Seven members of the family are each to pass through one of seven doors to complete a challenge. The first person can choose any door to activate. After completing the challenge, the adjacent left and right doors will be activated. The next person can choose any unchallenged door among the activated ones to complete their challenge. Upon completion, the adjacent left and right doors to the chosen one, if not yet activated, will also be activated. This process continues until all seven members have completed the challenge. The order in which the seven doors are challenged forms a seven-digit number. How many different possible seven-digit numbers are there?


If \(\alpha\) is a real root of the equation \(x^{5}-x^{3}+x-2=0\), find the value of \(\left\lfloor\alpha^{6}\right\rfloor\), where \(\lfloor x\rfloor\) is the greatest integer less than or equal to \(x\).


Two symmetrical coins are flipped. What is the probability that both coins show numbers on their upper sides?


There is a five-digit positive odd number \( x \). In \( x \), all 2s are replaced with 5s, and all 5s are replaced with 2s, while other digits remain unchanged. This creates a new five-digit number \( y \). If \( x \) and \( y \) satisfy the equation \( y = 2(x + 1) \), what is the value of \( x \)?
(Chinese Junior High School Mathematics Competition, 1987)


In triangle \( \triangle ABC \), the median \( BM \) is drawn. It is given that \( AB = 2BM \) and \( \angle MBA = 40^\circ \). Find \( \angle CBA \).


The plane angle at the vertex of a regular triangular pyramid is $90^{\circ}$. Find the ratio of the lateral surface area of the pyramid to the area of its base.


Calculate the volumes of solids formed by the rotation of figures bounded by the graphs of the functions around the x-axis.

$$
y=5 \cos x, \quad y=\cos x, \quad x=0, \quad x \geq 0
$$

Solution: \boxed{6 \pi^2}


As shown in the figure, in quadrilateral $ABCD$, $AB=7$, $BC=24$, $CD=20$, $DA=15$, and $\angle B=90^{\circ}$. What is the area of quadrilateral $ABCD$?


Find all natural numbers \( k \) such that \( 3^{k} + 5^{k} \) is a power of a natural number with an exponent \(\geq 2\).


From point \( O \), three rays \( OA, OB, OC \) are drawn, with the angles between each pair of rays being \( 60^{\circ}, 90^{\circ}, \) and \( 120^{\circ} \) respectively. What is the minimum angle between the angle bisectors of these three angles?


Given that the polynomial \(x^2 - x + a\) can divide \(x^{13} + x + 90\), determine the positive integer value of \(a\).


Find three numbers forming a geometric progression, knowing that their product is 64 and their arithmetic mean is \( \frac{14}{3} \).

Solution: \boxed{8, 4, 2}


The area of an equilateral triangle inscribed in a circle is 81 cm². Find the radius of the circle.

Solution: \boxed{6 \sqrt[4]{3}}


A certain project takes 24 days for person A to complete alone, 36 days for person B to complete alone, and 60 days for person C to complete alone. Given that all three teams worked for an exact integer number of days and completed the task within 18 days (including 18 days), how many days did person A work at the minimum?


In what ratio does the angle bisector of an acute angle of an isosceles right triangle divide the area of the triangle?

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


How many solutions does the equation $\left\lfloor\frac{x}{10}\right\rfloor = \left\lfloor\frac{x}{11}\right\rfloor + 1$ have in integers?


Let \( p \) be a prime number and the sequence \(\{a_n\}_{n \geq 0}\) satisfy \( a_0 = 0 \), \( a_1 = 1 \), and for any non-negative integer \( n \), \( a_{n+2} = 2a_{n+1} - p a_n \). If \(-1\) is a term in the sequence \(\{a_n\}\), find all possible values of \( p \).


Let \( X = \{0, a, b, c\} \) and \( M(X) = \{ f \mid f: X \rightarrow X \} \) be the set of all functions from \( X \) to itself. Define the addition operation \( \oplus \) on \( X \) as given in the following table:
\[
\begin{array}{|c|c|c|c|c|}
\hline
\oplus & 0 & a & b & c \\
\hline
0 & 0 & a & b & c \\
\hline
a & a & 0 & c & b \\
\hline
b & b & c & 0 & a \\
\hline
c & c & b & a & 0 \\
\hline
\end{array}
\]

1. Determine the number of elements in the set:
\[
S = \{ f \in M(X) \mid f((x \oplus y) \oplus x) = (f(x) \oplus f(y)) \oplus f(x), \forall x, y \in X \}.
\]

2. Determine the number of elements in the set:
\[
I = \{ f \in M(X) \mid f(x \oplus x) = f(x) \oplus f(x), \forall x \in X \}.
\]


Dima has 25 identical bricks of size $5 \times 14 \times 17$. Dima wants to construct a single tower using all of his bricks, each time adding one brick on top (each new brick adds 5, 14, or 17 to the current height of the tower). A number $n$ is called constructible if Dima can build a tower with a height of exactly $n$. How many constructible numbers exist?


A king has eight sons, and they are all fools. Each night, the king sends three of them to guard the golden apples from the Firebird. The princes cannot catch the Firebird and blame each other, so no two of them agree to go on guard together a second time. What is the maximum number of nights this can continue?


Given an arithmetic sequence where the sum of the first 4 terms is 26, the sum of the last 4 terms is 110, and the sum of all terms is 187, how many terms are in this sequence?


From the numbers $1,2,3, \cdots, 2014$, select 315 different numbers (order does not matter) to form an arithmetic sequence. Among these, the number of ways to form an arithmetic sequence that includes the number 1 is ___. The total number of ways to form an arithmetic sequence is ___.


The rectangular spiral shown in the diagram is constructed as follows. Starting at \((0,0)\), line segments of lengths \(1,1,2,2,3,3,4,4, \ldots\) are drawn in a clockwise manner. The integers from 1 to 1000 are placed, in increasing order, wherever the spiral passes through a point with integer coordinates (that is, 1 at \((0,0)\), 2 at \((1,0)\), 3 at \((1,-1)\), and so on). What is the sum of all of the positive integers from 1 to 1000 which are written at points on the line \(y=-x\)?


Let \( R \) be the rectangle in the Cartesian plane with vertices at \((0,0)\), \((2,0)\), \((2,1)\), and \((0,1)\). \( R \) can be divided into two unit squares. Pro selects a point \( P \) uniformly at random in the interior of \( R \). Find the probability that the line through \( P \) with slope \(\frac{1}{2}\) will pass through both unit squares.


Determine the seventy-third digit from the end in the square of the number consisting of 112 ones.


Two different natural numbers end with 6 zeros and have exactly 56 divisors. Find their sum.


An Ultraman is fighting a group of monsters. It is known that Ultraman has one head and two legs. Initially, each monster has two heads and five legs. During the battle, some monsters split, with each splitting monster creating two new monsters, each with one head and six legs (they cannot split again). At a certain moment, there are 21 heads and 73 legs on the battlefield. How many monsters are there at this moment?


Find the rational number which is the value of the expression

$$
2 \cos ^{6}\left(\frac{5 \pi}{16}\right) + 2 \sin ^{6}\left(\frac{11 \pi}{16}\right) + \frac{3 \sqrt{2}}{8}
$$


Let \( I \) be the center of the inscribed circle in triangle \( ABC \). It is given that \( CA + AI = BC \). Determine the value of the ratio \( \frac{\widehat{BAC}}{\widehat{CBA}} \).


There are 100 boxes numbered from 1 to 100. One of the boxes contains a prize, and the host knows where it is. The viewer can send the host a batch of notes with questions that require a "yes" or "no" answer. The host shuffles the notes in the batch and answers all of them honestly without announcing the questions out loud. What is the minimum number of notes that need to be sent to definitely find out where the prize is?


Xiao Ming places several chess pieces into a $3 \times 3$ grid of square cells. Each cell can have no pieces, one piece, or more than one piece. After calculating the total number of pieces in each row and each column, we obtain 6 different numbers. What is the minimum number of chess pieces required to achieve this?


In a cube \( A B C D - A_{1} B_{1} C_{1} D_{1} \), let the points \( P \) and \( Q \) be the reflections of vertex \( A \) with respect to the plane \( C_{1} B D \) and the line \( B_{1} D \), respectively. Find the sine of the angle between the line \( P Q \) and the plane \( A_{1} B D \).


Find the smallest natural number ending in the digit 4 that becomes 4 times larger when its last digit is moved to the beginning of the number.


Let \( A \) and \( B \) be two fixed positive real numbers. The function \( f \) is defined by
\[ f(x, y)=\min \left\{ x, \frac{A}{y}, y+\frac{B}{x} \right\}, \]
for all pairs \( (x, y) \) of positive real numbers.
Determine the largest possible value of \( f(x, y) \).

Solution: \boxed{\sqrt{A + B}}


Suppose that \(\{a_{n}\}_{n \geq 1}\) is an increasing arithmetic sequence of integers such that \(a_{a_{20}}=17\) (where the subscript is \(a_{20}\)). Determine the value of \(a_{2017}\).


\(\sin (x + 2\pi) \cos \left(2x - \frac{7\pi}{2}\right) + \sin \left(\frac{3\pi}{2} - x\right) \sin \left(2x - \frac{5\pi}{2}\right)\).

Solution: \boxed{\cos 3x}


In three out of the six circles of a diagram, the numbers 4, 14, and 6 are written. In how many ways can natural numbers be placed in the remaining three circles such that the product of the triplets along each of the three sides of the triangular diagram are the same?


Solution: \boxed{abc = 0}


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


Given an equilateral triangle with side length \( a \), find the segment connecting a vertex of the triangle to a point that divides the opposite side in the ratio \( 2:1 \).


Find the smallest natural number $A$ that satisfies the following conditions:

a) Its notation ends with the digit 6;

b) By moving the digit 6 from the end of the number to its beginning, the number increases fourfold.

Solution: \boxed{A = 153846}


These points form a new triangle with angles \(45^{\circ}, 60^{\circ},\) and \(75^{\circ}\). Find the ratio of the areas of the original and the new triangles.

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


There is a natural number written on the board. Five students made the following statements about this number:

- Petya: "This number is greater than 10."
- Kolya: "This number is at least 11."
- Vasya: "This number is greater than 12."
- Dima: "This number is less than 12."
- Oleg: "This number is at most 12."

Find the maximum possible number of correct statements.


(a) In Figure 1, trapezoid $ABED$ is formed by joining $\triangle BCE$ (which is right-angled at $C$) to rectangle $ABCD$. If $AB = 3$, $CE = 6$, and the area of trapezoid $ABED$ is 48, determine the length of $BE$.

(b) In Figure 2, circles with centres $P$ and $Q$ touch a line $\ell$ (i.e., are tangent to $\ell$) at points $S$ and $T$ respectively, and touch each other at point $X$. Because the circles touch at $X$, line segment $PQ$ passes through $X$ and so $PQ = PX + XQ$. Because the circles touch $\ell$ at $S$ and $T$, $PS$ and $QT$ are perpendicular to $\ell$. If the circle with centre $P$ has a radius of 25 and the circle with centre $Q$ has a radius of 16, determine the area of trapezoid $PQTS$.

(c) Figure 3 is formed by adding a circle of radius $r$ to Figure 2 that touches $\ell$ and each of the two other circles, as shown. Determine the value of $r$.


There are four weights of different masses. Katya weighs the weights in pairs. As a result, she gets $1700,1870,2110,2330,$ and 2500 grams. How many grams does the sixth pair of weights weigh?


Car A departs from point $A$ heading towards point $B$ and returns; Car B departs from point $B$ at the same time heading towards point $A$ and returns. After the first meeting, Car A continues for 4 hours to reach $B$, and Car B continues for 1 hour to reach $A$. If the distance between $A$ and $B$ is 100 kilometers, what is Car B's distance from $A$ when Car A first arrives at $B$?


There are 2018 playing cards on the table (2018 piles with one card each). Petka wants to combine them into one deck of 2018 cards in 2017 operations. Each operation involves combining two piles into one. When Petka combines piles of $a$ and $b$ cards, Vasily Ivanovich pays Petka $a \cdot b$ rubles. What is the maximum amount of money Petka can earn by performing all 2017 operations?


Calculate the number of zeros at the end of 2014!.


The sequence $\left\{a_{n}\right\}$ has its first term as 2, and the sum of the first $n$ terms is denoted by $S_{n}$. The sequence $\left\{S_{n}\right\}$ forms a geometric sequence with a common ratio of $\frac{1}{3}$.
1. Find the general term formula for the sequence $\left\{a_{n}\right\}$.
2. Let $b_{n}=a_{n} S_{n}$, find the sum of the terms of the sequence $\left\{S_{n}\right\}$.


Given a positive integer \( n \), find the number of ordered quadruples of integers \( (a, b, c, d) \) such that \( 0 \leq a \leq b \leq c \leq d \leq n \).


Determine all positive integer solutions \( x \) and \( y \) for the equation \( x^2 - 2 \cdot y! = 2021 \).

Solution: \boxed{(45, 2)}


In the coordinate plane, a square $K$ with vertices at points $(0,0)$ and $(10,10)$ is given. Inside this square, illustrate the set $M$ of points $(x, y)$ whose coordinates satisfy the equation

$$
[x] < [y]
$$

where $[a]$ denotes the integer part of the number $a$ (i.e., the largest integer not exceeding $a$; for example, $[10]=10,[9.93]=9,[1 / 9]=0,[-1.7]=-2$). What portion of the area of square $K$ does the area of set $M$ constitute?


Two circles touch each other internally. It is known that two radii of the larger circle, which form an angle of $60^\circ$ between them, are tangent to the smaller circle. Find the ratio of the radii of the circles.


An ant starts at the origin, facing in the positive \( x \)-direction. Each second, it moves 1 unit forward, then turns counterclockwise by \( \sin^{-1}\left(\frac{3}{5}\right) \) degrees. What is the least upper bound on the distance between the ant and the origin? The least upper bound is the smallest real number \( r \) that is at least as big as every distance that the ant ever is from the origin.


In triangle \( ABC \), \( AB = 33 \), \( AC = 21 \), and \( BC = m \), where \( m \) is a positive integer. If point \( D \) can be found on \( AB \) and point \( E \) can be found on \( AC \) such that \( AD = DE = EC = n \), where \( n \) is a positive integer, what must the value of \( m \) be?


The figure below shows a target on a wall that is fixed and cannot be rotated. It is divided into 10 parts, including a central circle, a smaller ring, and a larger (outer) ring. We must distribute the numbers 1 through 10, assigning one number to each part, which will correspond to the scores obtained by hitting each part.

a) In how many ways can we distribute the numbers among the parts of the target?

b) In how many ways can we distribute the numbers such that numbers closer to the center are not smaller than numbers farther from the center?

c) In how many ways can we distribute the numbers so that the sum of the numbers in the outer ring is equal to the sum of the numbers in the smaller ring?


Cat food is sold in large and small packages (with more food in the large package than in the small one). One large package and four small packages are enough to feed a cat for exactly two weeks. Is one large package and three small packages necessarily enough to feed the cat for 11 days?

Solution: \boxed{Yes}


Arseniy sat down at the computer between 16:00 and 17:00 when the hour and minute hands were pointing in opposite directions, and got up the same day between 22:00 and 23:00 when the hands overlapped. How long did Arseniy sit at the computer?

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


Once in winter, 43 children were throwing snowballs. Each of them threw exactly one snowball at someone else. It is known that:

- The first child threw a snowball at the one who threw a snowball at the second child,
- The second child threw a snowball at the one who threw a snowball at the third child,
- The forty-third child threw a snowball at the one who threw a snowball at the first child.

What is the number of the person who threw a snowball at the third child?


Given that \( g \) is a twice differentiable function over the positive real numbers such that
\[ g(x) + 2x^3 g'(x) + x^4 g''(x) = 0 \quad \text{for all positive reals } x \]
and
\[ \lim_{x \to \infty} x g(x) = 1, \]

find the real number \(\alpha > 1\) such that \( g(\alpha) = \frac{1}{2} \).


Find the integral \( \int \frac{dx}{\sin x \cos x} \).

Solution: \boxed{\ln |\tan x| + C}


Find the only value of \( x \) in the open interval \((- \pi / 2, 0)\) that satisfies the equation
$$
\frac{\sqrt{3}}{\sin x} + \frac{1}{\cos x} = 4.
$$


In a $3 \times 3$ grid, there are four $2 \times 2$ subgrids. Kasun wants to place an integer from 1 to 4 inclusive in each cell of the $3 \times 3$ grid so that every $2 \times 2$ subgrid contains each integer exactly once. For example, the grid below on the left satisfies the condition, but the grid below on the right does not. In how many ways can Kasun place integers in the grid so that they satisfy the condition?

| 1 | 2 | 3 |
| :--- | :--- | :--- |
| 3 | 4 | 1 |
| 1 | 2 | 3 |

| 1 | 3 | 2 |
| :--- | :--- | :--- |
| 2 | 4 | 1 |
| 1 | 3 | 3 |


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


Given a regular triangular pyramid \( S A B C \). Point \( S \) is the apex of the pyramid, \( AB = 1 \), \( AS = 2 \), \( BM \) is the median of triangle \( ABC \), and \( AD \) is the angle bisector of triangle \( SAB \). Find the length of segment \( DM \).


Find the last two digits of \(1032^{1032}\). Express your answer as a two-digit number.


The distance between the centers of two circles with radii 1 and 9 is 17. A third circle is tangent to these two circles and also to their common external tangent. Find the radius of the third circle.


Compute the volumes of the bodies bounded by the surfaces:

\[ z = 4x^2 + 9y^2 \]
\[ z = 6 \]


Mother decides to take Xiaohua to 10 cities for a vacation by car. Xiaohua checked the map and was surprised to find that any three of these 10 cities have either all highways between them or only two of the cities have no highway between them. How many highways are there at least between these 10 cities? (Note: There can be at most one highway between any two cities.)


Find the largest positive integer $n$ such that there exist $n$ real numbers $x_{1}, x_{2}, \cdots, x_{n}$ satisfying the inequality $\left(1+x_{i} x_{j}\right)^{2} \leqslant 0.99\left(1+x_{i}^{2}\right)\left(1+x_{j}^{2}\right)$ for any $1 \leq i < j \leq n$.


In a math class, each dwarf needs to find a three-digit number without any zero digits, divisible by 3, such that when 297 is added to the number, the result is a number with the same digits in reverse order. What is the minimum number of dwarfs that must be in the class so that there are always at least two identical numbers among those found?


The Small and Big Islands have a rectangular shape and are divided into rectangular counties. In each county, a road is laid along one of the diagonals. On each island, these roads form a closed path that does not pass through any point twice. Here is how the Small Island is arranged, where there are only six counties:

Draw how the Big Island can be arranged if it has an odd number of counties. How many counties did you get?


On September 10, 2005, the following numbers were drawn in the five-number lottery: 4, 16, 22, 48, 88. All five numbers are even, exactly four of them are divisible by 4, three by 8, and two by 16. In how many ways can five different numbers with these properties be selected from the integers ranging from 1 to 90?


On the base \(AB\) of an isosceles trapezoid \(ABCD\), point \(P\) is chosen such that the base is divided in the ratio \(AP : BP = 4 : 1\). Given that \(\angle CPD = \angle PAD\), find the ratio \(PD / PC\).


In triangle \( \triangle ABC \), with \( \angle A \leq \angle B \leq \angle C \), suppose
$$
\frac{\sin A + \sin B + \sin C}{\cos A + \cos B + \cos C} = \sqrt{3},
$$
find the value of \( \sin B + \sin 2B \).


Find the analytic function \( w = f(z) \) given its imaginary part \( v(x, y) = 3x + 2xy \) under the condition that \( f(-i) = 2 \).

Solution: \boxed{f(z) = z^2 + 3iz}


A triangle has a perimeter \( 2p \) and an inscribed circle. A tangent to this circle, parallel to one side of the triangle, is drawn. Find the maximum possible length of the segment of this tangent that lies within the triangle.


A square \(ABCD\) has a side length of 40 units. Point \(F\) is the midpoint of side \(AD\). Point \(G\) lies on \(CF\) such that \(3CG = 2GF\). What is the area of triangle \(BCG\)?


Let \([x]\) denote the greatest integer less than or equal to \(x\). Determine the number of elements in the set \(\{ [x] + [2x] + [3x] \mid x \in \mathbb{R} \} \cap \{1, 2, \ldots, 100\}\).


Determine the pairs of positive integers \((a_1, a_2)\) for which the sequence defined by the recurrence relation \(a_{n+2} = \frac{a_n + a_{n+1}}{\gcd(a_n, a_{n+1})} \, (n \geq 1)\) is periodic.

Solution: \boxed{(2, 2)}


The angle bisectors $\mathrm{AD}$ and $\mathrm{BE}$ of the triangle $\mathrm{ABC}$ intersect at point I. It turns out that the area of triangle $\mathrm{ABI}$ is equal to the area of quadrilateral $\mathrm{CDIE}$. Find the maximum possible value of angle $\mathrm{ACB}$.


Given an isosceles triangle \(ABC\) where \(AB = AC\) and \(\angle ABC = 53^\circ\), point \(K\) is such that \(C\) is the midpoint of segment \(AK\). Point \(M\) is chosen such that:

- \(B\) and \(M\) lie on the same side of line \(AC\);
- \(KM = AB\);
- the angle \(\angle MAK\) is the maximum possible.

How many degrees is the angle \(\angle BAM\)?


Find the volume common to two cylinders: \( x^{2} + y^{2} = a^{2} \) and \( y^{2} + z^{2} = a^{2} \) (bounded by these cylindrical surfaces).

Solution: \boxed{\frac{16}{3} a^3}


Let \( f \) be a function from \(\mathbb{R}\) to \(\mathbb{R}\) that satisfies the following functional equation for all real numbers \(x\) and \(y\):

$$
(y+1) f(x) + f(x f(y) + f(x+y)) = y
$$

1. Show that \( f \) is bijective.
2. Find all functions that satisfy this equation.

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


In a convex 13-gon, all diagonals are drawn. They divide it into polygons. Consider the polygon with the largest number of sides among them. What is the greatest number of sides that it can have?


Behind a thin lens with a focal length \( F \) and a diameter \( D \), a flat screen is placed perpendicularly to its optical axis at its focal point. A point light source is placed on the main optical axis at a distance \( d > F \) from the lens. Determine the diameter of the light spot on the screen.

Solution: \boxed{\frac{F D}{d}}


Find the integer closest to the sum $S$ obtained by taking the base 10 logarithm of each proper divisor of $1,000,000$ and adding these logarithm values together.


Let \(\mathbb{N}\) be the set of positive integers, and let \(f: \mathbb{N} \rightarrow \mathbb{N}\) be a function satisfying:
- \(f(1) = 1\)
- For \(n \in \mathbb{N}\), \(f(2n) = 2f(n)\) and \(f(2n + 1) = 2f(n) - 1\).

Determine the sum of all positive integer solutions to \(f(x) = 19\) that do not exceed 2019.


Given the positive integers \( x_{1}, x_{2}, x_{3}, x_{4}, x_{5} \). Any set of four numbers selected and summed results in the set \(\{44, 45, 46, 47\}\). Determine these five numbers.

Solution: \boxed{10, 11, 11, 12, 13}


What is the sum of all four-digit numbers that are equal to the cube of the sum of their digits (leading zeros are not allowed)?


Express the number one million using numbers that contain only the digit 9 and the algebraic operations of addition, subtraction, multiplication, division, powers, and roots. Determine at least three different solutions.


In an office, each computer was connected by cables to exactly 5 other computers. After a virus affected some of the computers, all the cables from the infected computers were disconnected (a total of 26 cables had to be disconnected). Now, each of the uninfected computers is connected by cables to only 3 other computers. How many computers were affected by the virus?


Let \( n \) be an integer greater than 3. Let \( R \) be the set of lattice points \( (x, y) \) such that \( 0 \leq x, y \leq n \) and \( |x-y| \leq 3 \). Let \( A_n \) be the number of paths from \( (0,0) \) to \( (n,n) \) that consist only of steps of the form \( (x, y) \rightarrow (x, y+1) \) and \( (x, y) \rightarrow (x+1, y) \) and are contained entirely within \( R \). Find the smallest positive real number that is greater than \( \frac{A_{n+1}}{A_n} \) for all \( n \).


There are six wooden sticks, each 50 cm long. They are to be connected end to end in sequence, with each connection section measuring 10 cm. After nailing them together, what is the total length of the wooden sticks? ( ) cm.


For the function \( f(x) \), the condition \( f(f(f(x))) + 3 f(f(x)) + 9 f(x) + 27 x = 0 \) is satisfied. Find \( f(f(f(f(2)))) \).


The sequence \( x_{n} \) is defined by the following conditions: \( x_{1}=3 \) and \( x_{1}+x_{2}+\ldots+x_{n-1}+\frac{4}{3}x_{n}=4 \). Find \( x_{2015} \).


\( A_1, A_2, A_3, A_4 \) are consecutive vertices of a regular \( n \)-gon. Given the equation \( \frac{1}{A_1A_2} = \frac{1}{A_1A_3} + \frac{1}{A_1A_4} \), what are the possible values of \( n \)?

Solution: \boxed{n = 7}


Vendelín lives between two bus stops, at three-eighths of their distance. Today he left home and discovered that whether he ran to one or the other stop, he would arrive at the stop at the same time as the bus. The average speed of the bus is $60 \mathrm{~km} / \mathrm{h}$.

What is the average speed at which Vendelín is running today?

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


Given the circle \( x^2 + y^2 = 1 \), a tangent line intersects the \( x \)-axis and \( y \)-axis at points \( A \) and \( B \) respectively. What is the minimum value of \( |AB| \)?


There is a point $A$, a line $b$ through it, the first image of a line $f$ (which also passes through $A$) is $f'$, an angle $\alpha$, and two distances $r$ and $2s$. Construct a triangle $ABC$ where one vertex is $A$, side $b$ lies on line $b$, the angle bisector lies on line $f$, one angle is $\alpha$, the perimeter is $2s$, and the circumradius is $r$.

Solution: \boxed{ABC}


Find the product of all the roots of the equation \( x^{4} + 4x^{3} - 2015x^{2} - 4038x + 2018 = 0 \).


Let \( A \) and \( B \) be points on the curve \( xy = 1 \) (where \( x > 0 \) and \( y > 0 \)) in the Cartesian coordinate system \( xOy \). Given the vector \( \vec{m} = (1, |OA|) \), find the minimum value of the dot product \( \vec{m} \cdot \overrightarrow{OB} \).

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


A pedestrian walks along a highway at a speed of 5 km/h. Buses travel along this highway in both directions at the same speed, meeting every 5 minutes. At 12 o'clock, the pedestrian noticed that the buses met near him and, continuing to walk, began counting the passing and overtaking buses. At 2 o'clock, the buses met again near him. It turned out that during this time, the pedestrian encountered four more buses than those overtaking him. Find the speed of the bus.

Solution: \boxed{30 \text{ km/h}}


Sergey, while being a student, worked part-time at a student cafe throughout the year. Sergey's salary was 9000 rubles per month. In the same year, Sergey paid 100000 rubles for his treatment at a healthcare institution and purchased medication prescribed by a doctor for 20000 rubles (eligible for deduction). The following year, Sergey decided to claim a social tax deduction. What amount of personal income tax paid is subject to refund to Sergey from the budget? (Answer as a whole number, without spaces or units of measurement.)


Calculate the limit of the function:

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


All natural numbers starting from 1 were written consecutively, forming a sequence of digits, as follows.

$$
1234567891011121314151617181920212223 \ldots
$$

What is the digit that appears in the 206788th position?


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?


Given the vectors \(\boldsymbol{a}\) and \(\boldsymbol{b}\) satisfy the following conditions:
$$
\begin{array}{l}
|a - b| = 3, \\
|a + 2b| = 6, \\
a^{2} + a \cdot b - 2b^{2} = -9.
\end{array}
$$

Find the magnitude of \(\boldsymbol{b}\), \(|b| =\).


In a tennis tournament, 512 schoolchildren participate. For a win, 1 point is awarded, for a loss, 0 points. Before each round, pairs are drawn from participants with an equal number of points (those who do not have a pair are awarded a point without playing). The tournament ends as soon as a sole leader is determined. How many schoolchildren will finish the tournament with 6 points?


Let real numbers \( x_{1}, x_{2}, \ldots, x_{1997} \) satisfy the following two conditions:
1. \( -\frac{1}{\sqrt{3}} \leq x_{i} \leq \sqrt{3} \quad(i=1, 2, \ldots, 1997) \);
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 explain the reasoning.


Solution: \boxed{2 \times 10^{2017}}


The function \( f(x) \) satisfies the condition: for any real numbers \( a \) and \( b \), the equality \( f\left(\frac{a+2 b}{3}\right)=\frac{f(a)+2 f(b)}{3} \) holds. Find the value of the function \( f(2021) \) if \( f(1) = 1 \) and \( f(4) = 7 \).


A six-digit number begins with digit 1 and ends with digit 7. If the digit in the units place is decreased by 1 and moved to the first place, the resulting number is five times the original number. Find this number.


Masha has 2 kg of "Lastochka" candies, 3 kg of "Truffle" candies, 4 kg of "Ptichye Moloko" candies, and 5 kg of "Citron" candies. What is the maximum number of New Year gifts she can make if each gift must contain 3 different types of candies, with 100 grams of each type?


Let \( A(x_{1}, y_{1}) \) and \( B(x_{2}, y_{2}) \) be two points on the curve \( C: x^{2}-y^{2}=2 \) (where \( x > 0 \)). Then the minimum value of \( f = \overrightarrow{OA} \cdot \overrightarrow{OB} \) is ______.


Four people, A, B, C, and D, each have a different number of playing cards.
A says: "I have 16 more cards than C."
B says: "D has 6 more cards than C."
C says: "A has 9 more cards than D."
D says: "If A gives me 2 cards, I will have 3 times as many cards as C."

It is known that the person with the fewest cards is lying, and the others are telling the truth. How many cards does D have?


Let \( x, y \) be nonnegative integers such that \( x + 2y \) is a multiple of 5, \( x + y \) is a multiple of 3, and \( 2x + y \geq 99 \). Find the minimum possible value of \( 7x + 5y \).


In a store, there are 9 headphones, 13 computer mice, and 5 keyboards for sale. Besides these, there are also 4 sets of "keyboard and mouse" and 5 sets of "headphones and mouse." How many ways can you buy three items: headphones, a keyboard, and a mouse? Answer: 646.


Appending three digits at the end of 2007, one obtains an integer \(N\) of seven digits. In order to get \(N\) to be the minimal number which is divisible by 3, 5, and 7 simultaneously, what are the three digits that one would append?


Determine the numeral system in which the following multiplication is performed: \(352 \cdot 31 = 20152\).


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


Given the sequence $\{a_{n}\}$, where $a_{n}$ are integers, and for $n \geq 3, n \in \mathbf{N}$, the relation $a_{n} = a_{n-1} - a_{n-2}$ holds. If the sum of the first 1985 terms of the sequence is 1000, and the sum of the first 1995 terms is 4000, then what is the sum of the first 2002 terms of the sequence?


Over the course of three years, Marina did not invest the funds in her Individual Investment Account (IIA) into financial instruments and therefore did not receive any income from it. However, she gained the right to an investment tax deduction for depositing her own money into the IIA.

The tax deduction is provided for the amount of money deposited into the IIA during the tax period, but not more than 400,000 rubles in total per year.

Marina is entitled to receive 13% of the amount deposited into the IIA as a refund of the Personal Income Tax (PIT) that she paid on her income.

The PIT amount deducted from Marina's annual salary is equal to \( 30,000 \text{ rubles} \times 12 \text{ months} \times 0.13 = 46,800 \text{ rubles} \).

The tax deduction for the first year is \( 100,000 \text{ rubles} \times 0.13 = 13,000 \text{ rubles} \). This amount does not exceed the PIT deducted from Marina's annual salary.

The tax deduction for the second year is \( 400,000 \text{ rubles} \times 0.13 = 52,000 \text{ rubles} \). This amount exceeds the PIT deducted from Marina's annual salary. Therefore, the tax deduction will be limited to the PIT paid for the second year, or 46,800 rubles.

The tax deduction for the third year is \( 400,000 \text{ rubles} \times 0.13 = 52,000 \text{ rubles} \). This amount also exceeds the PIT deducted from Marina's annual salary. Therefore, the tax deduction will be limited to the PIT paid for the third year, or 46,800 rubles.

The total amount of tax deduction for 3 years is \( 13,000 + 46,800 + 46,800 = 106,600 \text{ rubles} \).

The return on Marina's transactions over 3 years is \( \frac{106,600}{1,000,000} \times 100 \% = 10.66 \% \).

The annual return on Marina's transactions is \( \frac{10.66 \%}{3} = 3.55 \% \).


An event is a hit or a miss. The first event is a hit, the second is a miss. Thereafter, the probability of a hit equals the proportion of hits in the previous trials. For example, the probability of a hit in the third trial is 1/2. What is the probability of exactly 50 hits in the first 100 trials?


Determine the largest natural number \( n \) such that

\[ 4^{995} + 4^{1500} + 4^{n} \]

is a square number.


Determine the mass \( m \) of helium needed to fill an empty balloon with mass \( m_{1} = 10 \) g so that the balloon can ascend. Assume the temperature and pressure of the gas in the balloon are equal to the temperature and pressure of the air. The molar mass of helium \( M_{\mathrm{r}} = 4 \) g/mol and the molar mass of air \( M_{\mathrm{B}} = 29 \) g/mol.

Solution: \boxed{1.6 \, \text{g}}


The natural number \( A \) has the following property: the number \( 1+2+\cdots+A \) can be written (in base 10) as the number \( A \) followed by three other digits. Find \( A \).


A circle is inscribed in triangle \(ABC\). On the longest side \(AC\) of the triangle, points \(E\) and \(F\) are marked such that \(AE = AB\) and \(CF = CB\). The segment \(BE\) intersects the inscribed circle at points \(P\) and \(Q\), with \(BP = 1\) and \(PQ = 8\). What is the length of the segment \(EF\)?


Find all pairs of integers \((a, n)\) such that \(n\) divides \((a + 1)^{n} - a^{n}\).

Solution: \boxed{(a, 1)}


The corridors of a maze are the sides and diagonals of an n-sided convex polygon. What is the minimum number of lanterns we need to place in the maze in order to illuminate every corridor?


Let $b_{n}$ be the sum of the numbers in the $n$-th diagonal of Pascal's triangle. Show that $b_{n} = a_{n}$ where $a_{n}$ is the $n$-th term of the Fibonacci sequence

$$
a_{0} = 0, \quad a_{1} = 1, \quad a_{2} = 1, \quad a_{3} = 2, \ldots
$$

Solution: \boxed{b_n = a_n}


Evaluate the sum: $\sum_{m=1}^{\infty} \sum_{n=1}^{\infty} \frac{m^{2} n}{3^{m}\left(n \cdot 3^{m}+m \cdot 3^{n}\right)}$.


A right circular cone with a base radius \( R \) and height \( H = 3R \sqrt{7} \) is laid sideways on a plane and rolled in such a manner that its apex remains stationary. How many rotations will its base make until the cone returns to its original position?


Three cyclists started simultaneously: the first and second from point A, and the third towards them from point B. After 1.5 hours, the first cyclist was equidistant from the other two, and 2 hours after departure, the third cyclist was equidistant from the first and second. How many hours after departure was the second cyclist equidistant from the first and third?


Given three positive numbers \( a, b, \mathrm{and} c \) satisfying \( a \leq b+c \leq 3a \) and \( 3b^2 \leq a(a+c) \leq 5b^2 \), what is the minimum value of \(\frac{b-2c}{a}\)?


Several different points are marked on a line, and all possible line segments are constructed between pairs of these points. One of these points lies on exactly 80 of these segments (not including any segments of which this point is an endpoint). Another one of these points lies on exactly 90 segments (not including any segments of which it is an endpoint). How many points are marked on the line?


Let $ABC$ be a triangle. $K$ is the foot of the angle bisector of $\widehat{ABC}$ and $L$ is the foot of the angle bisector of $\widehat{CAB}$. Let $M$ be the point of intersection of the perpendicular bisector of the segment $[KB]$ and the line $(AL)$. The line parallel to $(KL)$ passing through $L$ intersects $(BK)$ at $N$. Show that $LN = NA$.

Solution: \boxed{LN = NA}


Estimate the number of 2016-digit squares. There are at least

$$
10^{2016 / 2} - 10^{2015 / 2} - 1 > 10^{1000}.
$$

The number of different sets of 2016 digits is not more than the sets where each digit appears no more than 2016 times, i.e., $2017^{10}$. Therefore, there exists such a set of 2016 digits, permutations of which can produce at least $10^{1000} / 2017^{10} > 10^{1000} / 10^{100} = 10^{900}$ squares, and certainly 2016.


Let \( a, b, c, \) and \( d \) be positive integers such that \( a^5 = b^4 \) and \( c^3 = d^2 \) with \( c - a = 19 \). Find \( d - b \).


\(A B C D\) and \(E F G H\) are squares of side length 1, and \(A B \parallel E F\). The overlapped region of the two squares has area \(\frac{1}{16}\). Find the minimum distance between the centers of the two squares.


Find the smallest number consisting only of zeros and ones that is divisible by 225.


If \( x_{1}=1, x_{2}=1-\mathrm{i}, x_{3}=1+\mathrm{i} \) (where \( \mathrm{i} \) is the imaginary unit) are the three solutions to the equation \( x^{3}+a x^{2}+b x+c=0 \), then find \( a+b-c \).


Given the quadratic function \( f(x) \) such that \( f(-1) = 0 \) and \( x \leq f(x) \leq \frac{1}{2}\left(x^2 + 1\right) \) holds for all real numbers \( x \). Determine the explicit form of the function \( f(x) \).


Let \( a \) and \( b \) be two distinct roots of the polynomial \( X^{3} + 3X^{2} + X + 1 \). Calculate \( a^{2}b + ab^{2} + 3ab \).


If one side of square \(ABCD\) lies on the line \(y = 2x - 17\) and two other vertices lie on the parabola \(y = x^2\), then the minimum area of the square is \(\quad\).


There are 2016 cards, each with a unique number from 1 to 2016. A certain number \( k \) of these cards are selected. What is the smallest \( k \) such that among these selected cards, there exist two cards with numbers \( a \) and \( b \) satisfying the condition \( |\sqrt[3]{a} - \sqrt[3]{b}| < 1 \)?


Calculate the area of the figure bounded by the lines defined by the equations:
$$
\begin{aligned}
& \left\{\begin{array}{l}
x=\sqrt{2} \cdot \cos t \\
y=2 \sqrt{2} \cdot \sin t
\end{array}\right. \\
& y=2(y \geq 2)
\end{aligned}
$$

Solution: \boxed{\pi - 2}


The nonzero numbers \( x, y \), and \( z \) satisfy the equations
\[
xy = 2(x + y), \quad yz = 4(y + z), \quad \text{and} \quad xz = 8(x + z).
\]
Solve for \( x \).


Petya places "+" and "-" signs in all possible ways into the expression $1 * 2 * 3 * 4 * 5 * 6$ at the positions of the asterisks. For each arrangement of signs, he calculates the resulting value and writes it on the board. Some numbers may appear on the board multiple times. Petya then sums all the numbers on the board. What is the sum that Petya obtains?


You have 5 identical buckets, each with a maximum capacity of some integer number of liters, and a 30-liter barrel containing an integer number of liters of water. All the water from the barrel was poured into the buckets, with the first bucket being half full, the second one-third full, the third one-quarter full, the fourth one-fifth full, and the fifth one-sixth full. How many liters of water were in the barrel?


Determine the largest integer $x$ for which $4^{27} + 4^{1010} + 4^{x}$ is a perfect square.


On weekdays, the Absent-minded Scientist commutes to work on the ring light of the Moscow Metro from "Taganskaya" station to "Kievskaya" station, and back in the evening. Upon entering the station, he catches the first arriving train. It is known that in both directions trains run at approximately equal intervals. Via the northern route (through "Belorusskaya"), the train travels from "Kievskaya" to "Taganskaya" or back in 17 minutes, and via the southern route (through "Paveletskaya") in 11 minutes.

The Scientist has calculated through many years of observations that:
- A train going counterclockwise arrives at "Kievskaya" on average 1 minute and 15 seconds after a train going clockwise arrives. The same is true for "Taganskaya".
- The trip from home to work on average takes 1 minute less than the trip from work to home.

Find the expected interval between trains running in one direction.

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


Find the relationship between the coefficients of the equation \(a x^{2}+b x+c=0\) if the ratio of the roots is 2.

Solution: \boxed{2b^2 = 9ac}


At 12 o'clock, the angle between the hour hand and the minute hand is 0 degrees. After that, at what time do the hour hand and the minute hand form a 90-degree angle for the 6th time? (12-hour format)


A sphere is inscribed in a cone, and the surface area of the sphere is equal to the area of the base of the cone. Find the cosine of the angle at the vertex in the axial section of the cone.


If \( p, q, \frac{2 p-1}{q}, \frac{2 q-1}{p} \) are all integers, and \( p > 1 \), \( q > 1 \), find the value of \( p+q \).


The perimeter of the parallelogram \(ABCD\) is 20. The bisector of angle \(B\) intersects the lines \(AD\) and \(CD\) at points \(K\) and \(L\) respectively. Find \(CL\), given that \(DK = 4\).


Compute the limit of the numerical sequence:

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


In the trapezoid \( A B C D \), sides \( A D \) and \( B C \) are parallel, and \( A B = B C = B D \). The height \( B K \) intersects the diagonal \( A C \) at point \( M \). Find \(\angle C D M\).


Pi Pi Lu wrote a 2020-digit number: \( 5368 \cdots \cdots \). If any four-digit number taken randomly from this multi-digit number is divisible by 11, what is the sum of the digits of this multi-digit number?


Given real numbers \(x_{1}, x_{2}, \cdots, x_{10}\) satisfying \(\sum_{i=1}^{10}\left|x_{i}-1\right| \leq 4\) and \(\sum_{i=1}^{10}\left|x_{i}-2\right| \leq 6\), find the average value \(\bar{x}\) of \(x_{1}, x_{2}, \cdots, x_{10}\).


The circles $k_{1}$ and $k_{2}$, both with unit radius, touch each other at point $P$. One of their common tangents that does not pass through $P$ is the line $e$. For $i>2$, let $k_{i}$ be the circle different from $k_{i-2}$ that touches $k_{1}$, $k_{i-1}$, and $e$. Determine the radius of $k_{1999}$.


The bases \(AB\) and \(CD\) of trapezoid \(ABCD\) are 155 and 13 respectively, and its lateral sides are mutually perpendicular. Find the scalar product of the vectors \(\overrightarrow{AC}\) and \(\overrightarrow{BD}\).


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

a) the expected number of matches;

b) the variance of the number of matches.


From the set \( M = \{1, 2, 3, \cdots, 2009\} \), remove all multiples of 3 and all multiples of 5. What is the number of remaining elements in \( M \)?


For any positive integer $n$, let $f_{1}(n)$ denote the square of the sum of the digits of $n$ plus $r+1$, where $r$ is the integer satisfying $n = 3q + r$ with $0 \leqslant r < 3$. For $k \geqslant 2$, let $f_{k}(n) = f_{1}(f_{k-1}(n))$. Find $f_{1990}(2345)$.


Let $ABC$ be an equilateral triangle and $D$ and $E$ be two points on segment $[AB]$ such that $AD = DE = EB$. Let $F$ be a point on $BC$ such that $CF = AD$. Find the value of $\widehat{CDF} + \widehat{CEF}$.


There is a 6 × 6 grid of lights. There is a switch at the top of each column and on the left of each row. A light will only turn on if the switches corresponding to both its column and its row are in the "on" position. Compute the number of different configurations of lights.


Given \( n > 2 \) natural numbers such that there are no three equal numbers among them, and the sum of any two of these numbers is a prime number. What is the largest possible value of \( n \)?


Given a natural number \( n \geq 3 \) and real numbers \( x_{1}, x_{2}, \cdots, x_{n} \) satisfying \( x_{1} + x_{2} + \cdots + x_{n} = n \) and \( x_{1}^{2} + x_{2}^{2} + \cdots + x_{n}^{2} = n^{2} \), find the minimum value of \( \sum_{i=1}^{n} x_{i}^{3} \) and the corresponding values of \( \left(x_{1}, x_{2}, \cdots, x_{n}\right) \) that achieve this minimum.

Solution: \boxed{-n^3 + 6n^2 - 4n}


In a convex quadrilateral inscribed around a circle, the products of opposite sides are equal. The angle between a side and one of the diagonals is $20^{\circ}$. Find the angle between this side and the other diagonal.


Of all the three-digit numbers formed by using the four digit cards $0$, $1$, $2$, and $2$, arrange them from largest to smallest. The 2nd number is $\qquad$. The difference between the 4th number and the 8th number is $\qquad$.


Let the four vertices of a regular tetrahedron be \( A, B, C, D \) with each edge length being 1 meter. A bug starts at point \( A \) and moves according to the following rule: at each vertex, it randomly chooses one of the three edges passing through that vertex and crawls to the end of that edge. What is the probability that, after crawling a total of 7 meters, the bug will be at vertex \( A \)?


Alloy $A$ of two metals has a mass of 6 kg, with the first metal being twice as abundant as the second metal. When placed in a container of water, it exerts a force of $30\ \mathrm{N}$ on the bottom. Alloy $B$ of the same metals has a mass of 3 kg, with the first metal being five times less abundant than the second metal. When placed in a container of water, it exerts a force of $10\ \mathrm{N}$ on the bottom. What force (in newtons) will the third alloy, obtained by combining the original alloys, exert on the bottom?

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


Given an isosceles triangle \(ABC\) with \(AB = AC\) and \(\angle ABC = 53^\circ\). Point \(K\) is such that \(C\) is the midpoint of segment \(AK\). Point \(M\) is chosen such that:

- \(B\) and \(M\) are on the same side of line \(AC\);
- \(KM = AB\);
- The angle \(\angle MAK\) is the largest possible.

How many degrees is the angle \(\angle BAM\)?


There are at least 150 boys in a school, and the number of girls is 15% more than the number of boys. When the boys went on a trip, 6 buses were required, with each bus carrying the same number of students. How many students are there in the school, given that the total number of students is not more than 400?


How many roots does the equation \(\sqrt{14-x^{2}}(\sin x-\cos 2x)=0\) have?


Rectangle \(ABCD\) has diagonal \(BD\) with endpoints \(B(4,2)\) and \(D(12,8)\). Diagonal \(AC\) lies on the line with equation \(x + 2y - 18 = 0\). Determine the area of \(ABCD\).

Solution: \boxed{20 \sqrt{5}}


A circle \( C \) touches three pairwise disjoint circles whose centers are collinear and none of which contains any of the others. Show that the radius of \( C \) must be larger than the radius of the middle of the three circles.

Solution: \boxed{R > r_2}


In the quadrilateral \(ABCD\), if \(\overrightarrow{AB} \cdot \overrightarrow{CD} = -3\) and \(\overrightarrow{AD} \cdot \overrightarrow{BC} = 5\), then \(\overrightarrow{AC} \cdot \overrightarrow{BD} =\) _______.


In a rectangle \(ABCD\) with an area of 1 (including its boundary), there are 5 points such that no three points are collinear. Determine the minimum number of triangles, formed by these 5 points as vertices, that have an area of at most \(\frac{1}{4}\).


Little Abel received a $2 \times n$ board and $n$ $2 \times 1$ tiles as a gift. For example, the figure below shows the case where $n=10$, that is, when Abel has a $2 \times 10$ board and 10 $2 \times 1$ tiles.

He plays by filling the board using the $n$ tiles. For example, for $n=10$ Abel could fill it in the ways illustrated below:

Note, however, that there are many other ways for Abel to fill his board.

a) Calculate the total number of ways in which Abel can fill his board for the cases where $n=1, n=2,$ and $n=3$, that is, when the boards have dimensions $2 \times 1, 2 \times 2,$ and $2 \times 3$.

b) Let $a_{n}$ be the number of ways in which Abel can fill a $2 \times n$ board using $n$ $2 \times 1$ tiles. Show that $a_{10}=a_{9}+a_{8}$.

c) Calculate the total number of ways in which Abel can fill his board when $n=10$.


As shown in Figure 2.10.5, let $\triangle ABC$ have a perimeter of $2p$. A tangent line to the incircle of the triangle, parallel to side $AC$, is drawn as $DE$. Find the maximum length of the segment of this tangent line intercepted by the other two sides of the triangle.


Given that \( a \), \( b \), and \( c \) are three distinct prime numbers, and \( a + b \times c = 37 \), what is the maximum value of \( a + b - c \)?


Let \(ABC\) be a triangle and \(\Gamma\) be its circumcircle. The tangent to \(\Gamma\) at \(A\) intersects \((BC)\) at \(D\). The bisector of \(\widehat{CDA}\) intersects \((AB)\) at \(E\) and \((AC)\) at \(F\). Show that \(AE = AF\).

Solution: \boxed{AE = AF}


The circle \(\gamma_{1}\) centered at \(O_{1}\) intersects the circle \(\gamma_{2}\) centered at \(O_{2}\) at two points \(P\) and \(Q\). The tangent to \(\gamma_{2}\) at \(P\) intersects \(\gamma_{1}\) at the point \(A\) and the tangent to \(\gamma_{1}\) at \(P\) intersects \(\gamma_{2}\) at the point \(B\), where \(A\) and \(B\) are distinct from \(P\). Suppose \(PQ \cdot O_{1}O_{2} = PO_{1} \cdot PO_{2}\) and \(\angle APB\) is acute. Determine the size of \(\angle APB\) in degrees.


How many four-digit numbers greater than 5000 can be formed from the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 if only the digit 4 may be repeated?


Find the number of eight-digit numbers where the product of the digits is 64827. Provide the answer as an integer.


Given that \( \mathrm{f}(x) \) is a polynomial of degree 2012, and that \( \mathrm{f}(k) = \frac{2}{k} \) for \( k = 1,2,3, \cdots, 2013 \), find the value of \( 2014 \times \mathrm{f}(2014) \).


Xiao Ming departs from his home to his grandmother's house on a bicycle at a speed of 12 km/h. After 2.5 hours, his father realizes that Xiao Ming forgot to bring his homework and starts chasing him on a motorcycle at a speed of 36 km/h. Half an hour after Xiao Ming reaches his grandmother's house, his father arrives. What is the distance between Xiao Ming's home and his grandmother's house in kilometers?

Solution: \boxed{36 \ \text{km}}


Find all natural numbers $p$ such that both $p$ and $5p + 1$ are prime.


Let \( f: \mathbb{Z}_{>0} \rightarrow \mathbb{Z} \) be a function with the following properties:

(i) \( f(1)=0 \),

(ii) \( f(p)=1 \) for all prime numbers \( p \),

(iii) \( f(xy)=y f(x)+x f(y) \) for all \( x, y \in \mathbb{Z}_{>0} \).

Determine the smallest integer \( n \geq 2015 \) that satisfies \( f(n)=n \).

(Gerhard J. Woeginger)


A segment can slide along a line; a circular arc can slide along a circle; do other plane curves exist that have this property?

Solution: \boxed{\text{No}}


On the sides \( AB \) and \( BC \) of an equilateral triangle \( ABC \), points \( P \) and \( Q \) are chosen such that \( AP:PB = BQ:QC = 2:1 \). Find \( \angle AKB \), where \( K \) is the intersection point of segments \( AQ \) and \( CP \).


Is it possible to find at least one cube of an integer among the numbers $2^{2^{n}}+1$ for $n=0,1,2, \ldots$?

Solution: \boxed{\text{No}}


If \( a, b, c, d \) are distinct positive integers such that \( a \times b \times c \times d = 357 \), then \( a + b + c + d = \) $\qquad$.


Petya and Masha take turns taking candy from a box. Masha took one candy, then Petya took 2 candies, Masha took 3 candies, Petya took 4 candies, and so on. When the number of candies in the box became insufficient for the next turn, all the remaining candies went to the person whose turn it was to take candies. How many candies did Petya receive if Masha got 101 candies?


In a checkers tournament, students from 10th and 11th grades participated. Each player played against every other player exactly once. A win earned a player 2 points, a draw earned 1 point, and a loss earned 0 points. The number of 11th graders was 10 times the number of 10th graders, and together they scored 4.5 times more points than all the 10th graders combined. How many points did the most successful 10th grader score?


On his birthday, the last guest to arrive was Yana, who gave Andrey a ball, and the second last was Eduard, who gave him a calculator. While using the calculator, Andrey noticed that the product of the total number of gifts he received and the number of gifts he had before Eduard arrived is exactly 16 more than the product of his age and the number of gifts he had before Yana arrived. How many gifts does Andrey have?


A true-false test has ten questions. If you answer five questions "true" and five "false," your score is guaranteed to be at least four. How many answer keys are there for which this is true?


Find all strictly positive integers $x, y, z$ such that $5^{x} - 3^{y} = z^{2}$.

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


A regular triangular pyramid \(SABC\) is given, with its base edge 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 lines containing these medians also contain edges of a certain cube. Find the length of the lateral edge of the pyramid.


In the first grade of the school, between 200 and 300 children enrolled. It was decided to form classes with 25 students each, but it turned out that there would be no place for 10 children. Then they formed classes with 30 students each, but one of the classes ended up having 15 fewer students. How many children enrolled in the first grade?


There is an urn with 10 balls, among which there are 2 red, 5 blue, and 3 white balls. Find the probability that a randomly drawn ball will be a colored one (event $A$).


In triangle \(ABC\), \(AC = 1\), \(AB = 2\), and \(O\) is the point where the angle bisectors intersect. A segment passing through point \(O\) parallel to side \(BC\) intersects sides \(AC\) and \(AB\) at points \(K\) and \(M\) respectively. Find the perimeter of triangle \(AKM\).


For any subset \( S \subseteq \{1, 2, \ldots, 15\} \), a number \( n \) is called an "anchor" for \( S \) if \( n \) and \( n+|S| \) are both members of \( S \), where \( |S| \) denotes the number of members of \( S \). Find the average number of anchors over all possible subsets \( S \subseteq \{1, 2, \ldots, 15\} \).


Given the expression \(1.24 \frac{\sqrt{\frac{a b c + 4}{a} + 4 \sqrt{\frac{b c}{a}}}}{\sqrt{a b c} + 2}\) where \( a = 0.04 \).


Let \( S = \{1, 2, \ldots, 2016\} \), and let \( f \) be a randomly chosen bijection from \( S \) to itself. Let \( n \) be the smallest positive integer such that \( f^{(n)}(1) = 1 \), where \( f^{(i)}(x) = f\left(f^{(i-1)}(x)\right) \). What is the expected value of \( n \)?


Amelia wrote down a sequence of consecutive positive integers, erased one integer, and scrambled the rest, leaving the sequence below. What integer did she erase?
$$
6,12,1,3,11,10,8,15,13,9,7,4,14,5,2
$$


Let \( x[n] \) denote \( x \) raised to the power of \( x \), repeated \( n \) times. What is the minimum value of \( n \) such that \( 9[9] < 3[n] \)?

(For example, \( 3[2] = 3^3 = 27 \); \( 2[3] = 2^{2^2} = 16 \).)


Given a triangle $ABC$ with $\angle BAC = 60^\circ$, let $H$, $I$, and $O$ denote its orthocenter, its incenter, and its circumcenter respectively. Show that $IO = IH$.

Solution: \boxed{IO = IH}


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


Find the largest six-digit number in which all digits are distinct, and each digit, except for the extreme ones, is equal either to the sum or the difference of its neighboring digits.


Let \( a, b, c \in \left[\frac{1}{2}, 1\right] \). Define \( s = \frac{a+b}{1+c} + \frac{b+c}{1+a} + \frac{c+a}{1+b} \). What is the range of possible values for \( s \)?

Solution: \boxed{[2, 3]}


Let \( p_{1}, p_{2}, \ldots, p_{97} \) be prime numbers (not necessarily distinct). What is the maximum integer value that the expression

$$
\sum_{i=1}^{97} \frac{p_{i}}{p_{i}^{2}+1}=\frac{p_{1}}{p_{1}^{2}+1}+\frac{p_{2}}{p_{2}^{2}+1}+\ldots+\frac{p_{97}}{p_{97}^{2}+1}
$$

can take?


Determine the prime numbers $p$ for which $p^{2}+8$ is also prime.


Given a triangle \( ABC \), a circle passing through \( B \) and \( C \) intersects \( AB \) and \( AC \) at \( B' \) and \( C' \) respectively. Let \( O \) be the center of the circumcircle of \( AB'C' \). Show that \( AO \) is perpendicular to \( BC \).

Solution: \boxed{AO \perp BC}


The bases of a trapezoid are 8 and 2. The angles adjacent to the larger base are each $45^{\circ}$. Find the volume of the solid formed by rotating the trapezoid about its larger base.


Let \( x_{1} \) and \( x_{2} \) be two real numbers that satisfy \( x_{1} x_{2} = 2013 \). What is the minimum value of \( (x_{1} + x_{2})^{2} \)?


174. \(n^{2}\) integers from 1 to \(n^{2}\) are written in a square table of size \(n \times n\): number 1 is in any position in the table; number 2 belongs to the row whose number is equal to the column number containing 1; number 3 belongs to the row whose number matches the column number containing 2, and so on. By how much does the sum of the numbers in the row containing number 1 differ from the sum of the numbers in the column containing the number \(n^{2}\)?

Solution: \boxed{n^2 - n}


Let $D$ be an interior point of the acute triangle $\triangle ABC$ such that $\angle ADB = \angle ACB + 90^{\circ}$ and $AC \cdot BD = AD \cdot BC$. Find the value of $\frac{AB \cdot CD}{AC \cdot BD}$.


Determine all functions \( f: \mathbb{N}^{*} \rightarrow \mathbb{N}^{*} \) such that for all \( m, n \in \mathbb{N}^{*} \), the number \( f(m) + f(n) - mn \) is nonzero and divides \( m f(m) + n f(n) \).

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


The sequence \(a_{1}, a_{2}, \ldots\) is defined by the equations

\[ a_{1}=100, \quad a_{n+1}=a_{n}+\frac{1}{a_{n}}, \quad n \in \mathbb{N} \]

Find the integer closest to \(a_{2013}\).


Given a complex number \( z \) such that \( |z| = 1 \), and \( u = z^4 - z^3 - 3z^2 i - z + 1 \). Find the maximum value of \( |u| \) and determine the complex number \( z \) when this maximum value is achieved.


Three rods (with negligible thickness) are fixed perpendicularly to each other at a common point at one end. The lengths of the rods are 1, 2, and 3. This construction is placed on a table in such a way that the free ends of the rods lie on the plane of the tabletop. Determine the exact height of the fixed point above the table.


The function \( f(x) = a^{2x} + 3a^x - 2 \) (where \( a > 0 \) and \( a \neq 1 \)) has a maximum value of 8 on the interval \( x \in [-1,1] \). Determine its minimum value on this interval.


Consider the set \( S = \{1, 2, \cdots, 15\} \). Let \( A = \{a_1, a_2, a_3\} \) be a subset of \( S \), with \( (a_1, a_2, a_3) \) satisfying:
\[
1 \leq a_1 < a_2 < a_3 \leq 15, \quad a_3 - a_2 \leq 6.
\]
Determine the number of subsets \( A \) that satisfy these conditions.


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


All natural numbers from 1 to 1000 inclusive are divided into two groups: even and odd. In which group is the sum of all the digits used to write the numbers greater and by how much?


$M$ is the midpoint of the height $B D$ in an isosceles triangle $A B C$. Point $M$ serves as the center of a circle with radius $M D$. Find the angular measure of the arc of the circle enclosed between the sides $B A$ and $B C$, given that $\angle B A C = 65^{\circ}$.


The perimeter of a trapezoid inscribed in a circle is 40. Find the midsegment of the trapezoid.


The sequence \(\left\{a_n\right\}\) satisfies \(a_1 = 1\) and \(a_{n+1} = \frac{(n^2 + n) a_n}{3 a_n + n^2 + n}\) for \(n \in \mathbb{N}^*\). Find the general term \(a_n\).


The perimeter of triangle \(ABC\) is \(2p\). What is the maximum length that a segment parallel to \(BC\) and tangent to the inscribed circle of \(ABC\) can have inside the triangle? For what triangle(s) is this value achieved?


Let \( x \) and \( y \) be real numbers with \( x > y \) such that \( x^{2} y^{2} + x^{2} + y^{2} + 2xy = 40 \) and \( xy + x + y = 8 \). Find the value of \( x \).

Solution: \boxed{3 + \sqrt{7}}


Calculate the limit of the function:

$$\lim _{x \rightarrow \pi} \frac{\sin ^{2} x-\operatorname{tg}^{2} x}{(x-\pi)^{4}}$$


Two straight roads intersect perpendicularly. On one road, a car starts from a distance of $l_{1}=80 \mathrm{~km}$ from the intersection and heads towards the intersection at a constant speed of $v_{1}=80 \mathrm{~km/h}$. On the other road, another car starts at the same time from a distance of $l_{2}=60 \mathrm{~km}$ from the intersection, traveling towards the intersection at a speed of $v_{2}=60 \mathrm{~km/h}$. How fast is the distance between the cars changing? What is the situation if the initial distances are $l_{1}=l_{2}^{\prime}=80 \mathrm{~km}$? How much time after departure will the two cars be closest to each other, and what is this distance in both cases?

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


A $3 \times 3 \times 3$ cube composed of 27 unit cubes rests on a horizontal plane. Determine the number of ways of selecting two distinct unit cubes from a $3 \times 3 \times 1$ block (the order is irrelevant) such that the line joining the centers of the two cubes makes a $45^{\circ}$ angle with the horizontal plane.


There is a magical tree with 46 fruits. On the first day, 1 fruit will fall from the tree. Every day, the number of fallen fruits increases by 1 compared to the previous day. However, if on any day the number of fruits on the tree is less than the number of fruits that should fall that day, then the process restarts by dropping 1 fruit, following the original pattern. Continue in this manner. On which day will all the fruits have fallen from the tree?


The clock shows 00:00, and the hour and minute hands coincide. Considering this coincidence to be number 0, determine after what time interval (in minutes) they will coincide the 19th time. If necessary, round the answer to two decimal places following the rounding rules.


When is the distance between the real image produced by a convex lens and the object the smallest?


Four people (A, B, C, D) are practicing passing a ball. The ball is initially passed by A, and each person who receives the ball has an equal probability of passing it to one of the other three people. Let \( p_{n} \) represent the probability that the ball returns to A after \( n \) passes. What is \( p_{6} \)?


The function \( f(x)=a x^{2}+b x+c \), where \( a \), \( b \), and \( c \) are integers, has two distinct roots in the interval \((0,1)\). Find the smallest positive integer \( a \) for which the given condition holds.


Losyash is walking to Sovunya's house along the river at a speed of 4 km/h. Every half-hour, he launches paper boats that travel to Sovunya at a speed of 10 km/h. What is the time interval at which the boats arrive at Sovunya's house?

Solution: \boxed{18 \text{ минут}}


If 10 points are placed at equal intervals on a line, they occupy a segment of length \( s \). If 100 points are placed, they occupy a segment of length \( S \). By what factor is \( S \) greater than \( s \)?


Can 1965 points be arranged inside a square with side 15 so that any rectangle of unit area placed inside the square with sides parallel to its sides must contain at least one of the points?

Solution: \boxed{Yes}


What is the maximum possible area of a triangle with sides \(a\), \(b\), and \(c\) that fall within the following ranges:

\[ 0 \leq a \leq 1, \quad 1 \leq b \leq 2, \quad 2 \leq c \leq 3? \]


Given a regular $n$-sided prism, where the area of the base is $S$. Two planes intersect all the lateral edges of the prism in such a way that the volume of the part of the prism between the planes is $V$. Find the sum of the lengths of the segments of the lateral edges of the prism that are enclosed between the planes, given that the planes do not have any common points inside the prism.

Solution: \boxed{\frac{nV}{S}}


The positive integers \( a \) and \( b \) are relatively prime. The greatest common divisor of the numbers \( A = 8a + 3b \) and \( B = 3a + 2b \) is not 1. What is the greatest common divisor of \( A \) and \( B \)?


Among 2017 natural numbers, there is at least one two-digit number, and for any two of these numbers, at least one is a three-digit number. Determine the number of three-digit numbers among these 2017 numbers.


Clara takes 2 hours to ride her bicycle from Appsley to Bancroft. The reverse trip takes her 2 hours and 15 minutes. If she travels downhill at 24 km/h, on level road at 16 km/h, and uphill at 12 km/h, what is the distance, in kilometres, between the two towns?

Solution: \boxed{34 \text{ km}}


Given \( w = \sqrt{2p - q} + \sqrt{3q - 2p} + \sqrt{6 - 2q} \), where \( p \) and \( q \) are real numbers that make \( w \) meaningful, determine the maximum value of \( w \).


In a regular triangular prism \(P-ABC\) where all edge lengths are \(1\), points \(L\), \(M\), and \(N\) are the midpoints of edges \(PA\), \(PB\), and \(PC\) respectively. Determine the area of the cross-section of the circumscribed sphere intersected by the plane \(LMN\).


A \(15 \times 15\) square is divided into \(1 \times 1\) small squares. From these small squares, several were chosen, and in each chosen square, one or two diagonals were drawn. It turned out that no two drawn diagonals have a common endpoint. What is the maximum number of diagonals that can be drawn? (In the solution, provide the answer, the method of drawing the diagonals, and proof that this number of diagonals is indeed the maximum possible.)


How many integers at minimum must be selected from the set $\{1, 2, \ldots, 20\}$ to ensure that this selection includes two integers \(a\) and \(b\) such that \(a - b = 2\)?


A fast-food chain offers summer jobs to students with a salary of 25,000 rubles per month. Those who perform well receive an additional monthly bonus of 5,000 rubles.

How much will a well-performing student working at the fast-food chain earn per month (take-home pay) after income tax deduction?

Provide only the number without units in the answer!


Distribute 16 identical books among 4 students so that each student gets at least one book, and each student gets a different number of books. How many distinct ways can this be done? (Answer with a number.)


Four points form a convex quadrilateral. If the smallest distance between any two of them is \( d \), show that there are two whose distance apart is at least \( d\sqrt{2} \).


What square is equal to the product of four consecutive odd numbers?


In Figure 1, \(ABCD\) is a square, \(M\) is the midpoint of \(AD\), and \(N\) is the midpoint of \(MD\). If \(\angle CBN : \angle MBA = P : 1\), find the value of \(P\).


Given any $n$, let

$$
a_{n}=\frac{1}{1 \cdot 2 \cdot 3}+\frac{1}{2 \cdot 3 \cdot 4}+\ldots+\frac{1}{n(n+1)(n+2)}
$$

Is the sequence $a_{n}$ convergent, and if it is, what is its limit?


A group of 9 boys and 9 girls is randomly paired up. Find the probability that at least one pair consists of two girls. Round your answer to two decimal places.


How many 8-digit numbers begin with 1, end with 3, and have the property that each successive digit is either one more or two more than the previous digit, considering 0 to be one more than 9?


On December 24, 1800, First Consul Bonaparte was heading to the Opera on Saint-Nicaise Street. A bomb exploded on his route with a delay of a few seconds. There were many killed and injured. Bonaparte blamed the republicans for the conspiracy; he deported 98 of them to the Seychelles and Guyana. Several people were executed.

Suppose the number of injured is equal to twice the number of killed (in the explosion) plus four thirds of the number of executed, and the sum of the number of killed or injured and the number of executed is slightly less than the number deported. Also, if you subtract 4 from the number of killed, you get exactly twice the number of executed.

Can you, without consulting history books, determine how many people Bonaparte executed after the assassination attempt on Saint-Nicaise Street?


Solution: \boxed{-\tan^4 \alpha}


Find the smallest natural number \( n \) for which \( (n+1)(n+2)(n+3)(n+4) \) is divisible by 1000.


In a two-story house that is inhabited in both floors as well as on the ground floor, 35 people live above someone and 45 people live below someone. One third of all the people living in the house live on the first floor.

How many people live in the house in total?


There are 2016 cards, each with a unique number from 1 to 2016. A certain number \( k \) of these cards are selected. What is the smallest \( k \) such that among these selected cards, there exist two cards with numbers \( a \) and \( b \) satisfying the condition \( |\sqrt[3]{a} - \sqrt[3]{b}| < 1 \)?


Find the first few terms of the series expansion in powers of $z$ for the function $f(z)=\operatorname{tg} z$ and determine the radius of convergence of the series.

Solution: \boxed{R = \frac{\pi}{2}}


On 2016 cards, the numbers from 1 to 2016 were written (each number exactly once). Then \( k \) cards were taken. What is the smallest \( k \) such that among them there will be two cards with numbers whose square root difference is less than 1?


An isosceles triangle has a side of length \(20 \text{ cm}\). Of the remaining two side-lengths, one is equal to two-fifths of the other. What is the length of the perimeter of this triangle?

Solution: \boxed{48 \text{ cm}}


The equation \(31 \cdot 38 \sqrt{x} + \sqrt{y} = \sqrt{2009}\) has integer solutions \((x, y)\). What is the maximum value of \(x + y\)?
(A) 1189
(B) 1517
(C) 1657
(D) 1749

(China, Shaanxi Province, Xi'an City, Middle School Mathematics Competition, 1990)


When 15 is added to a number \( x \), it becomes a square number. When 74 is subtracted from \( x \), the result is again a square number. Find the number \( x \).


Inside a square, 100 points are marked. The square is divided into triangles in such a way that the vertices of the triangles are only the marked 100 points and the vertices of the square, and for each triangle in the division, each marked point either lies outside this triangle or is a vertex of it (such divisions are called triangulations). Find the number of triangles in the division.


Given positive integers \( a, b, \c \), define the function \( f(x, y, z) = a x + b y + c z \), where \( x, y, z \in \mathbb{Z} \). Find the minimum positive integer value of \( f(x, y, z) \).

Solution: \boxed{\gcd(a, b, c)}


Find the equation of a line \( L \) such that the graph of the function

\[ y = x^4 + 4x^3 - 26x^2 \]

lies entirely on one side of this line, with two points of intersection.

Solution: \boxed{y = 60x - 225}


On a table, there is a certain quantity \( N \) of candies. Aline and Bruna agree that, alternately, each must eat at least one but no more than half of the existing quantity. The winner of the game is the one who eats the last candy. Aline always starts the game.

a) For \( N = 5 \), which of the two has the winning position? (A winning position is one in which the player can ensure a win regardless of the opponent's sequence of moves.)

b) For \( N = 20 \), which of the two has the winning position?

c) What are the values of \( N \), where \( 100 < N < 200 \), that give Bruna the winning position?


Find the Lagrange interpolation polynomial that takes the values $y_{0}=-5$, $y_{1}=-11$, $y_{2}=10$ at the points $x_{0}=-3$, $x_{1}=-1$, $x_{2}=2$, respectively.

Solution: \boxed{2x^2 + 5x - 8}


A $7 \times 7$ table is filled with zeroes. In one operation, the minimum number in the table is found (if there are several such numbers, any one of them is chosen), and one is added to it as well as to all the numbers in the cells adjacent to it by side or corner. What is the largest number that can appear in one of the cells of the table after 90 operations?

Answer: 40.


Bryce has 7 blue socks and 7 red socks mixed in a drawer. He plays a game with Sean. Blindfolded, Bryce takes two socks from the drawer. Sean looks at the socks, and if they have the same color, Sean gives Bryce 1 point. Bryce keeps drawing socks until the drawer is empty, at which time the game ends. The probability that Bryce's score is at most 2 can be written in the form \( \frac{p}{q} \), where \( p \) and \( q \) are relatively prime positive integers. Find \( p+q \).


Through the point \((1,2)\), two tangent lines to the circle \(x^2 + y^2 = 1\) are drawn. What is the area of the quadrilateral formed by these two tangent lines and the \(x\)-axis and \(y\)-axis?


There are 6 positive integers \( a, b, c, d, e, f \) arranged in order, forming a sequence, where \( a=1 \). If a positive integer is greater than 1, then the number that is one less than it must appear to its left. For example, if \( d > 1 \), then one of \( a, b, \) or \( c \) must have the value \( d-1 \). For instance, the sequence \( 1,1,2,1,3,2 \) satisfies the requirement; \( 1,2,3,1,4,1 \) satisfies the requirement; \( 1,2,2,4,3,2 \) does not satisfy the requirement. How many different permutations satisfy the requirement?


A food factory has made 4 different exquisite cards. Each bag of food produced by the factory randomly contains one card. If all 4 different cards are collected, a prize can be won. Xiaoming buys 6 bags of this food at once. What is the probability that Xiaoming will win the prize?


Find the largest real number \( m \) such that for all positive numbers \( a, b, \) and \( c \) satisfying \( a + b + c = 1 \),
$$
10\left(a^{3}+b^{3}+c^{3}\right)-m\left(a^{5}+b^{5}+c^{5}\right) \geqslant 1.
$$


In how many ways can an \( n \times n \) grid be filled with zeros and ones such that each row and each column contains an even number of ones? Each cell of the grid must contain either a zero or a one.


Find all three-digit numbers that are 11 times the sum of their digits.


Given that \( 990 \times 991 \times 992 \times 993 = \overline{966428 A 91 B 40} \), find the values of \( \overline{A B} \).


Find the largest real number \(\lambda\) such that for the real coefficient polynomial \(f(x) = x^3 + ax^2 + bx + c\) with all non-negative real roots, it holds that \(f(x) \geqslant \lambda(x - a)^3\) for all \(x \geqslant 0\). Additionally, determine when the equality in the expression is achieved.


Let \( n \) be a natural number less than 50. Find the sum of all possible values of \( n \) such that \( 4n + 5 \) and \( 7n + 6 \) have a common divisor greater than 1.


Given the quadratic equations in \( x \),
\[ x^{2} + kx - 12 = 0 \]
and
\[ 3x^{2} - 8x - 3k = 0, \]
having a common root, find all possible values of the real number \( k \).


Given two linear functions \( f(x) \) and \( g(x) \) such that the graphs \( y = f(x) \) and \( y = g(x) \) are parallel lines, and not parallel to the coordinate axes. Find the minimum value of the function \( (g(x))^2 + 8 f(x) \), if the minimum value of the function \( (f(x))^2 + 8 g(x) \) is -29.


In a group at a pioneer camp, there are children aged 10, 11, 12, and 13 years old. There are 23 people in total, and together they are 253 years old. How many 12-year-olds are in the group if it is known that there are one and a half times more 12-year-olds than 13-year-olds?


In the cube $ABCD-A_{1}B_{1}C_{1}D_{1}$, what is the measure of the dihedral angle $A-BD_{1}-A_{1}$?


A pedestrian is moving in a straight line towards a crosswalk at a constant speed of 3.6 km/h. Initially, the distance from the pedestrian to the crosswalk is 40 meters. The length of the crosswalk is 6 meters. What distance from the crosswalk will the pedestrian be after two minutes?

Solution: \boxed{74 \text{ m}}


Given a positive integer $n$, how many quadruples of ordered integers $(a, b, c, d)$ satisfy the condition
$$
0 \leqslant a \leqslant b \leqslant c \leqslant d \leqslant n ?
$$


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

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


Is it possible to replace stars with pluses or minuses in the following expression
\[ 1 \star 2 \star 3 \star 4 \star 5 \star 6 \star 7 \star 8 \star 9 \star 10 = 0 \]
so that to obtain a true equality?

Solution: \boxed{\text{No}}


What is the minimum number of points that can be chosen on a circle of length 1956 so that for each chosen point there is exactly one chosen point at a distance of 1 and exactly one chosen point at a distance of 2 (distances are measured along the circle)?


Given a regular triangular pyramid \( S-ABC \) with height \( SO=3 \) and base side length of 6. A perpendicular line is drawn from point \( A \) to the corresponding lateral face \( SBC \), with the foot of the perpendicular being \( D \). A point \( P \) is taken on \( AD \), such that \( \frac{AP}{PD} = 8 \). Find the area of the cross-section that passes through point \( P \) and is parallel to the base.


Given the parabola \( C: x^{2} = 2py \) with \( p > 0 \), two tangents \( RA \) and \( RB \) are drawn from the point \( R(1, -1) \) to the parabola \( C \). The points of tangency are \( A \) and \( B \). Find the minimum area of the triangle \( \triangle RAB \) as \( p \) varies.

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


Points \( K \) and \( M \) are marked on the sides \( BC \) and \( AD \) of the inscribed quadrilateral \( ABCD \), respectively, such that \( BK : KC = AM : MD \). A point \( L \) is chosen on the segment \( KM \) such that \( KL : LM = BC : AD \). Find the ratio of the areas of triangles \( ACL \) and \( BDL \), given that \( AC = p \) and \( BD = q \).

Solution: \boxed{\frac{p}{q}}


Given that \( n \) is a positive integer and \( S = 1 + 2 + 3 + \cdots + n \). Determine the units digits that \( S \) cannot have and find the sum of these forbidden digits.


In Class 3 (1), consisting of 45 students, all students participate in the tug-of-war. For the other three events, each student participates in at least one event. It is known that 39 students participate in the shuttlecock kicking competition and 28 students participate in the basketball shooting competition. How many students participate in all three events?


The diagonals \(AC\) and \(BD\) of a convex quadrilateral \(ABCD\), equal to 3 and 4 respectively, intersect at an angle of \(75^\circ\). What is the sum of the squares of the lengths of the line segments connecting the midpoints of opposite sides of the quadrilateral?


The function \( f(x) \) defined on the set of real numbers \( \mathbf{R} \) satisfies \( f(x+1) = \frac{1+f(x+3)}{1-f(x+3)} \). Determine the value of \( f(1) \cdot f(2) \cdots f(2008) + 2009 \).


Let \(M\) and \(N\) be the midpoints of the sides \(CD\) and \(DE\) of a regular hexagon \(ABCDEF\). Find the angle between the lines \(AM\) and \(BN\).


The points $E$ on the side $BC$ and $F$ on the side $AD$ of a convex quadrilateral $ABCD$ are such that $BE=2EC$ and $AF=2FD$. There is a circle with radius $r$ centered on segment $AE$, which is tangent to the sides $AB$, $BC$, and $CD$. There is another circle with the same radius $r$ centered on segment $BF$, which is tangent to the sides $AB$, $AD$, and $CD$. Find the area of the quadrilateral $ABCD$, given that the mentioned circles touch each other externally.


What is the value of \(a + b + c + d\) if

$$
\begin{gathered}
6a + 2b = 3848 \\
6c + 3d = 4410 \\
a + 3b + 2d = 3080
\end{gathered}
$$


Inflation over two years will be:

$$
\left((1+0,025)^{\wedge 2-1}\right)^{*} 100 \%=5,0625 \%
$$

The real interest rate of a bank deposit with reinvestment for the second year will be $(1.06 * 1.06 /(1+0,050625)-1) * 100=6,95 \%$


The calculation result of the expression \(\frac{\frac{2015}{1}+\frac{2015}{0.31}}{1+0.31}\) is


Let the sequence of non-negative integers $\left\{a_{n}\right\}$ satisfy:
$$
a_{n} \leqslant n \quad (n \geqslant 1), \quad \text{and} \quad \sum_{k=1}^{n-1} \cos \frac{\pi a_{k}}{n} = 0 \quad (n \geqslant 2).
$$

Find all possible values of $a_{2021}$.


\( \binom{n}{0} \binom{n}{\left\lfloor \frac{n}{2} \right\rfloor} + 2 \binom{n}{1} \binom{n-1}{\left\lfloor \frac{n-1}{2} \right\rfloor} + 2^2 \binom{n}{2} \binom{n-2}{\left\lfloor \frac{n-2}{2} \right\rfloor} + \ldots + 2^{n-1} \binom{n}{n-1} \binom{1}{\left\lfloor \frac{1}{2} \right\rfloor} + 2^n = \binom{2n+1}{n} \)

Show that the above equation holds true.


Two people, A and B, agree to meet at a certain location within 10 days. They agree that the person who arrives first will wait for the other person, but only for 3 days before leaving. Assuming that their arrivals at the destination are equally likely to occur within the given period, what is the probability that the two will meet?


The year 2009 has the following property: by rearranging the digits of the number 2009, it is not possible to obtain a smaller four-digit number (numbers do not start with zero). In which year will this property occur again for the first time?


In how many ways can an \( n \times n \) grid be filled with zeros and ones such that each row and each column contains an even number of ones? Each cell of the grid must contain either a zero or a one.


Let \(ABCD\) be a right trapezoid with bases \(AB\) and \(CD\), featuring right angles at \(A\) and \(D\). Given that the shorter diagonal \(BD\) is perpendicular to the side \(BC\), determine the minimum possible value for the ratio \(\frac{CD}{AD}\).


In triangle \(ABC\), it is known that \(AB = 3\), \(AC = 3\sqrt{7}\), and \(\angle ABC = 60^\circ\). The bisector of angle \(ABC\) is extended to intersect at point \(D\) with the circle circumscribed around the triangle. Find \(BD\).


In the figure, the rays \( O A, O B, O C, O D, O E, O F \) are such that:

- \( О B \) is the bisector of \(\angle A O C\)
- \( O E \) is the bisector of \(\angle D O F \)
- \(\angle A O F = 146^\circ\)
- \(\angle C O D = 42^\circ\)

How many degrees is the angle \( B O E \)?


Given \(\alpha \in \left[0, \frac{\pi}{2}\right]\), \(\beta \in \left[0, \frac{\pi}{2}\right]\), find the minimum value of \(\cos^{2} \alpha \sin \beta + \frac{1}{\sin \beta}\).


Find the sum of all three-digit numbers that are divisible by 7.


There are nine cards, each with the numbers $2, 3, 4, 5, 6, 7, 8, 9, 10$. Four people, A, B, C, and D, each draw two of these cards.

Person A says: "The two numbers I drew are relatively prime because they are consecutive."
Person B says: "The two numbers I drew are not relatively prime and are not multiples of each other."
Person C says: "The two numbers I drew are both composite numbers and they are relatively prime."
Person D says: "The two numbers I drew are in a multiple relationship and they are not relatively prime."

Assuming all four people are telling the truth, what is the number on the remaining card? $\quad \quad$


There are three water pipes, \(A\), \(B\), and \(C\), which can be used to fill a water truck. If pipes \(A\) and \(C\) are used simultaneously, the truck gets filled when pipe \(A\) has injected 4 tons of water. If pipes \(B\) and \(C\) are used simultaneously, the truck gets filled when pipe \(B\) has injected 6 tons of water. It is known that the water injection rate of pipe \(B\) is twice that of pipe \(A\). How many tons of water can the water truck hold at most?


If the eight digits \(1, 2, 3, 4, 5, 6, 7, 8\) are randomly permuted, what is the probability that the resulting eight-digit number is divisible by 8?


\(ABCD\) is a square-based pyramid with base \(ABCD\) and apex \(E\). Point \(E\) is directly above point \(A\), with \(AE = 1024\) units and \(AB = 640\) units. The pyramid is sliced into two parts by a horizontal plane parallel to the base \(ABCD\), at a height \(h\) above the base. The portion of the pyramid above the plane forms a new smaller pyramid. For how many integer values of \(h\) does the volume of this new pyramid become an integer?


Find the greatest common divisor of all nine-digit numbers in which each of the digits 1, 2, 3, ..., 9 appears exactly once.


The diagonal \( AC \) of a convex quadrilateral \( ABCD \) serves as the diameter of the circumscribed circle around it. Find the ratio of the areas of triangles \( ABC \) and \( ACD \), given that the diagonal \( BD \) divides \( AC \) in the ratio 2:5 (counting from point \( A \)), and \( \angle BAC = 45^\circ \).


Find the number of ways that 2010 can be written as a sum of one or more positive integers in non-decreasing order such that the difference between the last term and the first term is at most 1.


A unit square is called "colorful" if three of its four sides are colored in three different colors. Consider the $1 \times 3$ grid, which contains 10 unit length line segments. These 10 line segments are to be colored using red, yellow, or blue, such that all three unit squares are colorful. How many ways can this coloring be done? (Express the answer as a numerical value.)


Two right triangles \( \triangle AXY \) and \( \triangle BXY \) have a common hypotenuse \( XY \) and side lengths (in units) \( AX=5 \), \( AY=10 \), and \( BY=2 \). Sides \( AY \) and \( BX \) intersect at \( P \). Determine the area (in square units) of \( \triangle PXY \).


In the expansion of \((-xy + 2x + 3y - 6)^6\), what is the coefficient of \(x^4 y^3\)? (Answer with a specific number)

Solution: \boxed{ -21600}


In the country of Anchuria, a unified state exam takes place. The probability of guessing the correct answer to each question on the exam is 0.25. In 2011, to receive a certificate, one needed to answer correctly 3 questions out of 20. In 2012, the School Management of Anchuria decided that 3 questions were too few. Now, one needs to correctly answer 6 questions out of 40. The question is, if one knows nothing and simply guesses the answers, in which year is the probability of receiving an Anchurian certificate higher - in 2011 or in 2012?


There are 15 players participating in a Go tournament. Each pair of players needs to play one match. A win earns 2 points, a draw earns 1 point for each player, and a loss earns 0 points. If a player scores at least 20 points, they will receive a prize. What is the maximum number of players that can receive a prize?


(a) Determine the largest possible value \( M \) that \( x + y + z \) can take if \( x \), \( y \), and \( z \) are positive real numbers satisfying
\[
16xyz = (x + y)^2 (x + z)^2.
\]

(b) Show that there are infinitely many triples \( (x, y, z) \) of positive rational numbers for which
\[
16xyz = (x + y)^2 (x + z)^2 \quad \text{and} \quad x + y + z = M.
\]


Calculate the area of the regions bounded by the curves given in polar coordinates.

$$
r=3 \sin \phi, \quad r=5 \sin \phi
$$

Solution: \boxed{4 \pi}


Let \( p(x) \) be the product of the digits of the decimal number \( x \). Find all positive numbers \( x \) for which \( p(x) = x^2 - 10x - 22 \).


Mark on the line with red all points of the form $81x + 100y$, where $x, y$ are natural numbers, and paint the remaining integer points blue. Find a point on the line such that any symmetrical integer points relative to it are painted in different colors.


For the set $\{1,2,\cdots,n\}$ and each of its non-empty subsets, define a unique "alternating sum" as follows: Arrange the numbers in each subset in descending order, then start from the largest number and alternately subtract and add subsequent numbers to obtain the alternating sum (for example, the alternating sum of the set $\{1, 3, 8\}$ is $8 - 3 + 1 = 6$). For $n=8$, find the total sum of the alternating sums of all subsets.


In a trapezoid, the lengths of the bases are 5 and 15, and the lengths of the diagonals are 12 and 16. Find the area of the trapezoid.


Given two points on a plane and a line parallel to the line segment connecting the two points, along with the angle $\alpha$. Construct the segment on the parallel line that appears at an angle $\alpha$ from both points!

Solution: \boxed{CD}


Find all values of \( a \) for which the equation \( x^{2} + 2ax = 8a \) has two distinct integer roots. Record the product of all such \( a \), rounding to the nearest hundredth if necessary.


A battery of three guns fired a volley, and two shells hit the target. Find the probability that the first gun hit the target, given that the probabilities of hitting the target by the first, second, and third guns are $p_{1}=0,4$, $p_{2}=0,3$, and $p_{3}=0,5$, respectively.


In a chess-playing club, some of the players take lessons from other players. It is possible (but not necessary) for two players both to take lessons from each other. It so happens that for any three distinct members of the club, $A, B$, and $C$, exactly one of the following three statements is true: $A$ takes lessons from $B$; $B$ takes lessons from $C$; $C$ takes lessons from $A$. What is the largest number of players there can be?


Extend each side of the quadrilateral $ABCD$ in the same direction beyond itself, such that the new points are $A_{1}, B_{1}, C_{1}$, and $D_{1}$. Connect these newly formed points to each other. Show that the area of the quadrilateral $A_{1}B_{1}C_{1}D_{1}$ is five times the area of the quadrilateral $ABCD$.


A metro network has at least 4 stations on each line, with no more than three transfer stations per line. No transfer station has more than two lines crossing. What is the maximum number of lines such a network can have if it is possible to travel from any station to any other station with no more than two transfers?


Given the sequence $\left\{a_{n}\right\}$ that satisfies $a_{1}=1$ and $S_{n+1}=2 S_{n}-\frac{n(n+1)}{2}+1$, where $S_{n}=a_{1}+a_{2}+\cdots+a_{n}$ $(n=1,2, \cdots)$. If $\Delta a_{n}=a_{n+1}-a_{n}$, find the number of elements in the set $S=\left\{n \in \mathbf{N}^{*} \mid \Delta\left(\Delta a_{n}\right) \geqslant-2015\right\}$.


Consider equations of the form \( ax^{2} + bx + c = 0 \), where \( a, b, c \) are all single-digit prime numbers. How many of these equations have at least one solution for \( x \) that is an integer?


Find the sum of all \( x \) such that \( 0 \leq x \leq 360 \) and \( \cos 12 x^{\circ} = 5 \sin 3 x^{\circ} + 9 \tan^2 x^{\circ} + \cot^2 x^{\circ} \).


A and B play a number-changing game on a $5 \times 5$ grid: A starts and both take turns filling empty spaces, with A filling each space with the number 1 and B filling each space with the number 0. After the grid is completely filled, the sum of the numbers in each $3 \times 3$ square is calculated, and the maximum sum among these squares is denoted as $A$. A tries to maximize $A$, while B tries to minimize $A$. What is the maximum value of $A$ that A can achieve?

(The problem is from the 35th IMO selection test)


If a number \( N \) is chosen randomly from the set of positive integers, the probability of the units digit of \( N^4 \) being unity is \( \frac{P}{10} \). Find the value of \( P \).


The numbers \(2^{2000}\) and \(5^{2000}\) are written consecutively. How many digits are written in total?


A person is walking parallel to railway tracks at a constant speed. A train also passes by this person at a constant speed. The person noticed that depending on the direction of the train's movement, it takes either $t_{1}=1$ minute or $t_{2}=2$ minutes to pass by him. Determine how long it would take for the person to walk from one end of the train to the other.

Solution: \boxed{4 \text{ minutes}}


Gavrila found out that the front tires of a car last for 24,000 km, and the rear tires last for 36,000 km. Therefore, he decided to swap them at some point to maximize the total distance the car can travel. Find this maximum possible distance (in km).


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


A jar contains 97 marbles that are either red, green, or blue. Neil draws two marbles from the jar without replacement and notes that the probability that they would be the same color is \( \frac{5}{12} \). After Neil puts his marbles back, Jerry draws two marbles from the jar with replacement. Compute the probability that the marbles that Jerry draws are the same color.


Given the function
$$
f(x)=\left\{\begin{array}{ll}
\ln x, & \text{if } x>1 ; \\
\frac{1}{2} x + \frac{1}{2}, & \text{if } x \leq 1 .
\end{array}\right.
$$

If \( m < n \) and \( f(m) = f(n) \), find the minimum value of \( n - m \).

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


Initially, the numbers $1, 2, \ldots, n$ were written on the board. It is allowed to erase any two numbers on the board and write the absolute value of their difference instead. What is the smallest number that can be left on the board after $(n-1)$ such operations a) for $n=111$; b) for $n=110$?


In triangle \( ABC \), an incircle is inscribed with center \( I \) and points of tangency \( P, Q, R \) with sides \( BC, CA, AB \) respectively. Using only a ruler, construct the point \( K \) where the circle passing through vertices \( B \) and \( C \) is internally tangent to the incircle.

Solution: \boxed{K}


The area of two parallel plane sections of a sphere are $9 \pi$ and $16 \pi$. The distance between the planes is given. What is the surface area of the sphere?


In a forest, there live elves and dwarves. One day, 60 inhabitants of this forest stood in a line facing the same direction. At this moment, some of them might be wearing caps. (The number of elves could range from 0 to 60, and the number of inhabitants wearing caps could also range from 0 to 60.)

Each of the 60 inhabitants said one of the following phrases:

- "My neighbor to the right is an elf."
- "My neighbor to the right is wearing a cap."

It is known that elves without caps always tell the truth, while elves with caps always lie. For dwarves, it is the opposite: dwarves without caps always lie, while dwarves with caps always tell the truth.

(a) What is the maximum number of elves without caps that could be in the line?

(b) What is the maximum number of elves with caps that could be in the line?


Given that \(a, b, c\) are non-zero rational numbers and satisfy \(a b^{2}=\frac{c}{a}-b\), then \[\left(\frac{a^{2} b^{2}}{c^{2}}-\frac{2}{c}+\frac{1}{a^{2} b^{2}}+\frac{2 a b}{c^{2}}-\frac{2}{a b c}\right) \div\left(\frac{2}{a b}-\frac{2 a b}{c}\right) \div \frac{101}{c}=\]


Given a hyperbola \( H: x^{2}-y^{2}=1 \) with a point \( M \) in the first quadrant, and a line \( l \) tangent to the hyperbola \( H \) at point \( M \), intersecting the asymptotes of \( H \) at points \( P \) and \( Q \) (where \( P \) is in the first quadrant). If point \( R \) is on the same asymptote as \( Q \), then the minimum value of \( \overrightarrow{R P} \cdot \overrightarrow{R Q} \) is ______.


There are 1000 lights and 1000 switches. Each switch simultaneously controls all lights whose numbers are multiples of the switch's number. Initially, all lights are on. Now, if switches numbered 2, 3, and 5 are pulled, how many lights will remain on?


The product of five different integers is 12. What is the largest of the integers?


On a table, there are 100 identical-looking coins, of which 85 are counterfeit and 15 are real. You have a miracle tester at your disposal, in which you can place two coins and get one of three results - "both coins are real," "both coins are counterfeit," and "the coins are different." Can you find all the counterfeit coins in 64 such tests? (K. Knop)

Solution: \boxed{Yes}


Let \( n \) be a natural number. Determine the number of subsets \( A \subset \{1, 2, \ldots, 2n\} \) such that \( x + y = 2n + 1 \) does not hold for any two elements \( x, y \in A \).


Let $a$ be a natural number. Define $M$ as the set of all integers $x$ that satisfy $|x-a| < a + \frac{1}{2}$, and $N$ as the set of all integers $x$ that satisfy $|x| < 2a$. What is the sum of all the integers belonging to $M \cap N$?


Calculate the circulation of the vector field:

a) $\vec{A}=x^{2} y^{2} \vec{i}+\vec{j}+z \vec{k}$ along the circle $x^{2}+y^{2}=a^{2}, z=0$;

b) $\dot{A}=(x-2 z) \dot{i}+(x+3 y+z) \dot{j}+(5 x+y) \vec{k}$ along the perimeter of the triangle $A B C$ with vertices $A(1,0,0), B(0,1,0), C(0,0,1)$.


Calculate the sum \(2\left[h\left(\frac{1}{2009}\right)+h\left(\frac{2}{2009}\right)+\ldots+h\left(\frac{2008}{2009}\right)\right]\), where

\[
h(t)=\frac{5}{5+25^{t}}, \quad t \in \mathbb{R}
\]


Through the points \( R \) and \( E \), located on the sides \( AB \) and \( AD \) of the parallelogram \( ABCD \), such that \( AR = \frac{2}{3} AB \) and \( AE = \frac{1}{3} AD \), a line is drawn. Find the ratio of the area of the parallelogram to the area of the resulting triangle.


The functions \( f(x) \) and \( g(x) \) are defined for all \( x > 0 \). The function \( f(x) \) is equal to the greater of the numbers \( x \) and \( \frac{1}{x} \), and \( g(x) \) is equal to the smaller of the numbers \( x \) and \( \frac{1}{x} \). Solve the equation \( f(5x) \cdot g(8x) \cdot g(25x)=1 \). In the answer, specify the solution if it is unique, or the sum of the solutions if there are several. Round the answer to the nearest hundredth if necessary.


Let \( M = \{1, 2, \ldots, 10\} \), and let \( A_1, A_2, \ldots, A_n \) be distinct non-empty subsets of \( M \). For \( i \neq j \), the intersection \( A_i \cap A_j \) contains at most two elements. Find the maximum value of \( n \).