There are 8 lily pads in a pond numbered \(1, 2, \ldots, 8\). A frog starts on lily pad 1. During the \(i\)-th second, the frog jumps from lily pad \(i\) to \(i+1\), falling into the water with probability \(\frac{1}{i+1}\). The probability that the frog lands safely on lily pad 8 without having fallen into the water at any point can be written as \(\frac{m}{n}\), where \(m, n\) are positive integers and \(\operatorname{gcd}(m, n) = 1\). Find \(100m + n\).


Find the volume of a regular quadrilateral prism if its diagonal makes an angle of \(30^{\circ}\) with the plane of the lateral face, and the side of the base is equal to \(a\).

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


In the equation
$$
\frac{x^{2}+p}{x}=-\frac{1}{4},
$$
with roots \(x_{1}\) and \(x_{2}\), determine \(p\) such that:

a) \(\frac{x_{1}}{x_{2}}+\frac{x_{2}}{x_{1}}=-\frac{9}{4}\),

b) one root is 1 less than the square of the other root.


Given an ellipse $\frac{x^{2}}{5^{2}}+\frac{y^{2}}{3^{2}}=1$, find the minimum length of the line segment $AB$ where the line is a tangent to the ellipse and intersects the major and minor axes of the ellipse at points $A$ and $B$ respectively.


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


How many different lists \( a, b, c, d \) of distinct odd positive integers with \( a < b < c < d \) have the property that \( a + b + c + d = 24 \) ?


Find the generating functions for the following combinatorial problems and determine the corresponding solutions:
(1) Find the $k$-combinations of $\{a_{1}, a_{2}, \cdots, a_{n}\}$;
(2) Find the $k$-combinations of $\{\infty \cdot a_{1}, \infty \cdot a_{2}, \cdots, \infty \cdot a_{n}\}$;
(3) Find the 10-combinations of $\{3 \cdot a, 4 \cdot b, 5 \cdot c\}$, where $\{3 \cdot a, 4 \cdot b, 5 \cdot c\}$ represents a set consisting of 3 $a$'s, 4 $b$'s, and 5 $c$'s.


Anya has blue, green, and red paints. She wants to paint a wooden cube in such a way that each of the cube's faces is painted with one of these colors and there are exactly two faces of each color. In how many distinct ways can she do this? Note that painting arrangements that can be obtained by rotating the cube are considered identical.


In triangle \( \triangle ABC \), \(\angle A = 60^{\circ}\), \(AC = 16\), and the area of \(\triangle ABC\) is \(220\sqrt{3}\). Find \(BC\).


Find the maximum value of the expression \(a^{3} b - b^{3} a\) given that \(a^{2} + b^{2} = 1\).


Let \( A = 100101102103 \cdots 798799 \) be a 2100-digit positive integer formed by connecting the three-digit numbers from 100 to 799 in sequence. What is the remainder when \( A \) is divided by 126?


Points $A$ and $B$ are 600 kilometers apart. Two people, A and B, start bicycling from point $A$ to point $B$ simultaneously. Person A bicycles 40 kilometers every day, while person B bicycles 60 kilometers every other day, resting every other day. At the end of a certain day, the distance person B has left to cover to reach point $B$ is twice the distance person A has left to cover to reach point $B$.


In rectangle \(ABCD\), \(AB=2\) and \(BC=3\). Points \(E\) and \(F\) are the midpoints of \(AB\) and \(CD\) respectively. When \(\triangle FAB\) is rotated by \(90^\circ\) around axis \(EF\) to form \(\triangle FA'B'\), what is the volume of the tetrahedron \(A'B'C D\)?


A natural number has a digit sum of 2013. The next natural number has a smaller digit sum and is not divisible by 4. What is the digit sum of the next natural number?


Vanya wrote the number 1347 on the board.

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

Help Vasya solve Petya's task.


Let \( f \) be a function such that \( f(0) = 1 \), \( f'(0) = 2 \), and
\[ f''(t) = 4 f'(t) - 3 f(t) + 1 \]
for all \( t \). Compute the 4th derivative of \( f \), evaluated at 0.


Find the integral $\int \operatorname{tg} x \, dx$.

Solution: \boxed{-\ln |\cos x| + C}


Let \( x, y, z \) be nonzero real numbers. Use the listing method to represent the set of all possible values of the algebraic expression \( \frac{x}{|x|} + \frac{y}{|y|} + \frac{z}{|z|} + \frac{xy}{|xy|} + \frac{xyz}{|xyz|} \).

Solution: \boxed{\{-3, -1, 1, 5\}}


For four consecutive natural numbers, it is known that the largest of them is a divisor of the product of the other three. Find all possible values that the largest of these numbers can take.


An ellipse in the xy-plane has its foci at (9,20) and (49,55) and is tangent to the x-axis. Find the length of the major axis.


In a football championship with 20 teams, where each team plays with every other team exactly once, what is the minimum number of games that must be played to ensure that among any three teams, there will be two teams that have already played against each other?


The entire surface of a cube with dimensions $13 \times 13 \times 13$ was painted red, and then this cube was cut into $1 \times 1 \times 1$ cubes. All faces of the $1 \times 1 \times 1$ cubes that were not painted red were painted blue. By what factor is the total area of the blue faces greater than the total area of the red faces?


There is a four-digit number less than 2000 that has exactly 14 positive divisors (including 1 and itself). One of its prime divisors ends with the digit 1. Find this four-digit number.


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


Find all values of \( x \) for which the smaller of the numbers \( \frac{1}{x} \) and \( \sin x \) is greater than \( \frac{1}{2} \). In the answer, provide the total length of the resulting intervals on the number line, rounding to the nearest hundredth if necessary.


In the plane, 2013 red points and 2014 blue points are marked so that no three of the marked points are collinear. One needs to draw \( k \) lines not passing through the marked points and dividing the plane into several regions. The goal is to do it in such a way that no region contains points of both colors.

Find the minimal value of \( k \) such that the goal is attainable for every possible configuration of 4027 points.


Find the limits:
1) \(\lim_{x \rightarrow 1}\left(\frac{1}{\ln x} - \frac{1}{x-1}\right)\)
2) \(\lim_{\varphi \rightarrow \frac{\pi}{2}} (\sec \varphi - \operatorname{tg} \varphi)\)
3) \(\lim_{x \rightarrow 0} x \ln x\)
4) \(\lim_{\alpha \rightarrow \frac{\kappa}{2}} (\pi - 2\alpha) \operatorname{tg} \alpha\)


We consider \( n \) distinct points in a plane. For each segment formed by a pair of these points, we color the midpoint of the segment blue. How many blue points are needed, at minimum?

If two midpoints coincide, they count as a single point.

Solution: \boxed{2n - 3}


Based on the definition of the derivative, find \( f^{\prime}(0) \):

\[ f(x) = \begin{cases}
\operatorname{tg}\left(2^{x^{2} \cos \left( \frac{1}{8x} \right)} - 1 + x \right), & \text{if } x \neq 0 \\
0, & \text{if } x = 0
\end{cases} \]


If \( x \) is a real number, find the smallest value of \( \sqrt{x^{2}+4x+5}+\sqrt{x^{2}-8x+25} \).


How many numbers from 1 to 1000 (inclusive) cannot be represented as the difference between the squares of two integers?


Find the derivative.

\[ y = \frac{2x - 1}{4} \cdot \sqrt{2 + x - x^2} + \frac{9}{8} \cdot \arcsin \left( \frac{2x - 1}{3} \right) \]

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


Calculate: \( 25 \times 13 \times 2 + 15 \times 13 \times 7 = \)


Given the function
$$
\begin{aligned}
f(x)= & a \tan ^{2017} x+b x^{2017}+ \\
& c \ln \left(x+\sqrt{x^{2}+1}\right)+20,
\end{aligned}
$$

where \( a \), \( b \), and \( c \) are real numbers. If \( f\left(\ln \log _{2} 21\right)=17 \), find \( f\left(\ln \log _{21} 5\right) \).


Given a unit cube $A B C D-A' B' C' D'$ with a point $M$ on the side $A A' B' B$ such that the distance from $M$ to the lines $A B$ and $B' C'$ are equal. Find the minimum distance from $M$ to the point $C'$.


Given four Chinese idioms: "一家之言", "言扬行举", "举世皆知", and "知行合一", each character in these idioms represents one of 11 consecutive non-zero natural numbers. The same character represents the same number, and different characters represent different numbers. If the sum of the numbers represented by the four characters in each idiom is 21, what is the maximum number that "行" can represent?


When washing clothes by hand, you should apply detergent well, scrub thoroughly, and wring out the water. Of course, it's impossible to wring out all the water, and it's assumed that there will be 1 kilogram of water with impurities left in the clothes after wringing. Now, you have 18 kilograms of clean water available. Assume that each time you rinse, you use an integer number of kilograms of clean water, and after each rinse, the impurities are evenly distributed in the water.

Question:
1. If you rinse the clothes twice, what is the minimum fraction of impurities remaining compared to before rinsing?
2. To reduce the amount of impurities to less than $\frac{1}{300}$ of the initial amount before rinsing, how many rinses are required? Please provide a rinsing scheme that meets this condition and explain the reasoning.


Given three non-zero real numbers \(a, b, c\) such that the equations \(a x^{2}+b x+c=0\), \(b x^{2}+c x+a=0\), \(c x^{2}+a x+b=0\) each have two roots. How many of these roots can be negative?


If \( x \) is a real number, find the minimum value of \( |x+1| + 2|x-5| + |2x-7| + \left|\frac{x-11}{2}\right| \).


Mila and Zhenya each came up with a number and wrote down all the natural divisors of their numbers on the board. Mila wrote down 10 numbers, Zhenya wrote down 9 numbers, and the maximum number that was written on the board twice is 50. How many distinct numbers are written on the board in total?


The numbers \(2^n\) and \(5^n\) start with the digit \(a\). What is the value of \(a\)?


In a kindergarten's junior group, there are two identical small Christmas trees and five children. The teachers want to divide the children into two circles around each tree, with at least one child in each circle. The teachers distinguish the children but do not distinguish the trees: two such divisions into circles are considered the same if one can be obtained from the other by swapping the trees (along with their respective circles) and rotating each circle around its tree. In how many ways can the children be divided into the circles?


Solution: \boxed{x^{1/4} - 5}


Let \( a \) and \( b \) be real numbers such that \(\frac{a b}{a^{2}+b^{2}}=\frac{1}{4}\). Find all possible values of \(\frac{\left|a^{2}-b^{2}\right|}{a^{2}+b^{2}}\).


Real numbers \( r, s, t \) satisfy \( 1 \leq r \leq s \leq t \leq 4 \). Find the minimum value of \( (r-1)^{2}+\left(\frac{s}{r}-1\right)^{2}+\left(\frac{t}{s}-1\right)^{2}+\left(\frac{4}{t}-1\right)^{2} \).

Solution: \boxed{12 - 8\sqrt{2}}


There are \( n \) pieces of paper, each containing 3 different positive integers no greater than \( n \). Any two pieces of paper share exactly one common number. Find the sum of all the numbers written on these pieces of paper.


Find the maximum and minimum values of the expression \(2x + y\), given that \((x, y)\) are related by the equation \(\sqrt{x-1} + \sqrt{y-4} = 2\). For which \((x, y)\) are these values achieved?


The quadrilateral \(ABCD\) is inscribed in a circle. It is known that \(AC \perp BD\). Find the length of \(BC\) if the distance from the center of the circle to side \(AD\) is 2.


Given a sequence of positive integers \( a_1, a_2, \ldots, a_n \), a member of the sequence is picked at random, its value is \( a \). Another member is picked at random, independently of the first, its value is \( b \). Then a third member is picked, its value is \( c \). Show that the probability that \( a + b + c \) is divisible by 3 is at least \( \frac{1}{4} \).


(a) A bag contains 11 balls numbered from 1 to 11, 7 of which are black and 4 of which are gold. Julio removes two balls as described above. What is the probability that both balls are black?

(b) For some integer \( g \geq 2 \), a second bag contains 6 black balls and \( g \) gold balls; the balls are numbered from 1 to \( g+6 \). Julio removes two balls as described above. The probability that both balls are black is \( \frac{1}{8} \). Determine the value of \( g \).

(c) For some integer \( x \geq 2 \), a third bag contains \( 2x \) black balls and \( x \) gold balls; the balls are numbered from 1 to \( 3x \). Julio removes two balls as described above. The probability that both balls are black is \( \frac{7}{16} \). Determine the value of \( x \).

(d) For some integer \( r \geq 3 \), a fourth bag contains 10 black balls, 18 gold balls, and \( r \) red balls; the balls are numbered from 1 to \( r+28 \). This time, Julio draws three balls one after another. The probability that two of these three balls are black and one of these three balls is gold is at least \( \frac{1}{3000} \). What is the largest possible value of \( r \)?


A prime number \( p \) is such that the number \( p + 25 \) is the seventh power of a prime number. What can \( p \) be? List all possible options.


In an arm wrestling tournament, there are $2^{n}$ athletes, where $n$ is a natural number greater than 7. For each win, an athlete receives 1 point; for a loss, 0 points. Before each round, pairs are randomly formed from participants who have an equal number of points (those who cannot be paired receive a point automatically). After the seventh round, it turned out that exactly 42 participants had scored 5 points. What is the value of $n$?


For any \( x \in \mathbf{R} \), the function \( f(x) \) satisfies the equation \( f(x+2008) = f(x+2007) + f(x+2009) \) and given that \( f(1) = \log \frac{3}{2} \) and \( f(2) = \log 15 \), what is the value of \( f(2007) \)?


Let \( n \) be the smallest positive integer that satisfies the following conditions: (1) \( n \) is a multiple of 75; (2) \( n \) has exactly 75 positive integer factors (including 1 and itself). Find \(\frac{n}{75}\).


The median $AM$ of triangle $ABC$ divides the segment $PR$, parallel to the side $AC$, with endpoints on sides $AB$ and $BC$, into segments of lengths 5 cm and 3 cm, measured from side $AB$. What is the length of side $AC$?

Solution: \boxed{13 \ \text{cm}}


In a rectangular parallelepiped \( A B C D A_{1} B_{1} C_{1} D_{1} \), the edge lengths are given as \( A B = 48 \), \( A D = 24 \), and \( A A_{1} = 12 \). 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 \( A E \) and \( B F \).


Compute the limit of the numerical sequence:

$$
\lim _{n \rightarrow \infty} \frac{\sqrt{\left(n^{2}+5\right)\left(n^{4}+2\right)}-\sqrt{n^{6}-3 n^{3}+5}}{n}
$$


A truck left from the settlement of Mirny at a speed of 40 km/h. At the same time, a car left from the city of Tikhiy in the same direction as the truck. In the first hour of the journey, the car traveled 50 km, and every subsequent hour it traveled 5 km more than in the previous hour. In how many hours will the car catch up with the truck if the distance between the settlement and the city is 135 km?


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


A black bag contains 40 socks that feel identical but have different colors: 10 white, 12 red, and 18 black. If 3 socks are drawn from the bag, what is the probability that at least 1 pair of socks (2 socks of the same color) is drawn?


In order to purchase new headphones costing 275 rubles, Katya decided to save money by spending less on sports activities. Until now, she had bought a single-visit pass to the swimming pool, including a trip to the sauna, for 250 rubles to warm up. However, now that summer has arrived, there is no longer a need to visit the sauna. Visiting only the swimming pool costs 200 rubles more than visiting the sauna. How many times must Katya visit the swimming pool without the sauna to save enough to buy the headphones? (Give the answer as a whole number, without spaces and units of measurement.)


There are \( N \) lockers, labeled from 1 to \( N \), placed in clockwise order around a circular hallway. Initially, all lockers are open. Ansoon starts at the first locker and always moves clockwise. When she is at locker \( n \) and there are more than \( n \) open lockers, she keeps locker \( n \) open and closes the next \( n \) open lockers, then repeats the process with the next open locker. If she is at locker \( n \) and there are at most \( n \) lockers still open, she keeps locker \( n \) open and closes all other lockers. She continues this process until only one locker is left open. What is the smallest integer \( N > 2021 \) such that the last open locker is locker 1?


Every day at noon, a scheduled ship departs from Moscow to Astrakhan and from Astrakhan to Moscow. The ship traveling from Moscow takes exactly four days to reach Astrakhan, then stays there for two days, and at noon two days after its arrival in Astrakhan, it departs back to Moscow. The ship traveling from Astrakhan takes exactly five days to reach Moscow and, after resting for two days in Moscow, departs back to Astrakhan. How many ships must be operational on the Moscow - Astrakhan - Moscow route under the described travel conditions?

Solution: \boxed{13 \text{ ships}}


Given that $O$ is the circumcenter of acute $\triangle ABC$, $\angle BAC = 60^{\circ}$, extend $CO$ to intersect $AB$ at point $D$, and extend $BO$ to intersect $AC$ at point $E$. Find $\frac{BD}{CE}=$ ______.


In triangle \( \triangle ABC \), \( 3AB = 2AC \), and \( E \) and \( F \) are the midpoints of \( AC \) and \( AB \), respectively. If \( BE < t \cdot CF \) always holds, then the minimum value of \( t \) is \(\qquad\).


In the rectangular coordinate system, a circle $\Omega: x^{2}+y^{2}+d x+e y+f=0$ (where $d, e, f$ are real numbers) has a diameter $AB$ with $A(20,22)$ and $B(10,30)$. Find the value of $f$.


How many roots does the equation $\sin x = \frac{x}{100}$ have?


Two non-zero numbers are given. If each of them is increased by one, their product doubles. By how many times will their product increase if each of the original numbers is squared and then decreased by one?


Count the number of sequences \( 1 \leq a_{1} \leq a_{2} \leq \cdots \leq a_{5} \) of integers with \( a_{i} \leq i \) for all \( i \).


A lateral edge of a regular triangular pyramid forms an angle of $45^{\circ}$ with the plane of the base. Find the angle between the apothem of the pyramid and the plane of an adjacent lateral face.

Solution: \boxed{\arcsin \frac{3}{5}}


Calculate the double integral

$$
\iint_{\Omega}(x+y) \, dx \, dy
$$

where $\Omega$ is a triangle bounded by the lines:

a) $x=0$, $y=0$, $x+y=3$;

b) $x=3$, $y=0$, $y=x$.


In bridge, a standard 52-card deck is dealt to 4 players. Each hand is assigned a number of "points" based on the formula
\[ 4 \times(\# \text{A's}) + 3 \times(\# \text{K's}) + 2 \times(\# \text{Q's}) + 1 \times(\# \text{J's}). \]

Given that a particular hand has exactly 4 cards that are A, K, Q, or J, find the probability that its point value is 13 or higher.


Twelve candidates for mayor are participating in a televised debate. At some point, one of them says, "So far, we've lied once." A second then says, "Now it's twice." A third exclaims, "Three times now," and so on, up to the twelfth who claims that before him, they lied twelve times. The presenter then stops the discussion. Given that at least one of the candidates correctly stated how many times lies had been told before their turn to speak, determine how many candidates lied in total.


Using the 4 boards with 4 squares each from Figure 1, form the shape shown in Figure 2. If the 16 squares in Figure 2 are filled with the numbers 1, 3, 5, and 7 (each square filled with one number), such that each row, each column, and each board contains different numbers, then what is the average of the numbers in the $A, B, C, D$ squares?


Let \( A = \{1, 2, 3, 4, 5, 6\} \). Find the number of distinct functions \( f: A \rightarrow A \) such that \( f(f(f(n))) = n \) for all \( n \in A \).


Anton thought of a three-digit number, and Alex is trying to guess it. Alex successively guessed the numbers 109, 704, and 124. Anton observed that each of these numbers matches the thought number exactly in one digit place. What number did Anton think of?


The recruits stood in a line one behind the other, facing in the same direction. Among them were three brothers: Peter, Nikolai, and Denis. There were 50 people in front of Peter, 100 people in front of Nikolai, and 170 people in front of Denis. Upon the command "About-face!" everyone turned to face the opposite direction. It turned out that there were now four times as many people standing in front of one of the brothers as there were in front of another. How many recruits (including the brothers) could there be? List all possible options.


Suppose that \(a, b, c,\) and \(d\) are positive integers which are not necessarily distinct. If \(a^{2}+b^{2}+c^{2}+d^{2}=70\), what is the largest possible value of \(a+b+c+d?\)


Ten spheres of identical radius are stacked in the shape of a triangular pyramid such that each sphere touches at least three others. Find the radius of a sphere inscribed in the center of the pyramid of spheres, which touches six identical spheres, if the radius of the sphere into which the pyramid is inscribed is equal to $\sqrt{6}+1$.

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


For which positive integer values of $k$ can the product of the first $k$ prime numbers be expressed as the sum of two positive cube numbers?

Solution: \boxed{k = 1}


On the first day, Masha collected 25% fewer mushrooms than Vasya, and on the second day, she collected 20% more mushrooms than Vasya. Over two days, Masha collected 10% more mushrooms than Vasya. What is the minimum number of mushrooms they could have collected together?


In a ski race, Petya and Vasya started at the same time. Vasya ran the entire race at a constant speed of 12 km/h. Petya ran the first half of the distance at a speed of 9 km/h, falling behind Vasya. What should be Petya's speed on the second half of the distance in order to catch up with Vasya and finish simultaneously with him? Provide the answer in km/h.

Solution: \boxed{18 \text{ km/h}}


A certain intelligence station has four different passwords, $A$, $B$, $C$, and $D$. Each week, one of these passwords is used, and each week the password is chosen at random and equally likely from the three passwords that were not used in the previous week. If password $A$ is used in the first week, what is the probability that password $A$ will also be used in the seventh week? (Express your answer as a simplified fraction.)


\( p \) and \( q \) are primes such that the numbers \( p+q \) and \( p+7q \) are both squares. Find the value of \( p \).


An ice cream vendor offers only five-scoop ice creams and fourteen different flavors. One can choose the same flavor more than once. How many choices are there?


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

$$
5^{x^{2}-6 a x+9 a^{2}}=a x^{2}-6 a^{2} x+9 a^{3}+a^{2}-6 a+6
$$

have exactly one solution?

Solution: \boxed{a = 1}


Fill six numbers $1, 3, 5, 7, 9, 11$ into the circles (each circle containing only one number) so that the sum of three numbers on each side of the triangle equals 17. What is the sum of the three numbers in the circles at the vertices of the triangle?


In five pots arranged in a row, Rabbit poured three kilograms of honey (not necessarily into each pot and not necessarily evenly). Winnie-the-Pooh can take any two pots that are next to each other. What is the maximum amount of honey that Winnie-the-Pooh can be guaranteed to eat?

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


Let \(ABCD\) be a regular tetrahedron, and let \(O\) be the centroid of triangle \(BCD\). Consider the point \(P\) on \(AO\) such that \(P\) minimizes \(PA + 2(PB + PC + PD)\). Find \(\sin \angle PBO\).


Given that \( f(x) \) is an even function defined on \( \mathbf{R} \), and \( g(x) \) is an odd function such that \( g(x) = f(x-1) \), find the value of \( f(2007) \).


Let $\Gamma$ be the circumscribed circle of the equilateral triangle $ABC$. Let $M$ be a point on the arc between $B$ and $C$ that does not contain $A$. Show that $AM = BM + CM$.

Solution: \boxed{AM = BM + CM}


Let \( G \) be the centroid of \(\triangle ABC\). Given \( BG \perp CG \) and \( BC = \sqrt{2} \), find the maximum value of \( AB + AC \).


$n$ players participate in a chess tournament where each player plays a game with every other player. The winner of a game receives 1 point, the loser receives 0 points, and a draw awards each player 0.5 points. A tournament is said to have property $P(m)$ if, among any $m$ players, there is one player who has defeated the other $m-1$ players, and there is one player who has lost to the other $m-1$ players. For a given $m$ ($m \geq 4$), find the minimum value of $n$, denoted by $f(m)$, such that in any tournament with property $P(m)$, all $n$ players have different scores.

Solution: \boxed{f(m) = 2m - 3}


Find the maximum number of points that can be placed on a sphere of radius 1 such that the distance between any two points is: a) at least $\sqrt{2}$; b) more than $\sqrt{2}$.


Given that \( F_{1} \) and \( F_{2} \) are the left and right foci of the ellipse \(\frac{x^{2}}{4}+\frac{y^{2}}{3}=1\), and \( A \) is a moving point on the ellipse. Circle \( C \) is tangent to the extended line of \( F_{1}A \), the extended line of \( F_{1}F_{2} \), and segment \( AF_{2} \). If \( M(t, 0) \) is a point of tangency, then \( t=\_\_\_\_\_ \).


Given a set $S$ of $2n+1$ distinct irrational numbers, find the largest positive integer $m$ such that there exist $m$ different numbers in $S$ where the sum of any two of them is still an irrational number.


In the plane, there is a set of integer points \( S = \{ (a, b) \mid 1 \leqslant a, b \leqslant 5, a, b \in \mathbf{Z} \} \). Now consider a subset \( T \) of integer points in the plane such that for any point \( P \) in \( S \), there exists a distinct point \( Q \) in \( T \) such that the line segment \( PQ \) contains no other integer points except \( P \) and \( Q \). What is the minimum number of elements required in the set \( T \)?


Beatriz loves odd numbers. How many numbers between 0 and 1000 can she write using only odd digits?


Let \( n \) be a positive integer. Define a sequence by \( a_{0}=1 \), \( a_{2i+1}=a_{i} \), and \( a_{2i+2}=a_{i} + a_{i+1} \) for each \( i \geq 0 \). Determine, with proof, the value of \( a_{0} + a_{1} + a_{2} + \cdots + a_{2^{n}-1} \).

Solution: \boxed{\frac{3^n + 1}{2}}


In a game, there are three indistinguishable boxes; one box contains two red balls, one contains two blue balls, and the last contains one ball of each color. To play, Raj first predicts whether he will draw two balls of the same color or two of different colors. Then, he picks a box, draws a ball at random, looks at the color, and replaces the ball in the same box. Finally, he repeats this; however, the boxes are not shuffled between draws, so he can determine whether he wants to draw again from the same box. Raj wins if he predicts correctly; if he plays optimally, what is the probability that he will win?


Three fair six-sided dice, each numbered 1 through 6, are rolled. What is the probability that the three numbers that come up can form the sides of a triangle?


Given the quadratic function
\[ f(x) = x^2 + b x + 8 \quad (b \neq 0) \]

which has two distinct real roots \( x_1 \) and \( x_2 \), and \( x_1 + \frac{1}{x_2} \) and \( x_2 + \frac{1}{x_1} \) are the zero points of a quadratic function \( g(x) \) with the quadratic coefficient being 1. If \( g(1) = f(1) \), find all possible values of \( g(1) \).


The function \( f \) satisfies
\[
f(x) + f(2x + y) + 5xy = f(3x - y) + 2x^2 + 1
\]
for all real numbers \( x \) and \( y \). Determine the value of \( f(10) \).


Let the parabola $y = ax^2 + bx + c$ pass through the points $A(-1, -3)$, $B(4, 2)$, and $C(0, 2)$. Let $P$ be a moving point on the axis of symmetry of the parabola. If $P A + P C$ reaches its minimum value at the point $P$ with coordinates $(m, n)$, find $n$.


In how many ways can the number \( n \) be represented as the sum of several terms, each equal to 1 or 2? (Representations that differ in the order of terms are considered different.)


Calculate the arc lengths of the curves given by the parametric equations.

$$
\begin{aligned}
& \left\{\begin{array}{l}
x=\frac{1}{2} \cos t-\frac{1}{4} \cos 2 t \\
y=\frac{1}{2} \sin t-\frac{1}{4} \sin 2 t
\end{array}\right. \\
& \frac{\pi}{2} \leq t \leq \frac{2 \pi}{3}
\end{aligned}
$$

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


A merchant has two types of tea: Ceylon tea at 10 rubles per pound and Indian tea at 6 rubles per pound. To increase profit, the merchant decides to mix the two types but sell the mixture at the same price of 10 rubles per pound. In what proportion should he mix them to make an additional profit of 3 rubles per pound?


Find the smallest prime number $p$ such that $n^{2}+n+11$ is divisible by $p$ for some integer $n$.


Anya calls a date beautiful if all 6 digits of its record are different. For example, 19.04.23 is a beautiful date, while 19.02.23 and 01.06.23 are not. How many beautiful dates are there in the year 2023?


Construct on the plane the set of points whose coordinates satisfy the inequality \( |3x + 4| + |4y - 3| \leq 12 \). Indicate the area of the resulting figure.


The diagonals of a trapezoid are 6 and 8, and the midline is 5. Find the area of the trapezoid.


Given the points \( B_1 \) and \( B_2 \) are the lower and upper vertices of the ellipse \( C: \frac{x^2}{8} + \frac{y^2}{4} = 1 \), respectively. The line \( l \) passing through the point \( A(0, -2\sqrt{2}) \) intersects the ellipse \( C \) at points \( P \) and \( Q \) (different from \( B_1 \) and \( B_2 \)).

1. If \( \tan \angle B_1 P B_2 = 2 \tan \angle B_1 Q B_2 \), find the equation of the line \( l \).
2. Let \( R \) be the intersection point of the lines \( B_1 P \) and \( B_2 Q \). Find the value of \( \overrightarrow{A R} \cdot \overrightarrow{B_1 B_2} \).


There were 15 buns on a plate. Karlsson took three times as many buns as Little Boy, and Little Boy's dog Bimbo took three times fewer buns than Little Boy. How many buns are left on the plate? Explain your answer.


In triangle \( A B C \) with side \( A C = 8 \), a bisector \( B L \) is drawn. It is known that the areas of triangles \( A B L \) and \( B L C \) are in the ratio \( 3: 1 \). Find the bisector \( B L \), for which the height dropped from vertex \( B \) to the base \( A C \) will be the greatest.


Simplify the expression \(\left(\frac{2-n}{n-1}+4 \cdot \frac{m-1}{m-2}\right):\left(n^{2} \cdot \frac{m-1}{n-1}+m^{2} \cdot \frac{2-n}{m-2}\right)\) given that \(m=\sqrt[4]{400}\) and \(n=\sqrt{5}\).


What is the remainder when \(2^{99}\) is divided by 7?


A square with a side length of 100 was cut into two equal rectangles. These rectangles were then placed next to each other as shown in the picture. Find the perimeter of the resulting figure.


Find all prime numbers \( p \) not exceeding 1000 such that \( 2p + 1 \) is a perfect power (i.e., there exist natural numbers \( m \) and \( n \geq 2 \) such that \( 2p + 1 = m^n \)).


Calculate the surface area of the part of the paraboloid of revolution \( 3y = x^2 + z^2 \) that is located in the first octant and bounded by the plane \( y = 6 \).

Solution: \boxed{\frac{39 \pi}{4}}


A caterpillar is climbing a 20-meter pole. During the day, it climbs 5 meters, and during the night, it slides down 4 meters. How long will it take for the caterpillar to reach the top of the pole?

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


For what values of the parameter \( a \) does the equation \( x^{3} - 15x^{2} + ax - 64 = 0 \) have three distinct real roots that form a geometric progression?


Given a triangle \(ABC\) with a perimeter of 1. A circle is inscribed in angle \(BAC\), lying outside the triangle \(ABC\) and touching side \(BC\) (and the extensions of sides \(AB\) and \(AC\)). Find the length of segment \(AM\), where \(M\) is the point where the circle touches line \(AC\).


In triangle \( ABC \), \( AB = 14 \), \( BC = 6 \), and \( CA = 9 \). Point \( D \) lies on line \( BC \) such that \( BD:DC = 1:9 \). The circles inscribed in triangles \( ADC \) and \( ADB \) touch side \( AD \) at points \( E \) and \( F \) respectively. Find the length of segment \( EF \).


Four points are randomly chosen from the vertices of a regular 12-sided polygon. Find the probability that the four chosen points form a rectangle (including square).


How many 10-digit positive integers consisting only of the digits 0 and 1 are divisible by 11?


Given real numbers \( a, b, \) and \( c \) that satisfy
\[ f(x) = a \cos x + b \cos 2x + c \cos 3x \geq -1 \]
for any real number \( x \). What is the maximum value of \( a + b + c \)?


Let \( A = \{1, 2, \cdots, 10\} \). If the equation \( x^2 - bx - c = 0 \) satisfies \( b, c \in A \) and the equation has at least one root \( a \in A \), then the equation is called a "beautiful equation". Find the number of "beautiful equations".


Inside a convex $n$-gon there are 100 points positioned in such a way that no three of these $n+100$ points are collinear. The polygon is divided into triangles, each having vertices among any 3 of the $n+100$ points. For what maximum value of $n$ can no more than 300 triangles be formed?


\(ABC\) is a triangle with \(AB = 15\), \(BC = 14\), and \(CA = 13\). The altitude from \(A\) to \(BC\) is extended to meet the circumcircle of \(ABC\) at \(D\). Find \(AD\).


In a compartment, any \( m (m \geqslant 3) \) passengers have a unique common friend (if A is a friend of B, then B is also a friend of A, and no one is their own friend). How many friends does the person with the most friends have in this compartment?

Solution: \boxed{m}


Given $x, y \in \mathbb{N}$, find the maximum value of $y$ such that there exists a unique value of $x$ satisfying the following inequality:
$$
\frac{9}{17}<\frac{x}{x+y}<\frac{8}{15}.
$$


$a, b$ and $c$ are the lengths of the opposite sides $\angle A, $\angle B$ and $\angle C$ of the $\triangle ABC$ respectively. If $\angle C = 60^{\circ}$ and $\frac{a}{b+c} + \frac{b}{a+c} = P$, find the value of $P$.


Given a fixed triangle \( \triangle ABC \) and a point \( P \), find the maximum value of

\[
\frac{AB^{2} + BC^{2} + CA^{2}}{PA^{2} + PB^{2} + PC^{2}}
\]


How many ways can the integers from -7 to 7 be arranged in a sequence such that the absolute value of the numbers in the sequence is nondecreasing?


On the board, there are natural numbers from 1 to 1000, each written once. Vasya can erase any two numbers and write one of the following in their place: their greatest common divisor or their least common multiple. After 999 such operations, one number remains on the board, which is equal to a natural power of ten. What is the maximum value it can take?


There is a table of size $8 \times 8$ representing a chessboard. In each step, you are allowed to swap any two columns or any two rows. Is it possible, in several steps, to make the upper half of the table white and the lower half black?

Solution: \boxed{\text{No}}


A sequence of positive integers \(a_{1}, a_{2}, \ldots\) is such that for each \(m\) and \(n\) the following holds: if \(m\) is a divisor of \(n\) and \(m < n\), then \(a_{m}\) is a divisor of \(a_{n}\) and \(a_{m} < a_{n}\). Find the least possible value of \(a_{2000}\).


Given two sets of points \(A = \left\{(x, y) \mid (x-3)^{2}+(y-4)^{2} \leqslant \left(\frac{5}{2}\right)^{2}\right\}\) and \(B = \left\{(x, y) \mid (x-4)^{2}+(y-5)^{2} > \left(\frac{5}{2}\right)^{2}\right\}\), the number of lattice points (i.e., points with integer coordinates) in the set \(A \cap B\) is ...


Let \( x \in \mathbb{R} \). The function \( f(x)=|2x-1| + |3x-2| + |4x-3| + |5x-4| \). What is the minimum value of the function?


Let \( f: \mathbb{R} \rightarrow \mathbb{R} \) be an odd function, i.e., a function that satisfies \( -f(x) = f(-x) \) for all \( x \in \mathbb{R} \). Suppose that \( f(x+5) = f(x) \) for all \( x \in \mathbb{R} \) and that \( f(1/3) = 1 \). Determine the value of the sum:

\[
f(16/3) + f(29/3) + f(12) + f(-7)
\]


Find all non-empty finite sets \( S \) of positive integers such that if \( m, n \in S \), then \( \frac{m+n}{\gcd(m,n)} \in S \).


In the vertices of a unit square, perpendiculars are erected to its plane. On them, on one side of the plane of the square, points are taken at distances of 3, 4, 6, and 5 from this plane (in order of traversal). Find the volume of the polyhedron whose vertices are the specified points and the vertices of the square.


Find all functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) such that for all real numbers \( x \) and \( y \),

\[ f(f(x)+2y) = 6x + f(f(y)-x). \]

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


How many times longer is the staircase to the fourth floor of a building compared to the staircase to the second floor of the same building?


Let \(\pi\) be a randomly chosen permutation of the numbers from 1 through 2012. Find the probability that \(\pi(\pi(2012))=2012\).


The sum of all natural numbers not exceeding 200 that leave a remainder of 7 when divided by 11 and a remainder of 5 when divided by 7 is $\qquad$


Let \( x \) and \( y \) be positive real numbers, and let \( \theta \neq \frac{n \pi}{2} \) (where \( n \) is an integer). If \( \frac{\sin \theta}{x}=\frac{\cos \theta}{y} \), and \( \frac{\cos ^{4} \theta}{x^{4}}+\frac{\sin ^{4} \theta}{y^{4}}=\frac{97 \sin 2 \theta}{x^{3} y+y^{3} x} \), then find the value of \( \frac{y}{x}+\frac{x}{y} \).


Calculate the following:
$$\frac{1 \cdot 2 + 2 \cdot 3 + 3 \cdot 4 + \ldots + 2013 \cdot 2014}{(1 + 2 + 3 + \ldots + 2014) \cdot \frac{1}{5}}$$

If necessary, round the answer to two decimal places.


The side \( AD \) of the rectangle \( ABCD \) is equal to 2. On the extension of the side \( AD \) beyond point \( A \), a point \( E \) is taken such that \( EA = 1 \), and \( \angle BEC = 30^\circ \). Find \( BE \).


Calculate the limit of the function:

\[
\lim _{x \rightarrow 4} \frac{\sqrt{1+2 x}-3}{\sqrt{x}-2}
\]


As shown in the figure, first place 5 pieces (4 black and 1 white) on a circle. Then, place a white piece between two pieces of the same color, and place a black piece between two pieces of different colors, while removing the original 5 pieces. Continuously perform these operations. What is the maximum number of white pieces that can be on the circle among the 5 pieces?


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


To enter a park, a group of two men, four women, and two children paid 226 reais, while a group of three men, three women, and one child paid 207 reais.

a) How much would a group of 8 men, 10 women, and 4 children pay to enter the park?

b) If the ticket prices are all natural numbers, how many possible prices are there for the tickets?


A math professor asks 100 math students to guess an integer between 0 and 100, inclusive, such that their guess is 'two-thirds of the average of all the responses.' Each student who guesses the highest integer that is not higher than two-thirds of the average of all responses will receive a prize. Assuming that it is common knowledge that all students will write down the best response, and there is no communication between students, what single integer should every student write down?


In the expression \((x + y + z)^{2018} + (x - y - z)^{2018}\), the brackets were expanded, and like terms were combined. How many monomials \(x^{a} y^{b} z^{c}\) have a nonzero coefficient?


Mr. Rychlý and Mr. Louda started the same hiking route at the same time. Mr. Rychlý walked from a mountain hut down to the town, and Mr. Louda walked from the town up to the mountain hut. They passed each other at 10 AM. Mr. Rychlý hurried and reached his destination at 12 PM, while Mr. Louda moved slowly and reached the hut at 6 PM. At what time did they start their hike, given that each traveled at a constant speed?

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


Find all positive integers \( k \) such that the polynomial \( x^{2k+1} + x + 1 \) is divisible by \( x^k + x + 1 \). For each \( k \) that satisfies this condition, find the positive integers \( n \) such that \( x^n + x + 1 \) is divisible by \( x^k + x + 1 \).

(British Mathematical Olympiad, 1991)


There is a set of integers: two 2's, three 3's, four 4's, and five 5's (14 numbers in total). These numbers are placed in the circles of a star as shown in the diagram. Could it be such that the sums of the numbers along each segment (each with 4 circles) are equal to each other?

Solution: \boxed{\text{No}}


Find the limit as \( x \) approaches \( \frac{\pi}{2} \) of \((x - \frac{\pi}{2}) \cdot \tan(x)\).


60 explorers need to cross a river using a rubber boat that can carry 6 people (one trip across the river and back counts as two crossings). Each crossing takes 3 minutes. How many minutes will it take for all explorers to reach the other side of the river?


The Wolf and Ivan Tsarevich are 20 versts away from a source of living water, and the Wolf is taking Ivan Tsarevich there at a speed of 3 versts per hour. To revive Ivan Tsarevich, one liter of water is needed, which flows from the source at a rate of half a liter per hour. At the source, there is a Raven with unlimited carrying capacity; it must gather the water, after which it will fly towards the Wolf and Ivan Tsarevich at a speed of 6 versts per hour, spilling a quarter liter of water every hour. After how many hours will it be possible to revive Ivan Tsarevich?

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


Given that \( a, b, c \) are positive integers, and the parabola \( y = ax^2 + bx + c \) intersects the x-axis at two distinct points \( A \) and \( B \). If the distances from \( A \) and \( B \) to the origin are both less than 1, find the minimum value of \( a + b + c \).


The numbers \(2^{n}\) and \(5^{n}\) (in decimal notation) are written one after the other. How many digits does the resulting number have?


Calculate the limit

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


There are 37 people lined up in a row, and they are counting off one by one. The first person says 1, and each subsequent person says the number obtained by adding 3 to the previous person’s number. At one point, someone makes a mistake and subtracts 3 from the previous person's number instead. The sum of all the numbers reported by the 37 people is 2011. Which person made the mistake?


Find all sets of positive integers \((x, y, z)\) such that \(y\) is a prime number, neither \(y\) nor 3 divides \(z\), and \(x^3 - y^3 = z^2\).


During the vacation, for eight classes of a school, each with an equal number of students, a museum excursion was organized, while for the remaining students, who turned out to be $15\%$ more, a field trip to the puppet theater was arranged. How many students are in the school, given that there are no more than 520 of them, and more than 230 students participated in the excursion?


Find the strictly increasing functions \( f: \mathbb{N}^{*} \rightarrow \mathbb{N}^{*} \) such that \( f(2)=2 \) and for all \( n, m \geq 1 \), we have

\[ f(n m) = f(n) f(m). \]

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


A regular hexagon \( A B C D E K \) is inscribed in a circle of radius \( 3 + 2\sqrt{3} \). Find the radius of the circle inscribed in the triangle \( B C D \).


On the board, 34 ones are written. Each minute, Karlson erases two random numbers and writes their sum on the board, and then eats a number of candies equal to the product of the two erased numbers. What is the maximum number of candies he could eat in 34 minutes?


Brian has a 20-sided die with faces numbered from 1 to 20, and George has three 6-sided dice with faces numbered from 1 to 6. Brian and George simultaneously roll all their dice. What is the probability that the number on Brian's die is larger than the sum of the numbers on George's dice?


Let \( F_1 \) and \( F_2 \) be the foci of the ellipse \( \frac{x^{2}}{9}+\frac{y^{2}}{4}=1 \), and \( P \) be a point on the ellipse such that \( \left|P F_1\right|:\left|P F_2\right| = 2:1 \). Find the area of the triangle \( \triangle P F_1 F_2 \).


Find the minimum value of the function

$$
f(x)=x^{2}+(x-2)^{2}+(x-4)^{2}+\ldots+(x-100)^{2}
$$

If the result is a non-integer, round it to the nearest whole number.


Anya calls a date beautiful if all 6 digits in its recording are different. For example, 19.04.23 is a beautiful date, but 19.02.23 and 01.06.23 are not.

a) How many beautiful dates will there be in April 2023?

b) How many beautiful dates will there be in the entire year of 2023?


Let $\triangle ABC$ have three internal angles $\angle A, \angle B, \angle C$, and the side lengths opposite to these angles are $a, b, c$ respectively. Given that $a < b < c$, and
\[
\begin{cases}
\frac{b}{a} = \frac{\left|b^{2} + c^{2} - a^{2}\right|}{bc} \\
\frac{c}{b} = \frac{\left|c^{2} + a^{2} - b^{2}\right|}{ca} \\
\frac{a}{c} = \frac{\left|a^{2} + b^{2} - c^{2}\right|}{ab}
\end{cases}
\]
find the ratio of the radian measures of the angles $\angle A, \angle B, \angle C$.


The famous skater Tony Hawk rides a skateboard (segment \( AB \)) on a ramp, which is a semicircle with diameter \( PQ \). Point \( M \) is the midpoint of the skateboard, and \( C \) is the foot of the perpendicular dropped from point \( A \) to the diameter \( PQ \). What values can the angle \( \angle ACM \) take, given that the angular measure of arc \( AB \) is \( 24^\circ \)?


Four points \( A, O, B, O' \) are aligned in this order on a line. Let \( C \) be the circle centered at \( O \) with radius 2015, and \( C' \) be the circle centered at \( O' \) with radius 2016. Suppose that \( A \) and \( B \) are the intersection points of two common tangents to the two circles. Calculate \( AB \) given that \( AB \) is an integer \( < 10^7 \) and that \( AO \) and \( AO' \) are integers.


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


Given that \( m \) and \( n \) are both positive integers and satisfy \( 24m = n^4 \), find the minimum value of \( m \).


If a natural number \(N\) can be expressed as the sum of 3 consecutive natural numbers, the sum of 11 consecutive natural numbers, and the sum of 12 consecutive natural numbers, then the smallest value of \(N\) is \(\qquad\). (Note: The smallest natural number is 0.)


A porter needs to transport 200 steamed buns from the kitchen to the construction site (he is currently in the kitchen). He can carry 40 buns each time. However, because he is very greedy, he will eat 1 bun each time he travels from the kitchen to the construction site or from the construction site to the kitchen. What is the maximum number of buns he can transport to the construction site?


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


At a temperature of $13^{\circ}$, a 41% sugar solution in water becomes saturated. How many grams of sugar will not dissolve in water (and will precipitate) if (at a temperature of $15^{\circ}$) you thoroughly mix a glass of water (220 g) with 280 g of sugar?


For the "Skillful Hands" club, Boris needs to cut several identical pieces of wire (the length of each piece is an integer number of centimeters). Initially, Boris took a piece of wire 10 meters long and was able to cut only 14 required pieces from it. Then, Boris took a piece that was 50 centimeters longer, but it was also only enough for 14 pieces. What length did Boris need to cut the pieces? Express your answer in centimeters.


Find the remainder when \( 2^{1999} + 1 \) is divided by 17.


The largest divisor of a natural number \( N \), smaller than \( N \), was added to \( N \), producing a power of ten. Find all such \( N \).


Let \( x_{1}, x_{2}, \ldots, x_{100} \) be natural numbers greater than 1 (not necessarily distinct). In a \( 100 \times 100 \) table, numbers are placed as follows: at the intersection of the \( i \)-th row and the \( k \)-th column, the number \( \log _{x_{k}} \frac{x_{i}}{4} \) is written. Find the smallest possible value of the sum of all the numbers in the table.


There are 4 points \( A, B, C, D \) in space, satisfying \( AB = BC = CD \). Given that \( \angle ABC = \angle BCD = \angle CDA = 36^\circ \), what is the angle between line \( AC \) and line \( BD \)?


In $\triangle ACD$, $\angle A=45^{\circ}$, $CD=7$, $B$ is a point on $AD$, $CB=5$, $BD=3$, find $AC$.


Calculate the value of the expression \(\arccos \frac{\sqrt{6}+1}{2 \sqrt{3}} - \arccos \sqrt{\frac{2}{3}}\). Express the result in the form \(\frac{a \pi}{b}\), where \(a\) and \(b\) are integers that are coprime, and indicate the value of \(|a-b|\).


Evaluate the following limits:
1. $\lim_{x \rightarrow 0} \frac{1- \sqrt{x+1}}{x}$
2. $\lim_{x \rightarrow 0} \frac{\operatorname{tg} x}{1-\sqrt{1+\operatorname{tg} x}}$
3. $\lim_{x \rightarrow 1} \frac{2- \sqrt{x}}{3- \sqrt{2x+1}}$
4. $\lim_{x \rightarrow 1} \frac{1- \sqrt{x}}{1- \sqrt[3]{x}}$


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 \(\frac{BO}{OE}\).


An edge of an inclined parallelepiped is equal to $l$. It is bounded by two adjacent faces, whose areas are $m^{2}$ and $n^{2}$, and the planes of these faces form an angle of $30^{\circ}$. Calculate the volume of the parallelepiped.

Solution: \boxed{\frac{m^2 n^2}{2 l}}


Every third student in the sixth grade is a member of the math club, every fourth student is a member of the history club, and every sixth student is a member of the chemistry club. The rest of the students are members of the literature club. How many people are in the chemistry club if the number of members in the math club exceeds the number of members in the literature club by three?


The common difference of the arithmetic sequence $\left\{a_{n}\right\}$ is $d=2$, and the sum of the first 30 terms is $a_{1}+a_{2}+\cdots+a_{30}=100$. Find the sum of every third term from $a_{3}$ to $a_{30}$: $a_{3}+a_{6}+\cdots+a_{30}$.


Given a positive integer \( n \), and a set \( M = \{1, 2, \cdots, 2n\} \), find the smallest positive integer \( k \) such that for any \( k \)-element subset of \( M \), it must contain 4 different elements whose sum equals \( 4n + 1 \).

Solution: \boxed{n + 3}


A conveyor system produces on average 85% of first-class products. How many products need to be sampled so that, with a probability of 0.997, the deviation of the frequency of first-class products from 0.85 in absolute magnitude does not exceed 0.01?


Given \( n \in \mathbb{N}^{*} \). Find the number of \( n \)-digit numbers whose digits belong to \(\{2, 3, 7, 9\}\) and are divisible by 3.

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


In an arithmetic sequence \(\{a_{n}\}\), if \(\frac{a_{11}}{a_{10}} < -1\) and the sum of the first \(n\) terms \(S_{n}\) has a maximum value, then the value of \(n\) when \(S_{n}\) attains its smallest positive value is \(\qquad\).


There are two circles: one with center at point \( A \) and radius 6, and the other with center at point \( B \) and radius 3. Their common internal tangent touches the circles respectively at points \( C \) and \( D \). The lines \( AB \) and \( CD \) intersect at point \( E \). Find the length of \( CD \), given that \( AE = 10 \).


On a line \( r \), points \( A \) and \( B \) are marked, and on a line \( s \), parallel to \( r \), points \( C \) and \( D \) are marked so that \( A B C D \) forms a square. Point \( E \) is also marked on the segment \( C D \).

a) What is the ratio between the areas of triangles \( A B E \) and \( B C D \), if \( E \) is the midpoint of \( C D \)?

b) What is the ratio \( \frac{D E}{E C} \), for the area of triangle \( B F E \) to be twice the area of \( D F E \), where \( F \) is the intersection of segments \( A E \) and \( B D \)?


Find the number that becomes a perfect square either by adding 5 or by subtracting 11.


Determine all real values of \( A \) for which there exist distinct complex numbers \( x_{1} \) and \( x_{2} \) such that the following three equations hold:
\[
\begin{aligned}
x_{1}(x_{1}+1) &= A, \\
x_{2}(x_{2}+1) &= A, \\
x_{1}^{4} + 3x_{1}^{3} + 5x_{1} &= x_{2}^{4} + 3x_{2}^{3} + 5x_{2}.
\end{aligned}
\]


For all \( x \in \left(0, \frac{\pi}{2}\right) \), find the largest positive integer \( n \) such that the inequality \( \sin^{n} x + \cos^{n} x > \frac{1}{2} \) holds.


If \( a \) and \( b \) are prime numbers greater than 7, then the expression

$$
\left(a^{2}-1\right)\left(b^{2}-1\right)\left(a^{6}-b^{6}\right)
$$

is divisible by 290304.


A trapezoid with side lengths \( a \) and \( b \) is circumscribed around a circle. Find the sum of the squares of the distances from the center of the circle to the vertices of the trapezoid.

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


Show that as \( t \rightarrow \infty \), the limit of the variable \( x = \frac{6t^3 - 9t + 1}{2t^3 - 3t} \) is 3.


How many real solutions are there to the equation
\[ |||| x|-2|-2|-2|=|||| x|-3|-3|-3| ? \]


Let \(ABCD\) be a quadrilateral inscribed in a circle with diameter \(\overline{AD}\). If \(AB = 5\), \(AC = 6\), and \(BD = 7\), find \(CD\).


Let \( n \) be a natural number. For any real numbers \( x, y, z \), if the inequality \(\left(x^{2}+y^{2}+z^{2}\right) \leqslant n\left(x^{4}+y^{4}+z^{4}\right)\) always holds, then the smallest value of \( n \) is _____.


In how many ways can the numbers \(1, 2, 3, 4, 5, 6\) be arranged in a row so that for any three consecutive numbers \(a, b, c\), the expression \(ac - b^2\) is divisible by 7? Answer: 12.


Let \( X = \{1, 2, 3, \ldots, 17\} \). Find the number of subsets \( Y \) of \( X \) with odd cardinalities.


Jack Sparrow needed to distribute 150 piastres across 10 purses. After placing a certain amount of piastres in the first purse, he placed more piastres in each subsequent purse than in the previous one. As a result, the number of piastres in the first purse was not less than half the number of piastres in the last purse. How many piastres are in the 6th purse?


Meg started with the number 100. She increased this number by 20% and then increased the resulting number by 50%. Her final result was:

(A) 120
(B) 240
(C) 187.5
(D) 200
(E) 180


Formulate the equation of the plane \( Q \), which intercepts a segment \( OA = 3 \) on the \( OX \) axis and is perpendicular to the vector \( \bar{n} \{2, -3, 1\} \).

Solution: \boxed{2 x - 3 y + z = 6}


In the equation, $\overline{\mathrm{ABCD}}+\overline{\mathrm{EFG}}=2020$, different letters represent different digits. What is $A+B+C+D+E+F+G=$ $\qquad$?


Given \( n \in \mathbb{N} \) and \( a \in [0, n] \), under the condition
$$
\sum_{i=1}^{n} \sin ^{2} x_{i}=a
$$
find the maximum value of \( \left| \sum_{i=1}^{n} \sin 2 x_{i} \right| \).

Solution: \boxed{2 \sqrt{a(n - a)}}


In an isosceles trapezoid with bases \(a = 21\), \(b = 9\) and height \(h = 8\), find the radius of the circumscribed circle.


Petrov booked an apartment in a newly built house, which has five identical entrances. Initially, the entrances were numbered from left to right, and Petrov's apartment number was 636. Later, the developer changed the numbering to the opposite direction (right to left, as shown in the diagram). Then, Petrov's apartment number became 242. How many apartments are in the building? (The numbering of apartments within each entrance has not changed.)


It is required to make a box with a square base for placing boxes that are 9 cm wide and 21 cm long. What should be the minimum length of the side of the square base so that the boxes fit in the box snugly?

Solution: \boxed{63 \text{ cm}}


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


Find all positive integers \( n \) that satisfy

$$
n=2^{2x-1}-5x-3=\left(2^{x-1}-1\right)\left(2^x+1\right)
$$

for some positive integer \( x \).


Let $K$ be the incenter of $\triangle ABC$. Points $C_{1}$ and $B_{1}$ are the midpoints of sides $AB$ and $AC$, respectively. The line $AC$ intersects $C_{1}K$ at point $B_{2}$, and the line $AB$ intersects $B_{1}K$ at point $C_{2}$. If the area of $\triangle AB_{2}C_{2}$ equals the area of $\triangle ABC$, find $\angle CAB$.


Three numbers are given. If each of them is increased by 1, their product also increases by 1. If all the original numbers are increased by 2, their product also increases by 2. Find these numbers.

Solution: \boxed{-1, -1, -1}


Find the smallest natural number that is simultaneously twice an exact square and three times an exact cube.


How many digits are both prime and represented by a prime number of illuminated bars?
A 0
B 1
C 2
D 3
E 4


The sequence \( \{a_n\} \) satisfies: \( a_1 = 1 \), and for each \( n \in \mathbf{N}^{*} \), \( a_n \) and \( a_{n+1} \) are the roots of the equation \( x^2 + 3n x + b_n = 0 \). Find the value of \( \sum_{k=1}^{20} b_k \).


Let \(ABCD\) be a convex quadrilateral inscribed in a circle with center \(O\). Let \(P\) be the intersection point of the diagonals and \(Q\) the second intersection point of the circumcircles of triangles \(APD\) and \(BPC\).

Show that \(\widehat{OQP} = 90^\circ\).


At 9 a.m., ships "Anin" and "Vanin" departed from port O to port E. At the same moment, the ship "Sanin" set off from port E to port O. All three vessels are traveling along the same course (with "Sanin" heading towards "Anin" and "Vanin") at constant but different speeds. At 10 a.m., "Vanin" was equidistant from both "Anin" and "Sanin." At 10:30 a.m., "Sanin" was equidistant from both "Anin" and "Vanin." At what moment will "Anin" be exactly in the middle between "Vanin" and "Sanin"?


As shown in the figure, in a square \(ABCD\), \(AB=1\). Let the midpoints of \(AB\) and \(BC\) be \(E\) and \(F\) respectively. If line \(CE\) intersects line \(AF\) at point \(G\), find the area of quadrilateral \(GEBF\).


An experienced sawmiller, Harik, can make cuts. In one day of continuous work, he cuts 600 nine-meter logs into equal three-meter logs (the only difference from the original logs is their length). How much time will it take for the experienced sawmiller Harik to cut 400 twelve-meter logs (which only differ in length from the nine-meter logs) into the same three-meter logs?

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


Given a set $M$ of $n$ points on a plane, if every three points in $M$ form the vertices of an equilateral triangle, find the maximum value of $n$.


Solution: \boxed{x = 7, \; y = -2}


If the side length of an equilateral triangle \( \triangle ABC \) is 6 and the distances from its three vertices to the plane \( \alpha \) are 1, 2, and 3 respectively, find the distance from the centroid \( G \) of \( \triangle ABC \) to the plane \( \alpha \).


There are 20 numbers arranged in a circle. It is known that the sum of any six consecutive numbers is 24. What number is in the 12th position if the number in the 1st position is 1?


A prime number \( p \) is such that the number \( p + 25 \) is the seventh power of a prime number. What can \( p \) be? List all possible options.


Find the product of all divisors of the number \( N \), given that there are \( n \) divisors.


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


Given three points \( A, B, C \) on a plane such that \( |\overrightarrow{AB}| = 3 \), \( |\overrightarrow{BC}| = 4 \), \( |\overrightarrow{CA}| = 5 \), find the value of \( \overrightarrow{AB} \cdot \overrightarrow{BC} + \overrightarrow{BC} \cdot \overrightarrow{CA} + \overrightarrow{CA} \cdot \overrightarrow{AB} \).


The villages "Verkhnie Vasyuki" and "Nizhnie Vasyuki" are located on the riverbank. A steamboat travels the distance from Verkhnie Vasyuki to Nizhnie Vasyuki in one hour, while a motorboat covers the same distance in 45 minutes. It is known that the speed of the motorboat in still water is twice the speed of the steamboat (also in still water). Determine the time (in minutes) required for a raft to drift from Verkhnie Vasyuki to Nizhnie Vasyuki.

Solution: \boxed{90 \text{ minutes}}


Given \(a_{1} \geqslant a_{2} \geqslant \cdots \geqslant a_{n}\) with \(a_{n} \neq a_{n}\), \(\sum_{i=1}^{n} x_{i}=0\), \(\sum_{i=1}^{n}|x_{i}|=1\), find the minimum value of \(\lambda\) such that the inequality \(\left|\sum_{i=1}^{n} a_{i} x_{i}\right| \leqslant \lambda(a_{1}-a_{n})\) always holds.


Consider a 2 × 2 grid of squares. Each of the squares will be colored with one of 10 colors, and two colorings are considered equivalent if one can be rotated to form the other. How many distinct colorings are there?


Find the range of the function \( f(x) = 2 \sin \left( \left( \frac{\pi}{4} \right) \sin (\sqrt{x-2} + x + 2) - \frac{5\pi}{2} \right) \).

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


The numbers \( p, q, r, \) and \( t \) satisfy \( p < q < r < t \). When these numbers are paired, each pair has a different sum and the four largest sums are 19, 22, 25, and 28. What is the sum of the possible values for \( p \)?


Vijay chooses three distinct integers \(a, b, c\) from the set \(\{1,2,3,4,5,6,7,8,9,10,11\}\). If \(k\) is the minimum value taken on by the polynomial \(a(x-b)(x-c)\) over all real numbers \(x\), and \(l\) is the minimum value taken on by the polynomial \(a(x-b)(x+c)\) over all real numbers \(x\), compute the maximum possible value of \(k-l\).


Márcia is in a store buying a recorder she has wanted for a long time. When the cashier registers the price, she exclaims: "It's not possible, you have recorded the number backwards, you swapped the order of two digits, I remember that last week it cost less than 50 reais!" The cashier responds: "I'm sorry, but yesterday all our items had a 20% increase." What is the new price of the recorder?

Solution: \boxed{54 \text{ reais}}


Let $\left(u_{n}\right)_{n \in \mathbb{N}}$ be a sequence such that:

(i) $u_{0}=5$

(ii) $u_{n+1}=u_{n}+\frac{1}{u_{n}}$

Show that $u_{1000}>45$.

Solution: \boxed{u_{1000} > 45}


In the 100th year of his reign, the Immortal Treasurer decided to start issuing new coins. This year, he issued an unlimited supply of coins with a denomination of \(2^{100} - 1\), next year with a denomination of \(2^{101} - 1\), and so on. As soon as the denomination of a new coin can be obtained without change using previously issued new coins, the Treasurer will be removed from office. In which year of his reign will this happen?


A train took $X$ minutes ($0 < X < 60$) to travel from platform A to platform B. Find $X$ if it's known that at both the moment of departure from A and the moment of arrival at B, the angle between the hour and minute hands of the clock was $X$ degrees.


For every real number \( x \), let \( \lfloor x \rfloor \) denote the greatest integer less than or equal to \( x \). The fractional part of \( x \), denoted as \( \langle x \rangle \), is defined as \( \langle x \rangle = x - \lfloor x \rfloor \). How many real numbers \( x \) are there such that \( 1 \leqslant x \leqslant 10 \) and \( \langle x \rangle^{2} = \langle x^{2} \rangle \)?


Place the numbers $1,2,\cdots,n$ on a circle such that the absolute difference between any two adjacent numbers is either 3, 4, or 5. Find the smallest $n$ that satisfies these conditions.


Let \( M = \{1, 2, \cdots, 17\} \). If there exist four distinct numbers \( a, b, c, d \in M \) such that \( a + b \equiv c + d \pmod{17} \), then \( \{a, b\} \) and \( \{c, d\} \) are called a balanced pair of the set \( M \). Find the number of balanced pairs in the set \( M \).


Given that \( a, b, c \) are the lengths of the sides of a right triangle, and for any natural number \( n > 2 \), the equation \(\left(a^{n} + b^{n} + c^{n}\right)^{2} = 2\left(a^{2n} + b^{2n} + c^{2n}\right)\) holds, find \( n \).

Solution: \boxed{n = 4}


Find all functions $f: \mathbf{N} \rightarrow \mathbf{N}$ such that for all $m, n \in \mathbf{N}$, the equation $f(m^{2}+n^{2}) = f^{2}(m) + f^{2}(n)$ is satisfied and $f(1) > 0$.

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


A point is randomly thrown onto the segment [3, 8] and let $k$ be the resulting value. Find the probability that the roots of the equation $\left(k^{2}-2 k-3\right) x^{2}+(3 k-5) x+2=0$ satisfy the condition $x_{1} \leq 2 x_{2}$.


In a football championship, 16 teams participated. A team receives 2 points for a win; in case of a draw in regular time, both teams shoot penalty kicks, and the team that scores more goals receives one point. After 16 rounds, all teams have accumulated a total of 222 points. How many matches ended in a draw in regular time?

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


Carla wrote the integers from 1 to 21 on the blackboard. Diana wants to erase some of these numbers such that the product of the remaining numbers is a perfect square.

a) Show that Diana necessarily needs to erase the numbers 11, 13, 17, and 19 to achieve her goal.

b) What is the minimum number of numbers that Diana must erase to achieve her goal?


Given one hundred numbers: \(1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \ldots, \frac{1}{100}\). We compute 98 differences: \(a_{1} = 1 - \frac{1}{3}, a_{2} = \frac{1}{2} - \frac{1}{4}, \ldots, a_{98} = \frac{1}{98} - \frac{1}{100}\). What is the sum of all these differences?


If \(x\), \(y\), and \(z\) are distinct positive integers such that \(x^2 + y^2 = z^3\), what is the smallest possible value of \(x + y + z\)?


Let $H_{A}$, $H_{B}$, and $H_{C}$ be the feet of the altitudes from vertices $A$, $B$, and $C$ respectively in triangle $ABC$. Let $H$ be the orthocenter of triangle $ABC$, $P$ be the orthocenter of triangle $AH_{B}H_{C}$, and $Q$ be the orthocenter of triangle $CH_{A}H_{B}$. Show that $PQ = H_{C}H_{A}$.

Solution: \boxed{PQ = H_C H_A}


A smooth ball with a radius of 1 cm was dipped in red paint and then placed between two perfectly smooth concentric spheres with radii of 4 cm and 6 cm respectively (the ball ended up outside the smaller sphere but inside the larger one). When the ball comes into contact with both spheres, it leaves a red mark. During its motion, the ball traced a closed path, resulting in a region outlined by a red contour on the smaller sphere, with an area of 47 square cm. Find the area of the region outlined by the red contour on the larger sphere. Give the answer in square centimeters, rounded to hundredths if necessary.

Solution: \boxed{105.75 \text{ cm}^2}


In an acute-angled triangle \( ABC \), the altitude \( AA_1 \) is drawn. \( H \) is the orthocenter of triangle \( ABC \). It is known that \( AH = 3 \), \( A_1H = 2 \), and the radius of the circumcircle of triangle \( ABC \) is 4. Find the distance from the center of this circumcircle to \( H \).


Buses leave Moscow for Voronezh every hour, at 00 minutes. Buses leave Voronezh for Moscow every hour, at 30 minutes. The trip between cities takes 8 hours. How many buses from Voronezh will a bus leaving Moscow meet on its way?


Two medians of a triangle, measuring 18 and 24, are mutually perpendicular. Find the length of the third median of this triangle.


The second term of an infinite decreasing geometric progression is 3. Find the smallest possible value of the sum \( A \) of this progression, given that \( A > 0 \).


For what smallest natural number \( a \) are there exactly 50 perfect squares in the numerical interval \( (a, 3a) \)?


In an isosceles triangle $ABC$ with base $AB$, the bisector of angle $B$ is perpendicular to the median of side $BC$. Find the cosine of angle $C$.


Determine all real polynomials \(P\) such that \(P(0)=0\) and \(P\left(X^2+1\right)=P(X)^2+1\).

Solution: \boxed{P(X) = X}


A convex polygon has \( n \) sides. Each vertex is joined to a point \( P \) not in the same plane. If \( A, B, C \) are adjacent vertices of the polygon, consider the angle between the planes \( PBA \) and \( PBC \). The sum of the \( n \) such angles equals the sum of the \( n \) angles subtended at \( P \) by the sides of the polygon (such as the angle \( APB \)). Show that \( n = 3 \).

Solution: \boxed{n = 3}


If \(a\) is defined as a number that is not divisible by 4 but whose last digit is 4, then show that

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

is divisible by 480.


Find the minimum value of the expression

$$
\frac{|a-3b-2| + |3a-b|}{\sqrt{a^2 + (b+1)^2}}
$$

for \(a, b \geq 0\).


A triangle has two medians of lengths 9 and 12. Find the largest possible area of the triangle. (Note: A median is a line segment joining a vertex of the triangle to the midpoint of the opposite side.)


A tram ticket is called "lucky in Leningrad style" if the sum of its first three digits is equal to the sum of its last three digits. A tram ticket is called "lucky in Moscow style" if the sum of its digits in even positions is equal to the sum of its digits in odd positions. How many tickets are there that are both lucky in Leningrad style and lucky in Moscow style, including the ticket 000000?


Given three natural numbers 1, 2, 3, perform an operation by replacing one of these numbers with the sum of the other two. After performing this operation 9 times, what is the maximum possible value of the largest number among the resulting three natural numbers?


Bing Dwen Dwen cut 25 square pieces of paper, having a total of 100 corners. Xue Rong Rong cut a triangle from each square piece of paper. In the end, 50 pieces of paper had a total of 170 corners. How many more triangular pieces of paper are there than pentagonal pieces of paper among these 50 pieces?


Andrei was asked to name the apartment number his family received in a new building. He replied that this number is expressed by a number that is 17 times the digit in the units place of the number. What is this apartment number?


Find the smallest number, written using only ones and zeros, that would be divisible by 225.


A person picks \( n \) different prime numbers each less than 150 and finds that they form an arithmetic sequence. What is the greatest possible value of \( n \)?


Using the seven digits $1, 2, 3, 4, 5, 6, 7$ to appropriately arrange them into a 7-digit number so that it is a multiple of 11, how many such numbers can be formed?


Consider a table with \( m \) rows and \( n \) columns. In how many ways can this table be filled with all zeros and ones so that there is an even number of ones in every row and every column?


Let the sequence \(\{x_n\}\) be defined as follows: \(x_{1}=\frac{1}{2}\), and
\[ x_{k+1}=x_{k}+x_{k}^{2} \quad \text{for} \quad k=1,2, \ldots \]
Find the integer part of \(\frac{1}{x_{1}+1}+\frac{1}{x_{2}+1}+\cdots+\frac{1}{x_{100}+1}\).


An $8 \times 8$ chessboard is divided into $p$ rectangles such that no single square is cut in half. Each such division must satisfy the following conditions:

(a) Each rectangle must contain an equal number of white and black squares.

(b) If $a_{i}$ denotes the number of white squares in the $i$-th rectangle, the inequality $a_{1} < a_{2} < \ldots < a_{p}$ must hold.

Determine the largest possible value of $p$ for which such a division exists.

Additionally, generate the sequences $a_{1}, a_{2}, \ldots, a_{p}$ for this value of $p$.


In the Cartesian coordinate system, circle \( C_1 \) and circle \( C_2 \) intersect at points \( P \) and \( Q \), where the coordinates of point \( P \) are \( (3, 2) \). The product of the radii of the two circles is \( \frac{13}{2} \). If the line \( y = kx \) (where \( k > 0 \)) is tangent to both circles \( C_1 \) and \( C_2 \) and also tangent to the x-axis, find the value of \( k \).

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


In a triangle, the lengths of the three sides are integers \( l, m, n \), with \( l > m > n \). It is known that \( \left\{ \frac{3^{l}}{10^{4}} \right\} = \left\{ \frac{3^{m}}{10^{4}} \right\} = \left\{ \frac{3^{n}}{10^{4}} \right\} \), where \( \{x\} \) denotes the fractional part of \( x \) and \( [x] \) denotes the greatest integer less than or equal to \( x \). Determine the smallest possible value of the perimeter of such a triangle.


Let \( f(x) = 1 + x + x^2 + x^3 + \ldots \). Compute the formal power series \( g(x) \) such that \( f(x) g(x) = 1 \).

Solution: \boxed{g(x) = 1 - x}


Euler's inequality: Let $ \triangle ABC $ have a circumradius $ R $ and an inradius $ r $. Then $ R \geq 2r $.

Solution: \boxed{R \geq 2r}


Ludvík noticed in a certain division problem that when he doubles the dividend and increases the divisor by 12, he gets his favorite number as the result. He would get the same number if he reduced the original dividend by 42 and halved the original divisor. Determine Ludvík’s favorite number.


In triangle \( ABC \), it is known that \(\angle BAC = 75^\circ\), \( AB = 1 \), and \( AC = \sqrt{6} \). On the side \( BC \), a point \( M \) is chosen such that \(\angle BAM = 30^\circ\). The line \( AM \) intersects the circumcircle of triangle \( ABC \) at a point \( N \) different from \( A \). Find \( AN \).


Compose the equation of the plane that passes through the line of intersection of the planes \(x + 3y + 5z - 4 = 0\) and \(x - y - 2z + 7 = 0\) and is parallel to the y-axis.

Solution: \boxed{4x - z + 17 = 0}


Find all pairs of integers \((x, y)\) for which \(x^2 + xy = y^2\).


Let \( f: \mathbf{R} \rightarrow \mathbf{R} \) be a smooth function such that \( f^{\prime}(x)^{2} = f(x) f^{\prime \prime}(x) \) for all \( x \). Suppose \( f(0) = 1 \) and \( f^{(4)}(0) = 9 \). Find all possible values of \( f^{\prime}(0) \).

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


To obtain the summary of a number with up to 9 digits, you must write how many digits it has, then how many of those digits are odd, and finally how many are even. For example, the number 9103405 has 7 digits, 4 of which are odd and 3 are even, so its summary is 743.

a) Find a number whose summary is 523.

b) Find a number that is equal to its own summary.

c) For any number with up to 9 digits, we can calculate the summary of the summary of its summary. Show that this procedure always leads to the same result regardless of the initial number.


Let \( n \) be a positive integer, and let \( s \) be the sum of the digits of the base-four representation of \( 2^n - 1 \). If \( s = 2023 \) (in base ten), compute \( n \) (in base ten).


Martin is playing a game. His goal is to place tokens on an 8x8 chessboard such that there is at most one token per square, and each row and column contains at most 4 tokens.

a) How many tokens can Martin place at most?

b) If, in addition to the previous constraints, each of the two main diagonals can contain at most 4 tokens, how many tokens can Martin place at most?

The main diagonals of a chessboard are the two diagonals going from one corner of the board to the opposite corner.


Find all functions \( f: \mathbb{Q} \rightarrow \{-1, 1\} \) such that for all distinct \( x, y \in \mathbb{Q} \) satisfying \( xy = 1 \) or \( x + y \in \{0, 1\} \), we have \( f(x) f(y) = -1 \).

Intermediate question: Let \( f \) be a function having the above property and such that \( f(0) = 1 \). What is \( f\left(\frac{42}{17}\right) \) ?


Find the relationship between the coefficients \(a, b, c\) such that the system of equations
$$
\left\{\begin{array}{l}
a x^{2}+b x+c=0 \\
b x^{2}+c x+a=0 \\
c x^{2}+a x+b=0
\end{array}\right.
$$
has real solutions.

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


Calculate the volumes of solids formed by rotating the region bounded by the function graphs about the \( O y \) (y-axis).

$$
y = x^{2} + 1, \quad y = x, \quad x = 0, \quad x = 1
$$


How many numbers from 1 to 1000 (inclusive) cannot be represented as the difference of two squares of integers?


Let \( a, b, c \) be three pairwise distinct positive integers. If \( \{ a+b, b+c, c+a \} = \{ n^2, (n+1)^2, (n+2)^2 \} \), then what is the minimum value of \( a^2 + b^2 + c^2 \)?


Find the limits:
1) $\lim _{x \rightarrow 0} \frac{x}{\sin x}$
2) $\lim _{x \rightarrow 0} \frac{\sin 4 x}{\sin 5 x}$
3) $\lim _{x \rightarrow 0} \frac{1-\cos x}{x^{2}}$


It takes 60 grams of paint to paint a cube on all sides. How much paint is needed to paint a "snake" composed of 2016 such cubes? The beginning and end of the snake are shown in the diagram, and the rest of the cubes are indicated by ellipses.


Two players, Blake and Ruby, play the following game on an infinite grid of unit squares, all initially colored white. The players take turns starting with Blake. On Blake's turn, Blake selects one white unit square and colors it blue. On Ruby's turn, Ruby selects two white unit squares and colors them red. The players alternate until Blake decides to end the game. At this point, Blake gets a score, given by the number of unit squares in the largest (in terms of area) simple polygon containing only blue unit squares. What is the largest score Blake can guarantee?


Let's call a non-empty (finite or infinite) set $A$ consisting of real numbers complete if for any real numbers $a$ and $b$ (not necessarily distinct and not necessarily in $A$) such that $a+b$ lies in $A$, the number $ab$ also lies in $A$. Find all complete sets of real numbers.

Solution: \boxed{\mathbb{R}}


How many 12-element subsets \( T = \{a_1, a_2, \cdots, a_{12}\} \) from the set \( S = \{1, 2, 3, \cdots, 2009\} \) exist such that the absolute difference between any two elements is not 1?


Calculate the definite integral:

$$
\int_{-\pi / 2}^{0} 2^{8} \cdot \cos ^{8} x \, dx
$$

Solution: \boxed{35 \pi}


Given that \( a \) is a constant and \( x \in \mathbf{R} \), \( f(x)=\frac{f(x-a)-1}{f(x-a)+1} \). The period of \( f(x) \) is \_\_\_\_.


A positive integer with \( n+3 \) digits, \( 144\cdots430 \) (with \( n \) number of 4s), is a multiple of 2015. What is the smallest value of \( n \)?


If a square is divided into acute-angled triangles, what is the minimum number of parts that can be created?


For any positive integer \( n \), if \( f(n) \) is defined as:
\[ f(n) = \begin{cases}
\log_8 n & \text{if } \log_8 n \text{ is a rational number;} \\
0 & \text{if } \log_8 n \text{ is not a rational number.}
\end{cases} \]
Then evaluate the sum:
\[ \sum_{n=1}^{197} f(n) \]
Options:
A. \( \log_8 2047 \)
B. 6
C. \( \frac{55}{3} \)
D. \( \frac{58}{3} \)


A cleaner was mopping the stairs in a skyscraper. To make the work more enjoyable, she counted the cleaned steps. When she had cleaned exactly half of the steps, she took a break. After a while, she started working again and wanted to continue counting the steps. However, when trying to recall the number of steps she had already cleaned, she mistakenly read the correct three-digit number backward, creating a smaller number, and counted from there. After mopping all the stairs, she ended at the number 746. How many steps could she have actually cleaned?


Given an odd prime number \( p \). If there exists a positive integer \( k \) such that \( \sqrt{k^{2} - p k} \) is also a positive integer, then \( k = \) ______.


For what value of \(a\) does the inequality \(\log \frac{1}{\div}\left(\sqrt{x^{2}+a x+5}+1\right) \cdot \log _{5}\left(x^{2}+a x+6\right)+\log _{a} 3 \geqslant 0\) have exactly one solution?

Solution: \boxed{a = 2}


Suppose that \((a_{1}, b_{1}), (a_{2}, b_{2}), \ldots, (a_{100}, b_{100})\) are distinct ordered pairs of nonnegative integers. Let \(N\) denote the number of pairs of integers \((i, j)\) satisfying \(1 \leq i < j \leq 100\) and \(|a_{i} b_{j} - a_{j} b_{i}| = 1\). Determine the largest possible value of \(N\) over all possible choices of the 100 ordered pairs.


What are the values of \(p\) such that \(p\), \(p+2\), and \(p+4\) are all prime numbers?


Does there exist an angle \(\alpha \in (0, \pi / 2)\) such that \(\sin \alpha\), \(\cos \alpha\), \(\tan \alpha\), and \(\cot \alpha\), taken in some order, are consecutive terms of an arithmetic progression?

Solution: \boxed{\text{No}}


In a unit cube \( ABCD-A_{1}B_{1}C_{1}D_{1} \), points \( E, F, G \) are the midpoints of edges \( AA_{1}, C_{1}D_{1}, D_{1}A_{1} \) respectively. Find the distance from point \( B_{1} \) to the plane containing points \( E, F, G \).


Shuai Shuai memorized more than one hundred words in seven days. The number of words memorized in the first three days is $20\%$ less than the number of words memorized in the last four days, and the number of words memorized in the first four days is $20\%$ more than the number of words memorized in the last three days. How many words did Shuai Shuai memorize in total over the seven days?


In the cells of a \(75 \times 75\) table, pairwise distinct natural numbers are placed. Each of them has no more than three different prime divisors. It is known that for any number \(a\) in the table, there exists a number \(b\) in the same row or column such that \(a\) and \(b\) are not coprime. What is the maximum number of prime numbers that can be in the table?


Find the continuous functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) such that for all \( x \) and \( y \),

\[ f\left(\frac{x+y}{2}\right) = \frac{f(x) + f(y)}{2} \]

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


Consider the two hands of an analog clock, each of which moves with constant angular velocity. Certain positions of these hands are possible (e.g. the hour hand halfway between the 5 and 6 and the minute hand exactly at the 6), while others are impossible (e.g. the hour hand exactly at the 5 and the minute hand exactly at the 6). How many different positions are there that would remain possible if the hour and minute hands were switched?


Inside the square \(ABCD\), point \(M\) is chosen such that \(MA = 1\), \(MB = 2\), and \(MC = 3\). Find \(MD\).


Under what condition on \((x, y) \in \mathbb{N}^{2}\) is the integer \((x+y)^{2} + 3x + y + 1\) a perfect square?

Solution: \boxed{x = y}


We move on a grid by taking only one-unit steps to the right or upward. What is the number of ways to go from point \((0,0)\) to point \((m, n)\)?

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


Each chocolate costs 1 dollar, each licorice stick costs 50 cents, and each lolly costs 40 cents. How many different combinations of these three items cost a total of 10 dollars?


Given two linear functions \( f(x) \) and \( g(x) \) such that the graphs \( y=f(x) \) and \( y=g(x) \) are parallel lines not parallel to the coordinate axes. Find the minimum value of the function \( (g(x))^{2} + f(x) \) if the minimum value of the function \( (f(x))^{2} + g(x) \) is 4.

Solution: \boxed{- \frac{9}{2}}


Sergey Stanislavovich is 36 years, 36 months, 36 weeks, 36 days, and 36 hours old.

How many full years old is Sergey Stanislavovich?


Is a triangle necessarily isosceles if the center of its inscribed circle is equidistant from the midpoints of two of its sides?

Solution: \boxed{\text{No}}


Does the equation \( x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 = 0 \) have any negative roots?

Find the common fraction with a denominator of 21 that lies between the fractions \(\frac{5}{14}\) and \(\frac{5}{12}\).


Kevin has four red marbles and eight blue marbles. He arranges these twelve marbles randomly in a ring. Determine the probability that no two red marbles are adjacent.


Find all real solutions to the equation with 4 unknowns: \( x^{2} + y^{2} + z^{2} + t^{2} = x(y + z + t) \).

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


There are 2013 cards with the digit 1 and 2013 cards with the digit 2. Vasya arranges these cards to form a 4026-digit number. On each move, Petya can swap any two cards and pay Vasya 1 ruble. The process ends when Petya forms a number divisible by 11. What is the maximum amount of money Vasya can earn if Petya aims to pay as little as possible?


If the system of equations involving \( x \) and \( y \)

\[
\left\{
\begin{array}{l}
\sin x = m \sin^3 y, \\
\cos x = m \cos^3 y
\end{array}
\right.
\]

has real solutions, then what is the range of positive values for \( m \)?

Solution: \boxed{[1, 2]}


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


Given that $\alpha^{2005}+\beta^{2005}$ can be expressed as a two-variable polynomial in terms of $\alpha+\beta$ and $\alpha \beta$, find the sum of the coefficients of this polynomial.


Misha painted all the integers in several colors such that any two numbers whose difference is a prime number are painted in different colors. What is the minimum number of colors that could have been used by Misha? Justify your answer.


A random variable $X$ is distributed according to the normal law. The expected value $a=0$ and the standard deviation of this variable is $\sigma=0.5$. Find the probability that the deviation of the random variable $X$ in absolute value will be less than one.


In the sequence $\left\{a_{n}\right\}$, if $a_{n}^{2}-a_{n-1}^{2}=p$ (where $n \geq 2, n \in \mathbf{N^{*}}$, and $p$ is a constant), then $\left\{a_{n}\right\}$ is called an "equal variance sequence." Below are the judgments for the "equal variance sequence":

1. The sequence $\left\{(-1)^{n}\right\}$ is an equal variance sequence;
2. If $\left\{a_{n}\right\}$ is an equal variance sequence, then $ \left\{a_{n}^{2}\right\}$ is an arithmetic sequence;
3. If $\left\{a_{n}\right\}$ is an equal variance sequence, then $\left\{a_{k n}\right\}\left(k \in \mathbf{N}^{*}, k\right.$ being a constant) is also an equal variance sequence;
4. If $\left\{a_{n}\right\} $ is both an equal variance sequence and an arithmetic sequence, then this sequence is a constant sequence.

Among them, the correct proposition numbers are $\qquad$ (Fill in all the correct proposition numbers on the line).


Calculate the number of trailing zeroes in 2019!.


What is the smallest number of distinct integers needed so that among them one can select both a geometric progression and an arithmetic progression of length 5?


Find the maximum real number $\lambda$ such that for the real coefficient polynomial $f(x)=x^{3}+a x^{2}+b x+c$ having all non-negative real roots, it holds that $f(x) \geqslant \lambda(x-a)^{3}$ for all $x \geqslant 0$. Additionally, determine when equality holds in the given expression.


Given 100 numbers. Each number is increased by 2. The sum of the squares of the numbers remains unchanged. Each resulting number is then increased by 2 again. How has the sum of the squares changed now?


In how many ways can you select two letters from the word "УЧЕБНИК" such that one of the letters is a consonant and the other is a vowel?


Find all functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) such that

\[ \forall x, y \in \mathbb{R}, f(x^3) - f(y^3) = (x^2 + xy + y^2)(f(x) - f(y)) \]

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


In Anchuria, a checkers championship takes place over several rounds. The days and cities for hosting the rounds are determined by a draw. According to the championship rules, no two rounds can be held in the same city and no two rounds can be held on the same day. A lottery is held among the fans: the grand prize goes to the person who correctly guesses the cities and days for all the rounds before the championship starts. If no one guesses correctly, the grand prize is awarded to the championship organizing committee. There are eight cities in Anchuria, and the championship is scheduled to take place over eight days. How many rounds should there be in the championship to maximize the probability that the organizing committee wins the grand prize?


Find the smallest prime number that can be expressed as the sum of five different prime numbers.


The distance from platform $A$ to platform $B$ was covered by an electric train in $X$ minutes ($0<X<60$). Find $X$ if it is known that at both the moment of departure from $A$ and the moment of arrival at $B$, the angle between the hour and minute hands of the clock was $X$ degrees.


In a store, there are four types of nuts: hazelnuts, almonds, cashews, and pistachios. Stepan wants to buy 1 kilogram of nuts of one type and 1 kilogram of nuts of another type. He has calculated the cost of such a purchase depending on which two types of nuts he chooses. Five of Stepan's six possible purchases would cost 1900, 2070, 2110, 2330, and 2500 rubles. How many rubles is the cost of the sixth possible purchase?


A sphere is inscribed in a dihedral angle.

Point \( O \) is located in the section \( A A^{\prime} C^{\prime} C \) of a rectangular parallelepiped \( A B C D A^{\prime} B^{\prime} C^{\prime} D^{\prime} \) with dimensions \( 2 \times 6 \times 9 \) such that \( \angle O A B + \angle O A D + \angle O A A^{\prime} = 180^{\circ} \). The sphere with center at point \( O \) touches the planes \( A^{\prime} B^{\prime} C^{\prime} \) and \( A A^{\prime} B \) and does not intersect the plane \( A A^{\prime} D \). Find the distance from point \( O \) to this plane.


A three-digit number \(abc\) satisfies \(abc = a + b^2 + c^3\). How many such three-digit numbers \(\overline{abc}\) satisfy this condition?


In quadrilateral \(ABCD\), \(\angle ABD = 70^\circ\), \(\angle CAD = 20^\circ\), \(\angle BAC = 48^\circ\), \(\angle CBD = 40^\circ\). Find \(\angle ACD\).


Express \( x \) using the real parameter \( a \) if

$$
\sqrt{\log _{a} (a x) + \log _{x} (a x)} + \sqrt{\log _{a} \left(\frac{x}{a}\right) + \log _{x} \left(\frac{a}{x}\right)} = 2.
$$

Solution: \boxed{x = a}


At lunch, Abby, Bart, Carl, Dana, and Evan share a pizza divided radially into 16 slices. Each one takes one slice of pizza uniformly at random, leaving 11 slices. The remaining slices of pizza form "sectors" broken up by the taken slices; for example, if they take five consecutive slices then there is one sector, but if none of them take adjacent slices then there will be five sectors. What is the expected number of sectors formed?


If \( n \) is a positive integer such that \( n^{3} + 2n^{2} + 9n + 8 \) is the cube of an integer, find \( n \).


Two people travel toward each other from points \(A\) and \(B\) with speeds \(v_{1} = 6 \, \mathrm{m/s}\) and \(v_{2} = 4 \, \mathrm{m/s}\). At the moment they meet, one of them turns around and walks in the opposite direction, while the other does not change direction. The person who did not change direction reached their final destination \(t_{2} = 10\) minutes earlier than the person who turned around. Determine the time \(t_{1}\) that passed from the start of the journey until the meeting. (15 points)

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


Two runners, starting simultaneously and running at constant speeds, run in opposite directions on a circular track. One completes the circuit in 5 minutes, and the other in 8 minutes. Find the number of distinct meeting points of the runners on the track if they run for at least one hour.


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


Let \(\mathbb{N}\) be the set of positive integers, i.e., \(\mathbb{N}=\{1,2, \ldots\}\). Find all functions \(f: \mathbb{N} \rightarrow \mathbb{N}\) such that
$$
f(f(m)+f(n)) = m+n \text{ for all } m, n \in \mathbb{N}.
$$

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


Determine the functions \( f \) from \( \mathbb{Q}_{+}^{*} \) to \( \mathbb{Q}_{+}^{*} \) that satisfy \( f(x+1)=f(x)+1 \) and \( f\left(x^{3}\right)=f(x)^{3} \) for all \( x \in \mathbb{Q}_{+}^{*} \).

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


Given \( 0 < x < 1 \) and \( a, b \) are both positive constants, the minimum value of \( \frac{a^{2}}{x}+\frac{b^{2}}{1-x} \) is ______.

Solution: \boxed{(a + b)^2}


Find the smallest natural number greater than 1 that is at least 600 times greater than each of its prime divisors.


Three hedgehogs are dividing three pieces of cheese with masses of 5 g, 8 g, and 11 g. A fox comes to help them. She can simultaneously cut 1 g of cheese from any two pieces and eat it. Can the fox leave the hedgehogs with equal pieces of cheese?

Solution: \boxed{Yes}


In a closed bag, there are apples. Three friends tried to lift the bag and guess how many fruits were inside. The first friend guessed that there were 16 apples, the second guessed 19, and the third guessed 25. When they opened the bag, it turned out that one of them was off by 2, another by 4, and the last one by 5. How many apples were in the bag? Find all possible answers.


For a homework assignment, Tanya was asked to come up with 20 examples of the form \( * + * = * \), where different natural numbers need to be inserted in place of \( * \) (i.e., a total of 60 different numbers should be used). Tanya loves prime numbers very much, so she decided to use as many of them as possible while still getting correct examples. What is the maximum number of prime numbers Tanya can use?


What is the tens digit of the smallest positive integer that is divisible by each of 20, 16, and 2016?
(A) 0
(B) 2
(C) 4
(D) 6
(E) 8


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


A perpendicular dropped from the vertex of a rectangle to the diagonal divides the right angle into two parts in the ratio $1: 3$.

Find the angle between this perpendicular and the other diagonal.


The fare in Moscow with the "Troika" card in 2016 is 32 rubles for one trip on the metro and 31 rubles for one trip on ground transportation. What is the minimum total number of trips that can be made at these rates, spending exactly 5000 rubles?


You have a \(2 \times 3\) grid filled with integers between 1 and 9. The numbers in each row and column are distinct. The first row sums to 23, and the columns sum to 14, 16, and 17 respectively.

Given the following grid:
\[
\begin{array}{c|c|c|c|}
    & 14 & 16 & 17 \\
\hline
23 & a & b & c \\
\hline
    & x & y & z \\
\hline
\end{array}
\]

What is \(x + 2y + 3z\)?


What integer can be the sum of the lengths of the diagonals of a rhombus with side length 2?


Three candles were lit simultaneously. When the first candle burned out, $\frac{2}{5}$ of the second candle and $\frac{3}{7}$ of the third candle remained. What fraction of the third candle will remain when the second candle burns out?


Find the smallest real number $x$ such that $45\%$ of $x$ and $24\%$ of $x$ are natural numbers.


What is the probability that a randomly chosen positive integer is relatively prime to 6? What is the probability that at least one of two randomly chosen integers is relatively prime to 6?


Given that \(7,999,999,999\) has at most two prime factors, find its largest prime factor.


On a digital clock, the date is always displayed as an eight-digit number, such as January 1, 2011, which is displayed as 20110101. What is the last date in 2011 that is divisible by 101? This date is represented as $\overline{2011 \mathrm{ABCD}}$. What is $\overline{\mathrm{ABCD}}$?


From the natural numbers $1, 2, \ldots, 101$, select a group of numbers such that the greatest common divisor of any two numbers in the group is greater than two. What is the maximum number of such numbers in this group?


There are coins of denominations 50 kopecks, 1 ruble, 2 rubles, 5 rubles, and 10 rubles. In a wallet, there are several coins. It is known that no matter which 20 coins are taken out of the wallet, among them there will be at least one 1-ruble coin, at least one 2-ruble coin, and at least one 5-ruble coin. What is the maximum number of coins possible in the wallet such that this condition holds?


Let a positive integer \( k \) be called interesting if the product of the first \( k \) prime numbers is divisible by \( k \) (for example, the product of the first two prime numbers is \(2 \cdot 3 = 6\), and 2 is an interesting number).

What is the largest number of consecutive interesting numbers that can occur?


A ship travels downstream from point $A$ to point $B$ and takes 1 hour. On the return trip, the ship doubles its speed and also takes 1 hour. How many minutes will it take to travel from point $A$ to point $B$ if the ship initially uses double its speed?

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


In the plane Cartesian coordinate system $xOy$, given two points $M(-1, 2)$ and $N(1, 4)$, point $P$ moves along the $x$-axis. When the angle $\angle MPN$ reaches its maximum value, find the x-coordinate of point $P$.


Inside the cube \( ABCD A_1B_1C_1D_1 \), there is a center \( O \) of a sphere with a radius of 10. The sphere intersects the face \( AA_1D_1D \) along a circle of radius 1, the face \( A_1B_1C_1D_1 \) along a circle of radius 1, and the face \( CDD_1C_1 \) along a circle of radius 3. Find the length of the segment \( OD_1 \).


Josh takes a walk on a rectangular grid of \( n \) rows and 3 columns, starting from the bottom left corner. At each step, he can either move one square to the right or simultaneously move one square to the left and one square up. In how many ways can he reach the center square of the topmost row?


Determine the digits $A, B, C, D$ if

$$
A^{\overline{A B}}=\overline{C C B B D D C A}
$$

Solution: \boxed{A=2, B=5, C=3, D=4}