN : ND = 2 : 3 \), point \( F \) lies on side \( CD \) such that \( DF : FC = 1 : 4 \), and point \( K \) lies on side \( AB \) such that \( AK : KB = 1 : 4 \). Find the angle \( \angle KNF \).


In a sports complex, there is a rectangular area \(ABCD\) with the longer side \(AB\). The diagonals \(AC\) and \(BD\) intersect at an angle of \(60^\circ\). Runners train on the large circuit \(ACBDA\) or on the small track \(AD\). Mojmír ran ten times around the large circuit, and Vojta ran fifteen times around the small track, in one direction and then fifteen times in the opposite direction. Together they ran \(4.5 \text{ km}\). How long is the diagonal \(AC\)?


The bases \(AB\) and \(CD\) of trapezoid \(ABCD\) are 55 and 31 respectively, and its diagonals are mutually perpendicular. Find the dot product of vectors \(\overrightarrow{AD}\) and \(\overrightarrow{BC}\).


Calculate the integral

$$
\int_{C}(1+i-2 \bar{z}) d z
$$

along the paths connecting the points \( z_{1}=0 \) and \( z_{2}=1+i \):

1. Along a straight line.
2. Along the parabola \( y=x^{2} \).
3. Along the broken line \( z_{1} \rightarrow z_{3} \rightarrow z_{2} \), where \( z_{3}=1 \).


A right triangle was cut along a straight line into two parts and these parts were combined to form a square. What is the length of the shorter leg if the longer leg is 10?


We know the following about the city bus routes:

I. Each route has 3 stops.

II. From any route, it is possible to transfer to any other route, but only at one stop.

III. From any stop, you can reach any other stop, but only with one route.

How many bus routes are there in that city?


Find the range of negative values \(a\) such that the inequality \( \sin^2 x + a \cos x + a^2 \geq 1 + \cos x \) holds for all \( x \in \mathbb{R} \).

Solution: \boxed{(-\infty, -2]}


Given a trapezoid \(ABCD\) with \(BC \parallel AD\), it is known that \(\angle B = 150^\circ\), \(\angle D\) is a right angle, \(AB = 4\), and \(BC = 3 - \sqrt{3}\). Find the perimeter of the trapezoid.


In triangle \( \triangle ABC \), the three interior angles \( \angle A, \angle B, \angle C \) satisfy \( \angle A = 3 \angle B = 9 \angle C \). Find the value of
\[ \cos A \cdot \cos B + \cos B \cdot \cos C + \cos C \cdot \cos A = \quad . \]


Ksyusha runs twice as fast as she walks (both speeds are constant).

On Tuesday, when she left home for school, she first walked, and then, when she realized she was late, she started running. The distance Ksyusha walked was twice the distance she ran. As a result, she reached the school from home in exactly 30 minutes.

On Wednesday, Ksyusha left home even later, so she had to run twice the distance she walked. How many minutes did it take her to get from home to school on Wednesday?


Given \( x, y, z \) are 3 real numbers greater than or equal to 1, find the sum of the numerator and denominator of the simplest fraction representing the minimum value of

$$\left(\frac{\sqrt{x^{2}(y-1)^{2}+y^{2}}}{x y}+\frac{\sqrt{y^{2}(z-1)^{2}+z^{2}}}{y z}+\frac{\sqrt{z^{2}(x-1)^{2}+x^{2}}}{z x}\right)^{2}$$.


A pool contains one-eighteenth of its capacity in water. Two water pipes begin to fill the pool simultaneously. When the pool reaches two-ninths of its capacity, the first pipe starts to fill the pool alone and takes 81 minutes to fill an amount equal to what the second pipe had already added. Then, the second pipe fills the pool alone for 49 minutes, and at this point, the total amount of water added by both pipes is the same. After that, both pipes continue to fill the pool. How many additional minutes are needed for both pipes to completely fill the pool?

Solution: \boxed{231 \text{ minutes}}


In the interior of a triangle \( ABC \) with area 1, points \( D \), \( E \), and \( F \) are chosen such that \( D \) is the midpoint of \( AE \), \( E \) is the midpoint of \( BF \), and \( F \) is the midpoint of \( CD \). Find the area of triangle \( DEF \).


Let the quadratic function \( f(x) = ax^2 + bx + c \). When \( x = -2 \), it has a maximum value of \( a^2 \), and its graph passes through the point \((-1, 6)\). Find the value of \( \frac{a+c}{b} \).


In the following figure, \( AD = AB \), \(\angle DAB = \angle DCB = \angle AEC = 90^{\circ} \) and \( AE = 5 \). Find the area of the quadrangle \( ABCD \).


The angles \(A, B, C\) of triangle \(ABC\) are \(44^\circ, 66^\circ,\) and \(70^\circ\) respectively. The angle bisector of \(\angle ABC\) and the perpendicular bisector of side \(AC\) intersect at point \(D\). How many degrees is \(\angle ADC\)? (Write the answer without the degree symbol.)


Given the constants \( k \) and \( t \), consider the quadratic equation in \( x \):
\[ k x^{2}+(3-3 k) x+2 k-6=0 \]

with integer roots only, and the quadratic equation in \( y \):
\[ (k+3) y^{2}-15 y+t=0 \]

that has two positive integer roots \( y_{1} \) and \( y_{2} \).

(1) Find the value of \( k \).

(2) For what value of \( t \) does \( y_{1}^{2}+y_{2}^{2} \) have its minimum value?


On a table, there are 10 coins, all showing heads up. A move consists of flipping exactly 4 coins.

a) What is the minimum number of moves required to have all coins showing tails up?

b) If there were 11 coins, would it be possible to have all coins showing tails up?

Solution: \boxed{\text{No}}


In a quadrilateral pyramid \(S A B C D\):

- The lateral faces \(S A B\), \(S B C\), \(S C D\), and \(S D A\) have areas of 9, 9, 27, and 27 respectively.
- The dihedral angles at the edges \(A B\), \(B C\), \(C D\), and \(D A\) are equal.
- The quadrilateral \(A B C D\) is inscribed and has an area of 36.

Find the volume of pyramid \(S A B C D\).


Find the sum of all integer values of \(h\) for which the equation \(|r+h|-2|r|-3r=7|r-1|\) with respect to \(r\) has at most one solution.


Huahua washed 40 handkerchiefs and hung them on 3 clotheslines to dry. Each handkerchief must be clipped at both ends. Huahua used a total of ______ clips.


Given that \( x_{1} \) and \( x_{2} \) are distinct complex numbers and satisfy the following three equations:
$$
\begin{array}{l}
x_{1}(x_{1}+1)=A, \quad x_{2}(x_{2}+1)=A, \\
x_{1}^{4}+3 x_{1}^{3}+5 x_{1}=x_{2}^{4}+3 x_{2}^{3}+5 x_{2}.
\end{array}
$$

Find the value of \( A \) that satisfies the above equations.


Find all values of \(a\) for which the quadratic function \(f(x) = a x^2 + 4a x - 1\) takes values whose absolute value does not exceed 4 for all points in the interval \([-4, 0]\). Provide the total length of the intervals to which these found values of \(a\) belong.


Consider the two points \( A(4,1) \) and \( B(2,5) \). For each point \( C \) with positive integer coordinates, we define \( d_C \) to be the shortest distance needed to travel from \( A \) to \( C \) to \( B \) moving only horizontally and/or vertically. For example, for the point \( C(3,4) \), we compute \( d_C \) as follows:

- To get from \( A \) to \( C \) moving only horizontally and/or vertically, we can move 1 unit to the left then 3 units up for a total distance of \( 1+3=4 \).
- The shortest path from \( C \) to \( B \) moving only horizontally and/or vertically is to move 1 unit left and then 1 unit up (or 1 unit up then 1 unit left) for a total distance of \( 1+1=2 \).

Thus, for \( C(3,4) \), we have that \( d_C = 4 + 2 = 6 \).

The positive integer \( N \) has the property that there are exactly 2023 points \( C(x, y) \) with \( x > 0 \) and \( y > 0 \) and \( d_C = N \). What is the value of \( N \)?


Two people, A and B, each roll a six-sided die. When a "6" is rolled, they stop rolling. The probability that the number of rolls between the two people differs by at most one is denoted as $\mathrm{P}$. Find $\mathrm{P}$.


From the six digits \( 0, 1, 2, 3, 4, 5 \), select 2 odd numbers and 2 even numbers to form a 4-digit number with no repeating digits. What is the probability that the resulting 4-digit number is an even number?


In the figure, the diagonals \(AC\) and \(BD\) of quadrilateral \(ABCD\) intersect at point \(O\), and \(E\) is the midpoint of \(BC\). The area of triangle \(ABO\) is 45, the area of triangle \(ADO\) is 18, and the area of triangle \(CDO\) is 69. Find the area of triangle \(AED\).


Find the minimum value of the function
$$
f(x)=\sqrt{15-12 \cos x}+\sqrt{4-2 \sqrt{3} \sin x}+\sqrt{7-4 \sqrt{3} \sin x}+\sqrt{10-4 \sqrt{3} \sin x-6 \cos x}.
$$


In a school there are \( n \) students, each with a different student number. Each student number is a positive factor of \( 60^{60} \), and the H.C.F. of any two student numbers is not a student number in the school. Find the greatest possible value of \( n \).


A circle is circumscribed around a unit square \(ABCD\), and a point \(M\) is selected on the circle.

What is the maximum value that the product \(MA \cdot MB \cdot MC \cdot MD\) can take?


We have 64 small wooden cubes, each with an edge of $2 \mathrm{~cm}$, that are covered with paper. Then, these small cubes are used to construct a large cube. How many $\mathrm{cm}^{2}$ of paper would we save if we first assembled the large cube and then covered it with paper, applying only one layer of paper?

Solution: \boxed{1152 \text{ cm}^2}


Determine the number of positive divisors of 18800 that are divisible by 235.


Mowgli asked five monkeys to bring him some nuts. The monkeys gathered the same number of nuts each and carried them to Mowgli. On the way, they quarreled, and each monkey threw one nut at every other monkey. As a result, they brought Mowgli twice as few nuts as they gathered. How many nuts did Mowgli receive? Explain your answer.


Find the natural number \( n \) that is the product of the primes \( p, q, \) and \( r \), given that

\[
r - q = 2p \quad \text{and} \quad rq + p^2 = 676
\]


Given that the operation "※" satisfies the following equations:
\(4 ※ 2 = 14\),
\(5 ※ 3 = 22\),
\(3 ※ 5 = 4\),
\(7 ※ 18 = 31\).

According to these rules, what is \(6 ※ 8\)?


In a regular tetrahedron \( ABCD \) with side length \( \sqrt{2} \), it is known that \( \overrightarrow{AP} = \frac{1}{2} \overrightarrow{AB} \), \( \overrightarrow{AQ} = \frac{1}{3} \overrightarrow{AC} \), and \( \overrightarrow{AR} = \frac{1}{4} \overrightarrow{AD} \). If point \( K \) is the centroid of \( \triangle BCD \), then what is the volume of the tetrahedron \( KPQR \)?


As shown in Figure 1, in triangle \( \triangle ABC \), the circumcenter is \( O \) and the incenter is \( I \). It is given that \( OI \perp AI \) and \( AB = 10 \), \( AC = 18 \). Find the length of \( BC \).


A white ball is added to an urn initially containing two balls, then one ball is drawn from the urn at random. Find the probability that the drawn ball is white, assuming all initial compositions of the balls (in terms of color) are equally likely.


Using the digits $1, 2, 3, 4$ to form 6-digit numbers where each digit can appear no more than 2 times, how many different 6-digit numbers can be formed?


Find the number of integers between 1 and 200 inclusive whose distinct prime divisors sum to 16.


The diagonals of quadrilateral \(ABCD\) intersect at point \(O\). It is known that \(AB = BC = CD\), \(AO = 8\), and \(\angle BOC = 120^\circ\). What is the length of \(DO\)?

Solution: \boxed{DO = 8}


A sphere with radius \( \frac{3}{2} \) has its center at point \( N \). From point \( K \), which is at a distance of \( \frac{3 \sqrt{5}}{2} \) from the center of the sphere, two lines \( K L \) and \( K M \) are drawn, tangent to the sphere at points \( L \) and \( M \) respectively. Find the volume of the pyramid \( K L M N \), given that \( M L = 2 \).


Let the real numbers \(x_{1}, x_{2}, \cdots, x_{1997}\) satisfy the following two conditions:
1. \(-\frac{1}{\sqrt{3}} \leqslant x_{i} \leqslant \sqrt{3}\) for \(i=1,2,\cdots,1997\)
2. \(x_{1} + x_{2} + \cdots + x_{1997} = -318 \sqrt{3}\)

Find the maximum value of \(x_{1}^{12} + x_{2}^{12} + \cdots + x_{1997}^{12}\), and give a reason for your answer.


The teacher asked the students to calculate \(\overline{AB} . C + D . E\). Xiao Hu accidentally missed the decimal point in \(D . E\), getting an incorrect result of 39.6; while Da Hu mistakenly saw the addition sign as a multiplication sign, getting an incorrect result of 36.9. What should the correct calculation result be?


Given the set \( A=\{1,2, \cdots, 10\} \). Define a function \( f: A \rightarrow A \) that satisfies:
(1) For any \( x, y \in A \), if \( x \leq y \), then \( f(x) \leq f(y) \);
(2) \( f(3)=4 \).
How many such functions \( f \) are there?


In a frame of dimensions \(8 \times 8\) with a width of 2 cells, there are a total of 48 cells.

How many cells are in a frame of dimensions \(254 \times 254\) with a width of 2 cells?


As shown in the figure, given that the area of rectangle $ADEF$ is 16, the area of triangle $ADB$ is 3, and the area of triangle $ACF$ is 4, find the area of triangle $ABC$.


How many points of a body must be fixed at least for the entire body to remain stationary?


Given that sets \( A \), \( B \), \( C \) are subsets of \(\{1, 2, \ldots, 2020\}\), and \( A \subseteq C \), \( B \subseteq C \), find the number of ordered triples \((A, B, C)\).


Highlight the solution \( (0,0) \) by dividing both parts of the first equation by \( xy \) and the second by \( x^{2} y^{2} \). We get:

\[
\left\{\begin{array}{c}\left(\frac{x}{y}+\frac{y}{x}\right)(x+y)=15 \\ \left(\frac{x^{2}}{y^{2}}+\frac{y^{2}}{x^{2}}\right)\left(x^{2}+y^{2}\right)=85\end{array}\right.
\]

Let \( x + y = a \ and \ \frac{x}{y} + \frac{y}{x} = b \), then the first equation will be:

\[
ab = 15
\]

Considering that:

\[
\frac{x^{2}}{y^{2}} + \frac{y^{2}}{x^{2}} = b^{2} - 2 \ and \ xy = \frac{a^{2}}{2+b}
\]

Then, the second equation can be written as:

\[
(b^{2} - 2) a^{2} b = 85(2 + b)
\]

Substitute the expression for \( a \) from the first equation into this, we get:

\[
\frac{3}{b}(b^{2} - 2) 15 = 17(2 + b) \Rightarrow 14b^{2} - 17b - 45 = 0
\]

Then:

\[
b_{1} = -\frac{9}{7}, \ b_{2} = \frac{5}{2}
\]

Then:

\[
a_{1} = \frac{15}{b} = -\frac{15}{\frac{9}{7}} = -\frac{35}{3}, \ a_{2} = 6
\]

We have two systems:

1) \(\left\{\begin{array}{l}x + y = 6 \\ \frac{x}{y} + \frac{y}{x} = \frac{5}{2}\end{array}\right.\)

2) \(\left\{\begin{array}{l}x + y = -\frac{35}{3} \\ \frac{x}{y} + \frac{y}{x} = -\frac{9}{7}\end{array}\right.\)

Solve the first system:

\(\left\{\begin{array}{c}x + y = 6 \\ x^{2} + y^{2} = \frac{5}{2} xy\end{array} \Rightarrow \left\{\begin{array}{c} x + y = 6 \\ (x + y)^{2} = 4.5xy\end{array}\Rightarrow \left\{\begin{array}{c}x + y = 6 \\ 4.5xy = 36\end{array}\right.\right.\right.\)

\(\left\{\begin{array}{c}x + y = 6 \\ xy = 8\end{array}\right.\)

By Vieta's theorem:
\[
x = 2, \ y = 4 \ or \ x = 4, \ y = 2
\]

Solve the second system:

\(\left\{\begin{array}{l}x + y = -\frac{35}{3} \\ \frac{x}{y} + \frac{y}{x} + \frac{9}{7} = 0\ \left(\frac{y}{x}\right)^{2}+\frac{9}{7}\left(\frac{y}{x}\right)+1=0 \quad D<0\right.\}\) has no solutions.

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


We assume in this exercise and the following ones that $a b c = 1$. Show that

$$
\frac{1}{a^{3}(b+c)}+\frac{1}{b^{3}(c+a)}+\frac{1}{c^{3}(a+b)} \geq \frac{3}{2}
$$


A hospital has the following employees:

- Sara Dores da Costa: rheumatologist
- Iná Lemos: pulmonologist
- Ester Elisa: nurse
- Ema Thomas: traumatologist
- Ana Lisa: psychoanalyst
- Inácio Filho: obstetrician

a) In how many ways can the employees form a line?

b) In how many ways can the same employees sit at a round table? Remember that at a round table, if everyone moves to the seat to their left, the table looks the same!

c) And in how many ways can the employees form a committee consisting of a president, vice-president, and substitute?


What is the largest integer for which each pair of consecutive digits is a square?


How many roots does the equation

$$
\overbrace{f(f(\ldots f}^{10 \text{ times } f}(x) \ldots)) + \frac{1}{2} = 0
$$

have, where \( f(x) = |x| - 1 \)?


The motion of the fluid is defined by the complex potential \( f(z) = \ln \operatorname{sh} \pi z \). Find the flow rate \( N_{L} \) through the circle \( 2|z| = 3 \) and the circulation \( \Gamma_{L} \) around it.

Solution: \boxed{\Gamma_{L} = 0}


Elsa wrote the numbers $1, 2, \ldots, 8$ on the board. She then allows herself operations of the following form: she selects two numbers $a$ and $b$ such that $a+2 \leq b$, if such numbers exist, then she erases them, and writes the numbers $a+1$ and $b-1$ in their place.

(a) Demonstrate that Elsa must stop after a finite number of operations.

(b) What is the maximum number of operations Elsa can perform?


Fill each blank with a number not equal to 1, such that the equation holds true. The number of different ways to fill the blanks is $\qquad$.
$$
[\mathrm{A} \times(\overline{1 \mathrm{~B}}+\mathrm{C})]^{2}=\overline{9 \mathrm{DE} 5}
$$


The value of the product \(\cos \frac{\pi}{15} \cos \frac{2 \pi}{15} \cos \frac{3 \pi}{15} \cdots \cos \frac{7 \pi}{15}\).


Let \( \triangle ABC \) be an acute triangle with \( BC = 5 \). \( E \) is a point on \( AC \) such that \( BE \perp AC \). \( F \) is a point on \( AB \) such that \( AF = BF \). Moreover, \( BE = CF = 4 \). Find the area of the triangle.

Solution: \boxed{8 \sqrt{3} - 6}


In a triangle, two sides are given. For what value of the angle between them will the area of the triangle be the greatest?


A coin is flipped 2021 times. What is the probability that an even number of tails will result?


What amount can a worker, who is a tax resident, expect after paying personal income tax if they are credited with 45000 for their work?


Given a chessboard, it is allowed to recolor all the cells of any horizontal or vertical line in a different color. Is it possible to obtain a board with exactly one black cell?

Solution: \boxed{\text{No}}


What is the angular measure of an arc if the radius drawn to its endpoint forms an angle of $40^{\circ}$ with its chord?


From Moscow to city \( N \), a passenger can travel by train, taking 20 hours. If the passenger waits for a flight (waiting will take more than 5 hours after the train departs), they will reach city \( N \) in 10 hours, including the waiting time. By how many times is the plane’s speed greater than the train’s speed, given that the plane will be above this train 8/9 hours after departure from the airport and will have traveled the same number of kilometers as the train by that time?


Let \( A B C D \) be a parallelogram such that \( \angle B A D = 60^\circ \). Let \( K \) and \( L \) be the midpoints of \( B C \) and \( C D \), respectively. Assuming that \( A B K L \) is a cyclic quadrilateral, find \( \angle A B D \).


On the unit square \(ABCD\), the points \(A_1, B_1, C_1, D_1\) are marked such that \(AA_1 = BB_1 = CC_1 = DD_1 = \frac{1}{5}\). The points \(A\) and \(C_1\), \(B\) and \(D_1\), \(C\) and \(A_1\), \(D\) and \(B_1\) are then connected. What is the area of the small quadrilateral formed by these intersecting lines?


Find all ordered pairs \((m, n)\) of integers such that \(4^m - 4^n = 255\).


A string of 33 pearls has its middle pearl as the largest and most valuable. The values of the remaining pearls decrease by $3000 \mathrm{Ft}$ per pearl towards one end and by $4500 \mathrm{Ft}$ per pearl towards the other end. How much is the middle pearl worth if the total value of the string is 25 times the value of the fourth pearl from the middle on the more expensive side?


A table consists of several rows of numbers. From the second row onwards, each number in a row is equal to the sum of the two numbers directly above it. The last row contains only one number. The first row is made up of the first 100 positive integers arranged in ascending order. What is the number in the last row? (You may use exponential notation to express the number).

Solution: \boxed{101 \times 2^{98}}


Given a positive integer \(n > 1\). Let \(a_{1}, a_{2}, \cdots, a_{n}\) be a permutation of \(1, 2, \cdots, n\). If \(i < j\) and \(a_{i} < a_{j}\), then \(\left(a_{i}, a_{j}\right)\) is called an ascending pair. \(X\) is the number of ascending pairs in \(a_{1}, a_{2}, \cdots, a_{n}\). Find \(E(X)\).


What is the smallest positive odd number that has the same number of divisors as 360?


As shown in Diagram 2, in the cube $AC_{1}$, $M$ and $N$ are the midpoints of $D_{1}C_{1}$ and $AB$ respectively. Determine the distance from $C$ to the plane $MB_{1}ND$.


Find the positive integers $n$ such that $n^{2}+1$ divides $n+1$.


The pages in a book are numbered as follows: the first sheet contains two pages (numbered 1 and 2), the second sheet contains the next two pages (numbered 3 and 4), and so on. A mischievous boy named Petya tore out several consecutive sheets: the first torn-out page has the number 185, and the number of the last torn-out page consists of the same digits but in a different order. How many sheets did Petya tear out?


Given the function \( f_{1}(x) = \frac{2x - 1}{x + 1} \), for a positive integer \( n \), define \( f_{n+1}(x) = f_{1}\left[ f_{n}(x) \right] \). Find the explicit formula for \( f_{1234}(x) \).


Find all integers $n$ such that $2^{n} + 3$ is a perfect square. Same question with $2^{n} + 1$.


The dwarves painted cubic blocks with green and white paint so that each face was entirely painted with one of these two colors. After a while, they noticed that some painted blocks looked exactly the same after suitable rotations and began sorting them into groups based on this criterion (blocks that look the same are in the same group).

What is the maximum number of such groups they could get?

Hint: In what relationships can pairs of cube faces be?


The game takes place on a $9 \times 9$ grid of squared paper. Two players take turns; the player who starts the game places crosses in free cells while their partner places noughts. When all the cells are filled, the number of rows and columns $K$ in which crosses outnumber noughts, and the number of rows and columns $H$ in which noughts outnumber crosses, is counted. The difference $B=K-H$ is considered the winning score of the player who starts. Find the value of $B$ such that:

1) The first player can ensure a win of at least $B$, regardless of how the second player plays;
2) The second player can always ensure that the first player’s winning score is no more than $B$, regardless of how the first player plays.

Solution: \boxed{B = 2}


In triangle \(ABC\), let \(M\) be the midpoint of \(BC\), \(H\) be the orthocenter, and \(O\) be the circumcenter. Let \(N\) be the reflection of \(M\) over \(H\). Suppose that \(OA = ON = 11\) and \(OH = 7\). Compute \(BC^2\).


Points \( Q \) and \( R \) are located on the sides \( MN \) and \( MP \) of triangle \( MNP \), respectively, where \( MQ = 3 \) and \( MR = 4 \). Find the area of triangle \( MQR \), given \( MN = 4 \), \( MP = 5 \), and \( NP = 6 \).


How many six-digit multiples of 27 have only 3, 6, or 9 as their digits?


If the numbers \(a, b, c\) are pairwise coprime, then

$$
(a b + b c + a c, a b c) = 1
$$


Choose one vertex of a cube and consider the rays leading from it to the other vertices. How many different angles do we get when these rays are paired in all possible ways?


A sphere is the set of points at a fixed positive distance \( r \) from its center. Let \(\mathcal{S}\) be a set of 2010-dimensional spheres. Suppose that the number of points lying on every element of \(\mathcal{S}\) is a finite number \( n \). Find the maximum possible value of \( n \).


Given that \(\alpha\) and \(\beta\) satisfy \(\tan \left(\alpha + \frac{\pi}{3}\right) = -3\) and \(\tan \left(\beta - \frac{\pi}{6}\right) = 5\), find the value of \(\tan (\alpha - \beta)\).


Solution: \boxed{a^{1/3} + b^{1/3}}


Find the sum of the digits of the number $\underbrace{44 \ldots 4}_{2012 \text { times }} \cdot \underbrace{99 \ldots 9}_{2012 \text { times }}$.


In how many ways can we paint 16 seats in a row, each red or green, in such a way that the number of consecutive seats painted in the same colour is always odd?


Five guys join five girls for a night of bridge. Bridge games are always played by a team of two guys against a team of two girls. The guys and girls want to make sure that every guy and girl play against each other an equal number of times. Given that at least one game is played, what is the least number of games necessary to accomplish this?


If \( a > b > c > d \) is an arithmetic progression of positive reals and \( a > h > k > d \) is a geometric progression of positive reals, show that \( bc > hk \).

Solution: \boxed{bc > hk}


Let \( C \) be the complex numbers. \( f : C \rightarrow C \) satisfies \( f(z) + z f(1 - z) = 1 + z \) for all \( z \). Find \( f \).

Solution: \boxed{f(w) = 1}


In triangle \(ABC\), the angle bisector \(AL\) (where \(L \in BC\)) is drawn. Points \(M\) and \(N\) lie on the other two angle bisectors (or their extensions) such that \(MA = ML\) and \(NA = NL\). Given that \(\angle BAC = 50^\circ\).

Find the measure of \(\angle MAN\) in degrees.


Let \( G \) be the group \( \{ (m, n) : m, n \text{ are integers} \} \) with the operation \( (a, b) + (c, d) = (a + c, b + d) \). Let \( H \) be the smallest subgroup containing \( (3, 8) \), \( (4, -1) \) and \( (5, 4) \). Let \( H_{ab} \) be the smallest subgroup containing \( (0, a) \) and \( (1, b) \). Find \( a > 0 \) such that \( H_{ab} = H \).


For each positive integer \( n \), let \( \varphi(n) \) be the number of positive integers from 1 to \( n \) that are relatively prime to \( n \). Evaluate:
$$
\sum_{n=1}^{\infty} \frac{\varphi(n) 4^{n}}{7^{n}-4^{n}}
$$


Different numbers \(a\), \(b\), and \(c\) are such that the equations \(x^{2}+a x+1=0\) and \(x^{2}+b x+c=0\) have a common real root. In addition, the equations \(x^{2}+x+a=0\) and \(x^{2}+c x+b=0\) also have a common real root. Find the sum \(a+b+c\).


What is the minimum number of cells that need to be colored on a $6 \times 6$ board so that, no matter how a 4-cell L-shaped figure is placed on the board (it can be rotated or flipped), there is at least one colored cell in that figure?


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


The 3rd term of an arithmetic sequence is 14, and the 18th term is 23. Determine which term(s) among the first 2010 terms of the sequence are integers.


Determine the number of 7-combinations of the multiset $S = \{4 \cdot a, 4 \cdot b, 3 \cdot c, 3 \cdot d\}$.


Calculate (without using a calculator) \(\sqrt[3]{9+4 \sqrt{5}}+\sqrt[3]{9-4 \sqrt{5}}\), given that this number is an integer.


If the set \( S = \{1, 2, 3, \cdots, 16\} \) is arbitrarily divided into \( n \) subsets, there must exist some subset that contains elements \( a, b, \) and \( c \) (which can be the same) such that \( a + b = c \). Find the maximum value of \( n \).

**Note**: If the subsets \( A_1, A_2, \cdots, A_n \) of set \( S \) satisfy the following conditions:
1. \( A_i \neq \varnothing \) for \( i = 1, 2, \cdots, n \);
2. \( A_i \cap A_j = \varnothing \);
3. \( \bigcup_{i=1}^{n} A_i = S \),

then \( A_1, A_2, \cdots, A_n \) are called a partition of set \( S \).


Given $n$ rays in space such that any two rays form an obtuse angle, what is the maximum value of $n$?


Let \( x, y, z \) be strictly positive real numbers. Show that the following inequality is true:

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


Evaluate the sum
$$
\cos \left(\frac{2\pi}{18}\right) + \cos \left(\frac{4\pi}{18}\right) + \cdots + \cos \left(\frac{34\pi}{18}\right).
$$


In an isosceles triangle \(ABC\) where \(AB = AC\) and \(\angle BAD = 30^{\circ}\), \(D\) is an interior point on side \(BC\). Furthermore, triangle \(ADE\) is also isosceles, where \(E\) is an interior point on side \(AC\). What is the measure of \(\angle EDC\)?


Two fair octahedral dice, each with the numbers 1 through 8 on their faces, are rolled. Let \( N \) be the remainder when the product of the numbers showing on the two dice is divided by 8. Find the expected value of \( N \).


Let's call a natural number interesting if all its digits are distinct and the sum of any two adjacent digits is a square of a natural number. Find the largest interesting number.


Four points are independently chosen uniformly at random from the interior of a regular dodecahedron. What is the probability that they form a tetrahedron whose interior contains the dodecahedron's center?


Compute \( t(0)-t\left(\frac{\pi}{5}\right)+t\left(\frac{2\pi}{5}\right)-t\left(\frac{3\pi}{5}\right)+\ldots+t\left(\frac{8\pi}{5}\right)-t\left(\frac{9\pi}{5}\right) \), where \( t(x) = \cos 5x + * \cos 4x + * \cos 3x + * \cos 2x + *^2 \cos x + * \).

The coefficients indicated by * are missing. A math student claimed that he could compute the sum without knowing the values of the missing coefficients. Is he right?


ABC is a triangle. The points P and Q trisect BC, so BP = PQ = QC. A line parallel to AC meets the lines AB, AP, AQ at X, Y, and Z respectively. Show that YZ = 3 XY.

Solution: \boxed{YZ = 3XY}


Calculate the volume of the solid bounded by the surfaces.

$$
z=2 x^{2}+8 y^{2}, z=4
$$


It is known that in 3 out of 250 cases, twins are born, and in one out of three of these cases, the twins are identical (monozygotic) twins. What is the a priori probability that a particular pregnant woman will give birth to twins - a boy and a girl?


Let \( A = \{a_1, a_2, a_3\} \) and \( B = \{b_1, b_2, b_3, b_4\} \).
1. Write a function \( f: A \rightarrow B \) such that \( f \) is injective, and find the number of injective functions from \( A \) to \( B \).
2. Write a function \( f: A \rightarrow B \) such that \( f \) is not injective, and find the number of such functions.
3. Can a function from \( A \) to \( B \) be surjective?

Solution: \boxed{\text{No}}


In a $3 \times 3$ table, numbers are placed such that each number is 4 times smaller than the number in the adjacent cell to the right and 3 times smaller than the number in the adjacent cell above. The sum of all the numbers in the table is 546. Find the number in the central cell.


Let \( f: \mathbb{N} \rightarrow \mathbb{Q} \) be a function, where \( \mathbb{N} \) denotes the set of natural numbers, and \( \mathbb{Q} \) denotes the set of rational numbers. Suppose that \( f(1) = \frac{3}{2} \), and
\[ f(x+y) = \left(1 + \frac{y}{x+1}\right) f(x) + \left(1 + \frac{x}{y+1}\right) f(y) + x^2 y + xy + xy^2 \]
for all natural numbers \( x, y \). Find the value of \( f(20) \).


Find the derivative of the implicit function $y$ given by the equations:

1) $x^{2} + y^{2} + 2x - 6y + 2 = 0$

2) $x^{y} = y^{x}$

and calculate its value at $x = 1$.


In the company, there are elves, fairies, and dwarves. Each elf is friends with all fairies except for three of them, and each fairy is friends with twice as many elves. Each elf is friends with exactly three dwarves, and each fairy is friends with all the dwarves. Each dwarf is friends with exactly half of the total number of elves and fairies. How many dwarves are there in the company?


Find the increments of the argument and the function for \( y = 2x^2 + 1 \) when the argument \( x \) changes from 1 to 1.02.


Find the greatest natural number that cannot be expressed as the sum of two composite numbers.


Is \(65^{1000} - 8^{2001}\) greater than, less than, or equal to 0? (Fill in the blank with ">", "<", or "=").

Solution: \boxed{>}


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

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


Maria Ivanovna is a strict algebra teacher. She only gives grades of 2, 3, and 4, and never gives two consecutive 2s to the same student. It is known that she gave Vovochka 6 grades in a quarter. In how many different ways could she have done this?


The number \( N = 3^{16} - 1 \) has a divisor of 193. It also has some divisors between 75 and 85 inclusive. What is the sum of these divisors?


A group with 7 young men and 7 young women was divided into pairs randomly. Find the probability that at least one pair consists of two women. Round the answer to two decimal places.


Using the digits 1, 2, 3, 4 only once to form a 4-digit number, how many of them are divisible by 11?


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


A certain sports team has members with distinct numbers chosen from the positive integers 1 to 100. If the number of any member is neither the sum of the numbers of two other members nor twice the number of another member, what is the maximum number of members this sports team can have?


From an $m \times n$ chessboard, how many ways are there to select an L-shaped piece consisting of three squares?


Let \( S = \{1, 2, \ldots, 1963\} \). What is the maximum number of elements that can be chosen from \( S \) such that the sum of any two chosen numbers is not divisible by their difference?


On a board, the numbers $1, 2, 3, \ldots, 235$ were written. Petya erased several of them. It turned out that among the remaining numbers, no number is divisible by the difference of any two others. What is the maximum number of numbers that could remain on the board?


Compute the value of the infinite series
$$
\sum_{n=2}^{\infty} \frac{n^{4}+3 n^{2}+10 n+10}{2^{n} \cdot \left(n^{4}+4\right)}
$$


A fishing boat fishes in the territorial waters of a foreign country without permission. Each time the net is cast, it causes an equivalent-valued loss in the country's fish resources. The probability that the boat will be detained by the foreign coast guard during any net casting is $1/k$, where $k$ is a natural number specific to the country. Assume that the events of being detained or not during each net casting are independent of previous fishing activities. If the boat is detained by the coast guard, all previously caught fish are confiscated, and future fishing in those waters is prohibited. The captain plans to leave the foreign territorial waters after casting the net $n$ times.

Given that there is always a possibility of being detained by the foreign coast guard, making the fishing profit a random variable, determine the value of $n$ that maximizes the expected value of the fishing profit.

Solution: \boxed{n = k-1}


Find all functions \( f: \mathbb{R} \to \mathbb{R} \) such that for all real numbers \( x \) and \( y \), the following equation holds:

$$
f(y f(x+y) + f(x)) = 4x + 2y f(x+y)
$$

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


We are considering numbers that can be expressed as \(10 \times n + 1\), where \( \mathbf{n} \) is a positive integer. For example: \(11 = 10 \times 1 + 1\) and \(331 = 10 \times 33 + 1\). If such a number cannot be expressed as the product of two smaller numbers of the form \(10 \times n + 1\) (these two smaller numbers can be equal), we call this number a "Zhonghuan number". For example, \(341 = 11 \times 31\), since it can be expressed as the product of two numbers of the form \(10 \times n + 1\), it is not a "Zhonghuan number". However, 11 cannot be expressed as the product of two smaller numbers of the form \(10 \times n + 1\), so it is a "Zhonghuan number". How many "Zhonghuan numbers" are there among \(11, 21, 31, \ldots, 991\)?


On a ruler, three marks are indicated: 0, 2, and 5. How can you use this ruler to measure a segment equal to 6?


Let \( P \) be a polynomial with real coefficients such that

$$
\forall x \in \mathbb{R}, P(x) \geq 0.
$$

Show that there exist two polynomials \( A \) and \( B \) with real coefficients such that \( P = A^{2} + B^{2} \).

Solution: \boxed{P = A^2 + B^2}


Real numbers \(a, b, c\) and positive number \(\lambda\) make \(f(x) = x^3 + ax^2 + bx + c\) have three real roots \(x_1, x_2, x_3\), and satisfy:
1. \(x_2 - x_1 = \lambda\);
2. \(x_3 > \frac{1}{2}(x_1 + x_2)\).

Find the maximum value of \(\frac{2a^3 + 27c - 9ab}{\lambda^3}\).


For each natural number from 1 to 999, Damir subtracted the last digit from the first digit and wrote all the resulting 1000 differences on the board. For example, for the number 7, Damir wrote 0; for the number 105, he wrote (-4); for the number 61, he wrote 5.

What is the sum of all the numbers on the board?


In the pyramid \( A B C D \), the angle \( A B C \) is \( \alpha \). The orthogonal projection of point \( D \) onto the plane \( A B C \) is point \( B \). Find the angle between the planes \( A B D \) and \( C B D \).

Solution: \boxed{\alpha}


A circle is constructed on the side $BC$ of triangle $ABC$ as its diameter, and it intersects segment $AB$ at point $D$. Find the ratio of the areas of triangles $ABC$ and $BCD$, given that $AC = 15$, $BC = 20$, and $\angle ABC = \angle ACD$.


Let \(m\) be a positive integer, and let \(T\) denote the set of all subsets of \(\{1, 2, \ldots, m\}\). Call a subset \(S\) of \(T\) \(\delta\)-good if for all \(s_1, s_2 \in S\), \(s_1 \neq s_2\), \(|\Delta(s_1, s_2)| \geq \delta m\), where \(\Delta\) denotes symmetric difference (the symmetric difference of two sets is the set of elements that are in exactly one of the two sets). Find the largest possible integer \(s\) such that there exists an integer \(m\) and a \(\frac{1024}{2047}\)-good set of size \(s\).


Find the largest 5-digit number \( A \) that satisfies the following conditions:
1. Its 4th digit is greater than its 5th digit.
2. Its 3rd digit is greater than the sum of its 4th and 5th digits.
3. Its 2nd digit is greater than the sum of its 3rd, 4th, and 5th digits.
4. Its 1st digit is greater than the sum of all other digits.

(from the 43rd Moscow Mathematical Olympiad, 1980)


A regular hexagon is drawn on the plane with a side length of 1. Using only a ruler, construct a segment whose length is $\sqrt{7}$.


Write the result of the following expressions in decimal form.
(a) $7 \times \frac{2}{3} + 16 \times \frac{5}{12}$
(b) $5 - \left(2 \div \frac{5}{3}\right)$
(c) $1 + \frac{2}{1 + \frac{3}{1+4}}$


Egor wrote a number on the board and encrypted it according to the rules of letter puzzles (different letters correspond to different digits, identical letters - identical digits). The result was the word "GWATEMALA". How many different numbers could Egor have originally written if his number was divisible by 5?


Given that the real numbers \( x \) and \( y \) satisfy \( 4x^2 - 5xy + 4y^2 = 5 \), let \( S = x^2 + y^2 \). Determine the value of \( \frac{1}{S_{\max}} + \frac{1}{S_{\min}} \).


Given an infinite grid where each cell is either red or blue, such that in any \(2 \times 3\) rectangle exactly two cells are red, determine how many red cells are in a \(9 \times 11\) rectangle.


Find the flux of the vector field

$$
\vec{a} = x \vec{i} + y \vec{j} + z \vec{k}
$$

through the part of the surface

$$
x^2 + y^2 = 1
$$

bounded by the planes \( z = 0 \) and \( z = 2 \). (The normal vector is outward to the closed surface formed by these surfaces).


Find the largest three-digit number that is equal to the sum of its digits and the square of twice the sum of its digits.


Consider a \(2 \times 3\) grid where each entry is one of 0, 1, and 2. For how many such grids is the sum of the numbers in every row and in every column a multiple of 3? One valid grid is shown below:
$$
\left[\begin{array}{lll}
1 & 2 & 0 \\
2 & 1 & 0
\end{array}\right]
$$


When Vitya was a first grader, he had a set of 12 number cards: two cards with the number 1, two cards with the number 2, and so on up to the number 6. Vitya placed them on the table in a random order from left to right, and then removed the first 1, the first 2, the first 3, and so on. For example, if Vitya initially had the sequence 434653625112, the resulting sequence would be 436512. What is the probability that the sequence left on the table is 123456?


Given that positive integers \( a, b, c \) satisfy the equation \( a^{2} b + b^{2} c + a c^{2} + a + b + c = 2(a b + b c + a c) \), find \( \frac{c^{2017}}{a^{2016} + b^{2018}} \).


a) Derive the Leibniz formula from the identity in problem 140(c):

$$
\frac{\pi}{4}=1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\ldots
$$

b) What is the sum of the infinite series

$$
1+\frac{1}{3^{2}}+\frac{1}{5^{2}}+\frac{1}{7^{2}}+\ldots ?
$$


On the sides \( AB, BC \), and \( AC \) of triangle \( ABC \), points \( K, L \), and \( M \) are taken respectively, such that \( AK:KB = 2:3 \), \( BL:LC = 1:2 \), and \( CM:MA = 3:1 \). In what ratio does the segment \( KL \) divide the segment \( BM \)?


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


Adam the Ant started at the left-hand end of a pole and crawled $\frac{2}{3}$ of its length. Benny the Beetle started at the right-hand end of the same pole and crawled $\frac{3}{4}$ of its length. What fraction of the length of the pole are Adam and Benny now apart?


For \(x = \frac{\pi}{2n}\), find the value of the sum

$$
\cos ^{2}(x)+\cos ^{2}(2 x)+\cos ^{2}(3 x)+\ldots+\cos ^{2}(n x)
$$


The following numbers are written on a wall: \(1, 3, 4, 6, 8, 9, 11, 12, 16\). Four of these numbers were written by Vova, four were written by Dima, and one of the numbers is the house number of the local police officer. The officer discovered that the sum of the numbers written by Vova is three times the sum of the numbers written by Dima. Calculate the house number.


Find the smallest natural number that has exactly 12 different natural divisors, the largest prime divisor of which is the number 101, and the last digit is zero.


In the base of a regular triangular prism is a triangle \( ABC \) with a side length of \( a \). Points \( A_{1} \), \( B_{1} \), and \( C_{1} \) are taken on the lateral edges, and their distances from the base plane are \( \frac{a}{2} \), \( a \), and \( \frac{3a}{2} \) respectively. Find the angle between the planes \( ABC \) and \( A_{1} B_{1} C_{1} \).


Find all odd integers \( y \) such that there exists an integer \( x \) satisfying

\[ x^{2} + 2 y^{2} = y x^{2} + y + 1. \]

Solution: \boxed{y = 1}


Given a triangle \(ABC\) with \(AB=4\), \(BC=4\), and \(AC=1\). From point \(A\), an angle bisector is drawn that intersects the circumcircle of this triangle at point \(D\). Find the length \(DI\), where \(I\) is the incenter of triangle \(ABC\).


Several boys and girls are seated around a round table. It is known that to the left of exactly 7 girls sit girls, and to the left of 12 girls sit boys. It is also known that for 75% of the boys, girls sit to their right. How many people are seated at the table?


A cylinder with a volume of 9 is inscribed in a cone. The plane of the upper base of this cylinder truncates the original cone, forming a frustum with a volume of 63. Find the volume of the original cone.


Denote by \(\mathbb{N}\) the set of all positive integers. Find all functions \(f: \mathbb{N} \rightarrow \mathbb{N}\) such that for all positive integers \(m\) and \(n\), the integer \(f(m) + f(n) - mn\) is nonzero and divides \(mf(m) + nf(n)\).

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


How many two-digit numbers have the tens digit greater than the units digit?


1500 products are packed into 15 boxes for shipment. After sealing the boxes, it is discovered that one product was not packed into any box. Due to strict sealing requirements, boxes cannot be reopened for inspection. Someone suggests using a scale to weigh the boxes to find which box is missing a product (because the box missing a product will be lighter). Normally, it would take 14 weighings to find the box missing the product. Is there a way to reduce the number of weighings and still determine which box is missing a product?


Given a convex quadrilateral \(ABCD\) with \(\angle C = 57^{\circ}\), \(\sin \angle A + \sin \angle B = \sqrt{2}\), and \(\cos \angle A + \cos \angle B = 2 - \sqrt{2}\), find the measure of angle \(D\) in degrees.


The vertex \( A \) of the acute triangle \( ABC \) is equidistant from the circumcenter \( O \) and the orthocenter \( H \). Determine all possible values for the measure of angle \( A \).


There are three cars, $A$, $B$, and $C$, starting from the same location and traveling along the same road. They each catch up to a cyclist, Xiao Long, who is traveling in the same direction, at 5 minutes, 10 minutes, and 12 minutes respectively. Given that the speeds of $A$ and $B$ are 24 km/h and 20 km/h respectively, find the speed of car $C$, in km/h.


Given that \(a - b = 2 + \sqrt{3}\) and \(b - c = 2 - \sqrt{3}\), find the value of \(a^2 + b^2 + c^2 - ab - bc - ca\).


Arrange in ascending order the numbers that each contain exactly once each of the digits $1, 2, \ldots, 9$. Which number is in the 100,000th position?


Given the parabola \( y^{2} = 4ax \) where \( 0 < a < 1 \) with a focus at \( F \). Taking \( A(a + 4, 0) \) as the center of a circle, and \(|AF|\) as the radius, draw a circle above the x-axis that intersects the parabola at two distinct points \( M \) and \( N \). Let \( P \) be the midpoint of \( MN \).

(1) Find the value of \( |MF| + |NF| \).

(2) Does a value of \( a \) exist such that \( |MF| \), \( |PF| \), and \( |NF| \) form an arithmetic sequence? If such a value exists, find \( a \); if it does not, provide an explanation.


As shown in the diagram, \( AB \) is a path 28 meters long, and \( M \) is the midpoint of \( AB \). A dog starts running from a point on the left side of \( M \) along the path. The first run is 10 meters, the second run is 14 meters; the odd-numbered runs are 10 meters each, and the even-numbered runs are 14 meters each. The dog changes direction as follows: if after finishing a run \( M \) is to its right, it runs to the right next; if \( M \) is to its left, it runs to the left next. After running 20 times, the dog is 1 meter to the left of point \( B \). How far was the dog from point \( A \) when it started?


ABCDEF is a six-digit number. All of its digits are different and arranged in ascending order from left to right. This number is a perfect square.

Determine what this number is.


For how many ordered pairs of positive integers \((a, b)\) is \(1 < a + b < 22\)?


Find the pairs of positive integers \((a, b)\) such that \(\frac{a b^{2}}{a+b}\) is a prime number, where \(a \neq b\).


As shown in Figure 1, a line segment of length 1 is divided into two parts, $x$ and $y$. Then, the segment of length $x$ is bent into a semicircular arc $ACB$, and the segment of length $y$ is folded into a rectangle $ABDE$ along three sides $(BD, DE, EA)$, forming a closed "curved shape" $ACBDEA$. What is the maximum area of this curved shape?


Father has left several identical gold coins to his children. According to his will, the oldest child receives one coin and one-seventh of the remaining coins, the next child receives two coins and one-seventh of the remaining coins, the third child receives three coins and one-seventh of the remaining coins, and so on through the youngest child. If every child inherits an integer number of coins, find the number of children and the number of coins.


There were two decks of 36 cards each on the table. The first deck was shuffled and placed on top of the second deck. Then, for each card in the first deck, the number of cards between it and the same card from the second deck was counted (i.e., the number of cards between the sevens of hearts, between the queens of spades, etc.). What is the sum of these 36 counted numbers?


Given that the positive real numbers \( u, v, \) and \( w \) are all not equal to 1, if \(\log _{u} (v w)+\log _{v} w=5\) and \(\log _{v} u+\log _{w} v=3\), then find the value of \(\log _{w} u\).


Consider all possible 100-digit numbers where each digit is either 1 or 2. For each number, compute the remainder when divided by 1024. How many different remainders are there?


After walking $\frac{4}{9}$ of the bridge's length, a pedestrian notices a car catching up from behind, which has not yet entered the bridge. The pedestrian then turns back and meets the car at the start of the bridge. If the pedestrian had continued walking, the car would have caught up with them at the end of the bridge. Find the ratio of the car's speed to the pedestrian's speed.


Find all functions \( f: \mathbb{N} \rightarrow \mathbb{N} \) such that for all natural numbers \( a \) and \( b \):

\[ f(f(a) + f(b)) = a + b \]

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


Find natural numbers \(a, b, c\) such that for any \(n \in \mathbf{N}\) with \(n > 2\), the following inequality holds:
\[ b - \frac{c}{(n-2)!} < \frac{2^{3}-a}{2!} + \frac{3^{3}-a}{3!} + \cdots + \frac{n^{3}-a}{n!} < b. \]

Solution: \boxed{a = 5, b = 9, c = 4}


Decode the equation ${ }_{* *}+{ }_{* * *}={ }_{* * * *}$, given that both addends and the sum remain unchanged when read from right to left.

Solution: \boxed{22 + 979 = 1001}


Given the function \( f(x) = \left|\log_{2} x\right| \), if the real numbers \( a \) and \( b \) (where \( a < b \)) satisfy \( f(a) = f(b) \), then find the range of values for \( a + 2014b \).

Solution: \boxed{(2015, +\infty)}


A repunit is a positive integer, all of whose digits are 1's. Let \( a_{1} < a_{2} < a_{3} < \ldots \) be a list of all the positive integers that can be expressed as the sum of distinct repunits. Compute \( a_{111} \).


The following shows the graph of \( y = p x^{2} + 5 x + p \). The points \( A = (0, -2) \), \( B = \left(\frac{1}{2}, 0\right) \), \( C = (2, 0) \), and \( O = (0, 0) \) are given.

Find the value of \( p \).


Find the equation of the line passing through the point \( M(-1, 4) \) and perpendicular to the line \( x - 2y + 4 = 0 \).

Solution: \boxed{2x + y - 2 = 0}


If real numbers $\alpha, \beta, \gamma$ form a geometric sequence with a common ratio of 2, and $\sin \alpha, \sin \beta, \sin \gamma$ also form a geometric sequence, then $\cos \alpha = $ ?


We will call a natural number interesting if all its digits are different, and the sum of any two adjacent digits is the square of a natural number. Find the largest interesting number.


The numbers \(a_1, a_2, a_3, a_4,\) and \(a_5\) form a geometric progression. Among them, there are both rational and irrational numbers. What is the maximum number of terms in this progression that can be rational numbers?


Each face of a hexahedron and each face of a regular octahedron are equilateral triangles with side length \(a\). The ratio of the radii of the inscribed spheres (inradii) of the two polyhedra is a reduced fraction \(\frac{m}{n}\). What is the product \(m \cdot n\)?


How many six-digit numbers exist that have three even and three odd digits?


Is it possible to fill an $n \times n$ table with the numbers $-1, 0,$ and $1$ such that the sums of all rows, all columns, and both main diagonals are different? The main diagonals of the table are the diagonals drawn from the top-left corner to the bottom-right corner and from the top-right corner to the bottom-left corner.

Solution: \boxed{\text{No}}


Karlson filled a conical glass with lemonade and drank half of it by height (measured from the surface of the liquid to the top of the cone), and the other half was finished by Kid. By what factor did Karlson drink more lemonade than Kid?


Eli, Joy, Paul, and Sam want to form a company, and the company will have 16 shares to split among the 4 people. The following constraints are imposed:
- Every person must get a positive integer number of shares, and all 16 shares must be given out.
- No one person can have more shares than the other three people combined.

Assuming that shares are indistinguishable, but people are distinguishable, in how many ways can the shares be given out?


Integers \(1, 2, \ldots, n\) are written (in some order) on the circumference of a circle. What is the smallest possible sum of moduli of the differences of neighboring numbers?

Solution: \boxed{2n - 2}


We call the "last digit of a number" the digit that is most to the right. For example, the last digit of 2014 is the digit 4.

a) What is the last digit of \(11^{11}\)?

b) What is the last digit of \(9^{9}\)? And what is the last digit of \(9219^{9219}\)?

c) What is the last digit of \(2014^{2014}\)?


The sequence \( x_{n} \) has its first two elements as \( x_{1}=1001 \) and \( x_{2}=1003 \). For \( n \geq 1 \), the recurrence relation is given by:
\[ x_{n+2}=\frac{x_{n+1}-2004}{x_{n}}. \]
What is the sum of the first 2004 terms of the sequence?


The Red Sox play the Yankees in a best-of-seven series that ends as soon as one team wins four games. Suppose that the probability that the Red Sox win Game \( n \) is \(\frac{n-1}{6}\). What is the probability that the Red Sox will win the series?


What is the probability that in a group of 13 people, at least two were born in September? (For simplicity, assume that each month is equally likely.)


Buratino buried two ingots in the Field of Miracles - one gold and one silver. On days with good weather, the gold ingot increases by 30%, and the silver ingot increases by 20%. On days with bad weather, the gold ingot decreases by 30%, and the silver ingot decreases by 20%. After one week, it turns out that one ingot increased and the other decreased. How many days of good weather were there?


In the diagram, there is a karting track layout. Start and finish are at point $A$, but the go-kart driver can return to point $A$ any number of times and resume the lap.

The path from $A$ to $B$ and vice versa takes one minute. The loop also takes one minute to complete. The loop can only be traversed counterclockwise (the arrows indicate the possible direction of travel). The driver does not turn back halfway and does not stop. The racing duration is 10 minutes. Find the number of possible different routes (sequences of segments traversed).


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

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

Find \( g(-6) \).


Find all real functions such that

$$
(x+y)(f(x)-f(y))=(x-y) f(x+y)
$$

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


In a circle, mutually perpendicular diameters $AB$ and $CD$ are drawn, and a point $M$ is taken arbitrarily on the arc $AC$. Find $MB + MD$ if $MA + MC = a$.

Solution: \boxed{a(1 + \sqrt{2})}


Xiaoming and Xiaoliang are two stamp enthusiasts. Xiaoming exchanges two stamps with a face value of 1 yuan 6 jiao each (according to the face value of the stamps) for Xiaoliang's stamps with a face value of 2 jiao each. Before the exchange, the number of stamps Xiaoliang had was 5 times the number Xiaoming had. After the exchange, the number of stamps Xiaoliang has is 3 times the number Xiaoming has. Find the total number of stamps they have.


What is the area of the set of points $P(x ; y)$ in the right-angled coordinate system that satisfy the condition $|x+y|+|x-y| \leq 4?$


In the number \(2016^{* * * *} 02 *\), each of the 5 asterisks needs to be replaced with any of the digits \(0, 2, 4, 6, 7, 8\) (digits can be repeated) so that the resulting 11-digit number is divisible by 6. In how many ways can this be done?


Milan has a bag of 2020 red balls and 2021 green balls. He repeatedly draws 2 balls out of the bag uniformly at random. If they are the same color, he changes them both to the opposite color and returns them to the bag. If they are different colors, he discards them. Eventually, the bag has 1 ball left. Let \( p \) be the probability that it is green. Compute \( \lfloor 2021 p \rfloor \).


On the coordinate plane, the graphs of three reduced quadratic polynomials intersect the y-axis at the points $-15,-6,-27$ respectively. For each of the polynomials, the coefficient at $x$ is a natural number, and the larger root is a prime number. Find the sum of all roots of these polynomials.


Find the smallest positive integer that cannot be expressed in the form $\frac{2^{a}-2^{b}}{2^{c}-2^{d}}$, where $a$, $b$, $c$, and $d$ are all positive integers.


The first brigade of workers is paving a section of road, while the second brigade, which has 6 more workers, is paving another section of road that is three times larger in area. The productivity of all workers is the same. What is the minimum number of workers that could have been in the first brigade if they completed their work faster? If there are no solutions, then the answer should be 0.


Calculate the limit of the function:

$$
\lim _{x \rightarrow 3} \frac{\ln (2 x-5)}{e^{\sin \pi x}-1}
$$


In a class of 28 students, 4 prizes are distributed. How many ways can this be done,

a) if the prizes are identical and each student can receive at most one prize?

b) if the prizes are identical and each student can receive more than one prize?

c) if the prizes are distinct and each student can receive at most one prize?

d) if the prizes are distinct and each student can receive more than one prize?


Determine all functions from \(\mathbb{R}\) to \(\mathbb{R}\) satisfying:

\[ \forall(x, y) \in \mathbb{R}^{2}, f(x+y) = x f(x) + y f(y) \]

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


There are 10,001 students at a university. Some students join together to form several clubs (a student may belong to different clubs). Some clubs join together to form several societies (a club may belong to different societies). There are a total of \( k \) societies. Suppose that the following conditions hold:
1. Each pair of students are in exactly one club.
2. For each student and each society, the student is in exactly one club of the society.
3. Each club has an odd number of students. In addition, a club with \( 2m + 1 \) students (where \( m \) is a positive integer) is in exactly \( m \) societies.

Find all possible values of \( k \).


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


The circus arena is illuminated by \( n \) different spotlights. Each spotlight illuminates a certain convex figure. It is known that if any one spotlight is turned off, the arena will still be fully illuminated, but if any two spotlights are turned off, the arena will no longer be fully illuminated. For what values of \( n \) is this possible?

Solution: \boxed{n \geq 2}


Given \(\alpha\) and \(\beta\) satisfy the equations
\[
\begin{array}{c}
\alpha^{3}-3 \alpha^{2}+5 \alpha-4=0, \\
\beta^{3}-3 \beta^{2}+5 \beta-2=0 .
\end{array}
\]
find \(\alpha + \beta\).


Katya correctly solves a problem with a probability of $4 / 5$, and the magic pen solves a problem correctly without Katya's help with a probability of $1 / 2$. In a test containing 20 problems, solving 13 correctly is enough to get a "good" grade. How many problems does Katya need to solve on her own and how many should she leave to the magic pen to ensure that the expected number of correct answers is at least 13?


There are 200 computers in a computer center, some of which are connected by cables in pairs, with a total of 345 cables used. We call a "cluster" a set of computers such that any computer in this set can send a signal to all others through the cables. Initially, all computers formed one cluster. However, one night an evil hacker cut several cables, resulting in 8 clusters. Find the maximum possible number of cables that were cut.


Given a sequence of real numbers \( a_{1}, a_{2}, a_{3}, \ldots \) that satisfy
1) \( a_{1}=\frac{1}{2} \), and
2) \( a_{1} + a_{2} + \ldots + a_{k} = k^{2} a_{k} \), for \( k \geq 2 \).
Determine the value of \( a_{100} \).


Let $n$ be an integer such that $1 \leq n \leq 30$. Ariane and Bérénice are playing the following game. First, Ariane chooses $n$ distinct numbers from the integers $1, 2, \ldots, 30$. Then, based on Ariane's choices, Bérénice chooses an integer $d \geq 2$. Bérénice wins if $d$ divides at least two of the numbers chosen by Ariane. Otherwise, Ariane wins.

For which values of $n$ can Ariane ensure she will win?

Solution: \boxed{1 \leq n \leq 11}


From the 2015 positive integers 1, 2, ... , 2015, select $k$ numbers such that the sum of any two different selected numbers is not a multiple of 50. Find the maximum value of $k$.


Vanya wrote the number 1347 on the board.

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

Help Vasya solve Petya's task.


Given a square $ABCD$ with side length $a$, vertex $A$ lies in plane $\beta$, and the other vertices are on the same side of plane $\beta$. The distances from points $B$ and $D$ to plane $\beta$ are $1$ and $2$, respectively. If the angle between plane $ABCD$ and plane $\beta$ is $30^\circ$, find $a$.

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


Five friends - Katya, Polina, Alena, Lena, and Sveta - gather every day in the park after buying ice cream at the shop around the corner. One day, they had a conversation:

Polina: I was standing next to Alena.

Alena: I was the first in line!

Lena: No one was behind me.

Katya: There were five people in front of me.

Sveta: There was only one person behind me.

The girls are friends, so they do not lie to each other. How many people were there between Katya and Polina?


Calculate the sum

$$
S=\frac{2014}{2 \cdot 5}+\frac{2014}{5 \cdot 8}+\frac{2014}{8 \cdot 11}+\ldots+\frac{2014}{2012 \cdot 2015}
$$

In the answer, indicate the remainder when dividing by 5 the even number closest to the obtained value of $S$.


Natural numbers are placed in the cells of a chessboard such that each number is equal to the arithmetic mean of its neighbors. The sum of the numbers in the corners of the board is 16. Find the number in the cell at $e2$.


Find a sequence of maximal length consisting of non-zero integers in which the sum of any seven consecutive terms is positive and that of any eleven consecutive terms is negative.


Calculate the limit of the function:

$$
\lim _{x \rightarrow 0}\left(\frac{\operatorname{arctg} 3 x}{x}\right)^{x+2}
$$


Elsa uses ice blocks to make ice sculptures. One ice block can make a small sculpture, and three ice blocks can make a large sculpture. The leftover fragments from making 2 small sculptures or 1 large sculpture can perfectly form 1 new ice block. Given 30 ice blocks, and ensuring that the number of small sculptures made is greater than the number of large sculptures made, what is the maximum number of large sculptures that can be made?


Let $\{a_{n}\}$ be an integer sequence such that for any $n \in \mathbf{N}^{*}$, the condition \((n-1) a_{n+1} = (n+1) a_{n} - 2 (n-1)\) holds. Additionally, \(2008 \mid a_{2007}\). Find the smallest positive integer \(n \geqslant 2\) such that \(2008 \mid a_{n}\).


In square \( A B C D \), points \( F \) and \( E \) are the midpoints of sides \( A B \) and \( C D \), respectively. The point \( E \) is connected to vertices \( A \) and \( B \), and the point \( F \) is connected to vertices \( C \) and \( D \), as shown in the figure. Determine the area of the rhombus \( F G E H \) formed in the center if the side of the square \( A B = 4 \).


The largest diagonal of a regular hexagonal prism is \(d\) and forms a \(30^\circ\) angle with the lateral edge of the prism. Find the volume of the prism.

Solution: \boxed{\frac{9 d^3}{64}}


What is the smallest natural number that leaves a remainder of 2 when divided by 3, a remainder of 4 when divided by 5, and a remainder of 4 when divided by 7?


First, select $n$ numbers from the set $ \{1, 2, \cdots, 2020\} $. Then, from these $n$ numbers, choose any two numbers $a$ and $b$, such that $a$ does not divide $b$. Find the maximum value of $n$.


In the triangles \( \triangle ABC \) and \( \triangle AEF \), \( B \) is the midpoint of \( EF \). Given that \( AB = EF = 1 \), \( BC = 6 \), \( CA = \sqrt{33} \), and \( \overrightarrow{AB} \cdot \overrightarrow{AE} + \overrightarrow{AC} \cdot \overrightarrow{AF} = 2 \), find the cosine of the angle between \( \overrightarrow{EF} \) and \( \overrightarrow{BC} \).


In $\triangle ABC$, find the integer part of $S=\sqrt{3 \tan \frac{A}{2} \tan \frac{B}{2}+1}+\sqrt{3 \tan \frac{B}{2} \tan \frac{C}{2}+1}+\sqrt{3 \tan \frac{C}{2} \tan \frac{A}{2}+1}$.


In the NBA Finals between the Los Angeles Lakers and the Boston Celtics, the series is played in a best-of-seven format, meaning the first team to win 4 games will be crowned the champion. The games are divided into home and away games. Since the Los Angeles Lakers had a better regular-season record, games 1, 2, 6, and 7 are played in Los Angeles, while games 3-5 are played in Boston. The Lakers eventually win the championship on their home court. How many possible win-loss sequences are there during the series?


Cheburashka and Gena ate a cake. Cheburashka ate twice as slow as Gena, but he started eating one minute earlier. In the end, they ate an equal amount of cake. How long would it take for Cheburashka to eat the entire cake by himself?


ABC is a triangle. Show that the medians BD and CE are perpendicular if and only if \( b^2 + c^2 = 5a^2 \).

Solution: \boxed{b^2 + c^2 = 5a^2}


What could be the minimum length of a cycle in a graph where no single vertex is connected to all others, any two non-adjacent vertices have a common neighbor, and if the number of vertices is denoted by $n$, the sum of the squares of the degrees is $n^2 - n$?


A room is built in the shape of the region between two semicircles with the same center and parallel diameters. The farthest distance between two points with a clear line of sight is 12 meters. What is the area (in square meters) of the room?


Let \( p \) be an odd prime that satisfies \( p \equiv 1 \pmod{4} \). Compute the value of \( \sum_{k=1}^{p-1} \left\{ \frac{k^2}{p} \right\} \), where \( \{ x \} = x - \lfloor x \rfloor \) and \(\lfloor x \rfloor\) is the greatest integer less than or equal to \( x \).

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


Simplify the expression

$$
\frac{\sin 11^{\circ} \cos 15^{\circ}+\sin 15^{\circ} \cos 11^{\circ}}{\sin 18^{\circ} \cos 12^{\circ}+\sin 12^{\circ} \cos 18^{\circ}}
$$

Solution: \boxed{2 \sin 26^{\circ}}


Determine the maximum value of \( m^{2} + n^{2} \). Here, \( m \) and \( n \) are integers, and \( m, n \in \{1, 2, \cdots, 1981\} \), such that \((n^{2} - mn - m^{2})^{2} = 1\).


A single burger is not enough to satisfy a guy's hunger. The five guys go to Five Guys' Restaurant, which has 20 different meals on the menu. Each meal costs a different integer dollar amount between $1 and $20. The five guys have \$20 to split between them, and they want to use all the money to order five different meals. How many sets of five meals can the guys choose?


1. Call a year ultra-even if all of its digits are even. Thus \(2000, 2002, 2004, 2006\), and \(2008\) are all ultra-even years. They are all 2 years apart, which is the shortest possible gap. \(2009\) is not an ultra-even year because of the \(9\), and \(2010\) is not an ultra-even year because of the \(1\).

(a) In the years between the years 1 and 10000, what is the longest possible gap between two ultra-even years? Give an example of two ultra-even years that are far apart with no ultra-even years between them. Justify your answer.

(b) What is the second-shortest possible gap (that is, the shortest gap longer than 2 years) between two ultra-even years? Again, give an example, and justify your answer.


When choosing, without replacement, 4 out of 30 labelled balls that are marked from 1 to 30, how many combinations are possible? Find the value of \( r \).


The set $\{[x]+[2x]+[3x] \mid x \in \mathbf{R}\} \bigcap \{1, 2, \cdots, 100\}$ contains how many elements, where $[x]$ represents the greatest integer less than or equal to $x$.


In the triangle \(ABC\), \(AB = 8\), \(BC = 7\), and \(CA = 6\). Let \(E\) be the point on \(BC\) such that \(\angle BAE = 3 \angle EAC\). Find \(4AE^2\).


Calculate the limit of the numerical sequence:

\[
\lim _{n \rightarrow \infty}\left(\frac{3 n^{2}-5 n}{3 n^{2}-5 n+7}\right)^{n+1}
\]


Natural numbers \(a, b, c\) are chosen such that \(a < b < c\). It is also known that the system of equations \(2x + y = 2029\) and \(y = |x - a| + |x - b| + |x - c|\) has exactly one solution. Find the minimum possible value of \(c\).


Let \( T = \left\{ |9^k| \mid k \in \mathbf{Z}, 0 \leqslant k \leqslant 4000 \right\} \). It is known that \( 9^{4000} \) is a 3817-digit number, and its first digit is 9. How many numbers in \( T \) have 9 as their first digit?


For which natural numbers \( n \) is the number \( 3^{2n+1} - 2^{2n+1} - 6^n \) composite?

Solution: \boxed{n \geq 2}


Given positive integers \( a, b, c, \) and \( d \) such that \( a > b > c > d \) and \( a + b + c + d = 2004 \), as well as \( a^2 - b^2 + c^2 - d^2 = 2004 \), what is the minimum value of \( a \)?


From the set \( M = \{1, 2, \cdots, 2008\} \) of the first 2008 positive integers, a \( k \)-element subset \( A \) is chosen such that the sum of any two numbers in \( A \) cannot be divisible by the difference of those two numbers. What is the maximum value of \( k \)?


Simplify the following expression:

$$
\frac{\frac{a+b}{1-ab} + \frac{c-a}{1+ca}}{1-\frac{a+b}{1-ab} \cdot \frac{c-a}{1+ca}}
$$

Explain the result using trigonometric identities.


Suppose \(\overline{a_{1} a_{2} \ldots a_{2009}}\) is a 2009-digit integer such that for each \(i=1,2, \ldots, 2007\), the 2-digit integer \(\overline{a_{i} a_{i+1}}\) contains 3 distinct prime factors. Find \(a_{2008}\).

(Note: \(xyz \ldots\) denotes an integer whose digits are \(x, y, z, \ldots\))


For which (real) values of the variable $x$ are the following equalities valid:

a) $\sqrt{x+\sqrt{2x-1}}+\sqrt{x-\sqrt{2x-1}}=\sqrt{2}$,

b) $\sqrt{x+\sqrt{2x-1}}+\sqrt{x-\sqrt{2x-1}}=1$,

c) $\sqrt{x+\sqrt{2x-1}}+\sqrt{x-\sqrt{2x-1}}=2$.

(The square root without a sign always means the non-negative square root!)


Calculate the limit of the numerical sequence:

$$\lim _{n \rightarrow \infty} \frac{n \sqrt[6]{n}+\sqrt[3]{n^{10}+1}}{(n+\sqrt[4]{n}) \sqrt[3]{n^{3}-1}}$$

Solution: \boxed{\infty}


In a rectangular coordinate system, a circle centered at the point $(1,0)$ with radius $r$ intersects the parabola $y^2 = x$ at four points $A$, $B$, $C$, and $D$. If the intersection point $F$ of diagonals $AC$ and $BD$ is exactly the focus of the parabola, determine $r$.


Place the numbers $1-8$ on the eight vertices of a cube, then at the midpoint of each edge write the average of the two numbers at the endpoints of that edge. If the numbers at the four midpoints of the top face and the four midpoints of the bottom face are all integers, how many of the numbers at the other four midpoints are not integers?


Zhenya had 9 cards with numbers from 1 to 9. He lost the card with the number 7. Is it possible to arrange the remaining 8 cards in a row so that any two adjacent cards form a number divisible by 7?

Solution: \boxed{\text{No}}


The diameter of the red circle is 6 decimeters. What should be the minimum diameter (to the nearest $\frac{1}{2}$ decimeter) of five discs such that they can completely cover the circle?


Given the coordinates of the vertices of triangle $\triangle O A B$ are $O(0,0), A(4,4 \sqrt{3}), B(8,0)$, with its incircle center being $I$. Let the circle $C$ pass through points $A$ and $B$, and intersect the circle $I$ at points $P$ and $Q$. If the tangents drawn to the two circles at points $P$ and $Q$ are perpendicular, then the radius of circle $C$ is $\qquad$ .


Petya is creating a password for his smartphone. The password consists of 4 decimal digits. Petya wants the password to not contain the digit 7, and it must have at least two (or more) identical digits. How many ways can Petya create such a password?


How many ordered pairs of integers \((a, b)\) satisfy all of the following inequalities?
\[
\begin{array}{l}
a^{2}+b^{2}<16 \\
a^{2}+b^{2}<8a \\
a^{2}+b^{2}<8b
\end{array}
\]


In three-dimensional space, let \( S \) be the region of points \( (x, y, z) \) satisfying \( -1 \leq z \leq 1 \). Let \( S_{1}, S_{2}, \ldots, S_{2022} \) be 2022 independent random rotations of \( S \) about the origin \( (0,0,0) \). The expected volume of the region \( S_{1} \cap S_{2} \cap \cdots \cap S_{2022} \) can be expressed as \( \frac{a \pi}{b} \), for relatively prime positive integers \( a \) and \( b \). Compute \( 100a+b \).


On a table, there are three cones standing on their bases, touching each other. The radii of their bases are 10, 15, and 15. A truncated cone is placed on the table with its smaller base down, which has a common slant height with each of the other cones. Find the area of the smaller base of the truncated cone.


Given a parallelepiped $A B C D A_{1} B_{1} C_{1} D_{1}$, the point $X$ is chosen on the edge $A_{1} D_{1}$ and the point $Y$ is chosen on the edge $B C$. It is known that $A_{1} X=5$, $B Y=3$, and $B_{1} C_{1}=14$. The plane $C_{1} X Y$ intersects the ray $D A$ at point $Z$. Find $D Z$.


For which natural numbers \( n \) is the sum \( 5^n + n^5 \) divisible by 13? What is the smallest \( n \) that satisfies this condition?


A basket of apples is divided into two parts, A and B. The ratio of the number of apples in A to the number of apples in B is $27: 25$. Part A has more apples than Part B. If at least 4 apples are taken from A and added to B, then Part B will have more apples than Part A. How many apples are in the basket?


For a positive integer \( n \), let \( p(n) \) denote the product of the positive integer factors of \( n \). Determine the number of factors \( n \) of 2310 for which \( p(n) \) is a perfect square.


Given that \( f(x) \) is a function defined on \( \mathbf{R} \) with \( f(1) = 1 \) and for any \( x \in \mathbf{R} \), \( f(x+5) \geq f(x) + 5 \) and \( f(x+1) \leq f(x) + 1 \). If \( g(x) = f(x) + 1 - x \), find \( g(2002) \).


In triangle \( FGH \), angle \( G \) is right, \( FG = 8 \), \( GH = 2 \). Point \( D \) lies on side \( FH \), and \( A \) and \( B \) are the centroids of triangles \( FGD \) and \( DGH \) respectively. Find the area of triangle \( GAB \).


If the sum of the digits of a natural number \( n \) is subtracted from \( n \), the result is 2016. Find the sum of all such natural numbers \( n \).


A regular triangular prism \( A B C A_{1} B_{1} C_{1} \) is inscribed in a sphere. The base of the prism is \( A B C \), and the lateral edges are \( A A_{1}, B B_{1}, C C_{1} \). The segment \( C D \) is a diameter of this sphere, and the point \( K \) is the midpoint of the edge \( A A_{1} \). Find the volume of the prism, given that \( C K = 2 \sqrt{6} \) and \( D K = 4 \).


Given \( a \) and \( b \) are two orthogonal unit vectors, and \( c \cdot a = c \cdot b = 1 \), find the minimum value of \( \left|c+t a+\frac{1}{t} b\right| \) for any positive real number \( t \).


The librarian of a physics and mathematics high school noticed that if the number of geometry textbooks in the school library is increased by several (whole number) times and the obtained number is added to the number of algebra textbooks, the result is 2015. If the number of algebra textbooks is increased by the same number of times and the obtained number is added to the number of geometry textbooks, the result is 1580. How many algebra textbooks are in the library?


Today is January 30th, so we first write down 130. The rule for writing subsequent numbers is: if the last number written is even, divide it by 2 and then add 2; if the last number written is odd, multiply it by 2 and then subtract 2. Thus the sequence obtained is: $130, 67, 132, 68, \cdots$. What is the 2016th number in this sequence?


An isosceles trapezoid \(ABCD\) is inscribed in a circle with radius \(2 \sqrt{7}\), whereby its base \(AD\) is the diameter, and angle \(\angle BAD\) is \(60^\circ\). Chord \(CE\) intersects the diameter \(AD\) at point \(P\) such that the ratio \(AP:PD = 1:3\). Find the area of triangle \(BPE\).


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


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, determine the locus of the intersection point \( P \) of lines \( l \) and \( m \).

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


Masha has an integer multiple of toys compared to Lena, and Lena has the same multiple of toys compared to Katya. Masha gave 3 toys to Lena, and Katya gave 2 toys to Lena. After that, the number of toys each girl had formed an arithmetic progression. How many toys did each girl originally have? Provide the total number of toys the girls had initially.


Given a quadratic polynomial \( f(x) \) such that the equation \( (f(x))^3 - 4f(x) = 0 \) has exactly three solutions. How many solutions does the equation \( (f(x))^2 = 1 \) have?


Calculate the area of the figure enclosed by the lines given by the equations:

$$
\begin{aligned}
& \left\{\begin{array}{l}
x=6(t-\sin t) \\
y=6(1-\cos t)
\end{array}\right. \\
& y=6(0<x<12 \pi, y \geq 6)
\end{aligned}
$$

Solution: \boxed{18\pi + 72}


A girl, who is an acquaintance of a young man sitting by the window of a moving tram, was walking towards the tram. Eight seconds after she passed by the window, the young man got off the tram and started following her. How much time passed from this moment until he caught up with the girl? The speed of the young man is twice the speed of the girl and one-fifth of the speed of the tram.

Solution: \boxed{88 \text{ s}}


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


In triangle \(ABC\), \(AB = 13\) and \(BC = 15\). On side \(AC\), point \(D\) is chosen such that \(AD = 5\) and \(CD = 9\). The angle bisector of the angle supplementary to \(\angle A\) intersects line \(BD\) at point \(E\). Find \(DE\).


A regular triangular pyramid \(SABC\) is given, with the edge of its base equal to 1. Medians of the lateral faces are drawn from the vertices \(A\) and \(B\) of the base \(ABC\), and these medians do not intersect. It is known that the edges of a certain cube lie on the lines containing these medians. Find the length of the lateral edge of the pyramid.


In how many non-empty subsets of the set $\{1, 2, 3, \ldots, 10\}$ are there no two consecutive numbers?


In the diagram, the lengths of four of the sides of the figure are shown in terms of \( x \). Assuming that \( x \neq 0 \), what is the perimeter of the figure?
(A) \( 8x \)
(B) \( 9x \)
(C) \( 12x \)
(D) \( 11x \)
(E) \( 10x \)


An acute angle at the base of a trapezoid inscribed in a circle with a radius of 13 is $30^{\circ}$. The leg (non-parallel side) of the trapezoid is 10. Find the median (the midsegment) of the trapezoid.


Agronomist Bilbo noticed that if the length of his rectangular field were increased by 20 meters, then the perimeter of the field would be twice as much. However, if the width of the field were twice as much, then the perimeter of the field would be 18 meters more. What is the area of the field?


Three cones are standing on their bases on a table, touching each other. The radii of their bases are $2r$, $3r$, and $10r$. A truncated cone with the smaller base down is placed on the table, sharing a slant height with each of the other cones. Find $r$ if the radius of the smaller base of the truncated cone is 15.


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


Yura has an unusual clock with several minute hands moving in different directions. Yura calculated that over the course of an hour, the minute hands coincided exactly 54 times. What is the maximum number of minute hands that Yura's clock could have?


Given a sequence \(\left\{a_{n}\right\}\) of nonzero terms with the sum of the first \(k\) terms denoted by \(S_{k}\), where
\[ S_{k} = \frac{1}{2} a_{k} a_{k+1} \quad (k \in \mathbb{N}_{+}), \]
and \(a_{1} = 1\).

(1) Find the general term formula of the sequence \(\left\{a_{n}\right\}\).

(2) For any given positive integer \(n (n \geqslant 2)\), the sequence \(\left\{b_{n}\right\}\) satisfies:
\[ \frac{b_{k+1}}{b_{k}} = \frac{k-n}{a_{k+1}} \quad (k=1,2,3, \cdots, n), \quad b_{1}=1. \]
Find the sum \(b_{1}+b_{2}+\cdots+b_{n}\).


There are two squares. The side length of the larger square is 4 decimeters longer than that of the smaller square. The area of the larger square is 80 square decimeters greater than that of the smaller square. The sum of the areas of the larger and smaller squares is ____ square decimeters.


Cheburashka spent his money to buy as many mirrors from Galya's store as Gena bought from Shapoklyak's store. If Gena were buying from Galya, he would have 27 mirrors, and if Cheburashka were buying from Shapoklyak, he would have 3 mirrors. How many mirrors would Gena and Cheburashka have bought together if Galya and Shapoklyak agreed to set a price for the mirrors equal to the average of their current prices? (The average of two numbers is half of their sum, for example, for the numbers 22 and 28, the average is 25.)


Compute the lengths of the arcs of curves given by the parametric equations.

$$
\begin{aligned}
& \left\{\begin{array}{l}
x=\left(t^{2}-2\right) \sin t+2 t \cos t \\
y=\left(2-t^{2}\right) \cos t+2 t \sin t
\end{array}\right. \\
& 0 \leq t \leq \pi
\end{aligned}
$$


In a particular group of people, some always tell the truth, the rest always lie. There are 2016 in the group. One day, the group is sitting in a circle. Each person in the group says, "Both the person on my left and the person on my right are liars."
What is the difference between the largest and smallest number of people who could be telling the truth?


There are $n$ people standing in a circle, including two individuals $A$ and $B$. How many ways are there to arrange them so that there are exactly $r$ people standing between $A$ and $B$ $\left(r<\frac{n}{2} - 1\right)$?


In an isosceles trapezoid \(ABCD\), the angle bisectors of angles \(B\) and \(C\) intersect at the base \(AD\). Given that \(AB=50\) and \(BC=128\), find the area of the trapezoid.


Find the maximum value of the expression \(\cos(x+y)\) given that \(\cos x - \cos y = \frac{1}{4}\).


Let " $\sum$ " denote the cyclic sum. If $a, b, c$ are given distinct real numbers, then
$$
f(x)=\sum \frac{a^{2}(x-b)(x-c)}{(a-b)(a-c)}
$$
Simplify the expression to obtain...

Solution: \boxed{ x^2 }


Given a triangle \( \triangle KLM \), point \( A \) lies on the extension of \( LK \). Construct a rectangle \( ABCD \) such that points \( B \), \( C \), and \( D \) lie on the lines containing \( KM \), \( KL \), and \( LM \), respectively.

Solution: \boxed{ABCD}


Solution: \boxed{31 + i}


What is the minimum number of vertices in a graph that contains no cycle of length less than 6 and where every vertex has a degree of 3?


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


Find the minimum value of the function

$$
f(x)=x^{2}+(x-2)^{2}+(x-4)^{2}+\ldots+(x-102)^{2}
$$

If you obtain a non-integer number, round the result to the nearest whole number.


Let set \( A = \{a \mid a = 3k + 2, k \leqslant 2000, k \in \mathbf{N}_{+}\} \) and set \( B = \{b \mid b = 4k - 1, k \leqslant 2000, k \in \mathbf{N}_{+}\} \). How many elements are in \( A \cap B \)?


Given the equation with respect to \( x \)
\[
x^{2} - 34x + 34k - 1 = 0
\]
that has at least one positive integer root, find the values of the positive integer \( k \) that satisfy this condition.


A and B are jogging on a 300-meter track. They start at the same time and in the same direction, with A starting behind B. A overtakes B for the first time after 12 minutes, and for the second time after 32 minutes. Assuming both maintain constant speeds, how many meters behind B did A start?

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


The height of a cone is $h$. The development of the lateral surface of this cone is a sector with a central angle of $120^\circ$. Calculate the volume of the cone.

Solution: \boxed{\frac{\pi h^3}{24}}


People are standing in a circle - there are liars, who always lie, and knights, who always tell the truth. Each of them said that among the people standing next to them, there is an equal number of liars and knights. How many people are there in total if there are 48 knights?


The numbers \( x \) and \( y \) are such that the equalities \( \operatorname{ctg} x - \operatorname{ctg} y = 2 \) and \( 5 \sin (2x - 2y) = \sin 2x \sin 2y \) hold. Find \( \operatorname{tg} x \operatorname{tg} y \).


Find the smallest positive integer \( b \) such that \( 1111_b \) (1111 in base \( b \)) is a perfect square. If no such \( b \) exists, write "No solution".


We placed a cube on the center of a potter's wheel, with a natural number written on each of its faces. Just before we spun the wheel, from our point of view, we could see three faces of the cube with numbers totaling 42. After rotating the potter's wheel by $90^{\circ}$, we observed three faces with numbers totaling 34, and after another $90^{\circ}$ rotation, we still saw three numbers totaling 53.

1. Determine the sum of the three numbers we will see from our point of view if the wheel is rotated another $90^{\circ}$.
2. The cube always rested on the face with the number 6. Determine the maximum possible sum of all six numbers on the cube.


Find all ordered 4-tuples of integers \((a, b, c, d)\) (not necessarily distinct) satisfying the following system of equations:
\[
\begin{aligned}
a^2 - b^2 - c^2 - d^2 &= c - b - 2, \\
2ab &= a - d - 32, \\
2ac &= 28 - a - d, \\
2ad &= b + c + 31.
\end{aligned}
\]

Solution: \boxed{(5, -3, 2, 3)}


Let's call a number "small" if it is a 10-digit number and there is no smaller 10-digit number with the same sum of digits. How many small numbers exist?


Three marksmen shoot at a target independently of each other. The probability of hitting the target for the first marksman is 0.6, for the second marksman is 0.7, and for the third marksman is 0.75. Find the probability that at least one of them hits the target if each marksman makes one shot.


Let \( P \) be a moving point on the ellipse \(\frac{x^{2}}{16} + \frac{y^{2}}{9} = 1\) in the first quadrant. From point \( P \), two tangents \( PA \) and \( PB \) are drawn to the circle \( x^{2} + y^{2} = 9 \), with points of tangency \( A \) and \( B \) respectively. The line \( AB \) intersects the \( x \)-axis and \( y \)-axis at points \( M \) and \( N \) respectively. Find the minimum area of \( \triangle MON \).


Three tired cowboys entered a saloon and hung their hats on a bison horn at the entrance. Late at night, when the cowboys were leaving, they were unable to distinguish one hat from another and randomly took three hats. Find the probability that none of them took their own hat.


Given \( x \in \mathbf{R} \), find the maximum value of \( y = \frac{\sin x}{2 - \cos x} \).


In a meadow, ladybugs have gathered. If a ladybug has 6 spots on its back, it always tells the truth; if it has 4 spots, it always lies. There were no other types of ladybugs in the meadow. The first ladybug said, "Each of us has the same number of spots on our backs." The second said, "Altogether, we have 30 spots on our backs." The third countered, "No, altogether we have 26 spots on our backs." Each of the remaining ladybugs declared, "Exactly one of the three statements is true." How many ladybugs are in the meadow in total?


On the Cartesian plane, find the number of integer coordinate points (points where both x and y are integers) that satisfy the following system of inequalities:
\[
\begin{cases}
y \leq 3x, \\
y \geq \frac{1}{3}x, \\
x + y \leq 100.
\end{cases}
\]


How many three-digit natural numbers are there such that the sum of their digits is equal to 24?


Given the set
$$
M=\{1,2, \cdots, 2020\},
$$
for any non-empty subset $A$ of $M$, let $\lambda_{A}$ be the sum of the maximum and minimum numbers in the subset $A$. What is the arithmetic mean of all such $\lambda_{A}$?


In triangle \(ABC\), the angle bisector \(AL\) is drawn. Points \(E\) and \(D\) are marked on segments \(AB\) and \(BL\) respectively such that \(DL = LC\) and \(ED \parallel AC\). Find the length of segment \(ED\) if it is known that \(AE = 15\) and \(AC = 12\).


On the leg \(BC\) of the right triangle \(ABC\), a circle is constructed with \(BC\) as its diameter. This circle intersects the hypotenuse \(AB\) at point \(P\). Chord \(PQ\) is parallel to leg \(BC\). Line \(BQ\) intersects leg \(AC\) at point \(D\). Given that \(AC = b\) and \(DC = d\), find \(BC\).

Solution: \boxed{\sqrt{bd}}


Does there exist a fraction equivalent to $\frac{7}{13}$ such that the difference between the denominator and the numerator is 24?


All digits in the 6-digit natural numbers $a$ and $b$ are even, and any number between them contains at least one odd digit. Find the largest possible value of the difference $b-a$.


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


Let \( a \) and \( b \) be real numbers. The function \( f(x)=a x+b \) satisfies: For any \( x \in[0,1] \), \( |f(x)| \leqslant 1 \). Determine the maximum value of \( ab \).