The number of passengers on the bus remained constant from departure to the second stop. Then, it was observed that at each stop, as many people boarded as had boarded at the two previous stops, and as many people disembarked as had boarded at the previous stop. The tenth stop from departure is the final one, and at this stop, 55 passengers disembarked.

How many passengers were on the bus between the seventh and eighth stops?


Find all integers \( n \) such that \( n^{4} + 6n^{3} + 11n^{2} + 3n + 31 \) is a perfect square.


Given an arithmetic progression, the sum of its first 10 terms is 60, and the sum of its first 20 terms is 320. What can the 15th term of this progression be?


It is known that \(\frac{C+E+Д+b+M+O+\breve{И}}{K+Л+A+C+C}=\frac{22}{23}\),

where different letters represent different digits, and the same letters represent the same digits. What values can \(C\) take?


Find the number of positive integers \(x\) less than 100 for which
\[
3^{x} + 5^{x} + 7^{x} + 11^{x} + 13^{x} + 17^{x} + 19^{x}
\]
is prime.


There are 120 different five-digit numbers composed of the digits 1, 2, 3, 4, and 5. Arrange them in descending order. The 95th number is $\quad$


On the segment \( OA \) of length \( L \) of the number line \( OX \), two points are randomly placed: \( B(x) \) and \( C(y) \), with \( y \geq x \). Find the probability that the length of segment \( BC \) is less than the length of segment \( OB \). It is assumed that the probability of a point falling on a segment is proportional to the length of that segment and does not depend on its position on the number line.


Find the sum of all three-digit natural numbers that do not contain the digit 0 or the digit 5.


Point \( O \), located inside a convex quadrilateral with area \( S \), is reflected symmetrically with respect to the midpoints of its sides. Find the area of the quadrilateral formed by the reflection points.


Given a parallelogram \(ABCD\) with \(\angle B = 60^\circ\). Point \(O\) is the center of the circumcircle of triangle \(ABC\). Line \(BO\) intersects the bisector of the exterior angle \(\angle D\) at point \(E\). Find the ratio \(\frac{BO}{OE}\).


Given a trapezoid \(ABCD\) with bases \(AD = a\) and \(BC = b\). Points \(M\) and \(N\) lie on sides \(AB\) and \(CD\) respectively, with the segment \(MN\) parallel to the bases of the trapezoid. Diagonal \(AC\) intersects this segment at point \(O\). Find \(MN\), given that the areas of triangles \(AMO\) and \(CNO\) are equal.

Solution: \boxed{\sqrt{ab}}


Player \( A \) flips a coin \( n+1 \) times, and player \( B \) flips a coin \( n \) times. What is the probability that player \( A \) will have more heads than player \( B \) in the end?


It is known that all 'krakozyabrs' have either horns or wings (or both). From the results of a global census of 'krakozyabrs,' it was found that 20% of the 'krakozyabrs' with horns also have wings, and 25% of the 'krakozyabrs' with wings also have horns. How many 'krakozyabrs' are left in the world if it is known that their number is more than 25 but less than 35?


Find the remainder when \( 2^{100} \) is divided by 101.


Let \( P \in \mathbb{R}[X] \) be a monic polynomial of degree 2. Suppose that \( P(1) \geq P(0) + 3 \), and that \( P \) has two real roots (not necessarily distinct) \( x_1 \) and \( x_2 \). Find the smallest possible value of \( \left(x_1^2 + 1\right)\left( x_2^2 + 1\right) \).


1) \( \lim_{x \rightarrow 0} \frac{\sin 3x}{x} \)

2) \( \lim_{x \rightarrow 0} \frac{x^{2}}{1-\cos x} \)

3) \( \lim_{x \rightarrow 1} \frac{\cos \frac{\pi x}{2}}{1-x} \)

4) \( \lim_{x \rightarrow -2} \frac{x^{2}-4}{\operatorname{arctg}(x+2)} \)


Grass on an entire meadow grows uniformly dense and fast. It is known that 70 cows would eat all the grass in 24 days, and 30 cows would eat it in 60 days. How many cows would eat all the grass in 96 days?


We denote $\max(A, B, C)$ as the largest of the numbers $A, B, C$. Find the minimum value of $\max \left(x^{2} + |y|, (x + 2)^{2} + |y|, x^{2} + |y - 1|\right)$.


There are 10 boys and 15 girls sitting around a round table. It turned out that there are exactly 5 pairs of boys sitting next to each other.

How many pairs of girls are sitting next to each other?


Given 5 points, no three of which are collinear, find the probability that a randomly chosen pair of points will form the desired line.


A traveler visited a village where each person either always tells the truth or always lies. The villagers stood in a circle, and each one told the traveler whether their right-hand neighbor was truthful or deceitful. Based on these statements, the traveler uniquely determined the proportion of truthful villagers. Determine this proportion.


ABCD is a parallelogram. X is a point on the side AB and Y is a point on the side CD. AY and DX meet at R, and BY and CX meet at Q. The line QR meets BC at P and AD at S. Show that DS = BP.

Solution: \boxed{DS = BP}


On the radius $AO$ of a circle with center $O$, a point $M$ is chosen. On one side of $AO$, points $B$ and $C$ are chosen on the circumference such that $\angle AMB = \angle OMC = \alpha$. Find the length of $BC$ if the radius of the circle is $21$ and $\cos \alpha = \frac{4}{7}$.


Find all integer solutions \( x \) to the equation: \( x^{3}+(x+1)^{3}+(x+2)^{3}=(x+3)^{3} \)


In the isosceles triangle $ABC$, the angle $A$ at the base is $75^{\circ}$. The angle bisector of angle $A$ intersects the side $BC$ at point $K$. Find the distance from point $K$ to the base $AC$ if $BK = 10$.


Through the points \( A(0 ; 14) \) and \( B(0 ; 4) \), two parallel lines are drawn. The first line, passing through point \( A \), intersects the hyperbola \( y = \frac{1}{x} \) at points \( K \) and \( L \). The second line, passing through point \( B \), intersects the hyperbola \( y = \frac{1}{x} \) at points \( M \) and \( N \).

What is \( \frac{A L - A K}{B N - B M} \)?


Three trucks, "GAZ", "MAZ", and "KAMAZ", simultaneously departed from Zlatoust to Miass. KAMAZ, upon reaching Miass, immediately turned back and met MAZ at a point 18 km from Miass and met GAZ at a point 25 km from Miass. MAZ, upon reaching Miass, also immediately turned back and met GAZ at a point 8 km from Miass. What is the distance from Zlatoust to Miass?


Let \( n \) be an integer. Determine the largest possible constant \( C \) such that for all \( a_1, a_2, \ldots, a_n \geq 0 \) we have \(\left(\sum a_i\right)^{2} \geqslant C \sum_{i<j} a_i a_j\).


Let \( n \) be a natural number, and let \( a \) and \( b \) be positive real numbers such that \( a + b = 2 \). Find the minimum value of \( \frac{1}{1+a^{n}}+\frac{1}{1+b^{n}} \).


Triangle \(ABC\) has points \(E\) and \(F\) on sides \(AC\) and \(AB\), respectively. Line segments \(BE\) and \(CF\) intersect at \(X\). It is given that \(\frac{AF}{FB} = \left(\frac{AE}{EC}\right)^{2}\) and that \(X\) is the midpoint of \(BE\). Find the ratio \(CX/XF\).


What are the last two digits of the power \( 9^{h} \), given that starting from \( a=1 \):

\[
b=2^{a}, \quad c=3^{b}, \quad d=4^{c}, \quad e=5^{d}, \quad f=6^{e}, \quad g=7^{f}, \quad h=8^{g} ?
\]


When the integers \(1108 + a\), \(1453\), \(1844 + 2a\), and \(2281\) are divided by some positive integer \(n (> 1)\), they all get the same remainder \(b\). Determine the value of \(b\).


What is the largest \( n \), for which it is possible to place \( n \) points on the plane such that any 3 of them form the vertices of a right triangle?


Let \( x, y, z, A, B, C \in \mathbb{R} \), with
\[ A + B + C = \pi. \]

Define \( G_n = x^n \sin nA + y^n \sin nB + z^n \sin nC \) for \( n \in \mathbb{Z}_{+} \).

Given \( G_1 = G_2 = 0 \), find the value of \( G_n \).


A sphere and a cone have the same volume. The area of the lateral surface of the cone is \(80 \pi\) and the total surface area of the cone is \(144 \pi\). Determine the radius of the sphere.

The volume of a sphere with radius \(r\) is \(\frac{4}{3} \pi r^{3}\). The volume of a cone with radius \(r\) and height \(h\) is \(\frac{1}{3} \pi r^{2} h\). The surface area of a cone consists of two parts: its base and its lateral surface. The lateral surface area of a cone with radius \(r\) and slant height \(s\) is \(\pi r s\).


Find how many times the factorial of 25 (denoted as 25!) can be divided by 3 repeatedly until the result is no longer divisible by 3.


Calculate the area of the parallelogram formed by the vectors $a$ and $b$. Given the vectors and their magnitudes:

\[ a = 10p + q \]
\[ b = 3p - 2q \]
\[ |p| = 4 \]
\[ |q| = 1 \]
\[ (\widehat{p, q}) = \frac{\pi}{6} \]


Does there exist \(a, b \in \mathbf{R}\) such that the function \(f(x) = a x + b\) satisfies the inequality for all \(x \in [0, 2\pi]\):
$$
(f(x))^2 - \cos x \cdot f(x) < \frac{1}{4} \sin^2 x?
$$

Solution: \boxed{\text{No}}


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


In the expression \( x \cdot x + x \cdot x + \ldots + x \cdot x \) (50 terms), the placeholders have been replaced with numbers \( 1, \ldots, 100 \), each used exactly once. What is the maximum and minimum value that the resulting expression can have?


Find the minimum value of the function

$$
f(x)=4^{x}+4^{-x}-2^{x+1}-2^{1-x}+5
$$


Let \( ABCD \) be a convex quadrilateral with \( AC = 7 \) and \( BD = 17 \). Let \( M, P, N, Q \) be the midpoints of sides \( AB, BC, CD, DA \) respectively. Compute \( MN^{2} + PQ^{2} \).


Around the vertices of an equilateral triangle with a side length of 4 units, arcs of circles with radii of 1 and 5 units were drawn as shown in the figure.

What is the width of the narrowest strip between two parallel lines that can contain the resulting plane figure?

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


Find the maximum value of the function \( y = 3x + 4 \sqrt{1 - x^2} \).


Vasya thought of three natural numbers with a sum of 1003. After calculating their product, Vasya noticed that it ends in $N$ zeros. What is the maximum possible value of $N$?


In a middle school mathematics competition, three problems $A, B, C$ were posed and 25 students participated. Each student solved at least one problem. Among students who did not solve problem $A$, the number of students who solved problem $B$ is twice the number who solved problem $C$. Among students who solved problem $A$, the number of students who solved only problem $A$ is one more than the number of those who, in addition to problem $A$, solved other problems. Additionally, among students who solved only one problem, half of them did not solve problem $A$. How many students solved only problem $B$?


Given that \( x, y, z \in \mathbf{R}_{+} \) and \( x^{2} + y^{2} + z^{2} = 1 \), find the value of \( z \) when \(\frac{(z+1)^{2}}{x y z} \) reaches its minimum.

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


There are several points on a plane, all pairwise distances between which are different. Each of these points is connected to the nearest one. Can a closed polygonal chain result from this?

Solution: \boxed{\text{No}}


In a list of five numbers, the first number is 60 and the last number is 300. The product of the first three numbers is 810000, the product of the three in the middle is 2430000, and the product of the last three numbers is 8100000. Which number is third in the list?


Given a positive term sequence $\left\{a_{n}\right\}$ with the sum of the first $n$ terms being $S_{n}$, both $\left\{a_{n}\right\}$ and $\left\{\sqrt{S_{n}}\right\}$ are arithmetic sequences with a common difference $d$. Find $S_{n}$.


Find all the roots of the equation

\[ 1 - \frac{x}{1} + \frac{x(x-1)}{2!} - \frac{x(x-1)(x-2)}{3!} + \frac{x(x-1)(x-2)(x-3)}{4!} - \frac{x(x-1)(x-2)(x-3)(x-4)}{5!} + \frac{x(x-1)(x-2)(x-3)(x-4)(x-5)}{6!} = 0 \]

(Where \( n! = 1 \cdot 2 \cdot 3 \cdots n \))

In the answer, specify the sum of the found roots.


On the board, there are 101 numbers written: \(1^2, 2^2, \ldots, 101^2\). In one operation, it is allowed to erase any two numbers and write the absolute value of their difference instead.

What is the smallest number that can be obtained after 100 operations?


Find a four-digit number that is a perfect square, where the first two digits are the same and the last two digits are also the same.


What is the locus of the points of intersection of mutually perpendicular tangents drawn to the circle $x^{2}+y^{2}=32$?

Solution: \boxed{x^2 + y^2 = 64}


There are four distinct codes $A, B, C, D$ used by an intelligence station, with one code being used each week. Each week, a code is chosen randomly with equal probability from the three codes that were not used the previous week. Given that code $A$ is used in the first week, what is the probability that code $A$ is also used in the seventh week? (Express your answer as a simplified fraction.)


Consider seven-digit natural numbers in which each of the digits \(1, 2, 3, 4, 5, 6, 7\) appears exactly once.

(a) How many of these numbers have the digits from the first to the sixth in ascending order and the digits from the sixth to the seventh in descending order?

(b) How many of these numbers have the digits from the first to the fifth in ascending order and the digits from the fifth to the seventh in descending order?


Show that \(a^{2}+b^{2}+c^{2} \geq ab+bc+ca\). Find the cases of equality.

Solution: \boxed{a = b = c}


In triangle \(ABC\), \(AC < AB\), and the median \(AF\) divides the angle at \(A\) in the ratio \(1: 2\). The perpendicular from \(B\) to \(AB\) intersects line \(AF\) at point \(D\). Show that \(AD = 2 AC\).

Solution: \boxed{AD = 2AC}


In how many ways can the number 1500 be represented as the product of three natural numbers (variations where the factors are the same but in different orders are considered identical)?


Find the number of natural numbers \( k \), not exceeding 454500, such that \( k^2 - k \) is divisible by 505.


How many eight-digit numbers can be written using only the digits 1, 2, and 3 such that the difference between any two adjacent digits is 1?


Given the sequence \(\left\{a_{n}\right\}\) satisfying \(a_{1}=a > 2\), \(a_{2017} = 2017\), and for any positive integer \(n\), \(a_{n+1} = a_{n}^{2} - 2\). Compute \(\left\lfloor \frac{\sqrt{a-2}}{10^{6}} a_{1} a_{2} \cdots a_{2017} \right\rfloor\), where \(\lfloor x \rfloor\) denotes the greatest integer not exceeding the real number \(x\).


Young marketer Masha was supposed to interview 50 customers in an electronics store throughout the day. However, there were fewer customers in the store that day. What is the maximum number of customers Masha could have interviewed if, according to her data, 7 of those interviewed made an impulsive purchase, $75\%$ of the remaining respondents bought electronics influenced by advertising, and the number of customers who chose a product based on the advice of a sales consultant is one-third of the number who chose a product influenced by advertising?


The diagonals of the cyclic quadrilateral $ABCD$ intersect at point $O$. Within triangle $AOB$, a point $K$ is chosen such that line $KO$ is the angle bisector of $\angle CK$. Ray $DK$ intersects the circumcircle of triangle $COK$ again at point $L$, and ray $CK$ intersects the circumcircle of triangle $DOK$ again at point $M$. Find the ratio of the areas of triangles $ALO$ and $BMO$.


Given a sequence of positive integers $\left\{a_{n}\right\}$ satisfying
$$
a_{n+2}=a_{n+1}+a_{n}\ \ (n \in \mathbf{Z}_{+}),
$$
and knowing that $a_{11}=157$, find the value of $a_{1}$.


Given the sequence $\left\{a_{n}\right\}$ such that $a_{1}=1$ and $a_{n+1}=\frac{1}{8} a_{n}^{2}+m$ for $n \in \mathbf{N}^{*}$, if $a_{n}<4$ for any positive integer $n$, find the maximum value of the real number $m$.


Find all functions \( f: \mathbb{N}^{*} \rightarrow \mathbb{N}^{*} \) such that for all positive natural numbers \( m \) and \( n \),

$$
f\left(f^{2}(m)+2 f^{2}(n)\right)=m^{2}+2 n^{2}
$$

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


Find a natural number \( N \) that is divisible by 5 and 49, and has exactly 10 divisors, including 1 and \( N \).


One thousand points form the vertices of a convex 1000-sided polygon, with an additional 500 points inside such that no three of the 500 are collinear. This 1000-sided polygon is triangulated in such a way that all the given 1500 points are vertices of the triangles and these triangles have no other vertices. How many triangles are formed by this triangulation?


In $\triangle ABC$, $\angle A = 60^{\circ}$, $AB > AC$, point $O$ is the circumcenter, and the altitudes $BE$ and $CF$ intersect at point $H$. Points $M$ and $N$ lie on segments $BH$ and $HF$ respectively, such that $BM = CN$. Find the value of $\frac{MH + NH}{OH}$.


Let \( n \) be a natural number. Roger has a square garden of size \( (2n + 1) \times (2n + 1) \). He erects fences to divide it into rectangular beds. He wants exactly two horizontal \( k \times 1 \) beds and exactly two vertical \( 1 \times k \) beds for every even number \( k \) between 1 and \( 2n + 1 \), as well as a single square bed of size \( 1 \times 1 \) when he is finished. How many different ways can Roger divide his garden?


What is the minimum number of cells that need to be marked in a $7 \times 7$ grid so that in each vertical or horizontal $1 \times 4$ strip there is at least one marked cell?


A horse stands at the corner of a chessboard, on a white square. With each jump, the horse can move either two squares horizontally and one vertically or two vertically and one horizontally, like a knight moves. The horse earns two carrots every time it lands on a black square, but it must pay a carrot in rent to the rabbit who owns the chessboard for every move it makes. When the horse reaches the square on which it began, it can leave. What is the maximum number of carrots the horse can earn without touching any square more than twice?


Let \( P \) be any point inside a regular tetrahedron \( ABCD \) with side length \( \sqrt{2} \). The distances from point \( P \) to the four faces are \( d_1, d_2, d_3, d_4 \) respectively. What is the minimum value of \( d_1^2 + d_2^2 + d_3^2 + d_4^2 \)?


There is an $n \times n$ table in which $n-1$ cells contain ones, and the remaining cells contain zeros. The following operation is allowed with the table: choose a cell, subtract one from the number in that cell, and add one to all the other numbers in the same row or column as the chosen cell. Is it possible to obtain a table where all numbers are equal using these operations?

Solution: \boxed{\text{No}}


In quadrilateral \(ABCD\), points \(X, Y, Z\) are the midpoints of segments \(AB, AD, BC\), respectively. It is known that \(XY\) is perpendicular to \(AB\), \(YZ\) is perpendicular to \(BC\), and the measure of angle \(ABC\) is \(100^\circ\). Find the measure of angle \(ACD\).


An isosceles trapezoid \(ABCD\) is inscribed in a circle with diameter \(AD\) and center at point \(O\). A circle with center at point \(I\) is inscribed in the triangle \(BOC\). Find the ratio of the areas of triangles \(AID\) and \(BIC\) given that \(AD = 15\) and \(BC = 5\).


In the full permutation of 4 $x$, 3 $y$, and 2 $z$, determine the number of permutations where the patterns $x x x x$, $y y y$, and $z z$ do not appear. Let $A_{1}$ represent the set of permutations of 9 characters where the pattern $x x x x$ appears, $A_{2}$ represent the set where the pattern $y y y$ appears, and $A_{3}$ represent the set where the pattern $z z$ appears.


Into how many parts do the face planes divide the space of a) a cube; b) a tetrahedron?


Find all functions \( f: \mathbb{R} \to \mathbb{R} \) such that for all \( x \in \mathbb{R}^{*} \) (nonzero real numbers) and all \( y \in \mathbb{R} \),

$$
f\left(x^{2} + y\right) \geq \left(\frac{1}{x} + 1\right) f(y)
$$

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


Two Hungarian teams, Bolyai TC and Eötvös TK, have reached the top 16 teams in the European Cup. What is the probability that they will play against each other? (The competition follows a single-elimination format; in each match, one team progresses while the other is eliminated.)


Given a triangle \(ABC\) with the midpoints of sides \(BC\), \(AC\), and \(AB\) denoted as \(D\), \(E\), and \(F\) respectively, it is known that the medians \(AD\) and \(BE\) are perpendicular to each other, with lengths \(\overline{AD} = 18\) and \(\overline{BE} = 13.5\). Calculate the length of the third median \(CF\) of this triangle.


The function \( y = f(x) \) is defined on the set \( (0, +\infty) \) and takes positive values on it. It is known that for any points \( A \) and \( B \) on the graph of the function, the areas of the triangle \( AOB \) and the trapezoid \( ABH_BH_A \) are equal (where \( H_A \) and \( H_B \) are the bases of the perpendiculars dropped from points \( A \) and \( B \) onto the x-axis; \( O \) is the origin). Find all such functions.

Given \( f(1) = 4 \), find the value of \( f(4) \).


A perfect square greater than 1 sometimes has the property that the sum of all its positive divisors is also a perfect square. For example, the sum of all positive divisors of \(9^{2}\) (which are \(1, 3, 9, 27, 81\)) is 121, which equals \(11^{2}\). Find another perfect square with the aforementioned property.


Find all prime numbers \( p, q, r \) such that the equation \( p^3 = p^2 + q^2 + r^2 \) holds.

Solution: \boxed{p = 3, q = 3, r = 3}


Bag $A$ contains 2 ten-yuan bills and 3 one-yuan bills, and bag $B$ contains 4 five-yuan bills and 3 one-yuan bills. If two bills are randomly drawn from each bag, what is the probability that the sum of the denominations of the remaining bills in bag $A$ is greater than the sum of the denominations of the remaining bills in bag $B$?


Let \( S \) be the set of all rational numbers in \(\left(0, \frac{5}{8}\right)\). For each reduced fraction \(\frac{q}{p} \in S\) where \(\gcd(p, q) = 1\), define the function \( f(q p) = \frac{q+1}{p} \). Determine the number of solutions to \( f(x) = \frac{2}{3} \) in \( S \).


If the equation \( x^{3} - 3x^{2} - 9x = a \) has exactly two different real roots in the interval \([-2, 3]\), then the range of the real number \( a \) is \(\quad\) .

Solution: \boxed{[-2, 5)}


Let \( n \) (\( \geqslant 3 \)) be a positive integer, and \( M \) be an \( n \)-element set. Find the largest positive integer \( k \) that satisfies the following condition:

There exists a family \( \psi \) of \( k \) three-element subsets of \( M \) such that the intersection of any two elements in \( \psi \) (note: the elements of \( \psi \) are all three-element sets) is nonempty.


In an isosceles trapezoid \(ABCD\) (\(BC \parallel AD\)), the angles \(ABD\) and \(DBC\) are \(135^{\circ}\) and \(15^{\circ}\) respectively, and \(BD = \sqrt{6}\). Find the perimeter of the trapezoid.

Solution: \boxed{9 - \sqrt{3}}


Vitya and Masha were born in the same year in June. Find the probability that Vitya is at least one day older than Masha.


Neznaika wrote several different natural numbers on the board and divided (in his mind) the sum of these numbers by their product. After that, Neznaika erased the smallest number and divided (again in his mind) the sum of the remaining numbers by their product. The second result turned out to be 3 times larger than the first. What number did Neznaika erase?


Let the set
\[ \left\{\left.\frac{3}{a}+b \right\rvert\, 1 \leqslant a \leqslant b \leqslant 4\right\} \]
have maximum and minimum elements denoted by \( M \) and \( N \) respectively. Find \( M - N \).

Solution: \boxed{7 - 2\sqrt{3}}


Calculate:

$$\frac{\left(1+\frac{1}{2}\right)^{2} \times\left(1+\frac{1}{3}\right)^{2} \times\left(1+\frac{1}{4}\right)^{2} \times\left(1+\frac{1}{5}\right)^{2} \times \cdots \times\left(1+\frac{1}{10}\right)^{2}}{\left(1-\frac{1}{2^{2}}\right) \times\left(1-\frac{1}{3^{2}}\right) \times\left(1-\frac{1}{4^{2}}\right) \times\left(1-\frac{1}{5^{2}}\right) \times \cdots \times\left(1-\frac{1}{10^{2}}\right)}$$


Juquinha marks points on the circumference and draws triangles by connecting 3 of these points. The lengths of the arcs between 2 consecutive points are equal.

a) By marking 4 points on the circumference, how many triangles can he draw?

b) By marking 5 points on the circumference, how many equilateral triangles can he draw?

c) How many right triangles can he draw if he marks 6 points?


30 students from five courses created 40 problems for the olympiad, with students from the same course creating the same number of problems, and students from different courses creating different numbers of problems. How many students created exactly one problem?


There are 13 weights arranged in a row on a table, ordered by mass (the lightest on the left, the heaviest on the right). It is known that the mass of each weight is an integer number of grams, the masses of any two adjacent weights differ by no more than 5 grams, and the total mass of the weights does not exceed 2019 grams. Find the maximum possible mass of the heaviest weight under these conditions.


Given a real number \( t \), find all functions \( f: \mathbf{R} \rightarrow \mathbf{R} \) such that
\[ f(x + t + f(y)) = f(f(x)) + f(t) + y. \]

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


\[
\frac{\sqrt{c-d}}{c^{2} \sqrt{2c}} \cdot \left( \sqrt{\frac{c-d}{c+d}} + \sqrt{\frac{c^{2}+cd}{c^{2}-cd}} \right) \quad \text{where} \quad c=2 \quad \text{and} \quad d=\frac{1}{4}.
\]


Given a strictly increasing function \( f: \mathbb{N}_0 \to \mathbb{N}_0 \) (where \( \mathbb{N}_0 \) is the set of non-negative integers) that satisfies the relation \( f(n + f(m)) = f(n) + m + 1 \) for any \( m, n \in \mathbb{N}_0 \), find all possible values of \( f(2023) \).


In triangle \( \triangle ABC \), if \( \frac{\cos A}{\sin B} + \frac{\cos B}{\sin A} = 2 \), and the perimeter of \( \triangle ABC \) is 12, find the maximum possible value of its area.

Solution: \boxed{36(3 - 2\sqrt{2})}


Find the smallest integer \( n \) such that the expanded form of \( (xy - 7x - 3y + 21)^n \) has 2012 terms.


In a rectangular parallelepiped \( ABCD A_{1} B_{1} C_{1} D_{1} \), the edge lengths are given by \( AB = 42 \), \( AD = 126 \), and \( AA_{1} = 42 \). Point \( E \) is marked at the midpoint of edge \( A_{1}B_{1} \), and point \( F \) is marked at the midpoint of edge \( B_{1}C_{1} \). Find the distance between the lines \( AE \) and \( BF \).


The second term of a geometric sequence is \( b_{2} = 24.5 \), and the fifth term is \( b_{5} = 196 \). Find the third term and the sum of the first four terms.


The orthogonal projections of the triangle \(ABC\) onto two mutually perpendicular planes are equilateral triangles with sides of length 1. Find the perimeter of triangle \(ABC\), given that \(AB = \frac{\sqrt{5}}{2}\).

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


Find all quadratic trinomials \( p(x) \) that attain a minimum value of \(-\frac{49}{4}\) at \(x=\frac{1}{2}\), and the sum of the fourth powers of its roots equals 337.

Solution: \boxed{p(x) = x^2 - x - 12}


Given that \( c \) cannot be divided by the square of a prime number, if \( b^{2} c \) can be divided by \( a^{2} \), then \( b \) can be divided by \( a \).

Solution: \boxed{a \mid b}


A ball was thrown vertically upward from a balcony. It is known that it fell to the ground after 3 seconds. Knowing that the initial speed of the ball was 5 m/s, determine the height of the balcony. The acceleration due to gravity is $10 \mathrm{~m/s}^2$.

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


Show that the perimeter of a triangle is related to the perimeter of its pedal triangle as \( R : r \), where \( R \) is the radius of the circumcircle of the original triangle, and \( r \) is the radius of the inscribed circle of the triangle.

Solution: \boxed{K:k = R:r}


The regular octagon \( A B C D E F G H \) is inscribed in a circle. Points \( P \) and \( Q \) are on the circle, with \( P \) between \( C \) and \( D \), such that \( A P Q \) is an equilateral triangle. It is possible to inscribe a regular \( n \)-sided polygon, one of whose sides is \( P D \), in the circle. What is the value of \( n \)?


In an equilateral triangle $\triangle ABC$ with unit area, external equilateral triangles $\triangle APB$, $\triangle BQC$, and $\triangle CRA$ are constructed such that $\angle APB = \angle BQC = \angle CRA = 60^\circ$.
1. Find the maximum area of triangle $\triangle PQR$.
2. Find the maximum area of the triangle whose vertices are the incenters of $\triangle APB$, $\triangle BQC$, and $\triangle CRA$.


The real-valued function \( f \) is defined on the reals and satisfies \( f(xy) = x f(y) + y f(x) \) and \( f(x + y) = f(x \cdot 1993) + f(y \cdot 1993) \) for all \( x \) and \( y \). Find \( f(\sqrt{5753}) \).


On the coordinate plane with origin at point \( O \), a parabola \( y = x^2 \) is drawn. Points \( A \) and \( B \) are marked on the parabola such that \( \angle AOB \) is a right angle. Find the minimum possible area of triangle \( AOB \).


Given non-negative real numbers $a_{1}, a_{2}, \cdots, a_{2008}$ whose sum equals 1, determine the maximum value of $a_{1} a_{2} + a_{2} a_{3} + \cdots + a_{2007} a_{2008} + a_{2008} a_{1}$.


In the center of a circular field, there is a geologists' hut. From it, 6 straight roads radiate out, dividing the field into 6 equal sectors. Two geologists leave their hut at a speed of 4 km/h, each choosing a road at random. Determine the probability that the distance between them after one hour will be at least 6 km.


In tetrahedron \(ABCD\), \(\triangle ABD\) is an equilateral triangle, \(\angle BCD = 90^\circ\), \(BC = CD = 1\), \(AC = \sqrt{3}\). Points \(E\) and \(F\) are the midpoints of \(BD\) and \(AC\) respectively. Determine the cosine of the angle between lines \(AE\) and \(BF\).


Given that \( f(n) = \sin \frac{n \pi}{4} \), where \( n \) is an integer. If \( c = f(1) + f(2) + \ldots + f(2003) \), find the value of \( c \).


In how many ways can we place pawns on a \(4 \times 4\) chessboard such that each row and each column contains exactly two pawns?


How many ways are there to arrange the numbers {1, 2, 3, 4, 5, 6, 7, 8} in a circle so that every two adjacent elements are relatively prime? Consider rotations and reflections of the same arrangement to be indistinguishable.


What is the smallest possible sum of nine consecutive natural numbers if this sum ends in 3040102?


The equation \( x^{2}-x+1=0 \) has roots \(\alpha\) and \(\beta\). There is a quadratic function \( f(x) \) that satisfies \( f(\alpha) = \beta \), \( f(\beta) = \alpha \), and \( f(1) = 1 \). What is the expression for \( f(x) \)?

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


Let \( f(x) = \sin^6\left(\frac{x}{4}\right) + \cos^6\left(\frac{x}{4}\right) \) for all real numbers \( x \). Determine \( f^{(2008)}(0) \) (i.e., \( f \) differentiated 2008 times and then evaluated at \( x = 0 \)).


There are 2019 numbers written on the board. One of them occurs more frequently than the others - 10 times. What is the minimum number of different numbers that could be written on the board?


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

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

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


Tessa has a figure created by adding a semicircle of radius 1 on each side of an equilateral triangle with side length 2, with the semicircles oriented outwards. She then marks two points on the boundary of the figure. What is the greatest possible distance between the two points?


Let \( a \) and \( c \) be positive integers, and let \( b \) be a digit. Determine all triples \( (a, b, c) \) that satisfy the following conditions:

(1) \( (a, b b b \ldots)^{2}=c.777 \ldots \) (infinite decimals);

(2) \( \frac{c+a}{c-a} \) is an integer!

Solution: \boxed{(1, 6, 2)}


The captain's assistant, who had been overseeing the loading of the ship, was smoking one pipe after another from the very start of the process. When $2 / 3$ of the number of loaded containers became equal to $4 / 9$ of the number of unloaded containers, and noon struck, the old sea wolf started smoking his next pipe. When he finished this pipe, the ratio of the number of loaded containers to the number of unloaded containers reversed the ratio that existed before he began smoking this pipe. How many pipes did the second assistant smoke during the loading period (assuming the loading rate and the smoking rate remained constant throughout)?


We can label the squares of an 8 x 8 chessboard from 1 to 64 in 64! different ways. For each way, we find \( D \), the largest difference between the labels of two squares that are adjacent (orthogonally or diagonally). What is the smallest possible \( D \)?


If \(a, b,\) and \(c\) are all multiples of 5, \( a < b < c \), \( c = a + 10 \), then calculate \( \frac{(a-b)(a-c)}{b-c} \).


A polynomial \( P(x) \) of degree 10 with a leading coefficient of 1 is given. The graph of \( y = P(x) \) lies entirely above the x-axis. The polynomial \( -P(x) \) was factored into irreducible factors (i.e., polynomials that cannot be represented as the product of two non-constant polynomials). It is known that at \( x = 2020 \), all the resulting irreducible polynomials take the value -3. Find \( P(2020) \).


For what value of \( a \) does the equation \( |x-2|=a x-2 \) have an infinite number of solutions?


What is the radius of the smallest circle into which any system of points with a diameter of 1 can be enclosed?


Given $\cos \alpha = \tan \beta$, $\cos \beta = \tan \gamma$, $\cos \gamma = \tan \alpha$, find that $\sin^2 \alpha = \sin^2 \beta = \sin^2 \gamma = \cos^4 \alpha = \cos^4 \beta = \cos^4 \gamma = 4 \sin^2 18^\circ$.

Solution: \boxed{4 \sin^2 18^\circ}


The bases \( AB \) and \( CD \) of trapezoid \( ABCD \) are 101 and 20 respectively, and its diagonals are mutually perpendicular. Find the scalar product of the vectors \(\overrightarrow{AD}\) and \(\overrightarrow{BC}\).


Given a parallelogram \(ABCD\) where \(AB < AC < BC\). Points \(E\) and \(F\) are chosen on the circumcircle \(\omega\) of triangle \(ABC\) such that the tangents to \(\omega\) at these points pass through \(D\). Moreover, segments \(AD\) and \(CE\) intersect. It is found that \(\angle ABF = \angle DCE\). Find the angle \(\angle ABC\).


The base of a quadrilateral pyramid $S A B C D$ is a parallelogram $A B C D$. 1) Construct a cross-section of the pyramid by a plane passing through the midpoint of edge $A B$ and parallel to plane $S A D$. 2) Find the area of the resulting cross-section if the area of face $S A D$ is 16.


Let $2 \leq r \leq \frac{n}{2}$, and let $\mathscr{A}$ be a family of $r$-element subsets of $Z = \{1, 2, \ldots, n\}$. If $\mathscr{A} = \{A \mid A$ is an $r$-element subset of $Z$ that contains a fixed element $x \in Z\}$, then equality holds.

Solution: \boxed{ \binom{n-1}{r-1} }


A trapezoid $ABCD$ is circumscribed around a circle, with side $AB$ perpendicular to the bases, and $M$ is the point of intersection of the diagonals of the trapezoid. The area of triangle $CMD$ is $S$. Find the radius of the circle.

Solution: \boxed{R = \sqrt{S}}


A material point moves in such a way that its velocity is proportional to the distance traveled. At the initial moment, the point was at a distance of 1 meter from the origin, and after 2 seconds, it was at a distance of $e$ meters. Find the equation of motion for the material point.

Solution: \boxed{s = e^{\frac{t}{2}}}


In a grove, there are four types of trees: birches, spruces, pines, and aspens. There are 100 trees in total. It is known that among any 85 trees, there are trees of all four types. What is the smallest number of any trees in this grove that must include trees of at least three types?


The store purchased a batch of greeting cards at the price of 0.21 yuan each and sold them for a total of 14.57 yuan. If each card is sold at the same price and the selling price does not exceed twice the purchase price, how much did the store profit?


Let \(\Omega\) be a sphere of radius 4 and \(\Gamma\) be a sphere of radius 2. Suppose that the center of \(\Gamma\) lies on the surface of \(\Omega\). The intersection of the surfaces of \(\Omega\) and \(\Gamma\) is a circle. Compute this circle's circumference.

Solution: \boxed{\pi \sqrt{15}}


Solution: \boxed{x = 1}


It is known that \(\log_{10}\left(2007^{2006} \times 2006^{2007}\right) = a \times 10^k\), where \(1 \leq a < 10\) and \(k\) is an integer. Find the value of \(k\).


The number of lattice points (points with integer coordinates) inside the region bounded by the right branch of the hyperbola \( x^{2}-y^{2}=1 \) and the line \( x=100 \), excluding the boundary, is \(\qquad\) .


What is the smallest square number whose first five digits are 4 and the sixth digit is 5?


Given that \( f(x) \) is a periodic function on \(\mathbf{R}\) with the smallest positive period of 2, and when \( 0 \leq x < 2 \), \( f(x) = x^3 - x \), determine the number of points where the graph of the function \( f(x) \) intersects the x-axis in the interval \([0, 6]\).


Find the smallest natural number that ends with the digit 6 such that moving this digit to the front increases the number exactly fourfold.


Little children were eating candies. Each ate 7 candies less than all the others combined but still more than one candy.

How many candies were eaten in total?

Solution: \boxed{21 \text{ candies}}


A die is rolled three times. The first result is $a$, the second is $b$, and the third is $c$. Assuming each number from 1 to 6 is equally likely to appear:

1. Calculate the probability that $a b + 2 c \geq a b c$.
2. Calculate the probability that $a b + 2 c$ and $2 a b c$ are coprime.


What is the maximum number of cells that can be colored on a $6 \times 6$ board such that no four colored cells can be selected whose centers form a rectangle with sides parallel to the sides of the board?


Given that \(\log _{2} a \cdot \log _{2} b=1\) (where \(a>1\) and \(b>1\)), find the minimum value of \(a b\).


The remainder when a certain natural number \( n \) is divided by 22 is 7, and the remainder when \( n \) is divided by 33 is 18. Find the remainder when \( n \) is divided by 66.


The graphs \( y = 2 \cos 3x + 1 \) and \( y = - \cos 2x \) intersect at many points. Two of these points, \( P \) and \( Q \), have \( x \)-coordinates between \(\frac{17 \pi}{4}\) and \(\frac{21 \pi}{4}\). The line through \( P \) and \( Q \) intersects the \( x \)-axis at \( B \) and the \( y \)-axis at \( A \). If \( O \) is the origin, what is the area of \( \triangle BOA \)?


In triangle $ABC$, the angles $\angle B = 30^{\circ}$ and $\angle A = 90^{\circ}$ are known. Point $K$ is marked on side $AC$, and points $L$ and $M$ are marked on side $BC$ such that $KL = KM$ (point $L$ lies on segment $BM$).

Find the length of segment $LM$ if it is known that $AK = 4$, $BL = 31$, and $MC = 3$.


There is a unique quadruple of positive integers \((a, b, c, k)\) such that \(c\) is not a perfect square and \(a + \sqrt{b + \sqrt{c}}\) is a root of the polynomial \(x^4 - 20x^3 + 108x^2 - kx + 9\). Compute \(c\).


Which members of the sequence 101, 10101, 1010101, ... are prime?


For which values of the parameter \( a \) does the equation

$$
5^{x^{2}+2ax+a^{2}} = ax^{2}+2a^{2}x+a^{3}+a^{2}-6a+6
$$

have exactly one solution?

Solution: \boxed{a = 1}


Consider all functions \( f: \mathbb{Z} \rightarrow \mathbb{Z} \) satisfying

\[ f(f(x)+2x+20) = 15. \]

Call an integer \( n \) good if \( f(n) \) can take any integer value. In other words, if we fix \( n \), for any integer \( m \), there exists a function \( f \) such that \( f(n) = m \). Find the sum of all good integers \( x \).


Find the smallest term in the following sequence:
\[ a_{1} = 1993^{1094^{1995}}, \]
\[ a_{n+1} = \begin{cases}
\frac{1}{2} a_{n}, & \text{if } a_{n} \text{ is even}, \\
a_{n} + 7, & \text{if } a_{n} \text{ is odd}.
\end{cases} \]


Point $M$ lies on the leg $AC$ of the right triangle $ABC$ with a right angle at $C$, such that $AM=2$ and $MC=6$. Segment $MH$ is the altitude of triangle $AMB$. Point $D$ is located on the line $MH$ such that the angle $ADB$ is $90^{\circ}$, and points $C$ and $D$ lie on the same side of line $AB$. Find the length of segment $DC$ if the tangent of the angle $ACH$ is $1 / 7$. (16 points)


Solve the system of inequalities

$$
\left\{\begin{array}{l}
|2 x-3| \leqslant 3 \\
\frac{1}{x}<1
\end{array}\right.
$$

Solution: \boxed{(1, 3]}


There are 26 stamp collectors from different countries who want to exchange the latest commemorative stamps of each country through mutual correspondence. To ensure that each of the 26 individuals ends up possessing a complete set of the latest commemorative stamps from all 26 countries, at least how many letters need to be exchanged?


Ken is the best sugar cube retailer in the nation. Trevor, who loves sugar, is coming over to make an order. Ken knows Trevor cannot afford more than 127 sugar cubes, but might ask for any number of cubes less than or equal to that. Ken prepares seven cups of cubes, with which he can satisfy any order Trevor might make. How many cubes are in the cup with the most sugar?


The function \( f(x) \) satisfies \( f(x) + f(1-x) = \frac{1}{2} \) for all \( x \in \mathbb{R} \).

(1) Let the sequence \( \{a_n\} \) be defined as:
\[ a_n = f(0) + f\left(\frac{1}{n}\right) + f\left(\frac{2}{n}\right) + \cdots + f\left(\frac{n-1}{n}\right) + f(1) \]
Is the sequence \( \{a_n\} \) an arithmetic sequence? Please provide a proof.

(2) Let
\[ b_n = \frac{4}{4a_n - 1}, \quad T_n = b_1^2 + b_2^2 + b_3^2 + \cdots + b_n^2, \quad S_n = 32 - \frac{16}{n} \]
Compare the values of \( T_n \) and \( S_n \).

Solution: \boxed{T_n \leq S_n}


Given that \(ab = 1000\) with \(a > 1\) and \(b > 1\), what is the maximum value of \(\sqrt{1 + \lg a} + \sqrt{1 + \lg b}\)?


In the pentagon \( U V W X Y \), \(\angle U = \angle V = 90^\circ \), \( U Y = V W \), and \( Y X = X W \). Four equally spaced points are marked between \( U \) and \( V \), and perpendiculars are drawn through each point. The dark shaded region has an area of \( 13 \mathrm{~cm}^2 \) and the light shaded region has an area of \( 10 \mathrm{~cm}^2 \). What is the area, in \(\mathrm{cm}^2\), of the entire pentagon?
A) 45
B) 47
C) 49
D) 58
E) 60


Find all pairs \((a, b)\) of integers such that \(\sqrt{2010+2 \sqrt{2009}}\) is a solution of the quadratic equation \(x^{2} + ax + b = 0\).

Solution: \boxed{( - 2 ,  - 2008 )}


At the World Meteorological Conference, each participant successively announced the average monthly temperature in their hometown. Meanwhile, all others recorded the product of the temperatures in their and the speaking participant's hometown. In total, 42 positive and 48 negative numbers were recorded. What is the minimum number of times a positive temperature could have been announced?


On a $4 \times 4$ board, numbers from 1 to 16 must be placed in the squares, without repetition, so that the sum of the numbers in each row, column, and diagonal is the same. We call this sum the Magic Sum.

a) What is the Magic Sum for this board?

b) If the sum of the squares marked with $X$ on the board below is 34, what is the sum of the squares marked with $Y$?

| $Y$ |  |  | $Y$ |
| :--- | :--- | :--- | :--- |
|  | $X$ | $X$ |  |
|  | $X$ | $X$ |  |
| $Y$ |  |  | $Y$ |

c) If we fill the board with consecutive natural numbers from $k$ to $(k+15)$, so that the Magic Sum is 50, what is the value of $k$?


In tetrahedron \(ABCD\), \(AD = 2\sqrt{3}\), \(\angle BAC = 60^\circ\), \(\angle BAD = \angle CAD = 45^\circ\). If a sphere that is tangent to plane \(ABC\) at point \(D\) and is internally tangent to the circumsphere of the tetrahedron has a radius of 1, find the radius of the circumsphere of tetrahedron \(ABCD\).


263. Military Puzzle. An officer ordered the soldiers to form 12 rows with 11 people in each row in such a way that he could stand at a point equidistant from each row.

"Sir, there are only 120 of us," said one of the soldiers.

Is it possible to fulfill the officer's order?

Solution: \boxed{Yes}


We construct pyramids perpendicular to each face of a given cube. What is the volume of the resulting solid if all its vertices lie on a spherical surface and the length of each edge of the cube is $a$?

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


$E$ is the midpoint of side $BC$ of parallelogram $ABCD$. Line $AE$ intersects the diagonal $BD$ at point $G$. If the area of triangle $\triangle BEG$ is 1, find the area of parallelogram $ABCD$.


When the product
$$
\left(2021 x^{2021}+2020 x^{2020}+\cdots+3 x^{3}+2 x^{2}+x\right)\left(x^{2021}-x^{2020}+\cdots+x^{3}-x^{2}+x-1\right)
$$
is expanded and simplified, what is the coefficient of \(x^{2021}\)?


Determine the value of \( a \) such that the polynomial \( x^{n} - a x^{n-1} + a x - 1 \) is divisible by \( (x-1)^{2} \).


The germination rate of the seeds of a given plant is $90\%$. Find the probability that out of four planted seeds: a) three will germinate; b) at least three will germinate.


How many digits does the number \(2^{100}\) have? What are its last three digits? (Give the answers without calculating the power directly or using logarithms!) If necessary, how could the power be quickly calculated?


On the extensions of the medians \(A K\), \(B L\), and \(C M\) of triangle \(A B C\), points \(P\), \(Q\), and \(R\) are taken such that \(K P = \frac{1}{2} A K\), \(L Q = \frac{1}{2} B L\), and \(M R = \frac{1}{2} C M\). Find the area of triangle \(P Q R\) if the area of triangle \(A B C\) is 1.


A group of toddlers in a kindergarten collectively has 90 teeth. Any two toddlers together have no more than 9 teeth. What is the minimum number of toddlers that can be in the group?


Suppose that \( x, y \), and \( z \) are complex numbers of equal magnitude that satisfy
\[ x + y + z = -\frac{\sqrt{3}}{2} - i \sqrt{5} \]
and
\[ x y z = \sqrt{3} + i \sqrt{5}. \]
If \( x = x_{1} + i x_{2}, y = y_{1} + i y_{2} \), and \( z = z_{1} + i z_{2} \) for real \( x_{1}, x_{2}, y_{1}, y_{2}, z_{1} \), and \( z_{2} \), then
\[
\left(x_{1} x_{2} + y_{1} y_{2} + z_{1} z_{2}\right)^{2}
\]
can be written as \(\frac{a}{b}\) for relatively prime positive integers \( a \) and \( b \). Compute \( 100a + b \).


Suppose that \( f_{1}(x)=\frac{1}{2-x} \). For each positive integer \( n \geq 2 \), define \( f_{n}(x)=f_{1}\left(f_{n-1}(x)\right) \) for all real numbers \( x \) in the domain of \( f_{1}\left(f_{n-1}(x)\right) \). The value of \( f_{2019}(4) \) can be written as \( \frac{a}{b} \) where \( a \) and \( b \) are positive integers with no common divisor larger than 1. What is \( (a, b) \)?

Solution: \boxed{(6053, 6056)}


"That" and "this," and half of "that" and "this" - how many percent is this of three-quarters of "that" and "this"?


Let \( a, b, c \) be positive integers such that \( \frac{1}{a} + \frac{1}{b} = \frac{1}{c} \) and \( \operatorname{gcd}(a, b, c) = 1 \). Suppose \( a + b \leq 2011 \). Determine the largest possible value of \( a + b \).


The hares are sawing a log again, but this time both ends of the log are fixed. Ten intermediate pieces have fallen, while two end pieces remain fixed. How many cuts did the hares make?


A convex quadrilateral has three sides measuring $1 \mathrm{~cm}$, $4 \mathrm{~cm}$, and $8 \mathrm{~cm}$. Its diagonals are perpendicular to each other. What is the length of the fourth side?


Given a tetrahedron $ABCD$ with an internal point $P$, what is the minimum number of its edges that appear obtuse when viewed from point $P$?


There are eight ways to evaluate the expression " $\pm 2^{11} \pm 2^{5} \pm 2$ ". When these eight values are listed in decreasing order, what is the third value in the list?


We have $3^{2k}$ identical-looking coins, one of which is counterfeit and weighs slightly less than a real one. Additionally, we have three two-pan balances. It is known that two of the balances are accurate, while one is faulty (the result shown by the faulty balance has no connection to the actual weights of the coins placed on them and can be either correct or arbitrarily incorrect for different weighings). We do not know which balance is faulty. How can we identify the counterfeit coin in $3k + 1$ weighings?


Given the hyperbola \( P : \frac{x^{2}}{9}-\frac{y^{2}}{16}=1 \) with left and right foci \( B \) and \( C \), point \( A \) lies on \( P \). \( I \) is the incenter of triangle \( ABC \) and the line \( AI \) passes through the point \( (1,0) \). If \( \overrightarrow{A I}=x \overrightarrow{A B}+y \overrightarrow{A C} \), then \( x+y \) equals ____.


Find all functions \( f: \mathbf{Q} \rightarrow \mathbf{Q} \) (where \( \mathbf{Q} \) is the set of rational numbers) such that:
1. \( f(1) = 2 \)
2. For any \( x, y \in \mathbf{Q} \),
\[ f(xy) = f(x) f(y) - f(x + y) + 1 \]

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


As shown in the figure, the hyperbola function \( y = \frac{k}{x} \) where \( k > 0 \), passes through the midpoint \( D \) of the hypotenuse \( OB \) of the right triangle \( OAB \) and intersects the leg \( AB \) at point \( C \). If the area of \( \triangle OBC \) is 3, then find the value of \( k \).


Find the maximum constant \( k \) such that for all real numbers \( a, b, c, d \) in the interval \([0,1]\), the inequality \( a^2 b + b^2 c + c^2 d + d^2 a + 4 \geq k(a^3 + b^3 + c^3 + d^3) \) holds.


What is the solution of the equation \( 24 \div (3 \div 2) = (24 \div 3) \div m \)?


Ester goes to a stationery store to buy notebooks and pens. In this store, all notebooks cost $\mathrm{R} \$ 6.00$. If she buys three notebooks, she has R $4.00 left. If instead, her brother lends her an additional R $4.00, she will be able to buy two notebooks and seven pens, all of which are the same.

(a) How much does each pen cost?

(b) If she buys two notebooks and does not borrow money, how many pens can Ester buy?

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


In a certain kingdom, the king has decided to build 25 new towns on 13 uninhabited islands so that on each island there will be at least one town. Direct ferry connections will be established between any pair of new towns which are on different islands. Determine the least possible number of these connections.


In an exam, 153 people scored no more than 30 points, with an average score of 24 points. 59 people scored no less than 80 points, with an average score of 92 points. The average score of those who scored more than 30 points is 62 points. The average score of those who scored less than 80 points is 54 points. How many people participated in this exam?


An institute has 8 classes, each with 40 students. Three students are selected in such a way that each selected student is from a different class. What is the probability that a particular student will be among the selected ones?


In a circle with center $O$, a chord $AB$ intersects the diameter at point $M$ and forms an angle of $60^\circ$ with the diameter. Find $OM$ if $AM = 10$ cm and $BM = 4$ cm.

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


The non-negative integers \(a, b, c, d\) are such that

\[ a b + b c + c d + d a = 707 \]

What is the smallest possible value of the sum \(a + b + c + d\)?


On a horizontal plane, three points are 100 meters, 200 meters, and 300 meters away from the base point of an antenna. The sum of the angles of elevation to the antenna from these three points is $90^{\circ}$. What is the height of the antenna?


Fedya was 7 kopecks short of buying a portion of ice cream, and Masha was only 1 kopeck short. However, even when they combined all their money, it was still not enough to buy one portion of ice cream. How much did one portion of ice cream cost?


Four people, A, B, C, and D, move into four rooms numbered 1, 2, 3, and 4, with one person per room. Given that B does not live in room 2, and B and C must live in adjacent rooms, how many possible arrangements are there?


As shown in Figure 1-10, 5 beads are strung together to make a necklace using a string. If there are 3 different colors of beads available, how many different necklaces can be made?


If \( p \) is a real constant such that the roots of the equation \( x^{3} - 6p x^{2} + 5p x + 88 = 0 \) form an arithmetic sequence, find \( p \).


Let \( n > 1 \) and \( m \) be natural numbers. A parliament consists of \( m n \) members who have formed \( 2 n \) commissions, so that:

1. Each commission consists of \( m \) deputies.
2. Each Member of Parliament is a member of exactly 2 commissions.
3. No two commissions have more than one joint member.

Determine the largest possible value of \( m \) as a function of \( n \) so that this is possible.

Solution: \boxed{2n - 1}


Given that point \( P \) is a moving point on the line \( l: kx + y + 4 = 0 \) (where \( k > 0 \)), and \( PA \) and \( PB \) are two tangents to the circle \( C: x^2 + y^2 - 2y = 0 \), with \( A \) and \( B \) being the points of tangency. If the minimum area of the quadrilateral \( PACB \) is 2, find the value of \( k \).


Given the equation \( x y = 6(x + y) \), find all positive integer solutions \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right), \cdots,\left(x_{n}, y_{n}\right)\) and compute \(\sum_{k=1}^{n}\left(x_{k}+y_{k}\right)\).


Let \( m \) be the largest positive integer such that for every positive integer \( n \leqslant m \), the following inequalities hold:
\[
\frac{2n + 1}{3n + 8} < \frac{\sqrt{5} - 1}{2} < \frac{n + 7}{2n + 1}
\]
What is the value of the positive integer \( m \)?


Let \( f(x) \) be an increasing continuous function defined on the interval \([0, 3]\), and let \( g(x) \) be its inverse function, such that \( g(x) > f(x) \) for all positive \( x \) where both functions are defined. Additionally, \( f(0) = 0 \) and \( f(3) = 2 \).

The area under the graph of \( f(x) \) on the interval \([0, 3]\) is 2. Find the area of the figure bounded by the graphs of \( f(x) \) and \( g(x) \), as well as the segment connecting the points \((3, 2)\) and \((2, 3)\).


Two swimmers do their training in a rectangular quarry. The first swimmer finds it convenient to start at one corner of the quarry, so he swims along the diagonal to the opposite corner and back. The second swimmer finds it convenient to start from a point that divides one of the sides of the quarry in the ratio \(2018:2019\). He swims around a quadrilateral, visiting one point on each shore, and returns to the starting point. Can the second swimmer select points on the three other shores in such a way that his path is shorter than that of the first swimmer? What is the minimum value that the ratio of the length of the longer path to the shorter path can take?


How can you measure 50 cm from a string, which is $2 / 3$ meters long, without any measuring instruments?

Solution: \boxed{50 \text{ cm}}


The numbers from 1 to 9 are written in an arbitrary order. A move is to reverse the order of any block of consecutive increasing or decreasing numbers. For example, a move changes 91 653 2748 to 91 356 2748. Show that at most 12 moves are needed to arrange the numbers in increasing order.


Let \( f(a, b, c) = \frac{1}{\sqrt{1+2a}} + \frac{1}{\sqrt{1+2b}} + \frac{1}{\sqrt{1+2c}} \), where \( a, b, c > 0 \) and \( abc = 1 \). Find the minimum value of the constant \( \lambda \) such that \( f(a, b, c) < \lambda \) always holds.


In a convex pentagon \(ABCDE\), \(\angle A = 60^\circ\), and the other angles are equal to each other. It is known that \(AB = 6\), \(CD = 4\), and \(EA = 7\). Find the distance from point \(A\) to the line \(CD\).


For any real sequence $\left\{x_{n}\right\}$, define the sequence $\left\{y_{n}\right\}$ as follows:
$$
y_{1} = x_{1}, \quad y_{n+1} = x_{n+1} - \left(\sum_{i=1}^{n} x_{i}^{2}\right)^{\frac{1}{2}} \quad (n \geqslant 1).
$$

Find the smallest positive number $\lambda$ such that for any real sequence $\left\{x_{n}\right\}$ and any positive integer $m$, the following inequality holds:
$$
\frac{1}{m} \sum_{i=1}^{m} x_{i}^{2} \leqslant \sum_{i=1}^{m} \lambda^{m-i} y_{i}^{2}.
$$


A line passing through the focus \( F \) of a parabola intersects the parabola at points \( A \) and \( B \). If the projections of \( A \) and \( B \) onto the directrix of the parabola are \( A_1 \) and \( B_1 \) respectively, what is the measure of \( \angle A_1 F B_1 \)?


8 people are seated in two rows with 4 people in each row. Among them, 2 specific people must sit in the front row, and 1 specific person must sit in the back row.


As shown in the figure, in the square $ABCD$, an inscribed circle $\odot O$ is tangent to $AB$ at point $Q$ and to $CD$ at point $P$. Connect $PA$ and $PB$ to form an isosceles triangle $\triangle PAB$. Rotate these three shapes (the square, circle, and isosceles triangle) around their common axis of symmetry $PQ$ for one complete turn to form a cylinder, sphere, and cone. Find the ratio of their volumes.


The function \( g \) is defined on the set of triples of integers and takes real values. It is known that for any four integers \( a, b, c \), and \( n \), the following equalities hold: \( g(n a, n b, n c) = n \cdot g(a, b, c) \), \( g(a+n, b+n, c+n) = g(a, b, c) + n \), \( g(a, b, c) = g(c, b, a) \). Find \( g(14, 15, 16) \).


An ant starts at the point \((1,0)\). Each minute, it walks from its current position to one of the four adjacent lattice points until it reaches a point \((x, y)\) with \(|x|+|y| \geq 2\). What is the probability that the ant ends at the point \((1,1)\)?


Find the sum of the squares of the distances from the vertices of a regular $n$-gon, inscribed in a circle of radius $R$, to an arbitrary line passing through the center of the polygon.


Given positive numbers \( a \) and \( b \) that satisfy \( 2 + \log_{2} a = 3 + \log_{3} b = \log_{6}(a+b) \), find the value of \( \frac{1}{a} + \frac{1}{b} \).


Calculate the value of the following integral:

$$
\int_{0}^{\pi / 2^{n+1}} \sin x \cdot \cos x \cdot \cos 2 x \cdot \cos 2^{2} x \cdot \ldots \cdot \cos 2^{n-1} x \, dx
$$


Given the quadratic function \( f(x) = a x^{2} + b x + c \) where \( a, b, c \in \mathbf{R}_{+} \), if the function has real roots, determine the maximum value of \( \min \left\{\frac{b+c}{a}, \frac{c+a}{b}, \frac{a+b}{c}\right\} \).


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


Given a cube \( ABCD-A_{1}B_{1}C_{1}D_{1} \), find the value of \( \cos \theta \), where \( \theta \) is the angle between the planes \( AB_{1}D_{1} \) and \( A_{1}BD \), with \( 0^{\circ} \leq \theta \leq 90^{\circ} \).


Find the radius of a circle that is tangent to two concentric circles with radii 3 and 5.


Given that the number of positive divisors of 3600 is \(m\), and the number of positive divisors of 36 is \(n\), find the value of \(\frac{m}{n}\).


Density is the ratio of an object's mass to its volume. The volume after compaction is $V_{2} = 0.8 V_{1}$. Given that the mass did not change due to compaction, the density after compaction is $\rho_{2} = \frac{1}{0.8}\rho_{1} = 1.25\rho_{1}$, which means it increased by $25\%$.


The parabola with equation \(y = -\frac{1}{4} x^2 + 5x - 21\) has its vertex at point \(A\) and crosses the \(x\)-axis at \(B(b, 0)\) and \(F(f, 0)\) where \(b < f\). A second parabola with its vertex at \(B\) passes through \(A\) and crosses the \(y\)-axis at \(D\). What are the coordinates of \(D\)?

Solution: \boxed{(0, 9)}


There are 40 students in a class. Of these, 22 are involved in physical education clubs, 16 in a mathematics club, and 20 in artistic performance clubs. Additionally, 8 students are involved in both physical education and mathematics clubs, 6 in both artistic performance and mathematics clubs, 10 in both physical education and artistic performance clubs, and 2 in physical education, artistic performance, and mathematics clubs. How many students participate in only one club? How many do not participate in any clubs at all?


In trapezoid \(A B C D\), the base \(A D\) is four times larger than \(B C\). A line passing through the midpoint of diagonal \(B D\) and parallel to \(A B\) intersects side \(C D\) at point \(K\). Find the ratio \(D K : K C\).


Given that \( A \) and \( B \) are two subsets of \(\{1, 2, \ldots, 100\}\) such that \( |A| = |B| \), \( A \cap B = \emptyset \), and for any \( x \in A \), \( 2x + 2 \in B \). Find the maximum value of \( |A \cup B| \).


On one side of a right angle with vertex at point \( O \), two points \( A \) and \( B \) are taken, where \( O A = a \) and \( O B = b \). Find the radius of the circle passing through points \( A \) and \( B \) and tangent to the other side of the angle.

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


Using each of the nine digits exactly once, form prime numbers (numbers that are divisible only by 1 and themselves) such that their sum is minimized.


Given a convex quadrilateral \(ABCD\) with side \(AD\) equal to 3. The diagonals \(AC\) and \(BD\) intersect at point \(E\), and it is known that the areas of triangles \(ABE\) and \(DCE\) are both equal to 1. Find the side \(BC\), given that the area of \(ABCD\) does not exceed 4.


Two buildings in a residential area are 220 meters apart. If 10 trees are planted at equal intervals between them, what is the distance between the 1st tree and the 6th tree?

Solution: \boxed{100 \text{ meters}}


The mother squirrel collects pine nuts, gathering 20 each day on sunny days and only 12 each day on rainy days. Over several consecutive days, the squirrel collected a total of 112 pine nuts, with an average of 14 pine nuts per day. How many of these days were rainy?


Find the minimum value of the expression \((\sqrt{2(1+\cos 2x)} - \sqrt{3-\sqrt{2}} \sin x + 1) \cdot (3 + 2\sqrt{7-\sqrt{2}} \cos y - \cos 2y)\). If the answer is not an integer, round it to the nearest whole number.


\(\left(1-\operatorname{ctg}^{2}\left(\frac{3}{2} \pi - 2 \alpha\right)\right) \sin^{2}\left(\frac{\pi}{2} + 2 \alpha\right) \operatorname{tg}\left(\frac{5}{4} \pi - 2 \alpha\right) + \cos\left(4 \alpha - \frac{\pi}{2}\right)\).


90 students arrived at the camp. It is known that among any 10 students, there will always be two friends. A group of students is said to form a friendship chain if the children in the group can be numbered from 1 to \(k\) such that all the students can be divided into no more than 9 groups, each of which forms a friendship chain. (A group of one student also forms a friendship chain.)


Let \( S \) be the sum of the digits of the number \( 11^{2017} \). Find the remainder when \( S \) is divided by 9.


In the tetrahedron \(ABCD\), \(CD \perp BC\), \(AB \perp BC\), \(CD = AC\), \(AB = BC = 1\). The dihedral angle between the planes \(BCD\) and \(ABC\) is \(45^\circ\). Find the distance from point \(B\) to the plane \(ACD\).


Find all three-digit numbers $\overline{МГУ}$, comprised of different digits $M$, $\Gamma$, and $Y$, for which the equality $\overline{\text{МГУ}} = (M + \Gamma + Y) \times (M + \Gamma + Y - 2)$ holds.


The ring road is divided by kilometer posts, and it is known that the number of posts is even. One of the posts is painted yellow, another is painted blue, and the rest are painted white. The distance between posts is defined as the length of the shortest arc connecting them. Find the distance from the blue post to the yellow post if the sum of the distances from the blue post to the white posts is 2008 km.

Solution: \boxed{17 \text{ km}}


Find the largest value of \( c \), if \( c = 2 - x + 2 \sqrt{x - 1} \) and \( x > 1 \).


Given 5 distinct real numbers, any two of which are summed to yield 10 sums. Among these sums, the smallest three are 32, 36, and 37, and the largest two are 48 and 51. What is the largest of these 5 numbers?


Let \( A B C \) be a triangle. The midpoints of the sides \( B C, A C \), and \( A B \) are denoted by \( D, E \), and \( F \) respectively.

The two medians \( A D \) and \( B E \) are perpendicular to each other and have lengths \( \overline{A D} = 18 \) and \( \overline{B E} = 13.5 \).

Calculate the length of the third median \( CF \) of this triangle.


Oleg has four cards, each with a natural number on each side (a total of 8 numbers). He considers all possible sets of four numbers where the first number is written on the first card, the second number on the second card, the third number on the third card, and the fourth number on the fourth card. For each set of four numbers, he writes the product of the numbers in his notebook. What is the sum of the eight numbers on the cards if the sum of the sixteen numbers in Oleg’s notebook is $330?


The numbers $1,2, \ldots, 2016$ are grouped into pairs in such a way that the product of the numbers in each pair does not exceed a certain natural number $N$. What is the smallest possible value of $N$ for which this is possible?


The tetrahedron $A B C D$ has edge lengths of $7, 13, 18, 27, 36, 41$, with $A B = 41$. What is the length of $C D$?


A car was traveling at a speed of \( V \). Upon entering a city, the driver reduced the speed by \( x \% \), and upon leaving the city, increased it by \( 0.5 x \% \). It turned out that this new speed is \( 0.6 x \% \) less than the speed \( V \). Find the value of \( x \).


Find all pairs \((a, b)\) of positive integers such that

\[
1 + 5^a = 6^b .
\]


At a club, twenty gentlemen met. Some of them were wearing hats, and some were without hats. From time to time, one of the gentlemen took off his hat and put it on one of those who did not have a hat at that moment. In the end, ten gentlemen counted that each of them had given away a hat more times than they had received one. How many gentlemen came to the club wearing hats?


Given an integer \( n \geq 2 \), for any pairwise coprime positive integers \( a_1, a_2, \ldots, a_n \), let \( A = a_1 + a_2 + \ldots + a_n \). Denote by \( d_i \) the greatest common divisor (gcd) of \( A \) and \( a_i \) for \( i = 1, 2, \ldots, n \). Denote by \( D_i \) the gcd of the remaining \( n-1 \) numbers after removing \( a_i \). Find the minimum value of \( \prod_{i=1}^{n} \frac{A - a_i}{d_i D_i} \).


Given a positive integer \( n \), find the smallest real number \( \lambda \) such that there exist real numbers \( a_1, a_2, \ldots, a_n \) within the interval \([0, 1]\), for any real numbers \( x_1, x_2, \ldots, x_n \) satisfying \( 0 \leq x_1 \leq x_2 \leq \cdots \leq x_n \leq 1 \), we have
$$
\min_{1 \leq i \leq n} \left|x_i - a_i\right| \leq \lambda
$$


\( \Delta ABC \) is an isosceles triangle with \( AB = 2 \) and \( \angle ABC = 90^{\circ} \). Point \( D \) is the midpoint of \( BC \) and point \( E \) is on \( AC \) such that the area of quadrilateral \( AEDB \) is twice the area of triangle \( ECD \). Find the length of \( DE \).


As shown in the figure, Leilei uses 12 small wooden sticks to form a $3 \times 3$ square. Fanfan uses 9 small wooden sticks to cut it into 3 small $1 \times 2$ rectangles and 3 small $1 \times 1$ squares. If Leilei uses 40 small wooden sticks to form an $8 \times 12$ rectangle, then Fanfan needs to use $\qquad$ more small wooden sticks to cut it into 40 small rectangles, such that each small rectangle is either $1 \times 2$ or $1 \times 3$.


Two distinct numbers \( x \) and \( y \) (not necessarily integers) satisfy the equation \( x^2 - 2000x = y^2 - 2000y \). Find the sum of the numbers \( x \) and \( y \).


Let \( v(X) \) be the sum of elements of a nonempty finite set \( X \), where \( X \) is a set of numbers. Calculate the sum of all numbers \( v(X) \) where \( X \) ranges over all nonempty subsets of the set \(\{1,2,3, \ldots, 16\}\).


Anton, Vasya, Sasha, and Dima were driving from city A to city B, each taking turns at the wheel. The entire journey was made at a constant speed.

Anton drove the car for half the time Vasya did, and Sasha drove for as long as Anton and Dima together. Dima was at the wheel for only one-tenth of the distance. What fraction of the distance did Vasya drive? Provide your answer as a decimal.


Calculate: $\left(1+\frac{1}{2}\right) \times \left(1-\frac{1}{2}\right) \times \left(1+\frac{1}{3}\right) \times \left(1-\frac{1}{3}\right) \times \cdots \times \left(1+\frac{1}{10}\right) \times \left(1-\frac{1}{10}\right) = \ ?$


Márcio starts with a natural number and replaces it with the sum of its digits, repeating the process until he finally gets a single-digit number. For example, Márcio replaces 1784102 with 23, and then with 8. He also applies this process to lists of $N$ natural numbers, replacing each number in the list with the sum of its digits, continuing the process until he ends with a final list where each number is a single digit.

(a) Starting with $3^{2009}$, what is the final single-digit number?

(b) Starting with $17^{2009}$, what is the final single-digit number?

(c) Starting with the list of the first 20092009 natural numbers, does the final list have more 4s or 5s? How many 9s are in the final list?


In a cube $A B C D-A_{1} B_{1} C_{1} D_{1}$ with a side length of 1, points $E$ and $F$ are located on $A A_{1}$ and $C C_{1}$ respectively, such that $A E = C_{1} F$. Determine the minimum area of the quadrilateral $E B F D_{1}$.


Find the last digit of the number $1 \cdot 2 + 2 \cdot 3 + \dots + 999 \cdot 1000$.


Find \(\lim _{x \rightarrow \frac{\pi}{2}} \frac{\tan x}{\tan 3x}\).


Roman numerals I, V, X, L, C, D, M represent 1, 5, 10, 50, 100, 500, 1000 respectively. Calculate \(2 \times M + 5 \times L + 7 \times X + 9 \times I\).


During training shooting, each of the soldiers shot 10 times. One of them successfully completed the task and scored 90 points. How many times did he score 9 points if he scored four 10s, and the results of the other shots were 7s, 8s, and 9s, with no misses?


Let $ABC$ be an isosceles right triangle at $C$, and $D$ and $E$ be two points on $[CA]$ and $[CB]$ such that $CD = CE$. Then let $U$ and $V$ be two points on $[AB]$ such that $DU$ and $CV$ are perpendicular to $(AE)$. Show that $UV = VB$.

Solution: \boxed{UV = VB}


If there are 5 medical boxes, each containing a specific medicine common to exactly 2 of the boxes, and each type of medicine appears in exactly 2 of the boxes, how many types of medicine are there?


Find the smallest natural number that is divisible by 11, which after being increased by 1 is divisible by 13.


Each rational number is painted with one of two colors, white or red. A coloring is called "sanferminera" if for any two rational numbers \( x \) and \( y \) with \( x \neq y \), the following conditions are satisfied:
a) \( xy = 1 \),
b) \( x + y = 0 \),
c) \( x + y = 1 \),

then \( x \) and \( y \) are painted different colors. How many "sanferminera" colorings are there?


Liza found the total of the interior angles of a convex polygon. She missed one of the angles and obtained the result $2017^\circ$. Which of the following was the angle she missed?
A $37^\circ$
B $53^\circ$
C $97^\circ$
D $127^\circ$
E $143^\circ$


Given an isosceles triangle \( A B C \) where \( A B = A C \) and \( \angle A B C = 53^\circ \). Point \( K \) is such that \( C \) is the midpoint of \( A K \). Point \( M \) is chosen such that:

- \( B \) and \( M \) are on the same side of the line \( A C \);
- \( K M = A B \);
- the angle \( \angle M A K \) is the largest possible.

What is the measure of the angle \( \angle B A M \) in degrees?


Let's divide a sequence of natural numbers into groups:

\((1), (2,3), (4,5,6), (7,8,9,10), \ldots\)

Let \( S_{n} \) denote the sum of the \( n \)-th group of numbers. Find \( S_{16} - S_{4} - S_{1} \).


Calculate the limit of the function:

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


Let $\left\{a_{n}\right\}$ be a sequence of positive terms with the first term being 1, and satisfying the recurrence relation $(n+1) a_{n+1}^{2} - n a_{n}^{2} + a_{n} a_{n+1} = 0$ for $n=1, 2, 3, \ldots$. Find the general term $a_{n}$.


Find the maximum and minimum values of the function:

1) \( z = x^{2} - y^{2} + 2a^{2} \) in the circle \( x^{2} + y^{2} \leqslant a^{2} \);

2) \( v = 2x^{3} + 4x^{2} + y^{2} - 2xy \) in the closed region bounded by the lines \( y = x^{2} \) and \( y = 4 \).

Solution: \boxed{0 \text{ and } 32}


On a computer keyboard, the key for the digit 1 is not working. For example, if you try to type the number 1231234, only the number 23234 will actually print.

Sasha tried to type an 8-digit number, but only 202020 was printed. How many 8-digit numbers satisfy this condition?


On each side of a square, a point is taken. It turns out that these points are the vertices of a rectangle whose sides are parallel to the diagonals of the square. Find the perimeter of the rectangle if the diagonal of the square is 6.


Calculate as simply as possible the value of the expression

$$
a^{3}+b^{3}+3\left(a^{3} b+a b^{3}\right)+6\left(a^{3} b^{2}+a^{2} b^{3}\right)
$$

if $a+b=1$.


The positive integers \( x \) and \( y \), for which \( \gcd(x, y) = 3 \), are the coordinates of the vertex of a square centered at the origin with an area of \( 20 \cdot \operatorname{lcm}(x, y) \). Find the perimeter of the square.


In the language of the Ancient Tribe, the alphabet consists of only two letters: M and O. Two words are synonyms if one can be obtained from the other by

a) Removing the letter combinations МО or ООММ,

b) Adding the letter combination ОМ in any place

Are the words ОММ and МОО synonyms in the language of the Ancient Tribe?

Solution: \boxed{\text{No}}


Given that \(a\) is a digit from 1 to 9, and the repeating decimal \(0.1a = \frac{1}{a}\), find the value of \(a\).


At the "Economics and Law" congress, a "Best of the Best" tournament was held, in which more than 220 but fewer than 254 delegates—economists and lawyers—participated. During one match, participants had to ask each other questions within a limited time and record correct answers. Each participant played with each other participant exactly once. A match winner got one point, the loser got none, and in case of a draw, both participants received half a point each. By the end of the tournament, it turned out that in matches against economists, each participant gained half of all their points. How many lawyers participated in the tournament? Provide the smallest possible number as the answer.


Let the set \( \mathrm{S} = \{1, 2, 3, \ldots, 10\} \). The subset \( \mathrm{A} \) of \( \mathrm{S} \) satisfies \( \mathrm{A} \cap \{1, 2, 3\} \neq \emptyset \) and \( \mathrm{A} \cup \{4, 5, 6\} \neq \mathrm{S} \). Find the number of such subsets \( \mathrm{A} \).


Arrange the leftmost numbers of Pascal's triangle in the following manner:

Diagonals are a sequence of numbers connected by arrows in a slanted direction. For example, the first diagonal is " $1 \leftarrow 1$", the second diagonal is " $1 \longleftrightarrow \rightarrow 2$", and the third diagonal is " $1 \longleftrightarrow 3 \longleftrightarrow \rightarrow 1$".

Question: What is the sum of all the numbers in the 13th diagonal?


Let \( p \) be a function associated with a permutation. The order of this permutation is defined as the smallest integer \( k \) such that \( p^{(k)} = \text{Id} \).

What is the largest order for a permutation of size 11?


An exchange point conducts two types of operations:

1) Give 2 euros - receive 3 dollars and a candy as a gift;
2) Give 5 dollars - receive 3 euros and a candy as a gift.

When the wealthy Buratino came to the exchange point, he had only dollars. When he left, he had fewer dollars, he did not acquire any euros, but he received 50 candies. How many dollars did this "gift" cost Buratino?

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


Inside the tetrahedron \( ABCD \), points \( X \) and \( Y \) are given. The distances from point \( X \) to the faces \( ABC, ABD, ACD, BCD \) are \( 14, 11, 29, 8 \) respectively. The distances from point \( Y \) to the faces \( ABC, ABD, ACD, BCD \) are \( 15, 13, 25, 11 \) respectively. Find the radius of the inscribed sphere of tetrahedron \( ABCD \).


In a country with 15 cities, some of which are connected by airlines belonging to three different companies, it is known that even if any one of the airlines ceases operations, it will still be possible to travel between any two cities (possibly with transfers) using the remaining two companies' flights. What is the minimum number of airline routes in the country?


Find the limits:
1) $\lim _{x \rightarrow 4} \frac{x^{2}-16}{x^{2}-5 x+4}$
2) $\lim _{x \rightarrow 1} \frac{x^{3}-3 x+2}{x^{3}-x^{2}-x+1}$
3) $\lim _{x \rightarrow 0} \frac{x^{3}-6 x+6 \sin x}{x^{5}}$
4) $\lim _{x \rightarrow 0} \frac{\operatorname{tg} x-x}{x-\sin x}$
5) $\lim _{x \rightarrow \infty} \frac{a x^{2}+b}{c x^{2}+d}$
6) $\lim _{x \rightarrow 0} \frac{\ln \sin 5 x}{\ln \sin 2 x}$


In the parallelogram \(ABCD\), the longer side \(AD\) is 5. The angle bisectors of angles \(A\) and \(B\) intersect at point \(M\).

Find the area of the parallelogram, given that \(BM = 2\) and \(\cos \angle BAM = \frac{4}{5}\).


It is known that the sequence of numbers \(a_{1}, a_{2}, \ldots\) is an arithmetic progression, and the sequence of products \(a_{1}a_{2}, a_{2}a_{3}, a_{3}a_{4}, \ldots\) is a geometric progression. It is given that \(a_{1} = 1\). Find \(a_{2017}\).


A five-digit number $\overline{abcde}$ is given. When any two digits from this number are selected to form a two-digit number keeping the order as in the original five-digit number, there are 10 two-digit numbers: $33, 37, 37, 37, 38, 73, 77, 78, 83, 87$. What is the five-digit number $\overline{abcde}$?


What is the smallest positive integer that cannot be written as the sum of two nonnegative palindromic integers? (An integer is palindromic if the sequence of decimal digits is the same when read backward.)


Find \(\lim _{x \rightarrow 0}[(x-\sin x) \ln x]\).


Calculate the areas of the figures enclosed by the lines given in polar coordinates.

$$
r=\sin \phi, \quad r=2 \sin \phi
$$


For every positive integer \( N \), determine the smallest real number \( b_{N} \) such that, for all real \( x \),
\[
\sqrt[N]{\frac{x^{2N}+1}{2}} \leqslant b_{N}(x-1)^2 + x.
\]


Let \(ABC\) be a right triangle with \(\angle A = 90^\circ\). A circle \(\omega\) centered on \(BC\) is tangent to \(AB\) at \(D\) and \(AC\) at \(E\). Let \(F\) and \(G\) be the intersections of \(\omega\) and \(BC\) so that \(F\) lies between \(B\) and \(G\). If lines \(DG\) and \(EF\) intersect at \(X\), show that \(AX = AD\).

Solution: \boxed{AX = AD}


You are allowed to cut out any 18 squares from a $20 \times 20$ chessboard, and after that, place several rooks on the remaining squares such that no two rooks attack each other. Rooks attack each other if they are in the same row or column of the board and there are no cut-out squares between them. What is the maximum number of rooks that can be placed under these conditions?


The archipelago consists of $N \geqslant 7$ islands. Any two islands are connected by no more than one bridge. It is known that from each island, there are no more than 5 bridges, and among any 7 islands, there are always two islands connected by a bridge. What is the largest possible value of $N$?


Find all positive integers \( n \) such that \( n+1 \) divides \( 2n^2 + 5n \). Verify that the found \( n \) are solutions.


The sum of all numbers in the first row is equal to the sum of all numbers in the second row. What number should be placed in the position of the "?" in the second row?

\[
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|}
\hline 1 & 3 & 5 & 7 & 9 & 11 & 13 & 15 & 17 & 19 & 200 \\
\hline
\end{tabular}
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|}
\hline 1 & 4 & 7 & 10 & 13 & 16 & 19 & 22 & 25 & 28 & $?$ \\
\hline
\end{tabular}
\]


A locomotive is moving on a horizontal section of track at a speed of 72 km/h. In how much time and at what distance will it be stopped by the brake if the resistance to motion after the start of braking is equal to 0.2 of its weight?

Solution: \boxed{102 \ \text{meters}}


Indiana Jones reached an abandoned temple in the jungle and entered the treasury. There were 5 chests, with the knowledge that only one contains the treasure, and the others trigger a stone slab to fall on the head of anyone who tries to open them. The chests are numbered from left to right.

- The first, fourth, and fifth chests are made of cedar.
- The second and third chests are made of sandalwood.
- The inscriptions on the chests are as follows:
  - On the first chest: "The treasure is in me or in the 4th chest."
  - On the second chest: "The treasure is in the chest to the left of me."
  - On the third chest: "The treasure is in me or in the chest at the far right."
  - On the fourth chest: "There is no treasure in the chests to the left of me."
  - On the fifth chest: "All the inscriptions on other chests are false."

The last guardian of the temple, before dying, revealed to Indiana a secret: an equal number of false statements are written on both cedar and sandalwood chests.

In which chest is the treasure?


A natural number \( n \) was multiplied by the sum of the digits of the number \( 3n \), and the resulting number was then multiplied by 2. The result was 2022. Find \( n \).


Find an integer \( x \) such that \(\left(1+\frac{1}{x}\right)^{x+1}=\left(1+\frac{1}{2003}\right)^{2003}\).


Find the smallest positive integer \( n \) such that there exists a complex number \( z \), with positive real and imaginary part, satisfying \( z^{n} = (\bar{z})^{n} \).


The sequence \(\left\{x_{n}\right\}\) is defined as follows: \(x_{1}=\frac{1}{2}, x_{k+1}=x_{k}^{2}+x_{k}\). Calculate the integer part of the sum \(\frac{1}{x_{1}+1}+\frac{1}{x_{2}+1}+\cdots+\frac{1}{x_{100}+1}\).


Let $[x]$ denote the greatest integer less than or equal to the real number $x$. The equation $\lg ^{2} x-[\lg x]-2=0$ has how many real roots?


A villager A and a motorcycle with a passenger, villager B, set off at the same time from the village to the station along a single road. Before reaching the station, the motorcyclist dropped off villager B and immediately went back towards the village, while villager B walked to the station. Upon meeting villager A, the motorcyclist picked him up and took him to the station. As a result, both villagers arrived at the station simultaneously. What fraction of the journey from the village to the station did villager A travel by motorcycle, considering that both villagers walked at the same speed, which was 9 times slower than the speed of the motorcycle?


For a triangle with side lengths \(a\), \(b\), and \(c\), an area of \(\frac{1}{4}\), and a circumradius of 1, compare the magnitude of \(q = \sqrt{a} + \sqrt{b} + \sqrt{c}\) to \(t = \frac{1}{a} + \frac{1}{b} + \frac{1}{c}\).

Solution: \boxed{q < t}


Bob Barker decided to raise the intellectual level of The Price is Right by having contestants guess how many objects exist of a certain type, without going over. The number of points you will get is the percentage of the correct answer, divided by 10, with no points for going over (i.e., a maximum of 10 points).

The first object for our contestants is a table of shape \((5,4,3,2,1)\). This table is an arrangement of the integers 1 through 15 with five numbers in the top row, four in the next, three in the next, two in the next, and one in the last, such that each row and each column is increasing (from left to right, and top to bottom, respectively). For instance:
```
6
13 14
15
```
is one such table. How many tables are there?


Solution: \boxed{x = 0, x = -10}


Find all integers \( k \geq 1 \) so that the sequence \( k, k+1, k+2, \ldots, k+99 \) contains the maximum number of prime numbers.


Given that \(a, b, c, d\) are integers with \(ad \neq bc\), show that \(\frac{1}{(ax+b)(cx+d)}\) can be written in the form \(\frac{r}{ax+b} + \frac{s}{cx+d}\). Then, find the sum \(\frac{1}{1 \cdot 4} + \frac{1}{4 \cdot 7} + \frac{1}{7 \cdot 10} + \ldots + \frac{1}{2998 \cdot 3001}\).


Let \(\triangle ABC\) be a triangle with \(AB=7\), \(BC=1\), and \(CA=4\sqrt{3}\). The angle trisectors of \(C\) intersect \(\overline{AB}\) at \(D\) and \(E\), and lines \(\overline{AC}\) and \(\overline{BC}\) intersect the circumcircle of \(\triangle CDE\) again at \(X\) and \(Y\), respectively. Find the length of \(XY\).


There are some natural numbers such that when each of them is multiplied by 7, the last four digits of the product are 2012. The smallest of these natural numbers is $\qquad$ .


A polynomial \( P(x) \) of degree 10 with a leading coefficient of 1 is given. The graph of \( y = P(x) \) lies entirely above the x-axis. The polynomial \( -P(x) \) was factored into irreducible factors (i.e., polynomials that cannot be represented as the product of two non-constant polynomials). It is known that at \( x = 2020 \), all the resulting irreducible polynomials take the value -3. Find \( P(2020) \).


Mathematician Wiener, the founder of cybernetics, was asked about his age during his Ph.D. awarding ceremony at Harvard University because he looked very young. Wiener's interesting response was: "The cube of my age is a four-digit number, and the fourth power of my age is a six-digit number. These two numbers together use all the digits from 0 to 9 exactly once, with no repetition or omission." What is Wiener's age that year? (Note: The cube of a number \(a\) is equal to \(a \times a \times a\), and the fourth power of a number \(a\) is equal to \(a \times a \times a \times a\)).


A grid sheet of size \(5 \times 7\) was cut into \(2 \times 2\) squares, L-shaped pieces covering 3 cells, and \(1 \times 3\) strips. How many \(2 \times 2\) squares could be obtained?


The lengths of the edges of a regular tetrahedron \(ABCD\) are 1. \(G\) is the center of the base \(ABC\). Point \(M\) is on line segment \(DG\) such that \(\angle AMB = 90^\circ\). Find the length of \(DM\).


In square \( A B C D \), \( P \) and \( Q \) are points on sides \( C D \) and \( B C \), respectively, such that \( \angle A P Q = 90^\circ \). If \( A P = 4 \) and \( P Q = 3 \), find the area of \( A B C D \).


Fifty children went to the zoo, with 36 of them seeing the pandas, 28 seeing the giraffes, and 15 seeing the pandas but not the giraffes. How many children saw the giraffes but not the pandas?


Find the smallest prime number that cannot be expressed in the form $\left|3^{a}-2^{b}\right|$, where $a$ and $b$ are non-negative integers.


There are \( n \) people, and it is known that any two of them can call each other at most once. Among them, the total number of calls between any group of \( n-2 \) people is the same and is \( 3^{k} \) times, where \( k \) is a natural number. Find all possible values of \( n \).


Along a straight alley, there are 400 streetlights placed at equal intervals, numbered consecutively from 1 to 400. Alla and Boris start walking towards each other from opposite ends of the alley at the same time but with different constant speeds (Alla from the first streetlight and Boris from the four-hundredth streetlight). When Alla is at the 55th streetlight, Boris is at the 321st streetlight. At which streetlight will they meet? If the meeting occurs between two streetlights, indicate the smaller number of the two in the answer.


Given two positive integers \(x\) and \(y\), \(xy - (x + y) = \operatorname{HCF}(x, y) + \operatorname{LCM}(x, y)\), where \(\operatorname{HCF}(x, y)\) and \(\operatorname{LCM}(x, y)\) are respectively the greatest common divisor and the least common multiple of \(x\) and \(y\). If \(c\) is the maximum possible value of \(x + y\), find \(c\).


Find all values of \( a \) such that the roots \( x_1, x_2, x_3 \) of the polynomial
\[ x^3 - 6x^2 + ax + a \]
satisfy
\[ \left(x_1 - 3\right)^3 + \left(x_2 - 3\right)^3 + \left(x_3 - 3\right)^3 = 0. \]


In \(\triangle ABC\), \(\sin A = \frac{3}{5}\), \(\cos B = \frac{5}{13}\). What is the value of \(\cos C\)?


In the coordinate plane, a point whose x-coordinate and y-coordinate are both integers is called a lattice point. For any positive integer \( n \), connect the origin \( O \) with the point \( A_{n}(n, n+3) \). Let \( f(n) \) denote the number of lattice points on the line segment \( OA_{n} \) excluding the endpoints. Determine the value of \( f(1)+f(2)+\cdots+f(1990) \).


The first two digits of a natural four-digit number are either both less than 5 or both greater than 5. The same condition applies to the last two digits. How many such numbers are there in total? Justify your answer.


How many integers from 1 to 300 can be divided by 7 and also be divisible by either 2 or 5?


If we replace some of the " × " signs with " ÷ " signs in the expression $9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1$ such that the final result is still a natural number, denoted as $N$, what is the minimum value of $N$?


A polynomial of degree 10 has three distinct roots. What is the maximum number of zero coefficients it can have?


On an island, there are knights who always tell the truth and liars who always lie. Before a friendly match, 30 islanders gathered wearing shirts with numbers on them—random natural numbers. Each of them said, "I have a shirt with an odd number." After that, they exchanged shirts, and each said, "I have a shirt with an even number." How many knights participated in the exchange?


Calculate the limit of the numerical sequence:

$$\lim _{n \rightarrow \infty} \frac{3+6+9+\ldots+3n}{n^{2}+4}$$


Given a sequence \(\{a_n\}\) such that: \(a_1 = 1\) and \(a_{n+1} = \frac{a_n}{(n+1)(a_n + 1)}\) for \(n \in \mathbb{Z^+}\), find the value of \(\lim_{n \rightarrow +\infty} n! \cdot a_n\).


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


Find all functions \( f: \mathbb{Q} \rightarrow \mathbb{R} \) such that \( f(x + y) = f(x) + f(y) + 2xy \) for all \( x, y \) in \( \mathbb{Q} \) (the rationals).

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


Is there a natural number \( n \), and if so, what is it, such that the sum of the first \( n \) natural numbers is a three-digit number with identical digits?


Brothers Petya and Vasya decided to make a funny video and upload it to the internet. First, they filmed each of them walking from home to school; Vasya walked for 8 minutes, and Petya walked for 5 minutes. Then they came home and started editing the video on the computer: they started Vasya's video from the beginning and Petya's video from the end (in reverse) simultaneously; at the moment when the brothers met at the same point on the path in both videos, they spliced Petya's video into Vasya's. The result was a video where Vasya walks from home to school, and then at some point, suddenly transforms into Petya and walks home backward. What is the duration of the resulting video?

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


What is the largest integer \( n \) with the properties that \( 200 < n < 250 \) and \( 12n \) is a perfect square?


OKOSBANK's safe has several different locks. There are $n$ cashiers working in the bank, each having keys to some of the locks. (A cashier may have multiple keys, and multiple cashiers may have keys to the same lock.) We know that any $k$ cashiers together can always open the safe, but $k-1$ cashiers cannot. What is the minimum number of locks on the safe?


In triangle \( ABC \), the median \( AM \) (the segment connecting vertex \( A \) with the midpoint \( M \) of side \( BC \)) is extended beyond point \( M \) by a length equal to \( AM \). Find the distance from the resulting point to points \( B \) and \( C \) if the sides \( AB \) and \( AC \) are 5 and 4, respectively.

Solution: \boxed{NB = 4, \; NC = 5}


Triangles \(ABC\) and \(ABD\) are inscribed in a semicircle with diameter \(AB = 5\). A perpendicular from \(D\) to \(AB\) intersects segment \(AC\) at point \(Q\), ray \(BC\) at point \(R\), and segment \(AB\) at point \(P\). It is known that \(PR = \frac{27}{10}\), and \(PQ = \frac{5}{6}\). Find the length of segment \(DP\). If necessary, round the answer to hundredths.


Calculate the limit of the numerical sequence:

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

Solution: \boxed{\sqrt{e}}


As shown in the figure, $P$ is a point on the inscribed circle of square $ABCD$. Let $\angle APC = \alpha$ and $\angle BPD = \beta$. Then
$$
\tan^2 \alpha + \tan^2 \beta
$$
equals what?


Odd integers starting from 1 are grouped as follows: (1), (3,5), (7,9,11), (13,15,17,19), ... where the \( n \)-th group consists of \( n \) odd integers. How many odd integers are in the same group which 2007 belongs to?


Given a right triangle \(ABC\) with \(\angle B\) being the right angle, let \(M\) be a point on the hypotenuse that is equidistant from the two legs of the triangle. Could you find the value of the following expression:

\[
\begin{aligned}
E & =\sqrt{1830}\left(A C-\sqrt{A B^{2}+B C^{2}}\right) \\
& +1789-\frac{\frac{1}{A B}+\frac{1}{B C}-\frac{\sqrt{2}}{B M}}{(1848)^{3}} ?
\end{aligned}
\]


During the break, a fly entered the mathematics classroom and began crawling on a poster depicting the graph of a quadratic function $y=f(x)$ with a leading coefficient of 1. Initially, the fly moved exactly along the parabola to the point with an abscissa of 2, but then it started moving in a straight line until it again reached the parabola at the point with an abscissa of 4. Find $f(3)$, given that the line $y=2023x$ intersects the fly's path along the straight line segment at its midpoint.


Count the matches. A friend writes that he bought a small box of short matches, each one inch long. He found that he could arrange them in a triangle whose area in square inches was equal to the number of matches. Then he used 6 matches, and it turned out that from the remaining matches, he could form a new triangle with an area that contained as many square inches as there were matches left. After using another 6 matches, he was able to do the same thing again.

How many matches did he originally have in the box? This number is less than 40.


All the inhabitants of the island are either blondes or brunettes with either green or blue eyes. The proportion of brunettes among the blue-eyed is 65%. The proportion of blue-eyed among the blondes is 70%. Finally, the proportion of blondes among the green-eyed is 10%. What percentage of the island's population are green-eyed brunettes?


100 knights and 100 liars are arranged in a row (in some order). The first person was asked, "Are you a knight?", and the rest were sequentially asked, "Is it true that the previous person answered 'Yes'?" What is the greatest number of people who could have said "Yes"? Knights always tell the truth, and liars always lie.


If the remainders when 2017, 1029, and 725 are each divided by $d$ are all $r$, what is the maximum value of $d-r$?


Given the sequence $\left\{a_{n}\right\}$ defined by
$$
a_{n}=\left[(2+\sqrt{5})^{n}+\frac{1}{2^{n}}\right] \quad (n \in \mathbf{Z}_{+}),
$$
where $[x]$ denotes the greatest integer not exceeding the real number $x$, determine the minimum value of the constant $C$ such that for any positive integer $n$, the following inequality holds:
$$
\sum_{k=1}^{n} \frac{1}{a_{k} a_{k+2}} \leqslant C.
$$


For a natural number \( x \), five statements are made:

$$
3x > 91
$$

$$
\begin{aligned}
& x < 120 \\
& 4x > 37 \\
& 2x \geq 21 \\
& x > 7
\end{aligned}
$$

It is known that only three of these statements are true, and two are false. Determine \( x \).


Find the distance between the points of tangency of the circles inscribed in triangles \( ABC \) and \( CDA \) with side \( AC \), if

a) \( AB=5, BC=7, CD=DA \);

b) \( AB=7, BC=CD, DA=9 \).


Given numbers \(a, b, c\) satisfy \(a b c+a+c-b\). Then the maximum value of the algebraic expression \(\frac{1}{1+a^{2}}-\frac{1}{1+b^{2}}+\frac{1}{1+c^{2}}\) is


A, B, and C are guessing a two-digit number.

A says: The number has an even number of factors, and it is greater than 50.
B says: The number is odd, and it is greater than 60.
C says: The number is even, and it is greater than 70.

If each of them is only half correct, what is the number?


A quadrilateral \(ABCD\) with perpendicular diagonals is inscribed in a circle.

Perpendiculars dropped from vertices \(B\) and \(C\) to side \(AD\) intersect the diagonals \(AC\) and \(BD\) at points \(E\) and \(F\) respectively. Find \(EF\), given that \(BC = 1\).


There are people with masses $50, 51, 55, 57, 58, 59, 60, 63, 75,$ and $140$ kg standing in front of an elevator. The elevator has a load capacity of $180$ kg. What is the minimum number of trips necessary to ensure that all people can be transported?


Find all real-valued functions \( f(x) \) on the reals such that \( f(f(x) + y) = 2x + f(f(y) - x) \) for all \( x, y \).

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


Petya considered moments in his life to be happy when his digital clock showed the number of hours to be six times the number of minutes, or vice versa. Petya fell asleep and woke up at a happy moment in his life, without missing any such moment during his sleep. What is the maximum whole number of minutes Petya's sleep could have lasted?


The function \( f: \mathbb{R} \rightarrow \mathbb{R} \) satisfies \( f\left(x^{2}\right) f^{\prime \prime}(x) = f^{\prime}(x) f^{\prime}\left(x^{2}\right) \) for all real \( x \). Given that \( f(1) = 1 \) and \( f^{\prime \prime \prime}(1) = 8 \), determine \( f^{\prime}(1) + f^{\prime \prime}(1) \).


Find the greatest four-digit number where all digits are distinct, and which is divisible by each of its digits. Zero cannot be used.


The integers from 1 to 20 are written on the board. You can erase any two numbers, $a$ and $b$, and write the number $a+b-1$ in their place. What number will be written on the board after 19 operations? And what if instead of writing $a+b-1$, you wrote $a+b+a b$?

Solution: \boxed{21! - 1}


We have 3 locations labeled as 1, 2, and 3 where disks can be placed. Initially, there are \( n \) disks on position 1, with each disk having a smaller radius than the one below it. The goal is to move all these disks from position 1 to position 2 while respecting the following 2 rules:

1. At each step, exactly one disk can be moved from one position to another.
2. A disk cannot be placed on top of another disk with a smaller radius.

Show by induction that all the disks can be moved to position 2, and find a formula, also by induction, for the minimum number of steps required.

Additionally, consider an arithmetic-geometric sequence defined as \(\left(u_{n}\right)_{n \in \mathbb{N}}\) by the first term (denoted as \(u_{1}\)) and the recurrence relation \(u_{n+1}=a \cdot u_{n}+b\), where \(a \neq 1\) and \(b\) are fixed.

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