Consider the numbers \( k \) and \( n \) such that they are relatively prime, with \( n \geq 5 \) and \( k < \frac{n}{2} \). A regular \((n ; k)\)-star is defined as a closed broken line obtained by replacing every \( k \) consecutive sides of a regular \( n \)-gon with a diagonal connecting those endpoints. For example, a \((5 ; 2)\)-star has 5 self-intersection points as illustrated by the bold points in the figure. How many self-intersections does a \((2018 ; 25)\)-star have?


Let \( S \) be the set of integers that can be written in the form \( 50m + 3n \) where \( m \) and \( n \) are non-negative integers. For example, 3, 50, and 53 are all in \( S \). Find the sum of all positive integers not in \( S \).


Let \([x]\) be the largest integer not greater than \(x\). If \(a=\left[(\sqrt{3}-\sqrt{2})^{2009}\right]+16\), find the value of \(a\).


On the board are written 5 integers. By adding them in pairs, the following set of 10 numbers was obtained: $5, 8, 9, 13, 14, 14, 15, 17, 18, 23$. Determine which numbers are written on the board. In the answer, write their product.


Let \( x \) and \( y \) be positive integers such that

\[ x^2 + y^2 - 2017xy > 0 \]

and it is not a perfect square. Find the minimum value of \( x^2 + y^2 - 2017xy \).


A sequence of positive integers \( a_{1}, a_{2}, \ldots, a_{2017} \) has the property that for all integers \( m \) where \( 1 \leq m \leq 2017 \), \( 3\left(\sum_{i=1}^{m} a_{i}\right)^{2} = \sum_{i=1}^{m} a_{i}^{3} \). Compute \( a_{1337} \).


Let \( a, b, c, d, e \) be nonnegative integers such that \( 625a + 250b + 100c + 40d + 16e = 15^3 \). What is the maximum possible value of \( a + b + c + d + e \)?


Given that \( \triangle ABC \) has a circumcenter \( O \), and \( 2 \overrightarrow{OA} + 3 \overrightarrow{OB} + 4 \overrightarrow{OC} = 0 \), find \( \cos \angle BAC \).


José wants to fill the cells of a $2 \times n$ board with zeros and ones such that two identical neighboring numbers in the same row prevent the corresponding cells of the other row from being filled with identical numbers. For example, in the following configuration, the values of $A$ and $B$ cannot be the same.

| 0 | 1 | 0 | $\cdots$ | 1 | 1 | $\cdots$ |
| :--- | :--- | :--- | :--- | :--- | :--- | :--- |
| 1 | 1 | 0 | $\cdots$ | $A$ | $B$ | $\cdots$ |

a) Find all possible fillings of the following board:

| 0 | 0 | 1 |
| :--- | :--- | :--- |
| 1 |  |  |

b) On the board, all columns are filled except for the last two. In how many ways can the numbers in cells $A$ and $B$ be chosen?

| $\cdots$ | 0 | $A$ |
| :--- | :--- | :--- |
| $\cdots$ | 1 | $B$ |

c) In how many ways can José fill the board if $n=2020$?

Solution: \boxed{4 \times 3^{2019}}


The sequence $\left\{a_{n}\right\}$ satisfies:
$$
a_{1}=2, \quad a_{n}+a_{n+1}=1 \quad (\text{where } n \in \mathbf{Z}_{+}).
$$

Let $S_{n}$ be the sum of the first $n$ terms of the sequence $\left\{a_{n}\right\}$. Then find the value of
$$
S_{2017} - 2 S_{2018} + S_{2019}.
$$


Hooligan Vasya likes to run on the escalator in the subway. He runs down twice as fast as he runs up. If the escalator is not working, it takes Vasya 6 minutes to run up and down. If the escalator is moving downward, it takes him 13.5 minutes to run up and down. How many seconds will it take Vasya to run up and down the escalator if it is moving upward? (The escalator always moves at a constant speed.)


In a chess tournament, $n$ players played a round-robin tournament. Players A and B each played two games before withdrawing from the tournament. It is known that a total of 81 games were played. Determine the value of $n$.


On the extensions of the sides $DA, AB, BC$, and $CD$ of the convex quadrilateral $ABCD$, points $A_{1}, B_{1}, C_{1}, D_{1}$ are taken such that $\overrightarrow{D A_{1}}=2 \overrightarrow{D A}$, $\overrightarrow{A B_{1}}=2 \overrightarrow{A B}$, $\overrightarrow{B C_{1}}=2 \overrightarrow{B C}$, and $\overrightarrow{C D_{1}}=2 \overrightarrow{C D}$. Find the area of the resulting quadrilateral $A_{1} B_{1} C_{1} D_{1}$ if it is known that the area of quadrilateral $ABCD$ is $S$.


Randomly choose any pair of ordered integers \((b, c)\), where the absolute value of each integer is less than or equal to 5. Each such pair of ordered integers is equally likely to be chosen. What is the probability that the equation \(x^2 + bx + c = 0\) does not have two distinct positive real roots?
A. \(\frac{106}{121}\)
B. \(\frac{108}{121}\)
C. \(\frac{110}{121}\)
D. \(\frac{112}{121}\)
E. None of these.


How many unordered pairs of positive integers that are relatively prime to each other have a sum of $285?


What is the last digit of the sum $4.1 \times 1 + 2 \times 2 + 3 \times 3 + \ldots + 2011 \times 2011 + 2012 \times 2012$?


What is the area of a regular dodecagon inscribed in a circle with radius $r$?


Doug and Ryan are competing in the 2005 Wiffle Ball Home Run Derby. In each round, each player takes a series of swings. Each swing results in either a home run or an out, and an out ends the series. When Doug swings, the probability that he will hit a home run is \( \frac{1}{3} \). When Ryan swings, the probability that he will hit a home run is \( \frac{1}{2} \). In one round, what is the probability that Doug will hit more home runs than Ryan?


There are 49 ones written on a board. Each minute, Karlson erases any two numbers and writes their sum on the board, then he eats a number of candies equal to the product of the two erased numbers. What is the maximum number of candies he could eat in 49 minutes?


Consider an ellipse \( C: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0) \) with its left focus at point \( F \). A tangent to the ellipse at a point \( A \) intersects the \( y \)-axis at point \( Q \). Given that \( \angle Q F O = 45^{\circ} \) and \( \angle Q F A = 30^{\circ} \), determine the eccentricity \( e \) of the ellipse.


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


Calculate the limit of the numerical sequence:

$$
\lim _{n \rightarrow \infty} \frac{(n+1)^{3}-(n+1)^{2}}{(n-1)^{3}-(n+1)^{3}}
$$

Solution: \boxed{-\infty}


The last two digits of the integer $\left[\frac{10^{93}}{10^{31}+3}\right]$ are $\qquad$. (Write the tens digit first, followed by the units digit, where $[x]$ denotes the greatest integer less than or equal to $x$).


Two adjacent faces of a tetrahedron, which are isosceles right triangles with a hypotenuse of 2, form a dihedral angle of 60 degrees. The tetrahedron is rotated around the common edge of these faces. Find the maximum area of the projection of the rotating tetrahedron onto the plane containing the given edge.


The sum of the positive numbers \(a, b, c,\) and \(d\) does not exceed 4. Find the maximum value of the expression

$$
\sqrt[4]{2 a^{2}+a^{2} b} + \sqrt[4]{2 b^{2}+b^{2} c} + \sqrt[4]{2 c^{2}+c^{2} d} + \sqrt[4]{2 d^{2}+d^{2} a}
$$

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


Calculate the result of the expression \(201 \times 5 + 1220 - 2 \times 3 \times 5 \times 7\).


From point \(A\) to other locations, you can travel in two ways:

1. Start walking immediately.
2. Call a car, wait for a certain period, and then drive.

In each case, the method that requires less time is used. It turns out that:

\[
\begin{tabular}{|c|c|}
\hline Final point is & Total travel time required \\
\hline \(1 \text{ km}\) & 10 minutes \\
\(2 \text{ km}\) & 15 minutes \\
\(3 \text{ km}\) & \(17 \frac{1}{2}\) minutes \\
\hline
\end{tabular}
\]

The speeds of the pedestrian and the car, as well as the waiting time for the car, remain constant. How much time is needed to travel to a point 6 km away from \(A\)?

Solution: \boxed{25 \text{ minutes}}


Let \( S = \{1, 2, 3, 4, \ldots, 100000\} \). Find the least possible value of \( k \) such that any subset \( A \) of \( S \) with \( |A| = 2010 \) contains two distinct numbers \( a \) and \( b \) with \( |a - b| \leq k \).


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

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


Centipedes and three-headed dragons. In a certain herd of centipedes and three-headed dragons, there are a total of 26 heads and 298 legs. Each centipede has one head. How many legs does a three-headed dragon have?


As shown in the figure, in rectangle $ABCD$, $\triangle AED$ and $\triangle BFC$ are both isosceles right triangles, with $EF = AD = 2$. Find the area of rectangle $ABCD$.


Find the height of a right triangle dropped to the hypotenuse if it is known that the base of this height divides the hypotenuse into segments of 1 and 4.


Nine distinct positive integers summing to 74 are put into a \(3 \times 3\) grid. Simultaneously, the number in each cell is replaced with the sum of the numbers in its adjacent cells. (Two cells are adjacent if they share an edge.) After this, exactly four of the numbers in the grid are 23. Determine, with proof, all possible numbers that could have been originally in the center of the grid.


Through the point \( P(1,1) \), draw a line \( l \) such that the midpoint of the chord intercepted by the ellipse \(\frac{x^2}{9} + \frac{y^2}{4} = 1\) on this line is precisely \( P \). Find the equation of the line \( l \).

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


Solve the inequality

$$
x^{2}+y^{2}+z^{2}+3 < xy + 3y + 2z
$$

in integers \(x\), \(y\), and \(z\).

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


A poor woman brought a basket of eggs to the market. Someone bumped into her, causing her to drop the basket and all the eggs broke. The person who caused the accident wanted to compensate her for the loss and asked about the number of eggs. The woman replied that she did not know the exact number, but she remembered that when she grouped them by 2, there was 1 left over; the same happened when she grouped them by 3, 4, 5, or 6, but when she grouped them by 7, none remained. Question: Can you determine the number of eggs from this information?


In the isosceles trapezoid \(ABCD\) with bases \(AD\) and \(BC\), perpendiculars \(BH\) and \(DK\) are drawn from vertices \(B\) and \(D\) to the diagonal \(AC\). It is known that the feet of the perpendiculars lie on the segment \(AC\) and that \(AC = 20\), \(AK = 19\), and \(AH = 3\). Find the area of the trapezoid \(ABCD\).


$n$ mushroom pickers went into the forest and brought back a total of 200 mushrooms (possibly, some of the pickers did not bring any mushrooms home). A boy named Petya, upon learning this, stated: "Some two of them must have brought the same number of mushrooms!" What is the smallest $n$ for which Petya is certainly right? Don't forget to justify your answer.


Let \( f(x) \) be a polynomial of degree 2014 such that
\[ f(k) = \frac{1}{k} \quad (k=1, 2, \ldots, 2015). \]
Find \( f(2016) \).


Calculate the limit of the numerical sequence:

$$\lim _{n \rightarrow \infty} \frac{\sqrt{n\left(n^{5}+9\right)}-\sqrt{\left(n^{4}-1\right)\left(n^{2}+5\right)}}{n}$$


Show that every perfect number (except 6) is of the form $9n + 1$.

Solution: \boxed{9n + 1}


Let \(ABC\) be a triangle with incenter \(I\) and circumcenter \(O\). Let the circumradius be \(R\). What is the least upper bound of all possible values of \(IO\)?

Solution: \boxed{R}


There exist several solutions to the equation
$$
1+\frac{\sin x}{\sin 4 x}=\frac{\sin 3 x}{\sin 2 x},
$$
where \( x \) is expressed in degrees and \( 0^{\circ} < x < 180^{\circ} \). Find the sum of all such solutions.


Given that \( a \) and \( b \) are positive real numbers and \( a + b = 2 \). If \( S = \left( a + \frac{1}{a} \right)^2 + \left( b + \frac{1}{b} \right)^2 \), find the minimum value \( S \).


Given a quadrilateral \( ABCD \) such that \( AB = BD \), \( \angle ABD = \angle DBC \), and \( \angle BCD = 90^\circ \). Point \( E \) is marked on segment \( BC \) such that \( AD = DE \). What is the length of segment \( BD \) if it is known that \( BE = 7 \) and \( EC = 5 \)?


In the equation \(x^{2} + p x + 3 = 0\), determine \(p\) such that one root of the equation is three times the other root.

Solution: \boxed{\pm 4}


Let $n$ be a given integer greater than 2. There are $n$ indistinguishable bags, and the $k$-th bag $(k= 1,2, \cdots, n)$ contains $k$ red balls and $n-k$ white balls. After mixing these bags, one bag is chosen at random, and three balls are drawn consecutively without replacement. Find the probability that the third ball drawn is white.


In trapezoid \(ABCD\) with \(BC \parallel AD\), it is known that \(AD = 3 \cdot BC\). A line intersects the non-parallel sides of the trapezoid at points \(M\) and \(N\) such that \(AM:MB = 3:5\) and \(CN:ND = 2:7\). Find the ratio of the areas of quadrilaterals \(MBCN\) and \(AMND\).


Carlinhos likes writing numbers in his notebook. One day he wrote the numbers from 1 to 999, one after the other, to form the giant number:

$$
123456789101112 \ldots 997998999
$$

Based on this number, the following questions are asked:

(a) How many digits were written?

(b) How many times does the digit 1 appear?

(c) Considering that 1 occupies position 1, 2 occupies position 2, and 0 appears for the first time occupying position 11, which digit occupies position 2016?


Given that $\lg 2 = a$ and $\log_{2} 7 = b$, find $\lg 56$.


Juliana wants to assign each of the 26 letters $A, B, C, D, \ldots, W, X, Y, Z$ of the alphabet a nonzero numerical value, such that $A \times C = B, B \times D = C, C \times E = D$, and so on, up to $X \times Z = Y$.

a) If Juliana assigns the values 5 and 7 to $A$ and $B$ respectively, what will be the values of $C, D$, and $E$?

b) Show that $G = A$, regardless of the values Juliana assigns to $A$ and $B$.

c) If Juliana assigns values to $A$ and $B$ such that $A \times B = 2010$, what will be the value of the product $A \times B \times C \times D \times \cdots \times W \times X \times Y \times Z$?


On the sides \(AB\) and \(AC\) of an isosceles triangle \(ABC\) (\(AB = AC\)), points \(M\) and \(N\) are marked respectively such that \(AN > AM\). The lines \(MN\) and \(BC\) intersect at point \(K\). Compare the lengths of segments \(MK\) and \(MB\).

Solution: \boxed{MK > MB}


Three distinct diameters are drawn on a unit circle such that chords are drawn as shown. If the length of one chord is \(\sqrt{2}\) units and the other two chords are of equal lengths, what is the common length of these chords?


Construct a square \(A B C D\) with side length \(6 \text{ cm}\). Construct a line \(p\) parallel to the diagonal \(A C\) passing through point \(D\). Construct a rectangle \(A C E F\) such that vertices \(E\) and \(F\) lie on the line \(p\).

Using the given information, calculate the area of rectangle \(A C E F\).

Solution: \boxed{36 \text{ cm}^2}


In the trapezoid $ABCD$, the sides $AB$ and $CD$ are parallel. Let $M$ be the midpoint of the diagonal $AC$, $N$ the midpoint of the diagonal $BD$, and $P$ the midpoint of the side $AB$. Given that $AB = 15 \text{ cm}$, $CD = 24 \text{ cm}$, and the height of the trapezoid is $h = 14 \text{ cm}$:

a) Calculate the length of the segment $MN$.

b) Calculate the area of the triangle $MNP$.

Solution: \boxed{15.75 \text{ cm}^2}


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


Tanya sequentially wrote down numbers of the form \( n^{7} - 1 \) for natural numbers \( n = 2, 3, \ldots \) and noticed that for \( n = 8 \), the resulting number is divisible by 337. For what smallest \( n > 1 \) will she get a number divisible by 2022?


What is the minimum value that the product of two positive numbers \(a\) and \(b\) can take if it is known that \(ab = a + b\)?


If \( a > b > c > d \), and \( \frac{1}{a-b} + \frac{1}{h-c} + \frac{1}{c-d} \geq \frac{n}{a-d} \), then the maximum value of the integer \( n \) is ___


Let \( m \) and \( n \) be positive integers satisfying
\[ m n^{2} + 876 = 4 m n + 217 n. \]

Find the sum of all possible values of \( m \).


Find the number of natural numbers \( k \) not exceeding 267000 such that \( k^{2} -1 \) is divisible by 267.


Vasya cut a triangle out of cardboard and numbered its vertices with the digits $1, 2, 3$. It turned out that if Vasya rotates the triangle 12 times clockwise around its vertex numbered 1 by an angle equal to the angle at this vertex, it will return to its original position. If Vasya rotates the triangle 6 times clockwise around its vertex numbered 2 by an angle equal to the angle at this vertex, it will return to its original position. Vasya claims that if the triangle is rotated $n$ times around its vertex numbered 3 by an angle equal to the angle at this vertex, it will return to its original position. What is the minimum $n$ that Vasya might name so that his claim is true for at least some cardboard triangle?


Find the cosine of the angle between the vectors $\overrightarrow{AB}$ and $\overrightarrow{AC}$.

$A(-1, -2, 1), B(-4, -2, 5), C(-8, -2, 2)$


Angie has a jar that contains 2 red marbles, 2 blue marbles, and no other marbles. She randomly draws 2 marbles from the jar. If the marbles are the same color, she discards one and puts the other back into the jar. If the marbles are different colors, she discards the red marble and puts the blue marble back into the jar. She repeats this process a total of three times. What is the probability that the remaining marble is red?
(A) $\frac{1}{2}$
(B) $\frac{1}{4}$
(C) $\frac{2}{3}$
(D) $\frac{1}{3}$
(E) 0


Let \( u, v, w \) be positive real numbers, all different from 1. If
\[
\log_{u}(vw) + \log_{v}(w) = 5 \quad \text{and} \quad \log_{v}(u) + \log_{w}(v) = 3,
\]
find the value of \( \log_{w}(u) \).


Plot the set of points on the $(x, y)$ plane that satisfy the equation $|4x| + |3y| + |24 - 4x - 3y| = 24$ and find the area of the resulting figure.


How many ways are there to insert plus signs (+) between the digits of 1111111111111111 (fifteen 1's) so that the result will be a multiple of 30?


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


How many four-digit numbers contain the digit 9 followed immediately by the digit 5?


There are four piles of stones: one with 6 stones, two with 8 stones, and one with 9 stones. Five players numbered 1, 2, 3, 4, and 5 take turns, in the order of their numbers, choosing one of the piles and dividing it into two smaller piles. The loser is the player who cannot do this. State the number of the player who loses.


There are 36 students in a club. If any 33 of them attend a session, girls will always be in the majority. However, if 31 students attend, it might happen that boys are in the majority. How many girls are in the club?


Extract the square root of the polynomial:

$$
16ac + 4a^2 - 12ab + 9b^2 - 24bc + 16c^2
$$

Solution: \boxed{2a - 3b + 4c}


Through the origin, lines (including the coordinate axes) are drawn that divide the coordinate plane into angles of $1^{\circ}$. Find the sum of the x-coordinates of the points of intersection of these lines with the line $y = 100 - x$.


A school program will randomly start between 8:30AM and 9:30AM and will randomly end between 7:00PM and 9:00PM. What is the probability that the program lasts for at least 11 hours and starts before 9:00AM?


Eight singers sing songs at a festival. Each song is sung once by a group of four singers. Every pair of singers sings the same number of songs together. Find the smallest possible number of songs.


Find the largest real number \( x \) such that \( \sqrt[3]{x} + \sqrt[3]{4 - x} = 1 \).

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


The length of the road is 300 km. Car $A$ starts at one end of the road at noon and moves at a constant speed of 50 km/h. At the same time, car $B$ starts at the other end of the road with a constant speed of 100 km/h, and a fly starts with a speed of 150 km/h. Upon meeting car $A$, the fly turns around and heads towards car $B$.

1) When does the fly meet car $B$?
2) If upon meeting car $B$, the fly turns around, heads towards car $A$, meets it, turns around again, and continues flying back and forth between $A$ and $B$ until they collide, when do the cars crush the fly?

Solution: \boxed{2 \ \text{hours}}


On a $3 \times 3$ grid, there are knights who always tell the truth and liars who always lie. Each one stated: "Among my neighbors, exactly three are liars." How many liars are on the grid?

Neighbors are considered to be people located on cells that share a common side.


Given a $5 \times 5$ grid where the number in the $i$-th row and $j$-th column is denoted by \( a_{ij} \) (where \( a_{ij} \in \{0, 1\} \)), with the condition that \( a_{ij} = a_{ji} \) for \( 1 \leq i, j \leq 5 \). Calculate the total number of ways to fill the grid such that there are exactly five 1's in the grid.


The cube $A B C D-A_{1} B_{1} C_{1} D_{1}$ has an edge length of 1. What is the distance between the lines $A_{1} C_{1}$ and $B D_{1}$?


The list \( 11, 20, 31, 51, 82 \) is an example of an increasing list of five positive integers in which the first and second integers add to the third, the second and third add to the fourth, and the third and fourth add to the fifth. How many such lists of five positive integers have 124 as the fifth integer?
\begin{center}
(A) 10 \\
(B) 7 \\
(C) 9 \\
(D) 6 \\
(E) 8
\end{center}


If \( a + b + c = 1 \), what is the maximum value of \( \sqrt{3a+1} + \sqrt{3b+1} + \sqrt{3c+1} \)?


A circle is circumscribed around an isosceles triangle with a $45^\circ$ angle at the vertex. Another circle is inscribed within this triangle, touching the first circle internally and also touching the two equal sides of the triangle. The distance from the center of the second circle to this vertex of the triangle is 4 cm. Find the distance from the center of this second circle to the center of the circle inscribed in the triangle.

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


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


Using small cubes with edge length \( m \) to form a larger cube with an edge length of 12, the surface of the larger cube (6 faces) is painted red. The number of small cubes with only one face painted red is equal to the number of small cubes with two faces painted red. Find \( m \).


Professor Fernando's tree grows according to the following rule:

- In the first week, the tree starts to grow with only one branch;
- After growing for two weeks, this branch produces a new branch every week;
- Each new branch continues to grow and, after growing for two weeks, produces a new branch every week.

The figure below illustrates Professor Fernando's tree after five weeks:

(Note: The figure is omitted here)

Observe that after three weeks, there were two branches; after four weeks, there were three branches; and after five weeks, there were five branches.

a) How many branches will there be after six weeks?

b) How many branches will there be after seven weeks?

c) How many branches will there be after thirteen weeks?


Alice and Bob play on a $20 \times 20$ grid. Initially, all the cells are empty. Alice starts and the two players take turns placing stones on unoccupied cells. On her turn, Alice places a red stone on an empty cell that is not at a distance of $\sqrt{5}$ from any other cell containing a red stone. On his turn, Bob places a blue stone on an unoccupied cell. The game ends when a player can no longer place a stone. Determine the largest $K$ such that Alice can ensure to place at least $K$ red stones regardless of how Bob places his stones.


At a chamber music festival, six musicians have gathered. At each concert, some of the musicians perform while the others listen from the audience. What is the minimum number of concerts needed for each of the six musicians to listen (from the audience) to all the others?


Calculate the limit of the function:

$$\lim _{x \rightarrow 0} \frac{\sqrt{\cos x}-1}{\sin ^{2} 2 x}$$


As shown in the figure, a rectangle is divided into four smaller rectangles. The perimeters of rectangles $A$, $B$, and $C$ are 10 cm, 12 cm, and 14 cm, respectively. What is the maximum possible area of rectangle $D$ in square centimeters?

Solution: \boxed{16 \ \text{cm}^2}


Dragomir has 6 pairs of socks in a drawer. Each pair of socks is different from every other pair in the drawer. He randomly removes 4 individual socks from the drawer, one at a time. What is the probability that there is exactly 1 matching pair of socks among these 4 socks?


Tetrahedron \(ABCD\) has base \( \triangle ABC \). Point \( E \) is the midpoint of \( AB \). Point \( F \) is on \( AD \) so that \( FD = 2AF \), point \( G \) is on \( BD \) so that \( GD = 2BG \), and point \( H \) is on \( CD \) so that \( HD = 2CH \). Point \( M \) is the midpoint of \( FG \) and point \( P \) is the point of intersection of the line segments \( EH \) and \( CM \). What is the ratio of the volume of tetrahedron \( EBCP \) to the volume of tetrahedron \( ABCD \)?


Given an isosceles triangle \( ABC \) with \( AB = BC \) and the vertex angle equal to \( 102^\circ \). A point \( O \) is located inside the triangle \( ABC \) such that \( \angle OCA = 30^\circ \) and \( \angle OAC = 21^\circ \). Find the measure of angle \( \angle BOA \).


In a shooting competition, there are 8 clay targets arranged in 3 columns as shown in the diagram. A sharpshooter follows these rules to shoot down all the targets:
1. First, select a column from which one target will be shot.
2. Then, target the lowest remaining target in the selected column.

How many different sequences are there to shoot down all 8 targets?


The sequence \(\{a_n\}\) has consecutive terms \(a_n\) and \(a_{n+1}\) as the roots of the equation \(x^2 - c_n x + \left(\frac{1}{3}\right)^n = 0\), with initial term \(a_1 = 2\). Find the sum of the infinite series \(c_1, c_2, \cdots, c_n, \cdots\).


Grandpa prepared a pile of hazelnuts for his six grandchildren and told them to take some. First, Adam came, counted out half of the hazelnuts, took one extra hazelnut, and left. The second, Bob did the same, as did the third, Cyril, fourth, Dan, and fifth, Eda. Franta was left looking sadly at the empty table; there were no hazelnuts left for him.

How many hazelnuts were originally in the pile?

Hint: How many hazelnuts did Eda take?


A square has a tens digit of 7. What is the units digit?


The diagram below shows \( \triangle ABC \), which is isosceles with \( AB = AC \) and \( \angle A = 20^\circ \). The point \( D \) lies on \( AC \) such that \( AD = BC \). The segment \( BD \) is constructed as shown. Determine \( \angle ABD \) in degrees.


Find a point that is at distances $m$, $n$, and $p$ from three planes. How many solutions does this problem have?


In triangle \(ABC\), the sides opposite to angles \(A, B,\) and \(C\) are denoted by \(a, b,\) and \(c\) respectively. Given that \(c = 10\) and \(\frac{\cos A}{\cos B} = \frac{b}{a} = \frac{4}{3}\). Point \(P\) is a moving point on the incircle of triangle \(ABC\), and \(d\) is the sum of the squares of the distances from \(P\) to vertices \(A, B,\) and \(C\). Find \(d_{\min} + d_{\max}\).


In triangle \( \triangle ABC \), it is known that
\[ \cos C = \frac{2 \sqrt{5}}{5} \]
\[ \overrightarrow{A H} \cdot \overrightarrow{BC} = 0 \]
\[ \overrightarrow{AB} \cdot (\overrightarrow{CA} + \overrightarrow{CB}) = 0 \]

Determine the eccentricity of the hyperbola passing through point \( C \) with foci at points \( A \) and \( H \).

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


Inside the tetrahedron \(ABCD\), there is a point \(O\) such that the lines \(AO, BO, CO,\) and \(DO\) intersect the faces \(BCD, ACD, ABD,\) and \(ABC\) at the points \(A_{1}, B_{1}, C_{1},\) and \(D_{1}\) respectively. Given that \(\frac{AO}{OA_{1}}=\frac{BO}{OB_{1}}=\frac{CO}{OC_{1}}=\frac{DO}{OD_{1}}=k\), find the value of \(k\).


Find the number of pairs of integers $(a, b)$ such that $1 \leq a \leq 70, 1 \leq b \leq 50$, and the area $S$ of the figure defined by the system of inequalities

$$
\left\{\begin{array}{l}
\frac{x}{a}+\frac{y}{b} \geq 1 \\
x \leq a \\
y \leq b
\end{array}\right.
$$

is such that the number $2S$ is a multiple of 5.


Queenie and Horst play a game on a $20 \times 20$ chessboard. In the beginning the board is empty. In every turn, Horst places a black knight on an empty square in such a way that his new knight does not attack any previous knights. Then Queenie places a white queen on an empty square. The game gets finished when somebody cannot move.

Find the maximal positive $K$ such that, regardless of the strategy of Queenie, Horst can put at least $K$ knights on the board.


Find the sum of the first 10 elements that appear both in the arithmetic progression $\{5, 8, 11, 14, \ldots\}$ and in the geometric progression $\{20, 40, 80, 160, \ldots\}$.


Given that \( a b c = 8 \) with \( a, b, c \) being positive, what is the minimum value of \( a + b + c \)?


Without solving the equation \(3x^{2} - 5x - 2 = 0\), find the sum of the cubes of its roots.


Mr. Li goes to a toy store to buy balls. The money he has can buy exactly 60 plastic balls, or exactly 36 glass balls, or 45 wooden balls. Mr. Li decides to buy 10 plastic balls and 10 glass balls, and spends the remaining money on wooden balls. How many balls does Mr. Li buy in total?


Compute

$$
\int_{L} \frac{60 e^{z}}{z(z+3)(z+4)(z+5)} d z
$$

where \( L \) is a unit circle centered at the origin.

Solution: \boxed{2 \pi i}


On a line, 5 points \( P, Q, R, S, T \) are marked in that order. It is known that the sum of the distances from \( P \) to the other 4 points is 67, and the sum of the distances from \( Q \) to the other 4 points is 34. Find the length of segment \( PQ \).


Find the length of the arc of a cycloid defined by \( x = a(t - \sin t) \) and \( y = a(1 - \cos t) \).


$$
n < p_{Z_{k}^{k}}^{k} \text{,} $$
where $p_{2 k}$ denotes the $2k$-th prime number.

Solution: \boxed{n < p_{2k}^{k}}


Recall that the sum of the angles of a triangle is 180 degrees. In triangle $ABC$, angle $A$ is a right angle. Let $BM$ be the median of the triangle and $D$ be the midpoint of $BM$. It turns out that $\angle ABD = \angle ACD$. What are the measures of these angles?


Let \( n \) be a given positive integer. Find the smallest positive integer \( u_n \) such that, for every positive integer \( d \), the number of numbers divisible by \( d \) in any \( u_n \) consecutive positive odd numbers is no less than the number of numbers divisible by \( d \) in the set of odd numbers \( 1, 3, 5, \ldots, 2n-1 \).

Solution: \boxed{2n - 1}


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


Given a grid of size $m \times n$ (with $m, n > 1$). The centers of all the cells are marked. What is the maximum number of marked centers that can be chosen such that no three of them form the vertices of a right triangle?

Solution: \boxed{m + n - 2}


Find the value of \( n \) for which the following equality holds:

$$
\frac{1}{1+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{4}}+\ldots+\frac{1}{\sqrt{n}+\sqrt{n+1}}=2014
$$


Calculate the volumes of bodies bounded by the surfaces.

$$
\frac{x^{2}}{27}+y^{2}=1, z=\frac{y}{\sqrt{3}}, z=0(y \geq 0)
$$


If we can always select 4 numbers from any set of $\mathrm{n}$ numbers such that their sum is a multiple of 4, what is the smallest value of $n$?


A car license plate contains three letters and three digits, for example, A123BE. The allowable letters are А, В, Е, К, М, Н, О, Р, С, Т, У, Х (a total of 12 letters) and all digits except the combination 000. Katya considers a plate number lucky if the second letter is a consonant, the first digit is odd, and the third digit is even (there are no restrictions on the other characters). How many license plates does Katya consider lucky?


A student uses a calculator to find an answer but instead of pressing the \( x^2 \) key presses the \( \sqrt{x} \) key by mistake. The student's answer was 9. What should the answer have been?
(A) 243
(B) 81
(C) 729
(D) 3
(E) 6561


In the rhombus \(ABCD\), point \(M\) is chosen on side \(BC\). Perpendiculars are drawn from \(M\) to the diagonals \(BD\) and \(AC\), intersecting line \(AD\) at points \(P\) and \(Q\) respectively. It is known that lines \(PB\), \(QC\), and \(AM\) intersect at a single point. What is the ratio \(\frac{BM}{MC}\)?


Three candles can burn for 30, 40, and 50 minutes respectively (but they are not lit simultaneously). It is known that the three candles are burning simultaneously for 10 minutes, and only one candle is burning for 20 minutes. How many minutes are there when exactly two candles are burning simultaneously?

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


The secret object is a rectangle measuring $200 \times 300$ meters. Outside the object, there is one guard stationed at each of its four corners. An intruder approached the perimeter of the secret object from the outside, and all the guards ran towards the intruder using the shortest paths along the outer perimeter (the intruder remained stationary). Three guards collectively ran a total of 850 meters to reach the intruder. How many meters did the fourth guard run to reach the intruder?

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


On the clock tower at the train station, there is an electronic clock. Along the boundary of the circular clock face, there are small colorful lights at each minute mark. At 9:35:20 PM, there are how many small colorful lights in the acute angle formed by the minute hand and the hour hand?


Calculate the integral $\int_{1}^{4} x^{2} \, dx$ using Simpson's Rule.


There are 10 consecutive natural numbers written on the board. What is the maximum number of them that can have a digit sum equal to a perfect square?


Vanya received three sets of candies for New Year. Each set contains three types of candies: hard candies, chocolates, and gummy candies. The total number of hard candies in all three sets is equal to the total number of chocolates in all three sets, and also to the total number of gummy candies in all three sets. In the first set, there are equal numbers of chocolates and gummy candies, and 7 more hard candies than chocolates. In the second set, there are equal numbers of hard candies and chocolates, and 15 fewer gummy candies than hard candies. How many candies are in the third set if it is known that there are no hard candies in it?


Given a circle, inscribe an isosceles triangle with an acute angle. Above one of the legs of this triangle, construct another isosceles triangle whose apex is also on the circumference of the circle. Continue this process and determine the limit of the angles at the vertices of the triangles.


From head to tail of the zebra Hippotigris, there are 360 stripes of equal width. Fleas Masha and Dasha started crawling from the head to the tail of the zebra. At the same time, flea Sasha started crawling from the tail to the head. Flea Dasha crawls twice as fast as flea Masha. Before meeting flea Sasha, Masha covered exactly 180 stripes. How many stripes will Dasha cover before meeting flea Sasha?

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


The microcalculator MK-97 can perform only three operations on numbers stored in memory:

1) Check if two chosen numbers are equal,
2) Add the chosen numbers,
3) For chosen numbers \( a \) and \( b \), find the roots of the equation \( x^2 + ax + b = 0 \), and if there are no roots, display a message indicating so.

All results of actions are stored in memory. Initially, one number \( x \) is stored in the memory. Using the MK-97, how can you determine if this number is equal to one?

Solution: \boxed{x = 1}


The distance between the non-intersecting diagonals of two adjacent lateral faces of a cube is \( d \). Determine the total surface area of the cube.


Compute the definite integral:

$$
\int_{\pi / 4}^{\arccos (1 / \sqrt{26})} \frac{d x}{(6 - \tan x) \sin 2 x}
$$

Solution: \boxed{\frac{\ln 5}{6}}


Let \( a_{1}, a_{2}, \cdots, a_{21} \) be a permutation of \( 1, 2, \cdots, 21 \) that satisfies
\[ \left|a_{20} - a_{21}\right| \geq \left|a_{19} - a_{21}\right| \geq \left|a_{18} - a_{21}\right| \geq \cdots \geq \left|a_{1} - a_{21}\right|. \]
Determine the number of such permutations.


Inside the cube \(A B C D A_{1} B_{1} C_{1} D_{1}\) there is a center \(O\) of a sphere with radius 10. The sphere intersects the face \(A A_{1} D_{1} D\) creating a circle with radius 1, the face \(A_{1} B_{1} C_{1} D_{1}\) creating a circle with radius 1, and the face \(C D D_{1} C_{1}\) creating a circle with radius 3. Find the length of the segment \(O D_{1}\).


Circle \(\Omega\) has radius 13. Circle \(\omega\) has radius 14 and its center \(P\) lies on the boundary of circle \(\Omega\). Points \(A\) and \(B\) lie on \(\Omega\) such that chord \(AB\) has length 24 and is tangent to \(\omega\) at point \(T\). Find \(AT \cdot BT\).


If you flip a fair coin 1000 times, what is the expected value of the product of the number of heads and the number of tails?


Solve the following system of equations in the set of real numbers:

$$
\begin{gathered}
\log _{3} a+\log _{3} b+\log _{3} c=0 \\
3^{3^{a}}+3^{3^{b}}+3^{3^{c}}=81
\end{gathered}
$$

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


Analyzing natural numbers with 4 digits:

a) How many of them have all different digits?

b) How many have the digit 1 exactly once and all other digits are different?

c) How many have the digit 1?


The rational numbers \( x \) and \( y \), when written in lowest terms, have denominators 60 and 70, respectively. What is the smallest possible denominator of \( x + y \)?


Given two distinct numbers \(a\) and \(b\) such that \(\frac{a}{b} + a = \frac{b}{a} + b\), find \(\frac{1}{a} + \frac{1}{b}\).


The altitude drawn to the base of an isosceles triangle is $h$ and is twice its projection onto one of the legs. Find the area of the triangle.

Solution: \boxed{h^2 \sqrt{3}}


How can you cut a 5 × 5 square with straight lines so that the resulting pieces can be assembled into 50 equal squares? It is not allowed to leave unused pieces or to overlap them.


Write the equation of a plane passing through point $A$ and perpendicular to the vector $\overrightarrow{B C}$.

$A(2, 1, 7)$

$B(9, 0, 2)$

$C(9, 2, 3)$

Solution: \boxed{2y + z - 9 = 0}


On an island, there are only knights, who always tell the truth, and liars, who always lie. There are at least two knights and at least two liars. One day, each islander pointed to each of the others in turn and said either "You are a knight!" or "You are a liar!". The phrase "You are a liar!" was said exactly 230 times. How many times was the phrase "You are a knight!" said?


Let \( x \) and \( y \) be two non-zero numbers such that \( x^2 + xy + y^2 = 0 \) (where \( x \) and \( y \) are complex numbers, but that is not too important). Find the value of

\[ \left( \frac{x}{x+y} \right)^{2013} + \left( \frac{y}{x+y} \right)^{2013} \]


Each of the 12 knights sitting around a round table has chosen a number, and all the numbers are different. Each knight claims that the number they have chosen is greater than the numbers chosen by their neighbors on the right and left. What is the maximum number of these claims that can be true?


Determine the area of a triangle given its three sides:
1) \( a=13, b=14, c=15 \)
2) \( a=5, b=12, c=13 \)


The positive integers \( x, y \) for which \( \text{GCD}(x, y) = 5 \) are the coordinates of a vertex of a square centered at the origin with an area of \( \frac{169}{6} \cdot \text{LCM}(x, y) \). Find the side length of the square.


Solution: \boxed{(20, -22, 23)}


In the figure below, the numbers labeled at the 8 vertices are: \( a, b, c, d, e, f, g, h \). Each of these numbers is equal to one-third of the sum of the numbers at the three adjacent vertices. Find the value of \( (a + b + c + d) - (e + f + g + h) \).


Let \( s(n) \) denote the sum of the digits of a natural number \( n \). How many zeros are at the end of the number equal to the product \( s(1) \cdot s(2) \cdot \ldots \cdot s(100) \)?


Find $\lim _{x \rightarrow 0} \frac{x \sin 6 x}{(\operatorname{arctg} 2 x)^{2}}$.


Find, with proof, the largest possible value of

$$
\frac{x_{1}^{2}+\cdots+x_{n}^{2}}{n}
$$

where real numbers \(x_{1}, \ldots, x_{n} \geq -1\) are satisfying \(x_{1}^{3}+\cdots+x_{n}^{3}=0\).


Find all pairs of primes \((p, q)\) for which \(p - q\) and \(pq - q\) are both perfect squares.

Solution: \boxed{(3, 2)}


A triangular pyramid \( S-ABC \) has a base that is an equilateral triangle with a side length of 4. It is given that \( AS = BS = \sqrt{19} \) and \( CS = 3 \). Find the surface area of the circumscribed sphere of the triangular pyramid \( S-ABC \).


Let the sets be defined as follows:
$$
\begin{array}{l}
S=\left\{x \mid x^{2}-7x-30 < 0\right\}, \\
T=\left\{x \mid e^{x} > 1 - x, x \in \mathbf{Z}\right\}.
\end{array}
$$

Determine the number of non-empty proper subsets of \( S \cap T \).


There are two positive integers, one of which is a square number. If the sum of the two numbers is 2006 smaller than their product, find the difference between the two numbers.


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


Given that the sequence $\left\{a_{n}\right\}$ consists of positive terms, and for any positive integers $p$ and $q$, the property $a_{p+q} = a_{p} \cdot a_{q}$ holds, and it is also given that $a_{8} = 16$. What is $a_{10}$?


Let's call a point inside a triangle a good point if the three cevians passing through it are equal. In triangle \(ABC\), sides \(AB\) and \(BC\) are equal, and the number of good points is odd. What could it be equal to?


A calculator has a display, which shows a nonnegative integer \(N\), and a button which replaces \(N\) by a random integer chosen uniformly from the set \(\{0,1, \ldots, N-1\}\), provided that \(N>0\). Initially, the display holds the number \(N=2003\). If the button is pressed repeatedly until \(N=0\), what is the probability that the numbers 1, 10, 100, and 1000 will each show up on the display at some point?


The figure below shows a triangle \( \mathrm{ABC} \) where \( \mathrm{AB} = \mathrm{AC} \). \( \mathrm{D} \) and \( \mathrm{E} \) are points on sides \( AB \) and \( AC \), respectively, such that \( AB = 4DB \) and \( AC = 4AE \). If the area of the quadrilateral \( BCED \) is \( 52 \, \mathrm{cm}^{2} \) and the area of the triangle \( ADE \) is \( x \, \mathrm{cm}^{2} \), find \( x \).


For any two numbers \( x \) and \( y \), a new operation \(\otimes\) is defined as follows: \( x \otimes y = \frac{x \times y}{x + y \div 3} \). Calculate \( 0.36 \cdot \left(4 \otimes 1 \frac{1}{2}\right) \).


There are 4 people in the family. If Masha's scholarship is doubled, the total family income will increase by 5%. If the mother's salary is doubled instead, the total family income will increase by 15%. If the father's salary is doubled, the total family income will increase by 25%. By how many percent will the family's income increase if the grandfather's pension is doubled?


A car and a minivan drive from Alphaville to Betatown. The car travels at a constant speed of $40 \mathrm{~km} / \mathrm{h}$ and the minivan travels at a constant speed of $50 \mathrm{~km} / \mathrm{h}$. The minivan passes the car 10 minutes before the car arrives at Betatown. How many minutes pass between the time at which the minivan arrives in Betatown and the time at which the car arrives in Betatown?

Solution: \boxed{2 \text{ minutes}}


Solve the following system of inequalities:

$$
\left\{\begin{array}{l}
4x^{2} + 4xy + 19y^{2} \leq 2 \\
x - y \leq -1
\end{array}\right.
$$

Solution: \boxed{\text{No solution}}


The natural numbers 22, 23, and 24 have the following property: the exponents of the prime factors in their factorization are all odd:

$$
22=2^{1} \cdot 11^{1} ; \quad 23=23^{1} ; \quad 24=2^{3} \cdot 3^{1}
$$

What is the largest number of consecutive natural numbers that can have this property? Explain the reasoning for your answer.


Solve the equation \( n + S(n) = 1981 \), where \( S(n) \) denotes the sum of the digits of the natural number \( n \).


Let \( f(x) \) be a function satisfying the following conditions:
1. If \( x > y \) and \( f(x) + x \geq w \geq f(y) + y \), then there exists a real number \( z \in [y, x] \) such that \( f(z) = w - z \);
2. The equation \( f(x) = 0 \) has at least one solution, and among the solutions of this equation, there is one that is not greater than all other solutions;
3. \( f(0) = 1 \);
4. \( f(-2007) \leq 2008 \);
5. \( f(x)f(y) = f[x f(y) + y f(x) + x y] \).

Find the value of \( f(-2007) \).

Solution: \boxed{f(-2007) = 2008}


In the right triangle \(PQR\), a circle is constructed on the leg \(PR\) as its diameter, which intersects the hypotenuse \(PQ\) at point \(T\). A tangent to the circle is drawn through point \(T\), and it intersects the leg \(RQ\) at point \(S\). Find the length of \(SQ\) if \(PT = 15\) and \(QT = 5\).


The five books "Poetry," "Documents," "Rites," "Changes," and "Spring and Autumn" all have different numbers of pages. The differences in the number of pages between the books are as follows:
1. "Poetry" and "Documents" differ by 24 pages.
2. "Documents" and "Rites" differ by 17 pages.
3. "Rites" and "Changes" differ by 27 pages.
4. "Changes" and "Spring and Autumn" differ by 19 pages.
5. "Spring and Autumn" and "Poetry" differ by 15 pages.

Determine the difference in the number of pages between the book with the most pages and the book with the least pages.


a) Find the number of natural divisors of \( N = 1 \underbrace{00 \ldots 0}_{999} \) that are not perfect squares (i.e., not squares of natural numbers).

b) Find the number of natural divisors of \( N = 1 \underbrace{00 \ldots 0}_{999} \) that are neither perfect squares nor perfect cubes.


The triangle ABC satisfies \(0 \leq AB \leq 1 \leq BC \leq 2 \leq CA \leq 3\). What is the maximum area it can have?


The perimeter of triangle \( ABC \) is \( 2p \). Points \( M \) and \( N \) are taken on sides \( AB \) and \( AC \) respectively such that \( MN \parallel BC \) and \( MN \) touches the incircle of triangle \( ABC \). Find the maximum length of segment \( MN \).


In a school, there are 300 boys and 300 girls, divided into 5 classes, each with the same number of students. It is known that there are at least 33 boys and 33 girls in each class. A boy and a girl from the same class may form a group to enter a contest, and each student may only belong to one group. What is the maximum number of groups that can be guaranteed to form?


Assume that all angles of a triangle \(ABC\) are acute. Let \(D\) and \(E\) be points on the sides \(AC\) and \(BC\) of the triangle such that \(A, B, D,\) and \(E\) lie on the same circle. Further suppose the circle through \(D, E,\) and \(C\) intersects the side \(AB\) in two points \(X\) and \(Y\). Show that the midpoint of \(XY\) is the foot of the altitude from \(C\) to \(AB\).

Solution: \boxed{M = H}


Find the number of bijections \( f \) with the following properties: (i) \( f \) maps the set \( A = \{1, 2, 3, 4, 5\} \) to itself; (ii) Exactly 2 elements in \( A \) are fixed points (i.e., their images under \( f \) are themselves).


What is the smallest number of points that need to be marked inside a convex \( n \)-gon so that every triangle with vertices at the vertices of the \( n \)-gon contains at least one marked point?


Anton ran down a moving escalator and counted 30 steps. Then he decided to run up the same escalator at the same speed relative to the escalator and counted 150 steps. How many steps did he count while descending with a policeman on a stationary escalator?


Two circles with radii 4 and 3 touch each other externally. Common external tangents \(PQ\) and \(RS\) are drawn to these circles such that points \(P\) and \(S\) belong to the circle with the larger radius, and points \(Q\) and \(R\) belong to the circle with the smaller radius. Find the radius of the circle that is tangent to the segments \(RS\), \(SP\), and \(PQ\).


In triangle \(ABC\), the altitude \(BM\), the angle bisector \(BN\), and the median \(BL\) are drawn. It is known that \(AM = MN = NL\). Find the tangent of angle \(A\) of this triangle.


Let \( x \) and \( y \) be non-negative integers such that \( 69x + 54y \leq 2008 \). Find the greatest possible value of \( xy \).


Does there exist 7 distinct lines in a plane such that among their intersection points, there are at least 6 points where exactly 3 lines intersect, and at least 4 points where exactly 2 lines intersect?

Solution: \boxed{\text{No}}


Let \( M \) and \( N \) be two points on the Thales' circle of segment \( AB \), distinct from \( A \) and \( B \). Let \( C \) be the midpoint of segment \( NA \), and \( D \) be the midpoint of segment \( NB \). The circle is intersected at the point \( E \) a second time by the line \( MC \), and at point \( F \) by the line \( MD \). What is the value of the expression

\[
MC \cdot CE + MD \cdot DF
\]

if \( AB = 2 \) units?


On the bank of a river 100 meters wide, there is an electric plant situated. On the opposite bank, and 500 meters upstream, a factory is being constructed. Given that the river is straight between the plant and the factory, the cost of laying cables along the bank is 9 € per meter, and the cost of laying cables underwater is 15 € per meter, what is the length of the most economical cable route between the electric plant and the factory?

Solution: \boxed{550 \text{ meters}}


Find the limits of the following sequences:

a) \(\left\{a_{n}\right\}=\left\{\left(\frac{3+4 i}{7}\right)^{n}, n \in \mathbb{N}\right\}\);

b) \(\left\{a_{n}\right\}=\left\{\frac{(1+i)^{n}}{2+(1+i)^{n}}, \quad n \in \mathbb{N}\right\}\);

c) \(\left\{a_{n}\right\}=\left\{\left|\frac{z^{n}}{z^{n}}\right| \frac{(\cos i-i \sin i)^{n}}{(\cos i+i \sin i)^{n}}-2 \operatorname{ch} 2 n\right\}, z \neq 0+0 i\).


Find all integer triplets $(x, y, z)$ such that $x^{2}+y^{2}+z^{2}-2xyz=0$.

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


Find the limit of the sequence
$$
\lim _{n \rightarrow \infty} \frac{3 n^{5}-6 n^{4}+3 n+5}{(n+1)^{6}-(n-2)^{6}}
$$


Three concentric circles have radii of 1, 2, and 3 units, respectively. Points are chosen on each of these circles such that they are the vertices of an equilateral triangle. What can be the side length of this equilateral triangle?


In the figure below, \( \triangle ABC \) is a triangle with \( AB = 10 \text{ cm} \) and \( BC = 40 \text{ cm} \). Points \( D \) and \( E \) lie on side \( AC \) and point \( F \) on side \( BC \) such that \( EF \) is parallel to \( AB \) and \( DF \) is parallel to \( EB \). Given that \( BE \) is an angle bisector of \( \angle ABC \) and that \( AD = 13.5 \text{ cm} \), find the length of \( CD \) in cm.


A point is randomly thrown onto the segment [12, 17] and let \( k \) be the resulting value. Find the probability that the roots of the equation \((k^{2} + k - 90)x^{2} + (3k - 8)x + 2 = 0\) satisfy the condition \( x_{1} \leq 2x_{2} \).


A rectangle \(ADEC\) is circumscribed around a right triangle \(ABC\) with legs \(AB = 5\) and \(BC = 6\). What is the area of \(ADEC\)?


Find the positive integer values of \( m \) for which there exists a function \( f \) from the set of integers to itself such that \( f^{(m)}(n) = n + 2017 \), where \( f^{(m)} \) denotes the function \( f \) applied \( m \) times.

Solution: \boxed{1 \text{ and } 2017}


On the segment \( AB = 2R \) taken as the diameter, a circle is constructed. A second circle, whose radius is half the radius of the first circle, is tangent to the first circle from the inside at point \( A \). A third circle is tangent to the first circle from the inside, to the second circle from the outside, and also tangent to the segment \( AB \). Find the radius of the third circle.


Calculate the area of the figure bounded by the lines given by the equations.

$$
\begin{aligned}
& \left\{\begin{array}{l}
x=4 \sqrt{2} \cdot \cos ^{3} t \\
y=2 \sqrt{2} \cdot \sin ^{3} t
\end{array}\right. \\
& x=2(x \geq 2)
\end{aligned}
$$

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


Using rectangular wooden blocks that are 5 cm long, 4 cm wide, and 3 cm high, how many of these rectangular blocks are needed to form a cube?


Fill a $2 \times 4$ grid with the first eight positive integers, one number per cell, such that in each row the numbers increase from left to right and in each column the numbers increase from bottom to top. How many different ways can this be done?


In the number $2 * 0 * 1 * 6 * 0 * 2 *$, each of the 6 asterisks must be replaced by any of the digits $0, 2, 4, 5, 7, 9$ (digits can be repeated) so that the resultant 12-digit number is divisible by 75. In how many ways can this be done?


On sides \( BC \) and \( AC \) of triangle \( ABC \), points \( D \) and \( E \) are chosen respectively such that \( \angle BAD = 50^\circ \) and \( \angle ABE = 30^\circ \). Find \( \angle BED \) if \( \angle ABC = \angle ACB = 50^\circ \).


A random variable $X$ has a variance $D(X)=0.001$. What is the probability that the random variable $X$ differs from $M(X)=a$ by more than 0.1?


If \(\alpha\), \(\beta\), and \(\gamma\) are the roots of the equation \(x^{3}-x-1=0\), find the value of
$$
\frac{1+\alpha}{1-\alpha} + \frac{1+\beta}{1-\beta} + \frac{1+\gamma}{1-\gamma}.
$$


Given the function \( f(x) = a x^{3} + b x^{2} + c x + d \) (where \( a \neq 0 \)), and knowing that \( \left| f^{\prime}(x) \right| \leq 1 \) when \( 0 \leq x \leq 1 \), find the maximum value of \( a \).


Inside a circular pancake with a radius of 10, a coin with a radius of 1 has been baked. What is the minimum number of straight cuts needed to surely hit the coin?


Represent the number 19 as the difference of the cubes of natural numbers. Show that this representation is unique.

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


Real numbers \(a, b, c,\) and \(d\) satisfy the condition \(a^{6}+b^{6}+c^{6}+d^{6}=64\). Find the maximum value of the expression \(a^{7}+b^{7}+c^{7}+d^{7}\).


Among all fractions of the form $\frac{m}{n}$, where $m$ and $n$ are four-digit numbers with the same sum of digits, the greatest one was selected. Find it. Record your answer as an improper fraction (do not round or simplify!).


30 people are arranged in six rows of five people each. Each of them is either a knight, who always tells the truth, or a liar, who always lies, and they all know who among them is a knight or a liar. A journalist asked each of them: "Is it true that there are at least 4 rows in which more than half are liars?" What is the maximum number of "yes" answers he could hear?


Several people completed the task of planting 2013 trees, each person planting the same number of trees. If 5 people do not participate in planting, the remaining people each plant 2 more trees and cannot complete the task, but if each plants 3 more trees, they can exceed the task. How many people participated in planting the trees?


For a sequence \(a_1, a_2, a_3, \ldots\) of real numbers, it is known that
\[ a_n = a_{n-1} + a_{n+2} \quad \text{for} \ n = 2, 3, 4, \ldots \]
What is the largest number of its consecutive elements that can all be positive?


Given two arbitrary natural numbers, determine the probability that their product will be an even number; also, determine the probability that it will be an odd number.


\( \triangle ABC \) is an equilateral triangle with side length 1, and \( PA \perp \) plane \( ABC \) with \( PA = \frac{\sqrt{6}}{4} \). Let \( A' \) be the reflection of point \( A \) across the plane \( PBC \). Find the angle between the line \( A'C \) and \( AB \).


Given that a line $l$ with slope $k$ passes through the point $P(3,0)$ and intersects the right branch of the hyperbola $C: x^{2}-\frac{y^{2}}{3}=1$ at points $A$ and $B$. The point $F$ is the right focus of the hyperbola $C$, and it is known that $|AF| + |BF| = 16$. Find the value of $k$.

Solution: \boxed{k = \pm 3}


Given that the volume of the tetrahedron \(ABCD\) is 40, \(AB = 7\), \(BC = 8\), \(AC = 9\), and the orthogonal projection of vertex \(D\) onto the plane \(ABC\) is precisely the incenter \(H\) of triangle \(ABC\), what is the surface area of the tetrahedron?

Solution: \boxed{60 + 12 \sqrt{5}}


Pairs of numbers are written on a board. Initially, the pair of numbers \((1, 2)\) is written. If a pair of numbers \((a, b)\) is on the board, it is possible to add the pair \((-a, -b)\) or the pair \((-b, a+b)\) to the board. Additionally, if pairs \((a, b)\) and \((c, d)\) are on the board, it is possible to add the pair \((a+c, b+d)\) to the board. Could the pair \((2022, 2023)\) possibly appear on the board after some time? The order of numbers in the pair is significant; for example, the pairs \((1, 2)\) and \((2, 1)\) are considered different.

Solution: \boxed{\text{No}}


Given that \(\alpha\), \(\beta\), and \(\gamma\) are acute angles such that \(\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1\), find the minimum value of \(\tan \alpha \cdot \tan \beta \cdot \tan \gamma\).


Given the sequence \(\{a_n\}\) defined by
\[ a_1 = 0, \quad a_{n+1} = a_n + 4 \sqrt{a_n + 1} + 4 \quad (n \geq 1), \]
find \(a_n\).

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


Simplify $\frac{1+\sin \theta+\cos \theta}{1+\sin \theta-\cos \theta}+\frac{1-\cos \theta+\sin \theta}{1+\cos \theta+\sin \theta}$.

Solution: \boxed{2 \csc \theta}


Find the smallest natural number whose digit sum is 2017. In the answer, specify the first digit from the left multiplied by the number of digits.


Sven wrote five different one-digit positive integers on a blackboard. He discovered that none of the sums of two different numbers on the board equaled 10. Which of the following numbers did Sven definitely write on the blackboard?
A 1
B 2
C 3
D 4
E 5


Two circles on a plane are given, one inside the other. Construct a point \( O \) such that one circle is obtained from the other by a homothety with respect to the point \( O \) (in other words, so that scaling the plane from the point \( O \) with a certain coefficient transforms one circle into the other).

Solution: \boxed{O}


There were cards with numbers from 1 to 20 in a bag. Vlad pulled out 6 cards and asserted that all these cards can be divided into pairs such that the sums of the numbers in each pair were the same. Lena managed to sneak a peek at 5 of Vlad's cards: they had the numbers $2, 4, 9, 17, 19$ written on them. Which number on the sixth card did Lena not manage to see? (It is sufficient to provide one suitable answer.)


Mark writes the expression \(\sqrt{d}\) for each positive divisor \(d\) of \(8!\) on the board. Rishabh then simplifies each expression to the form \(a \sqrt{b}\), where \(a\) and \(b\) are integers, and \(b\) is not divisible by the square of a prime number. For example, \(\sqrt{20}, \sqrt{16}\), and \(\sqrt{6}\) simplify to \(2 \sqrt{5}, 4 \sqrt{1}\), and \(1 \sqrt{6}\), respectively. Compute the sum of \(a + b\) across all expressions that Rishabh writes.


Let \( f(x) \) be a quadratic polynomial with integer coefficients. Given that \( f(\sqrt{3}) - f(\sqrt{2}) = 4 \), find \( f(\sqrt{10}) - f(\sqrt{7}) \).


Calculate the limit of the function:

\[
\lim _{x \rightarrow \frac{\pi}{2}} \frac{e^{\operatorname{tg} 2 x} - e^{-\sin 2 x}}{\sin x - 1}
\]


We consider a set with $n$ elements.

How many subsets with an odd number of elements are there?


The function \( f(x) \) is a strictly increasing function defined on \(\mathbf{N}\) and taking values in \(\mathbf{N}\) (a function \( f(x) \) is called a strictly increasing function on a set \(A\) if for any \( x_1, x_2 \in A \), when \( x_1 < x_2 \), it holds that \( f(x_1) < f(x_2) \)). Additionally, it satisfies the condition \( f(f(k)) = 3k \). Find the value of \( f(1) + f(9) + f(96) \).


Let \( p = 2^{24036583} - 1 \), the largest prime currently known. For how many positive integers \( c \) do the quadratics \( \pm x^2 \pm p x \pm c \) all have rational roots?


In triangle \(PQR\) with side \(PQ = 3\), a median \(PM = \sqrt{14}\) and an altitude \(PH = \sqrt{5}\) are drawn from vertex \(P\) to side \(QR\).

Find side \(PR\), given that \(\angle QPR + \angle PRQ < 90^\circ\).


Find the number of pairs of integers $(x ; y)$ that satisfy the equation $6x^{2} - 7xy + y^{2} = 10^{100}$.


The quadratic function \(y=f(x)\) has a leading coefficient of 1. The coefficients of the linear term and the constant term are integers. Given that \(f(f(x))=0\) has four distinct real roots, and these roots can be arranged in an arithmetic sequence, find the function \(f(x)\) such that the sum of its coefficients is minimized.

Solution: \boxed{x^2 + 22x + 105}


In the acute-angled triangle \(ABC\), the altitude \(BH\) is drawn. It turns out that \(CH = AB + AH\). How many degrees is the angle \(BAC\) if \(\angle ABC = 84^\circ\)?


In the plane $xOy$, the lines $y = 3x - 3$ and $x = -1$ intersect at point $\mathrm{B}$. A line passing through the point $M(1, 2)$ intersects the given lines at points $\mathrm{A}$ and $\mathrm{C}$, respectively. For which positive value of the $x$-coordinate of point $\mathrm{A}$ will the area of triangle $\mathrm{ABC}$ be minimized?

(12 points)


For how many positive integers \( n \) less than 200 is \( n^n \) a cube and \( (n+1)^{n+1} \) a square?


The number \( n^2 \) ends with \( m \) equal non-zero digits (in base 10). What is the largest possible value of \( m \)?


Find the integer part of the number \(a + \frac{9}{b}\), where \(a\) and \(b\) are the integer and fractional parts, respectively, of the number \(\sqrt{76 - 42 \sqrt{3}}\).


The number 12320 is written on the board. Petya appended $10n + 1$ threes to it, where $n$ is a nonnegative integer. Vasya thought this was a base-4 number representing a natural number $x$, and he factored $x$ into prime factors. It turned out that there were exactly two distinct prime factors among them. For which $n$ is this possible?

Solution: \boxed{n = 0}


At the bus stop near Absent-minded Scientist's house, buses of two routes stop: No. 152 and No. 251. Both go to the metro station. The interval between buses No. 152 is exactly 5 minutes, and the interval between buses No. 251 is exactly 7 minutes. The intervals are strictly adhered to, but these two routes are not coordinated between each other and their schedules do not depend on each other. At a completely random moment in time, the Absent-minded Scientist comes to the bus stop and gets on the first bus that arrives to get to the metro. What is the probability that the Scientist will get on bus No. 251?


Given that \( x_{1}, x_{2}, \cdots, x_{2021} \) are values either equal to \( 1 \) or \( -\mathrm{i} \), denote

\[
\begin{aligned}
S= & x_{1} x_{2} x_{3}+x_{2} x_{3} x_{4}+\cdots+x_{2019} x_{2020} x_{2021}+ \\
& x_{2020} x_{2021} x_{1}+x_{2021} x_{1} x_{2} .
\end{aligned}
\]

Find the smallest non-negative value that \( S \) can achieve.


Given a triangle \( A B C \) with sides \( A B = \sqrt{17} \), \( B C = 5 \), and \( A C = 4 \). Point \( D \) is taken on the side \( A C \) such that \( B D \) is the altitude of triangle \( A B C \). Find the radius of the circle passing through points \( A \) and \( D \) and tangent at point \( D \) to the circumcircle of triangle \( B C D \).


For a positive integer \(a\) and integers \(b\), \(c\), the quadratic equation \(a x^{2} + b x + c = 0\) has two roots \(\alpha, \beta\) that satisfy \(0 < \alpha < \beta < 1\). Find the minimum value of \(a\).


a) Consider the $4 \times 4$ board shown in the figure below. The letters and numbers are used to help locate the squares. For example, the square $A1$ is at the top left corner and the $D4$ is at the bottom right corner.

Show a way to choose 12 squares such that exactly 3 squares are selected in each row and each column.

b) Determine the number of different ways to choose 12 squares on the $4 \times 4$ board such that exactly 3 squares are selected in each row and each column.

c) Consider an $8 \times 8$ board with a similar row and column identification as the previous board, but colored alternately in black and white.

Determine the number of ways to choose 56 squares from the board such that all the black squares are chosen and exactly 7 squares are chosen in each row and each column.


Let \( f(n) \) denote the number of 0's in the decimal representation of the positive integer \( n \). For example, \( f(10001123) = 3 \) and \( f(1234567) = 0 \). Define the expression:
\[
M = f(1) \times 2^{f(1)} + f(2) \times 2^{f(2)} + f(3) \times 2^{f(3)} + \cdots + f(99999) \times 2^{f(99999)}
\]
Find the value of \( M - 100000 \).


\(\sin ^{2} \varphi - \cos ^{2}(\alpha - \varphi) + 2 \cos \alpha \cos \varphi \cos (\alpha - \varphi) = \cos ^{2} \alpha\).

Solution: \boxed{\cos^2 \alpha}


A fast train and a slow train are moving towards each other. The length of the fast train is 315 meters, and the length of the slow train is 300 meters. A person sitting on the slow train observes that the fast train passes by in 21 seconds. How long does it take for a person sitting on the fast train to see the slow train pass by in seconds?


The ratio of the sides of a rectangle is \(3:4\), and its diagonal is 9. Find the shorter side of the rectangle.


In a tournament with 2017 participating teams, each round consists of three randomly chosen teams competing, with exactly one team surviving from each round. If only two teams remain, a one-on-one battle determines the winner. How many battles must take place to declare a champion?


Given \( n \) new students such that among any 3 students, there are at least 2 students who know each other, and among any 4 students, there are at least 2 students who do not know each other. Determine the maximum value of \( n \).


There are 5 integers written on a board. By summing these numbers in pairs, the following set of 10 numbers is obtained: $-1, 2, 6, 7, 8, 11, 13, 14, 16, 20$. Determine which numbers are written on the board. Write their product as the answer.


Let $S$ be the set of seven-element ordered arrays $\left(a_{1}, a_{2}, \ldots, a_{7}\right)$, where $a_i = 1$ or 0. For any two elements $a = \left(a_{1}, a_{2}, \ldots, a_{7}\right)$ and $b = \left(b_{1}, b_{2}, \ldots, b_{7}\right)$ in $S$, define the distance between $a$ and $b$ as $\sum_{i=1}^{7}\left|a_{i}-b_{i}\right|$. Let $T$ be a subset of $S$ such that the distance between any two elements is at least 3. Find the maximum value of $|T|$.


Tanya had a set of identical sticks. She arranged them into a large triangle, each side of which consists of 11 sticks, and created a pattern inside the triangle such that the triangle was divided into small triangles with a side of 1 stick. How many sticks did Tanya use in total?


How many numbers from 1 to 100 are divisible by 3, but do not contain the digit 3?


Compute the circumradius of cyclic hexagon \( ABCDEF \), which has side lengths \( AB = BC = 2 \), \( CD = DE = 9 \), and \( EF = FA = 12 \).


Let $[x]$ denote the greatest integer not exceeding $x$. Find the last two digits of $\left[\frac{1}{3}\right]+\left[\frac{2}{3}\right]+\left[\frac{2^{2}}{3}\right]+\cdots+\left[\frac{2^{2014}}{3}\right]$.


What is the smallest three-digit positive integer which can be written in the form \( p q^{2} r \), where \( p, q \), and \( r \) are distinct primes?


Suppose that \( a = \cos^4 \theta - \sin^4 \theta - 2 \cos^2 \theta \), find the value of \( a \).

If \( x^y = 3 \) and \( b = x^{3y} + 10a \), find the value of \( b \).

If there is (are) \( c \) positive integer(s) \( n \) such that \( \frac{n+b}{n-7} \) is also a positive integer, find the value of \( c \).

Suppose that \( d = \log_{4} 2 + \log_{4} 4 + \log_{4} 8 + \ldots + \log_{4} 2^{c} \), find the value of \( d \).


Determine the value of

$$
z=a \sqrt{a} \sqrt[4]{a} \sqrt[8]{a} \ldots \sqrt[2^{n}]{a} \ldots
$$

if \( n \) is infinitely large.


Given the function \( f(x) = A \sin (\omega x + \varphi) \) where \( A \neq 0 \), \( \omega > 0 \), \( 0 < \varphi < \frac{\pi}{2} \), if \( f\left(\frac{5\pi}{6}\right) + f(0) = 0 \), find the minimum value of \( \omega \).


Given the system of equations involving \( x \) and \( y \):
$$
\left\{\begin{array}{l}
\sin x = m \sin^3 y, \\
\cos x = m \cos^3 y
\end{array}\right.
$$
If real solutions exist, what is the range of values for the positive real number \( m \)?

Solution: \boxed{ [1,2] }


Let \(\mathbb{N}\) be the set of all positive integers. A function \( f: \mathbb{N} \rightarrow \mathbb{N} \) satisfies \( f(m + n) = f(f(m) + n) \) for all \( m, n \in \mathbb{N} \), and \( f(6) = 2 \). Also, no two of the values \( f(6), f(9), f(12) \), and \( f(15) \) coincide. How many three-digit positive integers \( n \) satisfy \( f(n) = f(2005) \) ?


Given \( f(x) = x^{5} - 10x^{3} + ax^{2} + bx + c \), and knowing that the equation \( f(x) = 0 \) has all real roots, let \( m \) be the largest of these 5 real roots. Find the maximum value of \( m \).


A rubber band is 4 inches long. An ant begins at the left end. Every minute, the ant walks one inch rightwards along the rubber band, but then the band is stretched uniformly by one inch. For what value of \( n \) will the ant reach the right end during the \( n \)-th minute?


Find all functions \( f: \mathbb{Z} \longrightarrow \mathbb{Z} \) such that \( f(p) > 0 \) for every prime \( p \) and for every prime \( p \) and every \( x \in \mathbb{Z} \):

$$
p \mid (f(x)+f(p))^{f(p)} - x
$$

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


Find the measure of the angle

$$
\delta = \arccos \left( \left( \sin 2195^\circ + \sin 2196^\circ + \cdots + \sin 5795^\circ \right)^{\cos 2160^\circ} + \cos 2161^\circ + \cdots + \cos 5760^\circ \right)
$$


There are 1990 piles of stones, with the number of stones in each pile being $1, 2, \cdots, 1990$. You can perform the following operation: in each step, you can choose any number of piles and remove the same number of stones from each of the chosen piles. What is the minimum number of steps required to remove all the stones from all the piles?


For natural numbers \(a > b > 1\), define the sequence \(x_1, x_2, \ldots\) by the formula \(x_n = \frac{a^n - 1}{b^n - 1}\). Find the smallest \(d\) such that for any \(a\) and \(b\), this sequence does not contain \(d\) consecutive terms that are prime numbers.


Suppose a die is rolled. There are two opportunities to roll: you can choose to roll just once, or twice, but the result is determined by the last roll. What is the expected value of the outcome in this scenario? If there are three opportunities, still taking the value of the last roll, what is the expected value of the outcome?


Juca has fewer than 800 marbles. He likes to separate the marbles into groups of the same size. He noticed that if he forms groups of 3 marbles each, exactly 2 marbles are left over. If he forms groups of 4 marbles, 3 marbles are left over. If he forms groups of 5 marbles, 4 marbles are left over. Finally, if he forms groups of 7 marbles each, 6 marbles are left over.

(a) If Juca formed groups of 20 marbles each, how many marbles would be left over?

(b) How many marbles does Juca have?


The numbers \( a, b, c, d \) belong to the interval \([-5, 5]\). Find the maximum value of the expression \( a + 2b + c + 2d - ab - bc - cd - da \).


Little Rabbit and Little Turtle start from point $A$ to the Forest Amusement Park simultaneously. Little Rabbit jumps forward 36 meters in 1 minute and rests after every 3 minutes of jumping. The first rest period is 0.5 minutes, the second rest period is 1 minute, the third rest period is 1.5 minutes, and so on, with the $k$th rest period being $0.5k$ minutes. Little Turtle does not rest or play on the way. It is known that Little Turtle reaches the Forest Amusement Park 3 minutes and 20 seconds earlier than Little Rabbit. The distance from point $A$ to the Forest Amusement Park is 2640 meters. How many meters does Little Turtle crawl in 1 minute?


Given that the equation \( a x(x+1) + b x(x+2) + c (x+1)(x+2) = 0 \) has roots 1 and 2, and that \( a + b + c = 2 \), find the value of \( a \).


The sequence $\mathrm{Az}\left(a_{n}\right)$ is defined as follows:

$$
a_{0}=a_{1}=1, \quad a_{n+1}=a_{n}+\frac{1}{a_{n-1}}
$$

Show that $a_{n} \geq \sqrt{n}$.

Solution: \boxed{a_n \geq \sqrt{n}}


The number \( n \) has exactly six divisors (including 1 and itself). These divisors are arranged in ascending order. It turns out that the third divisor is seven times larger than the second one, and the fourth divisor is 10 larger than the third one. What is \( n \)?


The graph of the function \( f(x)=\sin \left(\frac{\pi}{4} x-\frac{\pi}{6}\right)-2 \cos ^{2} \frac{\pi}{8} x+1 \) is symmetrical with the graph of the function \( y=g(x) \) about the line \( x=1 \). When \( x \in\left[0, \frac{4}{3}\right] \), the maximum value of \( g(x) \) is \(\qquad\) .


A ring has beads placed at positions $0^{\circ}, 90^{\circ}, 180^{\circ}, 270^{\circ}$ in a clockwise direction. The beads can be either red or blue. How many distinct equivalence classes of arrangements are there? In other words, how many different arrangements exist, considering rotations of the ring as equivalent?

This problem can also be viewed as the case of four slots on a square (as shown in Figure 4-8), where the square is a transparent glass panel and is colored with either red or blue. How many coloring schemes are there?


Let $(x, y, z, t) \in\left(\mathbb{N}^{*}\right)^{4}$ be a quadruple satisfying $x+y=z+t$ and $2xy=zt$, with $x \geq y$. Find the largest value of $m$ such that $m \leq x / y$ for sure.

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


Let

$$
2^{x}=\left(1+\tan 0.01^{\circ}\right)\left(1+\tan 0.02^{\circ}\right)\left(1+\tan 0.03^{\circ}\right) \ldots\left(1+\tan 44.99^{\circ}\right)
$$

Find \( x \). If necessary, round the answer to the nearest 0.01.


Points $A, B, C, D, X$ are located on a plane. Some segment lengths are known: $AC = 2$, $AX = 5$, $AD = 11$, $CD = 9$, $CB = 10$, $DB = 1$, $XB = 7$. Find the length of segment $CX$.


Given 95 numbers \(a_{1}, a_{2}, \cdots, a_{95}\), where each can only take the values +1 or -1, what is the minimum positive value of the sum of their pairwise products \(a_{1} a_{2}+a_{1} a_{3}+\cdots+a_{94} a_{95}\)?


A three-stage launch vehicle consists of cylindrical stages. All these cylinders are similar to each other. The length of the middle stage is half the sum of the lengths of the first and third stages. When fueled, the mass of the middle stage is $13 / 6$ times less than the combined mass of the fueled first and third stages. Find the ratio of the lengths of the first and third stages. The masses of the engines, instruments, and the shells of the stages can be neglected.


A scatterbrained scientist in his laboratory has developed a unicellular organism, which has a probability of 0.6 of dividing into two identical organisms and a probability of 0.4 of dying without leaving any offspring. Find the probability that after some time the scatterbrained scientist will have no such organisms left.


There are 12 shapes made from matches - 3 triangles, 4 squares, and 5 pentagons. The shapes do not share sides. Petya and Vasya take turns removing one match at a time. Vasya wants as few untouched shapes as possible to remain, while Petya wants as many untouched shapes as possible to remain. How many shapes will remain after 10 turns? Each player makes 5 moves, and Petya starts first.


In a winter camp, Vanya and Grisha share a room. Each night they draw lots to determine who will turn off the light before bed. The switch is near the door, so the loser has to walk back to bed in complete darkness, bumping into chairs.

Usually, Vanya and Grisha draw lots without any particular method, but this time Grisha has come up with a special way:

- "Let's flip a coin. If heads come up on an even-numbered throw, we stop flipping the coin: I win. If tails come up on an odd-numbered throw, you win."

a) What is the probability that Grisha wins?

b) Find the expected number of coin flips until the outcome is decided.


Find the probability of a simultaneous occurrence of a specific digit when tossing two coins once.


Construct polynomials \( f(x) \) of degree no higher than 2 that satisfy the conditions:
a) \( f(0)=1, f(1)=3, f(2)=3 \)

b) \( f(-1)=-1, f(0)=2, f(1)=5 \)

c) \( f(-1)=1, f(0)=0, f(2)=4 \).


Determine which is greater without using a calculator or logarithm table: \(200!\) or \(100^{200}\).


For a positive integer $n$, define $n! = n(n-1)(n-2) \cdots 2 \cdot 1$. Let $S_{n}=n!\left[\frac{1}{2!}+\frac{2}{3!}+\cdots+\frac{n}{(n+1)!}-1\right]$. Find the value of $S_{2017} = \quad$.


From eleven school students and three teachers, a school committee needs to be formed consisting of eight people. At least one teacher must be included in the committee. In how many ways can the committee be formed?


Let integers \( x \) and \( y \) satisfy \( x^2 + y^2 < 16 \), and \( xy > 4 \). Determine the maximum value of \( x^2 - 2xy - 3y \).


Find the smallest prime \( p > 100 \) for which there exists an integer \( a > 1 \) such that \( p \) divides \( \frac{a^{89} - 1}{a - 1} \).


A two-digit number is divided by the sum of its digits. The result is a number between 2.6 and 2.7. Find all of the possible values of the original two-digit number.


Previously, on an old truck, I traveled from village $A$ through $B$ to village $C$. After five minutes, I asked the driver how far we were from $A$. "Half as far as from $B," was the answer. Expressing my concerns about the slow speed of the truck, the driver assured me that while the truck cannot go faster, it maintains its current speed throughout the entire journey.

$13$ km after $B$, I inquired again how far we were from $C$. I received exactly the same response as my initial inquiry. A quarter of an hour later, we arrived at our destination. How many kilometers is the journey from $A$ to $C$?

Solution: \boxed{26 \text{ km}}


In the rectangular parallelepiped $ABCDA_{1}B_{1}C_{1}D_{1}$, the lengths of the edges are given as $AB=18$, $AD=30$, and $AA_{1}=20$. Point $E$ is marked on the midpoint of edge $A_{1}B_{1}$, and point $F$ is marked on the midpoint of edge $B_{1}C_{1}$. Find the distance between the lines $AE$ and $BF$.


One of two parallel lines is tangent to a circle of radius $R$ at point $A$, while the other intersects this circle at points $B$ and $C$. Express the area of triangle $ABC$ as a function of the distance $x$ between the parallel lines.

Solution: \boxed{x \sqrt{2R x - x^2}}


Compute the definite integral:

$$
\int_{0}^{\pi} 2^{4} \cdot \sin ^{4} x \cos ^{4} x \, dx
$$


$M$ is a moving point on the circle $x^{2}+y^{2}-6 x-8 y=0$, $O$ is the origin, and $N$ is a point on the ray $O M$. If $|O M| \cdot |O N| = 150$, find the equation of the locus of point $N$.

Solution: \boxed{3 x + 4 y = 75}


Given that point \( F \) is the common focus of the parabola \( C: x^{2}=4y \) and the ellipse \( \frac{y^{2}}{a^{2}}+\frac{x^{2}}{b^{2}}=1 \) (\(a > b > 0\)), and the maximum distance from point \( M \) on the ellipse to point \( F \) is 3:

1. Find the equation of the ellipse.
2. Draw two tangent lines from point \( M \) to the parabola \( C \), with points of tangency labeled \( A \) and \( B \). Find the maximum area of \( \triangle MAB \).

Solution: \boxed{8 \sqrt{2}}


In triangle \( \triangle ABC \), \( BD \) is a median, point \( P \) lies on \( BD \) with the ratio \( |BP| : |PD| = 3:1 \), and \( AP \) intersects \( BC \) at \( K \). Find the ratio of the area of \( \triangle ABK \) to the area of \( \triangle ACK \).


Let \( a_{1}, a_{2}, \cdots, a_{2006} \) be 2006 positive integers (they can be the same) such that \( \frac{a_{1}}{a_{2}}, \frac{a_{2}}{a_{3}}, \cdots, \frac{a_{2005}}{a_{2006}} \) are all different from each other. What is the minimum number of distinct numbers in \( a_{1}, a_{2}, \cdots, a_{2006} \)?


Triangle \(ABC\) is isosceles \((AB = BC)\). Segment \(AM\) divides it into two isosceles triangles with bases \(AB\) and \(MC\). Find angle \(B\).


Solve the cryptarithm:

$$
C, B A + A, A A = B, A
$$

(Different letters represent different digits, and the same letters represent the same digits.)

Solution: \boxed{A=5, B=9, C=3}


On the lateral sides \( A B \) and \( C D \) of trapezoid \( A B C D \), points \( P \) and \( Q \) are marked such that line \( P Q \) is parallel to \( A D \), and segment \( P Q \) is divided by the diagonals of the trapezoid into three equal parts. Find the length of the base \( B C \), given that \( A D = a \), \( P Q = m \), and the intersection point of the diagonals of the trapezoid lies inside the quadrilateral \( B P C Q \).

Solution: \boxed{\frac{am}{3a - 2m}}


Given a sequence of $n$ positive integers $x_{1}, x_{2}, \cdots, x_{n}$ whose sum is 2009, and which can be partitioned into 41 groups with equal sums as well as 49 groups with equal sums, find the smallest value of $n$.


\(AB\) is the diameter of a circle, \(BC\) is a tangent, and \(CDA\) is a secant. Find the ratio \(CD : DA\) if \(BC\) is equal to the radius of the circle.


Calculate \(\sin (\alpha-\beta)\) if \(\sin \alpha - \cos \beta = \frac{3}{4}\) and \(\cos \alpha + \sin \beta = -\frac{2}{5}\).


In a right triangle $ABC$ (right angle at $C$), the bisector $BK$ is drawn. Point $L$ is on side $BC$ such that $\angle C K L = \angle A B C / 2$. Find $KB$ if $AB = 18$ and $BL = 8$.


Consider the cards $A, 2, \cdots, J, Q, K$ as the numbers $1, 2, \cdots, 11, 12, 13$. If we take the 13 cards of spades and 13 cards of hearts together and randomly draw 2 cards, what is the probability that the two cards are of the same suit and the product of the two numbers is a perfect square?


The side \(A D\) of rectangle \(A B C D\) is three times longer than side \(A B\). Points \(M\) and \(N\) divide \(A D\) into three equal parts. Find \(\angle A M B + \angle A N B + \angle A D B\).


How can you find the lightest and heaviest stones among $2N$ stones, where any two stones have different weights, with $3N-2$ weighings? All weighings are done using a two-pan balance scale without any weights.


Let \( N \) be a natural number whose base-2016 representation is \( ABC \). Working now in base-10, what is the remainder when \( N - (A + B + C + k) \) is divided by 2015, if \( k \in \{ 1, 2, \ldots, 2015 \} \)?

Solution: \boxed{2015 - k}


A regular octahedron has a side length of 1. What is the distance between two opposite faces?


A particular coin has a $\frac{1}{3}$ chance of landing on heads (H), $\frac{1}{3}$ chance of landing on tails (T), and $\frac{1}{3}$ chance of landing vertically in the middle (M). When continuously flipping this coin, what is the probability of observing the continuous sequence HMMT before HMT?


In the city of Perpendicularka, new multi-story houses are planned to be built, with some potentially being single-story houses, so that the total number of floors is equal to 30. The city’s architect Parallelnikov proposed a project according to which, if you climb to the roof of each new house, count the number of new houses that are shorter, and add all such numbers, the resulting sum will be maximized. What is this sum? How many houses are proposed to be built, and what are their floor counts?


In triangle ABC, angle CAB is 30 degrees, and angle ABC is 80 degrees. The point M lies inside the triangle such that angle MAC is 10 degrees and angle MCA is 30 degrees. Find angle BMC in degrees.


Show that if two integers \( m \) and \( n \) are such that \( p(m) = p(n) \), then \( m = n \).

Solution: \boxed{m = n}


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

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

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

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


In rectangle \( J K L M \), the bisector of angle \( K J M \) cuts the diagonal \( K M \) at point \( N \). The distances between \( N \) and sides \( L M \) and \( K L \) are \( 8 \) cm and \( 1 \) cm respectively. The length of \( K L \) is \( (a+\sqrt{b}) \) cm. What is the value of \( a+b \)?


There are 52 cards numbered from 1 to 52. It is considered that 1 is older than 52, and in all other pairs, the card with the higher number is older. The cards are lying face down on the table in a random order. In one question, Petya can find out which of any two cards is older. Can Petya guarantee to find the card numbered 52 in no more than 64 questions?

Solution: \boxed{Yes}


The rank of a rational number \( q \) is the unique \( k \) for which \( q=\frac{1}{a_{1}}+\cdots+\frac{1}{a_{k}} \), where each \( a_{i} \) is the smallest positive integer such that \( q \geq \frac{1}{a_{1}}+\cdots+\frac{1}{a_{i}} \). Let \( q \) be the largest rational number less than \( \frac{1}{4} \) with rank 3, and suppose the expression for \( q \) is \( \frac{1}{a_{1}}+\frac{1}{a_{2}}+\frac{1}{a_{3}} \). Find the ordered triple \( \left( a_{1}, a_{2}, a_{3} \right) \).

Solution: \boxed{(5, 21, 421)}


In the following figure, the square \(ABCD\) is divided into three rectangles of equal area.

If the length of the segment \(BM\) is equal to 4, calculate the area of the square \(ABCD\).


The smallest four two-digit primes are written in different squares of a \(2 \times 2\) table. The sums of the numbers in each row and column are calculated. Two of these sums are 24 and 28. The other two sums are \(c\) and \(d\), where \(c<d\). What is the value of \(5c + 7d\)?


Positive integers \( m \) and \( n \) satisfy \( mn = 5000 \). If \( m \) is not divisible by 10 and \( n \) is not divisible by 10, what is the value of \( m + n \)?


The accrued salary of a citizen from January to June inclusive was 23,000 rubles per month, and from July to December, it was 25,000 rubles. In August, the citizen, participating in a poetry contest, won a prize and was awarded an e-book worth 10,000 rubles. What amount of personal income tax needs to be paid to the budget? (Provide the answer as a whole number, without spaces or units of measurement.)

Answer: 39540.


a) Find the last digit of the numbers \(9^{(9^8)}\) and \(2^{(3^4)}\).

b) Find the last two digits of the numbers \(2^{999}\) and \(3^{999}\).

c) * Find the last two digits of the number \(14^{(14^4)}\).


Given \( A=\left\{x \mid \log _{3}\left(x^{2}-2 x\right) \leqslant 1\right\}, B=(-\infty, a] \cup(b,+\infty) \), where \( a < b \), if \( A \cup B=\mathbf{R} \), what is the minimum value of \( a - b \) ?


Quadrilateral \(ABCD\) is such that \(\angle BAC = \angle CAD = 60^\circ\) and \(AB + AD = AC\). It is also known that \(\angle ACD = 23^\circ\). How many degrees is \(\angle ABC\)?


\( a \) and \( b \) are positive integers. After rounding to three decimal places, the expression \(\frac{a}{5} + \frac{b}{7} = 1.51\). Find \( a + b \).


Calculate the limit of the function:

\[ \lim_{x \to 0} \frac{e^{3x} - e^{2x}}{\sin 3x - \tan 2x} \]


A metallic weight has a mass of 20 kg and is an alloy of four metals. The first metal in this alloy is one and a half times the amount of the second metal. The mass of the second metal relates to the mass of the third metal as $3:4$, and the mass of the third metal to the mass of the fourth metal as $5:6$. Determine the mass of the fourth metal. Give your answer in kilograms, rounding to the nearest hundredth if necessary.

Solution: \boxed{5.89 \text{ kg}}


There were cards with the digits from 1 to 9 (a total of 9 cards) on the table. Katya selected four cards such that the product of the digits on two of them was equal to the product of the digits on the other two. Then Anton took one more card from the table. As a result, the cards left on the table were 1, 4, 5, and 8. Which digit was on the card that Anton took?


A nine-digit number is allowed to undergo the following operation: any digit of the number can be replaced with the last digit of the sum of the digits of the number. Is it possible to obtain the number 123456789 from the number 133355555 using such operations?

Solution: \boxed{\text{No}}


A dandelion blossoms in the morning, blooms yellow for three days, turns white on the morning of the fourth day, and by the evening of the fifth day, its seeds disperse. On Monday afternoon, there were 20 yellow and 14 white dandelions in the meadow, and on Wednesday, there were 15 yellow and 11 white. How many white dandelions will be in the meadow on Saturday?


In a convex quadrilateral \(ABCD\), side \(AB\) is equal to diagonal \(BD\), \(\angle A=65^\circ\), \(\angle B=80^\circ\), and \(\angle C=75^\circ\). What is \(\angle CAD\) (in degrees)?


Let \( M = \{1, 2, \cdots, 1995\} \). \( A \) is a subset of \( M \) that satisfies the condition: if \( x \in A \), then \( 15x \notin A \). What is the maximum number of elements in \( A \)?


A random variable \(\xi\) is uniformly distributed on the interval \([0 ; 6]\). Find the probability that the inequality \(x^{2}+ (2 \xi + 1)x + 3 - \xi \leq 0\) holds for all \(x \in [-2 ;-1]\).


The number \( a \) is a root of the equation \( x^{11} + x^{7} + x^{3} = 1 \). Specify all natural values of \( n \) for which the equality \( a^{4} + a^{3} = a^{n} + 1 \) holds.


A student plotted the graph of the function \( y = ax^2 + bx + c \) (\(a \neq 0\)) by choosing 7 different values for the variable \( x \): \( x_1 < x_2 < \cdots < x_7 \). They set \( x_2 - x_1 = x_3 - x_2 = \cdots = x_7 - x_6 \) and calculated the corresponding \( y \) values, creating the following table:
\[
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
$x$ & $x_1$ & $x_2$ & $x_3$ & $x_4$ & $x_5$ & $x_6$ & $x_7$ \\
\hline
$y$ & 51 & 107 & 185 & 285 & 407 & 549 & 717 \\
\hline
\end{tabular}
\]

However, due to a calculation error, one of the \( y \) values is incorrect. Which \( y \) value is incorrect, what should the correct value be, and why?


Beginner millionaire Bill buys a bouquet of 7 roses for $20. Then, he can sell a bouquet of 5 roses for $20 per bouquet. How many bouquets does he need to buy to "earn" a difference of $1000?


Two smaller equal circles are tangent to a given circle, one from the inside and one from the outside, such that the arc between the points of tangency is $60^{\circ}$. The radii of the smaller circles are $r$, and the radius of the larger circle is $R$. Find the distance between the centers of the smaller circles.

Solution: \boxed{\sqrt{R^2 + 3r^2}}


Compute the limit of the function:

\[
\lim _{x \rightarrow 3} \frac{\sqrt{x+13}-2 \sqrt{x+1}}{\sqrt[3]{x^{2}-9}}
\]


Compute
$$
\sum_{\substack{a+b+c=12 \\ a \geq 6, b, c \geq 0}} \frac{a!}{b!c!(a-b-c)!},
$$
where the sum runs over all triples of nonnegative integers \((a, b, c)\) such that \(a+b+c=12\) and \(a \geq 6\).


Six consecutive numbers were written on a board. When one of them was crossed out and the remaining were summed, the result was 10085. What number could have been crossed out? Specify all possible options.


A smooth ball with a radius of 1 cm was dipped in blue paint and placed between two absolutely smooth concentric spheres with radii of 4 cm and 6 cm, respectively (the ball ended up outside the smaller sphere but inside the larger one). When touching both spheres, the ball leaves a blue mark. During its motion, the ball traveled along a closed route, resulting in a blue-bordered region on the smaller sphere with an area of 27 square cm. Find the area of the region bordered in blue on the larger sphere. Provide your answer in square centimeters, rounded to two decimal places if necessary.


It is known that the numbers \( x, y, z \) form an arithmetic progression with a common difference of \( \alpha = \arccos \left(-\frac{2}{5}\right) \), and the numbers \( 3 + \sin x, 3 + \sin y, 3 + \sin z \) form a non-constant geometric progression. Find \( \sin y \).


In how many different ways can a chess king move from square $e1$ to square $h5$, if it is only allowed to move one square to the right, upward, or diagonally right-upward?


In the regular tetrahedron \(ABCD\), take the midpoint \(M\) on the altitude \(AH\), and connect \(BM\) and \(CM\). Then \(\angle BMC =\)


If \(\sin \theta - \cos \theta = \frac{\sqrt{6} - \sqrt{2}}{2}\), find the value of \(24\left(\sin ^{3} \theta - \cos ^{3} \theta\right)^{2}\).


A regular hexagon \(ABCDEF\) has a side length of 2. A laser beam is fired inside the hexagon from point \(A\) and hits line segment \(\overline{BC}\) at point \(G\). The laser then reflects off \(\overline{BC}\) and hits the midpoint of \(\overline{DE}\). Find the length of \(BG\).


Numbers 1, 2, 3, ..., 10 are written in a circle in some order. Petya calculated the sums of all triplets of neighboring numbers and wrote the smallest of these sums on the board. What is the largest number that could have been written on the board?


Is it possible to place the numbers $1, -1, 0$ in the cells of an $8 \times 8$ square table such that all the sums in each column, each row, and each of the two diagonals are different?

Solution: \boxed{\text{No}}


In the movie "Monkey King: Hero Is Back," there is a scene where Sun Wukong battles mountain demons. Some of the demons are knocked down, such that the number of knocked-down demons is one-third more than the standing demons. After a while, 2 more demons are knocked down, but 10 demons stand back up. At this point, the number of standing demons is one-fourth more than the knocked-down demons. How many demons are standing now?


From the digits $1, 2, 3, 4, 5, 6, 7, 8, 9$, nine nine-digit numbers (not necessarily different) are formed with each digit used exactly once in each number. What is the largest number of zeros that the sum of these nine numbers can end with?


Determine all prime numbers \( p \) such that \( p^2 - 6 \) and \( p^2 + 6 \) are both prime numbers.


Suppose \( x, y \), and \( z \) are real numbers greater than 1 such that
\[
\begin{aligned}
x^{\log _{y} z} & = 2, \\
y^{\log _{y} x} & = 4, \\
z^{\log _{x} y} & = 8 .
\end{aligned}
\]
Compute \(\log _{x} y\).


Vasya replaced the same digits in two numbers with the same letters, and different digits with different letters. It turned out that the number ZARAZA is divisible by 4, and ALMAZ is divisible by 28. Find the last two digits of the sum ZARAZA + ALMAZ.


How many different two-digit numbers can be composed using the digits: a) $1, 2, 3, 4, 5, 6$; b) $0, 1, 2, 3, 4, 5, 6$?


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


In a plane rectangular coordinate system, the coordinates of two vertices of square $OABC$ are $O(0,0)$ and $A(4,3)$, and point $C$ is in the fourth quadrant. Find the coordinates of point $B$. $\qquad$

Solution: \boxed{(7, -1)}


$O A B C$ is a tetrahedron with $O A, O B$ and $O C$ being mutually perpendicular. Given that $O A = O B = O C = 6x$.

1. If the volume of $O A B C$ is $a x^{3}$, find $a$.
2. If the area of $\triangle A B C$ is $b \sqrt{3} x^{2}$, find $b$.
3. If the distance from $O$ to $\triangle A B C$ is $c \sqrt{3} x$, find $c$.
4. If $\theta$ is the angle of depression from $C$ to the midpoint of $A B$ and $\sin \theta = \frac{\sqrt{d}}{3}$, find $d$.


Calculate the limit of the function:

\[
\lim _{x \rightarrow \frac{\pi}{2}}(\cos x+1)^{\sin x}
\]


From a natural number, subtract the sum of its digits, then subtract the sum of the digits of the resulting difference. If you continue this process, with which number will the calculations end?


Call a positive integer \( N \geq 2 \) "special" if for every \( k \) such that \( 2 \leq k \leq N \), \( N \) can be expressed as a sum of \( k \) positive integers that are relatively prime to \( N \) (although not necessarily relatively prime to each other). How many special integers are there less than 100?


In an archery competition held in a certain city, after ranking the scores, the average score of the top seven participants is 3 points lower than the average score of the top four participants. The average score of the top ten participants is 4 points lower than the average score of the top seven participants. How many points more is the sum of the scores of the fifth, sixth, and seventh participants compared to the sum of the scores of the eighth, ninth, and tenth participants?


Find the point \( M^{\prime} \) that is symmetric to the point \( M \) with respect to the line.

$$
\begin{aligned}
& M(0, -3, -2) \\
& \frac{x-1}{1} = \frac{y+1.5}{-1} = \frac{z}{1}
\end{aligned}
$$

Solution: \boxed{M' = (1, 1, 1)}


In the parallelogram \(ABCD\), points \(E\) and \(F\) are located on sides \(AB\) and \(BC\) respectively, and \(M\) is the point of intersection of lines \(AF\) and \(DE\). Given that \(AE = 2BE\) and \(BF = 3CF\), find the ratio \(AM : MF\).


Five integers were written on the board. By summing them in pairs, the following ten numbers were obtained:

$$
0, 2, 4, 4, 6, 8, 9, 11, 13, 15
$$

What are the five numbers that were written on the board? Is it possible to obtain the following ten numbers by summing five integers in this way:

$$
12, 13, 14, 15, 16, 16, 17, 17, 18, 20?
$$

Solution: \boxed{\text{No}}


Find the maximum value of the expression \( (\sin 2x + \sin y + \sin 3z)(\cos 2x + \cos y + \cos 3z) \).


There is a magical grid paper of dimensions \(2000 \times 70\), initially all cells are grey. A painter starts at a certain cell and paints it red. Every second, the painter takes two steps: one cell to the left and one cell downwards, and paints the cell he lands on red. If the painter stands in the leftmost column and needs to step left, he teleports to the rightmost cell of the same row; if he stands in the bottom row and needs to step downwards, he teleports to the top cell of the same column. After several moves, the painter returns to the cell he started from. How many red cells are there on the grid at that moment?


Two sisters were picking currants: the older one was using a 4-liter bucket, and the younger one was using a 3.5-liter bucket. The older sister was always working faster than the younger one. When the older sister had collected three-quarters of her bucket, and the younger sister had collected more than half of hers, the girls exchanged buckets. Continuing to work at the same speeds as before exchanging buckets, they finished at the same time. How many times faster was the older sister picking berries compared to the younger sister?


A line parallel to the bases of a trapezoid passes through the intersection point of its diagonals. Find the length of the segment of this line that is enclosed between the non-parallel sides of the trapezoid, given that the lengths of the bases of the trapezoid are 4 cm and 12 cm.

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