Let \( x \) be a non-zero real number such that

\[ \sqrt[5]{x^{3}+20 x}=\sqrt[3]{x^{5}-20 x} \].

Find the product of all possible values of \( x \).


The manager of a store checked the sales price in 2006 of a television from the brand VejoTudo. He found a partially faded invoice that read: "lot of 72 VejoTudo TVs sold for R$ \ldots 679 \ldots reais," where the digits of the unit and the ten thousand place were illegible. What was the sales price in 2006 of each of those televisions?


\( N \) is a 5-digit number formed by 5 different non-zero digits, and \( N \) is equal to the sum of all 3-digit numbers that can be formed using any 3 of these 5 digits. Find all such 5-digit numbers \( N \).


Suppose that \( x - y = 1 \). Find the value of \( x^4 - xy^3 - x^3 y - 3x^2 y + 3xy^2 + y^4 \).


Given the natural numbers $1,2,3,\ldots,10,11,12$, divide them into two groups such that the quotient of the product of all numbers in the first group by the product of all numbers in the second group is an integer and takes on the smallest possible value. What is this quotient?


Given a sequence of positive integers $\left\{a_{n}\right\}$ defined by $a_{0}=m$ and $a_{n+1}=a_{n}^{5}+487$ for $n \in \mathbf{N}$, find the value of $m$ such that the sequence $\left\{a_{n}\right\}$ contains the maximum number of perfect squares.


Find the smallest positive integer that can be expressed as the sum of 9 consecutive integers, 10 consecutive integers, and 11 consecutive integers.


Each cell in a \(5 \times 5\) grid contains a natural number written in invisible ink. It is known that the sum of all the numbers is 200, and the sum of any three numbers inside any \(1 \times 3\) rectangle is 23. What is the value of the central number in the grid?


Masha was given a box with differently colored beads (each bead having a unique color, with a total of $\mathrm{n}$ beads in the box). Masha chose seven beads for her dress and decided to try out all possible combinations of these beads on the dress (thus, Masha considers sewing one, two, three, four, five, six, or seven beads, with the order of the beads not mattering to her). Then she counted the total number of combinations and was very surprised that the number turned out to be odd.
1) What number did Masha get?
2) Is it true that when selecting from a set with an even number of beads, Masha could get an even number of combinations?
3) Is it true that if the order of the beads sewn onto the dress mattered to Masha, she could have ended up with either an even or odd number of combinations?

Solution: \boxed{\text{Yes}}


Let \( a \) be a strictly positive integer. Suppose that \( 4(a^n + 1) \) is the cube of an integer for every positive integer \( n \). Find \( a \).


January 1st of a certain non-leap year fell on a Saturday. How many Fridays are there in this year?


Given a right triangle \( \triangle ABC \) with vertices \( A, B, \) and \( C \) on the parabola \( y = x^2 \), and the hypotenuse \( AB \) parallel to the x-axis, find the length of the altitude from \( C \) to the hypotenuse \( AB \).


Find the smallest natural number \( n \) for which the number \( A = n^3 + 12n^2 + 15n + 180 \) is divisible by 23.


What is the smallest number of participants that could have been in the school drama club if the fifth graders made up more than \(22\%\) but less than \(27\%\) of the club, the sixth graders made up more than \(25\%\) but less than \(35\%\), and the seventh graders made up more than \(35\%\) but less than \(45\%\) (there were no participants from other grades)?


Suppose we keep rolling a fair 2014-sided die (whose faces are labelled 1, 2, ..., 2014) until we obtain a value less than or equal to the previous roll. Let E be the expected number of times we roll the die. Find the nearest integer to 100E.


If the complex number \( z \) satisfies
\[ |z-1| + |z-3-2i| = 2\sqrt{2}, \]
then the minimum value of \( |z| \) is ______ .


Find the equation of the normal line to the given curve at the point with the abscissa $x_{0}$.

$y=\sqrt{x}-3 \sqrt[3]{x}, x_{0}=64$

Solution: \boxed{x = 64}


Given a cube with a side length of 4, is it possible to entirely cover three of its faces, which share a common vertex, using 16 paper rectangular strips each measuring \(1 \times 3\)?

Solution: \boxed{\text{No}}


Compute the limit of the numerical sequence:

\[
\lim _{n \rightarrow \infty} \frac{\sqrt{n+1}-\sqrt[3]{n^{3}+1}}{\sqrt[4]{n+1}-\sqrt[5]{n^{5}+1}}
\]


Three three-digit numbers, with all digits except zero being used in their digits, sum up to 1665. In each number, the first digit was swapped with the last digit. What is the sum of the new numbers?


Let \(a\) and \(b\) be coprime integers. Show that \(a + b\) and \(ab\) are coprime.


What number must be added to both terms of a fraction to obtain the reciprocal of that same fraction?

Solution: \boxed{- (a + b)}


Yvan and Zoé play the following game. Let \( n \in \mathbb{N} \). The integers from 1 to \( n \) are written on \( n \) cards arranged in order. Yvan removes one card. Then, Zoé removes 2 consecutive cards. Next, Yvan removes 3 consecutive cards. Finally, Zoé removes 4 consecutive cards.

What is the smallest value of \( n \) for which Zoé can ensure that she can play her two turns?


Two cars, Car A and Car B, simultaneously depart from point $A$ to point $B$. At the start, Car A’s speed is 2.5 km/h faster than Car B’s speed. After 10 minutes, Car A slows down; another 5 minutes later, Car B also slows down, and at this time, Car B is 0.5 km/h slower than Car A. After another 25 minutes, both cars arrive at point $B$ at the same time. By how many kilometers per hour did Car A's speed decrease?

Solution: \boxed{10 \text{ km/h}}


Place a known mathematical symbol between the numbers 4 and 5 such that the result is a number greater than 4 but less than 5.


The wristwatch is 5 minutes slow per hour; 5.5 hours ago, it was set to the correct time. It is currently 1 PM on a clock that shows the correct time. How many minutes will it take for the wristwatch to show 1 PM?

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


Given a sequence \(\{a_n\} (n \geq 0)\) that satisfies \(a_0 = 0\), \(a_1 = 1\), and for all positive integers \(n\), \(a_{n+1} = 2a_n + 2007a_{n-1}\), find the smallest positive integer \(n\) such that \(2008\) divides \(a_n\).


The figure \( A B C D E F \) is a regular hexagon. Evaluate the quotient
$$
\frac{\text { Area of hexagon } A B C D E F}{\text { Area of triangle } A C D}.
$$


Points $A, B$, and $C$ lie in that order on line $\ell$, such that $AB = 3$ and $BC = 2$. Point $H$ is such that $CH$ is perpendicular to $\ell$. Determine the length $CH$ such that $\angle AHB$ is as large as possible.


Find the distance between the face diagonal \( A_{1}B \) and the face diagonal \( B_{1}D_{1} \) in a cube \( ABCD-A_{1}B_{1}C_{1}D_{1} \) with a side length of 1.


Howard chooses \( n \) different numbers from the list \{2, 3, 4, 5, 6, 7, 8, 9, 10, 11\}, so that no two of his choices add up to a square. What is the largest possible value of \( n \)?


Given \( x, y, z \in \mathbf{Z} \) such that \(\left\{\begin{array}{l}x+ y + z = 3 \\ x^3 + y^3 + z^3 = 3\end{array}\right.\), determine the set of all possible values of \( x^2 + y^2 + z^2 \).

Solution: \boxed{\{3, 57\}}


In a certain school, there are 100 students in the second year of high school who are excellent in at least one of the three subjects: mathematics, physics, or chemistry. Among them, 70 students are excellent in mathematics, 65 in physics, and 75 in chemistry. There are 40 students who are excellent in both mathematics and physics, 45 in both mathematics and chemistry, and 25 students who are excellent in all three subjects. How many students are excellent in both physics and chemistry but not in mathematics?


Let \(\mathbb{Z}_{>0}\) be the set of positive integers. Find all functions \(f: \mathbb{Z}_{>0} \rightarrow \mathbb{Z}_{>0}\) such that
$$
m^{2}+f(n) \mid m f(m)+n
$$
for all positive integers \(m\) and \(n\).

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


There are 158 children standing in a line. Starting from the first child on the left, every other child receives an apple (the first child receives an apple). Starting from the first child on the right, every third child receives a banana (the first child receives a banana). Find the number of children who do not receive any fruit.


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?


How many positive integers less than 1998 are relatively prime to 1547? (Two integers are relatively prime if they have no common factors besides 1.)


Given a triangle \( \triangle ABC \) with circumradius \( R \), it is known that

\[
\frac{a \cos \alpha + b \cos \beta + c \cos \gamma}{a \sin \beta + b \sin \gamma + c \sin \alpha} = \frac{a + b + c}{9R},
\]

where \( a, b, c \) are the lengths of the sides of \( \triangle ABC \), and \( \alpha, \beta, \gamma \) are the measures of angles \( \angle A, \angle B, \angle C \) respectively. Determine \( \alpha, \beta, \gamma \).


Given that point \( P \) moves on the circle \( x^{2} + (y - 4)^{2} = 1 \) and point \( Q \) moves on the ellipse \( \frac{x^{2}}{9} + y^{2} = 1 \), find the maximum value of \( |PQ| \).

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


Let \( a < b < c < d \) be odd natural numbers such that \( ad = bc \) and \( a + d \) and \( b + c \) are powers of 2. Show that \( a = 1 \).


The series below includes the consecutive even integers from 2 to 2022 inclusive, where the signs of the terms alternate between positive and negative:

$$
S=2-4+6-8+10-\cdots-2016+2018-2020+2022
$$

What is the value of $S$?


Let \( a \) be a real number, and let \( m(a) \) denote the minimum value of the function \( f(x) = x^2 - ax + \frac{a}{2} \) for \( 0 \leq x \leq 1 \). Find the maximum value of \( m(a) \) as \( a \) varies.


Compute the limit of the function:

$$\lim _{x \rightarrow 1} \frac{\cos (2 \pi x)}{2+\left(e^{\sqrt{x-1}}-1\right) \operatorname{arctg} \frac{x+2}{x-1}}$$


All three-digit natural numbers, whose first digits are odd and greater than 1, are written on the board. What is the maximum number of quadratic equations of the form \( a x^{2} + b x + c = 0 \) that can be created by using these numbers as \( a \), \( b \), and \( c \), each no more than once, such that all these equations have roots?


Given \( a_{n}=\log _{n}(n+1) \), determine \( \sum_{n=2}^{1023} \frac{1}{\log _{a_{n}} 100}=\frac{q}{p} \), where \( p \) and \( q \) are positive integers such that \( (p, q)=1 \). Find \( p + q \).


There are points \( A, B, C, D \) marked on a piece of paper. A recognition device can perform two types of operations with absolute accuracy: a) measure the distance between any two given points in centimeters; b) compare two given numbers. What is the minimum number of operations needed for this device to definitively determine whether the quadrilateral \( ABCD \) is a rectangle?


In how many different ways can 1,000,000 be represented as the product of three natural numbers? Products that differ only in the order of the factors are considered the same.


Consider an equilateral triangle \(ABC\), where \(AB = BC = CA = 2011\). Let \(P\) be a point inside \(\triangle ABC\). Draw line segments passing through \(P\) such that \(DE \parallel BC\), \(FG \parallel CA\), and \(HI \parallel AB\). Suppose \(DE : FG : HI = 8 : 7 : 10\). Find \(DE + FG + HI\).


How many ordered pairs \((x, y)\) of positive integers, where \(x < y\), satisfy the equation
$$
\frac{1}{x} + \frac{1}{y} = \frac{1}{2007}?
$$


As shown in the figure, in parallelogram $ABCD$, the ratio $\frac{AE}{ED}=\frac{9}{5}$ and the ratio $\frac{BF}{FC}=\frac{7}{4}$. Find the ratio of the areas of $\triangle ACE$ and $\triangle BDF$.


Given \( f(x) = \lg (x+1) - \frac{1}{2} \log_{3} x \).
(1) Solve the equation \( f(x) = 0 \);
(2) Determine the number of subsets of the set \( M = \{ n \mid f(n^2 - 214n - 1998) \geq 0, n \in \mathbf{Z} \} \).


In a quiz, no two people had the same score and the score of each participant is equal to \( n + 2 - 2k \) where \( n \) is a constant and \( k \) is the rank of the participant. If the total score of all participants is 2009, find the smallest possible value of \( n \).


Find \( k \) such that, for all \( n \), the following expression is a perfect square:

$$
4 n^{2} + k n + 9
$$


Draw a rectangle on each side of a given rectangle such that the height of each new rectangle is one $n$-th of the length of the corresponding side of the original rectangle. Starting with rectangles of equal perimeter, can you choose a value of $n$ such that the area of the resulting shape, which consists of the original rectangle and the four added rectangles, is always the same?


In some 16 cells of an $8 \times 8$ board, rooks are placed. What is the minimum number of pairs of rooks that can attack each other in this configuration?


In 12 days, it will be the year 2016. Haohau remarked: Up to now, I have only experienced 2 leap years, and the year I was born is a multiple of 9. How old will Haohau be in 2016?


How many ways are there to color every integer either red or blue such that \( n \) and \( n+7 \) are the same color for all integers \( n \), and there does not exist an integer \( k \) such that \( k, k+1 \), and \( 2k \) are all the same color?


A plane is drawn through each face of a cube. Into how many parts do these planes divide the space?


On the coordinate plane, the points $A(9, 1)$, $B(2, 0)$, $D(1, 5)$, and $E(9, 7)$ are given. Find the area of the pentagon $ABCDE$, where $C$ is the intersection point of the lines $AD$ and $BE$.


Let \(ABCD\) be a square of side length 13. Let \(E\) and \(F\) be points on rays \(AB\) and \(AD\), respectively, so that the area of square \(ABCD\) equals the area of triangle \(AEF\). If \(EF\) intersects \(BC\) at \(X\) and \(BX=6\), determine \(DF\).


If \( x \) and \( y \) are both greater than 0, and \( x + 2y = 1 \), then what is the minimum value of \( u = \left( x + \frac{1}{x} \right) \left( y + \frac{1}{4y} \right) \)?


Let points \( C \) and \( D \) be the trisection points of \( AB \). At 8:00, person \( A \) starts walking from \( A \) to \( B \) at a constant speed. At 8:12, person \( B \) starts walking from \( B \) to \( A \) at a constant speed. After a few more minutes, person \( C \) starts walking from \( B \) to \( A \) at a constant speed. When \( A \) and \( B \) meet at point \( C \), person \( C \) reaches point \( D \). When \( A \) and \( C \) meet at 8:30, person \( B \) reaches \( A \) exactly at that time. At what time did person \( C \) start walking?


A column of cars, moving uniformly at the same speed, has a length of $5 \mathrm{km}$. In the last car is the head of the column, next to whom is a motorcyclist. On the orders of the leader, the motorcyclist increased his speed, caught up with the front car, delivered a package, immediately turned around and, with the same speed with which he went forward, went back to his place. While the motorcyclist completed the task, the column moved forward by 5 km. How many kilometers did the motorcyclist travel?

Solution: \boxed{5(1 + \sqrt{2})}


In November, Panteleimon and Gerasim each received 20 grades. Panteleimon received as many fives as Gerasim received fours, as many fours as Gerasim received threes, as many threes as Gerasim received twos, and as many twos as Gerasim received fives. Additionally, their average grades for November are the same. How many twos did Panteleimon receive in November?


What is the maximum integer number of liters of water that can be heated to boiling temperature using the heat obtained from burning solid fuel, if during the first 5 minutes of burning the fuel produces 480 kJ, and during each subsequent five-minute period $25\%$ less than during the previous one. The initial temperature of the water is $20^{\circ} \mathrm{C}$, the boiling temperature is $100^{\circ} \mathrm{C}$, and the specific heat capacity of water is 4.2 kJ.


In cyclic pentagon \(ABCDE\), \(\angle ABD = 90^\circ\), \(BC = CD\), and \(AE\) is parallel to \(BC\). If \(AB = 8\) and \(BD = 6\), find \(AE^2\).


Given a rectangle with dimensions \(100 \times 101\), divided by grid lines into unit squares. Find the number of segments into which the grid lines divide its diagonal.


How many paths are there on an $n \times n$ grid to go from point $(0, 0)$ to point $(n, n)$ using only moves of 1 up or 1 to the right?


Given a trapezoid \(ABCD\) with bases \(AB\) and \(CD\), and angles \(\angle C = 30^\circ\) and \(\angle D = 80^\circ\). Find \(\angle ACB\), given that \(DB\) is the angle bisector of \(\angle D\).

Solution: \boxed{10^\circ }


Given that \( w>0 \) and that \( w-\frac{1}{w}=5 \), find the value of \( \left(w+\frac{1}{w}\right)^{2} \).


Given the sequence $\left\{a_{n}\right\}$ with the general term $a_{n}=\log _{3}\left(1+\frac{2}{n^{2}+3 n}\right)$, find $\lim _{n \rightarrow \infty}\left(a_{1}+a_{2}+\cdots+a_{n}\right)$.


Find the coordinates of the center of gravity of a homogeneous plane figure bounded by the curves \( y = \frac{1}{2} x^2 \) and \( y = 2 \).

Solution: \boxed{(0, 1.2)}


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


Let \( A \) be a subset of \(\{1, 2, 3, \ldots, 2019\}\) having the property that the difference between any two of its elements is not a prime number. What is the largest possible number of elements in \( A \)?


Fill the six numbers $1, 3, 5, 7, 9, 11$ into the circles in the given diagram (each circle contains one number) so that the sum of the three numbers on each side equals 19. Then, find the sum of the three numbers in the circles that form the vertices of the triangle.


In the cells of a 9 × 9 square, there are non-negative numbers. The sum of the numbers in any two adjacent rows is at least 20, and the sum of the numbers in any two adjacent columns does not exceed 16. What can be the sum of the numbers in the entire table?


Compute the limit of the function:

\[
\lim _{x \rightarrow 0}\left(\frac{1 + x \cdot 2^{x}}{1 + x \cdot 3^{x}}\right)^{\frac{1}{x^{2}}}
\]


At a sumo wrestling tournament, 20 sumo wrestlers participated. After weighing, it was found that the average weight of the wrestlers is 125 kg. What is the maximum possible number of wrestlers weighing more than 131 kg, given that according to the rules, individuals weighing less than 90 kg cannot participate in sumo wrestling?


Determine the number of quadruples of natural numbers \((a, b, c, d)\) satisfying \(a \cdot b \cdot c \cdot d = 98\).

Note: \((98, 1, 1, 1)\) and \((1, 1, 98, 1)\) are considered different quadruples.


How many five-digit numbers are roots of the equation \( x = [\sqrt{x} + 1][\sqrt{x}] \)? The symbol \([a]\) denotes the integer part of the number \(a\), which is the largest integer not exceeding \(a\).


If a sequence of numbers \(a_{1}, a_{2}, \cdots\) satisfies, for any positive integer \(n\),
$$
a_{n}=\frac{n^{2}+n-2-\sqrt{2}}{n^{2}-2},
$$

then what is the value of \(a_{1} a_{2} \cdots a_{2016}\)?

Solution: \boxed{2016\sqrt{2} - 2015}


Problems A, B, and C were posed in a mathematical contest. 25 competitors solved at least one of the three. Amongst those who did not solve A, twice as many solved B as solved C. The number solving only A was one more than the number solving A and at least one other. The number solving just A equalled the number solving just B plus the number solving just C. How many solved just B?


In the equation \( x^{3} + ax^{2} + bx + 6 = 0 \), determine \( a \) and \( b \) such that one root of the equation is 2 and another root is 3. What is the value of the third root?


Let \( n \) be the smallest positive integer with exactly 2015 positive factors. What is the sum of the (not necessarily distinct) prime factors of \( n \)? For example, the sum of the prime factors of 72 is \( 2 + 2 + 2 + 3 + 3 = 14 \).


Given a positive integer \( n \). What fraction of the non-empty subsets of \( \{1, 2, \ldots, 2n\} \) has an odd smallest element?


In the square \(ABCD\), \(E\) and \(F\) are trisection points of the diagonal \(BD\). The line extending \(AE\) intersects \(BC\) at \(G\), and the line extending \(GF\) intersects \(AD\) at \(H\). Given that the area of \(\triangle DHF\) is 12, find the area of the square \(ABCD\).


Find all natural numbers \( x \) that satisfy the following conditions: the product of the digits of \( x \) is equal to \( 44x - 86868 \), and the sum of the digits is a perfect cube.


While waiting for customers, a watermelon seller sequentially weighed 20 watermelons (weighing 1 kg, 2 kg, 3 kg, ..., 20 kg), balancing each watermelon on one side of the scale with one or two weights on the other side (possibly identical weights). The seller recorded on a piece of paper the weights of the scales he used. What is the smallest number of different weights that could appear in his records if the mass of each weight is an integer kilogram?


For an industrial internship, 30 students are provided with 15 spots in Moscow, 8 spots in Tula, and 7 spots in Voronezh. What is the probability that two specific students will be assigned to the same city for their internship?


In 60 chandeliers (each with 4 shades), the shades need to be replaced. Each electrician takes 5 minutes to replace one shade. A total of 48 electricians will be working. No more than one shade can be replaced in a chandelier at the same time. What is the minimum time required to replace all the shades in all the chandeliers?


Given a triangle $ABC$ and a point $O$ inside it. Construct a segment with a midpoint at point $O$, whose endpoints lie on the boundary of triangle $ABC$. What is the maximum number of solutions this problem can have?


Three boys \( B_{1}, B_{2}, B_{3} \) and three girls \( G_{1}, G_{2}, G_{3} \) are to be seated in a row according to the following rules:
1) A boy will not sit next to another boy and a girl will not sit next to another girl,
2) Boy \( B_{1} \) must sit next to girl \( G_{1} \).
If \( s \) is the number of different such seating arrangements, find the value of \( s \).


Given the ellipse \( C \) passing through the point \( M(1,2) \) with foci at \((0, \pm \sqrt{6})\) and the origin \( O \) as the center, a line \( l \) parallel to \( OM \) intersects the ellipse \( C \) at points \( A \) and \( B \). Find the maximum area of \( \triangle OAB \).


Find all integers \( n \) such that \( 2^n + 3 \) is a perfect square. Similarly, find all integers \( n \) such that \( 2^n + 1 \) is a perfect square.


In how many ways can 4 purple balls and 4 green balls be placed into a 4x4 grid such that every row and column contains one purple ball and one green ball? Only one ball may be placed in each box, and rotations and reflections of a single configuration are considered different.


Given the function \(f(x)=\sin ^{4} \frac{k x}{10}+\cos ^{4} \frac{k x}{10}\), where \(k\) is a positive integer, if for any real number \(a\), it holds that \(\{f(x) \mid a<x<a+1\}=\{f(x) \mid x \in \mathbb{R}\}\), find the minimum value of \(k\).


The lengths of two parallel sides of a rectangle are 1 cm. Additionally, it is known that the rectangle can be divided by two perpendicular lines into four smaller rectangles, three of which have an area of at least \(1 \, \mathrm{cm}^2\), and the fourth has an area of at least \(2 \, \mathrm{cm}^2\). What is the minimum possible length of the other two sides of the rectangle?


A circle with radius 3 passes through vertex $B$, the midpoints of sides $AB$ and $BC$, and touches side $AC$ of triangle $ABC$. Given that angle $BAC$ is acute and $\sin \angle BAC = \frac{1}{3}$, find the area of triangle $ABC$.

Solution: \boxed{16 \sqrt{2}}


The angle formed by the bisector of angle \( ABC \) with its sides is 6 times smaller than the angle adjacent to angle \( ABC \). Find angle \( ABC \).


Does there exist an infinite sequence of natural numbers such that for any natural number \( k \), the sum of any \( k \) consecutive terms of this sequence is divisible by \( k+1 \)?

Solution: \boxed{\text{No}}


Find all natural numbers \( n \) such that \( n^{4} + 4^{n} \) is prime.


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

Solution: \boxed{\text{No}}


There is a basket of apples. If Group A divides the apples, each person gets 3 apples and there are 10 apples left. If Group B divides them, each person gets 4 apples and there are 11 apples left. If Group C divides them, each person gets 5 apples and there are 12 apples left. What is the minimum number of apples in the basket?


Let $ABC$ be a triangle with circumcircle $\Gamma$. A line intersects $[AB], [BC], [CA]$ at $X, D, Y$ respectively. The circumcircle of $ADX$ intersects $\Gamma$ at $Z$. Let $V = \Gamma \cap (ZD)$ and $W = \Gamma \cap (YZ)$. Show that $AB = VW$.

Solution: \boxed{AB = VW}


Let $S$ be a set composed of positive integers with the property that for any $x \in S$, removing $x$ from $S$ results in the remaining numbers having an arithmetic mean that is a positive integer. Additionally, $1 \in S$ and $2016$ is the largest element in $S$. Find the maximum value of $|S|$.


There are 5555 children, numbered 1 to 5555, sitting around a circle in order. Each child has an integer in hand: the child numbered 1 has the integer 1, the child numbered 12 has the integer 21, the child numbered 123 has the integer 321, and the child numbered 1234 has the integer 4321. It is known that the sum of the integers held by any 2005 consecutive children is equal to 2005. What is the integer held by the child numbered 5555?


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


The integers \( n \) and \( m \) satisfy the inequalities \( 3n - m < 5 \), \( n + m > 26 \), and \( 3m - 2n < 46 \). What values can \( 2n + m \) take? List all possible options.


Let \( AB \) and \( CD \) be perpendicular segments intersecting at point \( P \). Suppose that \( AP = 2 \), \( BP = 3 \), and \( CP = 1 \). If all the points \( A \), \( B \), \( C \), and \( D \) lie on a circle, find the length of \( DP \).


On the fields $A$, $B$, and $C$ in the bottom left corner of a chessboard, there are white rooks. You can make moves according to the usual chess rules. However, after any move, each rook must be protected by another rook. Is it possible, in several moves, to rearrange the rooks so that each lands on the square marked with the same letter in the top right corner?

Solution: \boxed{\text{No}}


Find the cosine of the angle between vectors $\overrightarrow{A B}$ and $\overrightarrow{A C}$.

$A(0, 2, -4), B(8, 2, 2), C(6, 2, 4)$


A cuckoo clock rings "cuckoo" every hour, with the number of rings corresponding to the hour shown by the hour hand (e.g., at 7:00, it rings 7 times). One morning, Maxim approached the clock at 9:05 and started moving the minute hand until 7 hours had passed. How many times did the clock ring "cuckoo" during this period?


Determine the smallest positive integer \( n \) such that \( n \) is divisible by 20, \( n^2 \) is a perfect cube, and \( n^3 \) is a perfect square.


In the village where Glafira lives, there is a small pond that is filled by springs at the bottom. Glafira discovered that a herd of 17 cows completely drank this pond in 3 days. After some time, the springs refilled the pond, and then 2 cows drank it in 30 days. How many days will it take for one cow to drink this pond?


27 identical dice were glued together to form a $3 \times 3 \times 3$ cube in such a way that any two adjacent small dice have the same number of dots on the touching faces. How many dots are there on the surface of the large cube?


Given the function \( f(x) = a x - \frac{3}{2} x^2 \) has a maximum value of no more than \( \frac{1}{6} \), and when \( x \in \left[ \frac{1}{4}, \frac{1}{2} \right] \), \( f(x) \geqslant \frac{1}{8} \), find the value of \( a \).


Given \( x > y > 0 \) and \( xy = 1 \), find the minimum value of \( \frac{3x^3 + 125y^3}{x-y} \).


In the finals of a beauty contest among giraffes, there were two finalists: the Tall one and the Spotted one. There are 135 voters divided into 5 districts, each district is divided into 9 precincts, and each precinct has 3 voters. Voters in each precinct choose the winner by majority vote; in a district, the giraffe that wins in the majority of precincts is the winner; finally, the giraffe that wins in the majority of districts is declared the winner of the final. The Tall giraffe won. What is the minimum number of voters who could have voted for the Tall giraffe?


A merchant had 10 barrels of sugar, which he arranged into a pyramid as shown in the illustration. Each barrel, except one, was numbered. The merchant accidentally arranged the barrels such that the sum of the numbers along each row equaled 16. Could you rearrange the barrels such that the sum of the numbers along each row equals the smallest possible number? Note that the central barrel (which happened to be number 7 in the illustration) does not count in the sum.


Find all positive integers \( n \) for which \( x^n + (x+2)^n + (2-x)^n = 0 \) has an integral solution.


Let \( a_{1}, a_{2}, a_{3}, a_{4} \) be any permutation of \{1, 2, 3, 4\\}. Let \( f \) be a mapping from \{1, 2, 3, 4\} to \{1, 2, 3, 4\} such that \( f(i) \neq i \) for all \( i \). Consider the table \(\left[ \begin{array}{cccc} a_{1} & a_{2} & a_{3} & a_{4} \\ f(a_{1}) & f(a_{2}) & f(a_{3}) & f(a_{4}) \end{array} \right] \). Two tables \( M \) and \( N \) are said to be different if they differ in at least one corresponding entry. Determine the number of different tables satisfying these conditions.


The lateral edge of a regular triangular pyramid is \(\sqrt{5}\), and the height of the pyramid is 1. Find the dihedral angle at the base.


The mathematician Fibonacci, while studying the problem of rabbit reproduction, discovered a sequence of numbers: \(1, 1, 2, 3, 5, 8, 13, \cdots\). The characteristic of this sequence is that the first two numbers are both 1, and starting from the third number, each number is equal to the sum of the two preceding numbers. A sequence formed in this way is called the Fibonacci sequence, denoted as \(\{a_n\}\). Calculate the following expression using the Fibonacci sequence:
$$
\begin{array}{l}
\left(a_{1} a_{3}+a_{2} a_{4}+a_{3} a_{5}+\cdots+a_{2019} a_{2021}\right)- \\
\left(a_{2}^{2}+a_{3}^{2}+a_{4}^{2}+\cdots+a_{2020}^{2}\right) \\
=
\end{array}
$$


Let \( S \) be a subset of \(\{1, 2, 3, \ldots, 12\}\) such that it is impossible to partition \( S \) into \( k \) disjoint subsets, each of whose elements sum to the same value, for any integer \( k \geq 2 \). Find the maximum possible sum of the elements of \( S \).


Let \( A = \{1, 2, \ldots, 2002\} \) and \( M = \{1001, 2003, 3005\} \). For any non-empty subset \( B \) of \( A \), \( B \) is called an \( M \)-free set if the sum of any two elements in \( B \) does not belong to \( M \). If \( A = A_1 \cup A_2 \), \( A_1 \cap A_2 = \varnothing \), and both \( A_1 \) and \( A_2 \) are \( M \)-free sets, then the ordered pair \(\left(A_{1}, A_{2}\right)\) is called an \( M \)-partition of \( A \). Find the number of all \( M \)-partitions of \( A \).


Alison folds a square piece of paper in half along the dashed line. After opening the paper out again, she then folds one of the corners onto the dashed line. What is the value of $\alpha$?
A) 45
B) 60
C) 65
D) 70
E) 75


Determine all quadruples $(p, q, r, n)$ of prime numbers $p, q, r$ and positive integers $n$ for which

$$
p^{2}=q^{2}+r^{n}
$$

is satisfied.

Solution: \boxed{(5, 3, 2, 4)}


The set of all positive integers can be divided into two disjoint subsets, $\{f(1), f(2), \cdots, f(n), \cdots\}$ and $\{g(1), g(2), \cdots, g(n), \cdots\}$, where $f(1)<f(2)<\cdots<f(n)<\cdots$ and $g(1)<g(2)<\cdots<g(n)<\cdots$ satisfy the condition $g(n)=f[f(n)]+1$ for $n \geq 1$. Find $f(240)$.


Two circles intersect at points $A$ and $B$. Through these points, two chords are drawn in arbitrary directions, intersecting the first circle at points $C$ and $D$, and the second circle at points $C^{\prime}$ and $D^{\prime}$. Show that the chords $CD$ and $C^{\prime}D^{\prime}$ are parallel.

Solution: \boxed{CD \parallel C'D'}


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

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

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

a) What is the probability that Grisha wins?

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


Compute the definite integral:

$$
\int_{1 / 24}^{1 / 3} \frac{5 \sqrt{x+1}}{(x+1)^{2} \sqrt{x}} \, dx
$$


How many natural numbers less than 1000 are multiples of 4 and do not contain the digits 1, 3, 4, 5, 7, 9 in their representation?


There are fewer than 30 students in a class. The probability that a randomly selected girl is an honor student is $\frac{3}{13}$, and the probability that a randomly selected boy is an honor student is $\frac{4}{11}$. How many honor students are there in the class?


A dragon has 40 piles of gold coins, with the number of coins in any two piles differing. After the dragon plundered a neighboring city and brought back more gold, the number of coins in each pile increased by either 2, 3, or 4 times. What is the minimum number of different piles of coins that could result?


On Ming's way to the swimming pool, there are 200 trees. On his round trip, Ming marked some trees with red ribbons. On his way to the swimming pool, he marked the 1st tree, the 6th tree, the 11th tree, and so on, marking every 4th tree. On his way back, he marked the 1st tree he encountered, the 9th tree, the 17th tree, and so on, marking every 7th tree. How many trees are unmarked when he returns home?


Given the sequence \(\left\{a_{n}\right\}\) that satisfies:
\[
\begin{array}{l}
a_{1}=\frac{9}{4}, \\
2 a_{n+1} a_{n}-7 a_{n+1}-3 a_{n}+12=0 \text{ for } n \geq 1.
\end{array}
\]
(1) Let \(c_{n}=a_{n}-2\). Find the general term formula for the sequence \(\left\{c_{n}\right\}\).
(2) Let \([x]\) denote the greatest integer less than or equal to the real number \(x\), and define \(b_{n}=\frac{n^2}{n+1} a_{n}\). Find the largest positive integer \(n\) such that \(\sum_{k=1}^{n}\left[b_{k}\right] \leq 2019\).


Let \(\mathbb{N}\) denote the set of positive integers. Let \(\varphi: \mathbb{N} \rightarrow \mathbb{N}\) be a bijective function and assume that there exists a finite limit

$$
\lim _{n \rightarrow \infty} \frac{\varphi(n)}{n} = L
$$

What are the possible values of \(L\)?


Find the maximum volume of a tetrahedron inscribed in a cylinder with a base radius \( R \) and height \( h \).

Solution: \boxed{\frac{2}{3} R^2 h}


The base of a hexagonal prism is a regular hexagon, and the side edges are perpendicular to the base. It is known that all the vertices of the hexagonal prism lie on the same spherical surface, and the volume of the hexagonal prism is \(\frac{9}{8}\). The perimeter of the base is 3. Calculate the volume of the sphere.

Solution: \boxed{\frac{4}{3} \pi}


For how many two-digit natural numbers \( n \) are exactly two of the following three statements true: (A) \( n \) is odd; (B) \( n \) is not divisible by 3; (C) \( n \) is divisible by 5?


A taxi driver wasn't very polite, and an upset Mr. Wilkins asked for his number.

- "Do you want to know my number?" said the driver. "Well, okay. If you divide it by 2, 3, 4, 5, or 6, you'll get a remainder of 1, but it divides evenly by 11. I'll also say that, among all drivers who could say the same about their number, mine is the smallest."

What number did the driver have?


Note that if you flip a sheet with numbers written on it, the digits $0, 1, 8$ remain unchanged, $6$ and $9$ swap places, and the rest lose their meaning. How many nine-digit numbers exist that remain unchanged when the sheet is flipped?


For the real numbers \(a, b, c, d\) the following equations apply:

\[ a = \sqrt{45 - \sqrt{21 - a}}, \]
\[ b = \sqrt{45 + \sqrt{21 - b}}, \]
\[ c = \sqrt{45 - \sqrt{21 + c}}, \]
\[ d = \sqrt{45 + \sqrt{21 + d}}. \]

Show that \(abcd = 2004\).


Tetrahedron \(ABCD\) has side lengths \(AB = 6\), \(BD = 6\sqrt{2}\), \(BC = 10\), \(AC = 8\), \(CD = 10\), and \(AD = 6\). The distance from vertex \(A\) to face \(BCD\) can be written as \(\frac{a \sqrt{b}}{c}\), where \(a, b, c\) are positive integers, \(b\) is square-free, and \(\operatorname{gcd}(a, c) = 1\). Find \(100a + 10b + c\).


A motorboat departed from dock A downstream. Simultaneously, a boat departed from dock B upstream towards the motorboat. After some time, they met. At the moment of their meeting, a second boat departed from B and after some time met the motorboat. The distance between the points of the first and second meetings is $\frac{56}{225}$ of the distance AB. The flow velocities, the motorboat, and the boats' speeds are constant (the boats' speeds are the same), and the ratio of the motorboat's speed to the boat's speed is $2:3$. What fraction of the distance AB had the motorboat traveled by the time it met the second boat?

Answer: $\frac{161}{225}$ or $\frac{176}{225}$.


Given the acute angles \( \alpha \) and \( \beta \) satisfy
\[ \sin \beta = m \cos (\alpha + \beta) \cdot \sin \alpha \quad (m > 0, \alpha + \beta \neq \frac{\pi}{2}) \]
let \( x = \tan \alpha \) and \( y = \tan \beta \).

(1) Find the expression of \( y = f(x) \);

(2) Under the condition in (1), determine the maximum value of the function \( y \) when \( \alpha \in \left[\frac{\pi}{4}, \frac{\pi}{2}\right) \).

Solution: \boxed{\frac{m}{m + 2}}


How many three-digit numbers are there in which any two adjacent digits differ by 3?


Given \( x, y, z \in \mathbf{R}^{+} \) and \( x + y + z = 1 \), find the minimum value of \( \frac{1}{x} + \frac{4}{y} + \frac{9}{z} \).


In how many ways can you choose $k$ items from $n$ items?

Solution: \boxed{\frac{n!}{k!(n-k)!}}


In a trapezoid, the longer base is 5, and one of the non-parallel sides is 3. It is known that one diagonal is perpendicular to this non-parallel side, while the other diagonal bisects the angle between this non-parallel side and the base. Find the area of the trapezoid.


Solution: \boxed{757 \frac{1}{380}}


Given point \( P \), circle \( k \), and the secant line \( AB \) passing through \( P \) such that \( PA = AB = 1 \). The tangents from \( P \) to circle \( k \) touch the circle at points \( C \) and \( D \). The intersection of \( AB \) and \( CD \) is point \( M \). What is the distance \( PM \)?


Triangle $ABC$ is equilateral. Point $M$ is marked on side $AC$, and point $N$ is marked on side $BC$, with $MC = BN = 2AM$. Segments $MB$ and $AN$ intersect at point $Q$. Find the angle $CQB$.


Given that \(x, y, z\) are all positive real numbers and satisfy \(x + y + z = 1\), find the minimum value of the function \(f(x, y, z) = \frac{3x^2 - x}{1 + x^2} + \frac{3y^2 - y}{1 + y^2} + \frac{3z^2 - z}{1 + z^2}\), and provide a proof.


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


Calculate: \(2011 - (9 \times 11 \times 11 + 9 \times 9 \times 11 - 9 \times 11) =\)


There are several (more than one) consecutive natural numbers written on a board, the sum of which equals 2016. What can be the smallest of these numbers?


On a rectangular sheet of paper, a picture in the shape of a "cross" is drawn from two rectangles $ABCD$ and $EFGH$, with sides parallel to the edges of the sheet. It is known that $AB=9$, $BC=5$, $EF=3$, and $FG=10$. Find the area of the quadrilateral $AFCH$.


At a physical education class, 27 seventh graders attended, some of whom brought one ball each. Occasionally during the class, a seventh grader would give their ball to another seventh grader who did not have a ball.

At the end of the class, \( N \) seventh graders said: "I received balls less often than I gave them away!" Find the maximum possible value of \( N \) given that nobody lied.


Ilya takes a triplet of numbers and transforms it following the rule: at each step, each number is replaced by the sum of the other two. What is the difference between the largest and the smallest numbers in the triplet after the 1989th application of this rule, if the initial triplet of numbers was \(\{70, 61, 20\}\)? If the question allows for multiple solutions, list them all as a set.


Vera bought 6 notebooks less than Misha and Vasya together, and Vasya bought 10 notebooks less than Vera and Misha together. How many notebooks did Misha buy?


Two parallel chords of a circle have lengths 24 and 32 respectively, and the distance between them is 14. What is the length of another parallel chord midway between the two chords?


Pasha, Masha, Tolya, and Olya ate 88 candies, with each of them eating at least one candy. Masha and Tolya together ate 57 candies, but Pasha ate the most candies. How many candies did Olya eat?

Solution: \boxed{O = 1}


Inside the triangle \( ABC \), a point \( M \) is taken such that the following property is known: if the sum of the squares of all sides of the triangle is added to three times the sum of all squares of the distances from point \( M \) to the vertices of the triangle, the resulting value does not exceed \( 24 \cdot x \). Find the side of the triangle \( y \), given that the area of the triangle \( ABC \) is at least \( \sqrt{3} \cdot x \). If necessary, round the found value to two decimal places.

Solution: \boxed{y = 2 \sqrt{x}}


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


Masha was given a box with differently colored beads (each bead having a unique color, with a total of $\mathrm{n}$ beads in the box). Masha chose seven beads for her dress and decided to try out all possible combinations of these beads on the dress (thus, Masha considers sewing one, two, three, four, five, six, or seven beads, with the order of the beads not mattering to her). Then she counted the total number of combinations and was very surprised that the number turned out to be odd.
1) What number did Masha get?
2) Is it true that when selecting from a set with an even number of beads, Masha could get an even number of combinations?
3) Is it true that if the order of the beads sewn onto the dress mattered to Masha, she could have ended up with either an even or odd number of combinations?

Solution: \boxed{\text{Yes}}


A triangle was cut into two triangles. Find the maximum value of $N$ such that among the 6 angles of these two triangles, exactly $N$ are the same.


Let the function \( f(x) = \sin^4 \left( \frac{kx}{10} \right) + \cos^4 \left( \frac{kx}{10} \right) \), where \( k \) is a positive integer. If for any real number \( a \), the set \(\{ f(x) \mid a < x < a+1 \} = \{ f(x) \mid x \in \mathbf{R} \}\), then find the minimum value of \( k \).


Construct spheres that are tangent to 4 given spheres. If we accept the point (a sphere with zero radius) and the plane (a sphere with infinite radius) as special cases, how many such generalized spatial Apollonian problems exist?


Two siblings sold their flock of sheep. Each sheep was sold for as many florins as the number of sheep originally in the flock. They divided the revenue by giving out 10 florins at a time. First, the elder brother took 10 florins, then the younger brother, then the elder again, and so on. In the end, the younger brother received less than 10 florins, so the elder brother gave him his knife, making their earnings equal. How much is the knife worth in florins?


Inside the triangle $ABC$, a point $D$ is chosen such that $\angle BAD = 60^\circ$ and $\angle ABC = \angle BCD = 30^\circ$. It is given that $AB = 15$ and $CD = 8$. Find the length of segment $AD$. If necessary, round the answer to 0.01 or write the answer as a common fraction.


Given a rectangular playing field of size \( 13 \times 2 \) and an unlimited number of dominoes of sizes \( 2 \times 1 \) and \( 3 \times 1 \), the playing field is to be covered completely without gaps and without overlapping, and no domino should extend beyond the playing field. Additionally, all dominoes must be oriented the same way, i.e., their long sides must be parallel to each other.

How many such coverings are possible?


The set $M$ consists of $n$ numbers, where $n$ is odd and $n > 1$. It is such that if any element is replaced by the sum of the remaining $n-1$ elements of $M$, the sum of all $n$ elements does not change. Find the product of all $n$ elements in the set $M$.


If we select $a_{1}, a_{2}, a_{3}$ from the numbers $1, 2, \cdots, 14$ in ascending order such that both $a_{2} - a_{1} \geqslant 3$ and $a_{3} - a_{2} \geqslant 3$ are satisfied, how many different ways are there to select these numbers?


The village council of the secret pipeline is gathering around a round table, where each arriving member can sit in any available seat. How many different seating arrangements are possible if 7 participants join the council? (Two arrangements are considered identical if the same people are sitting to the left and right of each participant, and empty seats are not considered.)


Two players, A and B, take turns removing stones from a pile of \( n \) stones. Player A starts first and can take any number of stones, but must take at least one and not all of them. Following this, each player must take a number of stones that is a divisor of the number of stones the other player took in the previous turn. The player who takes the last stone wins. What is the smallest value of \( n > 1992 \) for which player B has a winning strategy?


Consider a regular hexagon \(ABCDEF\). A frog starts at vertex \(A\). Each time it jumps, it can move to one of the two adjacent vertices. If the frog reaches vertex \(D\) within 5 jumps, it stops jumping. If it does not reach vertex \(D\) within 5 jumps, it stops after making 5 jumps. How many different ways can the frog jump from the start until it stops?


In a lathe workshop, parts are turned from steel blanks, one part from one blank. The shavings left after processing three blanks can be remelted to get exactly one blank. How many parts can be made from nine blanks? What about from fourteen blanks? How many blanks are needed to get 40 parts?


There are 20 chairs in a room, colored either blue or red. Each chair is occupied by either a knight or a liar. Knights always tell the truth and liars always lie. Initially, each occupant claimed that they were sitting on a blue chair. Later, they switched seats somehow, after which half of them claimed to be sitting on blue chairs and the other half claimed to be sitting on red chairs. How many knights are now sitting on red chairs?


How many solutions does the equation

$$
\left\lfloor\frac{x}{20}\right\rfloor = \left\lfloor\frac{x}{17}\right\rfloor
$$

have over the set of positive integers?

Here, $\lfloor a\rfloor$ denotes the greatest integer less than or equal to \(a\).


Let \( ABC \) be a right triangle with the hypotenuse \( BC \) measuring \( 4 \) cm. The tangent at \( A \) to the circumcircle of \( ABC \) meets the line \( BC \) at point \( D \). Suppose \( BA = BD \). Let \( S \) be the area of triangle \( ACD \), expressed in square centimeters. Calculate \( S^2 \).


A cashier from Aeroflot has to deliver tickets to five groups of tourists. Three of these groups live in the hotels "Druzhba", "Rossiya", and "Minsk". The fourth group's address will be given by tourists from "Rossiya", and the fifth group's address will be given by tourists from "Minsk". In how many ways can the cashier choose the order of visiting the hotels to deliver the tickets?


In a right triangle \(ABC\) with hypotenuse \(AC\) equal to 2, medians \(AM\) and \(CN\) are drawn. A circle can be circumscribed around quadrilateral \(ANMC\). Find the radius of this circle.


Let \( S = \{1, 2, \cdots, 2005\} \). Find the minimum value of \( n \) such that any set of \( n \) pairwise coprime elements from \( S \) contains at least one prime number.


A chessboard has its rows and columns numbered as illustrated in the figure. In each cell, the number written is the sum of the numbers of the row and column of that cell. For example, in the cell located in row 4 and column 5, the number written is $4+5=9$.

a) What is the sum of the numbers written in all the cells of the chessboard?

b) Let $S_{\text {black}}$ be the sum of all the numbers written in the black cells and $S_{\text {white}}$ be the sum of all the numbers written in the white cells. What is the value of the difference $S_{\text {black}}-S_{\text {white}}$?

c) What is the value of $S_{\text {black}}$?


Each of the lateral edges of a pyramid is equal to 269/32. The base of the pyramid is a triangle with sides 13, 14, 15. Find the volume of the pyramid.


How many zeros are at the end of the number "100!" (i.e., the product of all integers from 1 to 100)?


A circle is inscribed in a right triangle. One of the legs is divided by the point of tangency into segments measuring 6 and 10, starting from the vertex of the right angle. Find the area of the triangle.


A binary operation $\star$ satisfies $(a \star b) \star c = a + b + c$ for all real numbers $a, b$, and $c$. Show that $\star$ is the usual addition.

Solution: \boxed{a \star b = a + b}


In the regular quadrilateral pyramid $P-ABCD$, $G$ is the centroid of $\triangle PBC$. Find the value of $\frac{V_{G-PAD}}{V_{G-PAB}}$.


If the sum of roots of \(5 x^{2} + a x - 2 = 0\) is twice the product of roots, find the value of \(a\).

Given that \(y = a x^{2} - b x - 13\) passes through \((3,8)\), find the value of \(b\).

If there are \(c\) ways of arranging \(b\) girls in a circle, find the value of \(c\).

If \(\frac{c}{4}\) straight lines and 3 circles are drawn on a paper, and \(d\) is the largest number of points of intersection, find the value of \(d\).


Let \( \triangle ABC \) be an acute triangle with circumcenter \( O \) such that \( AB = 4 \), \( AC = 5 \), and \( BC = 6 \). Let \( D \) be the foot of the altitude from \( A \) to \( BC \), and \( E \) be the intersection of \( AO \) with \( BC \). Suppose that \( X \) is on \( BC \) between \( D \) and \( E \) such that there is a point \( Y \) on \( AD \) satisfying \( XY \parallel AO \) and \( YO \perp AX \). Determine the length of \( BX \).


Calculate the limit of the function:

\[
\lim _{x \rightarrow 1} \frac{1+\cos \pi x}{\operatorname{tg}^{2} \pi x}
\]


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

1. Check if two selected numbers are equal.
2. Add selected numbers.
3. Given selected numbers \(a\) and \(b\), find the roots of the equation \(x^2 + ax + b = 0\), and indicate if there are no roots.

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

Solution: \boxed{x = 1}


Given positive real numbers \( x \) and \( y \) (\( x > y \)) satisfying \( x y = 490 \) and \( (\lg x - \lg 7)(\lg y - \lg 7) = -\frac{143}{4} \), determine the number of digits in the integer part of \( x \) in decimal representation.


The gardener Fedya has a miracle tree with seven branches in his garden. On each branch, there can either grow 6 apples, 5 pears, or 3 oranges. Fedya discovered that the tree has fruit of all types, with the most pears and the fewest apples.

How many fruits in total grew on the miracle tree?


Consider the integer sequence \( a_{1}, a_{2}, \cdots, a_{10} \) satisfying:
\[
a_{10} = 3a_{1}, \quad a_{2} + a_{8} = 2a_{5}
\]
and \( a_{i+1} \in \{1 + a_{i}, 2 + a_{i}\} \) for \(i = 1, 2, \cdots, 9\). Find the number of such sequences.


The sum and difference of 44 and 18 consist of the same digits, just in reverse order. How many pairs of two-digit numbers have this property?


On side $BC$ of a triangle $ABC$ with an obtuse angle at $C$, a point $M$ is marked. A point $D$ is chosen such that triangle $BCD$ is acute, and points $A$ and $D$ lie on opposite sides of line $BC$. Circles $\omega_{B}$ and $\omega_{C}$ are circumscribed around triangles $BMD$ and $CMD$ respectively. Side $AB$ intersects circle $\omega_{B}$ again at point $P$, and ray $AC$ intersects circle $\omega_{C}$ again at point $Q$. Segment $PD$ intersects circle $\omega_{C}$ again at point $R$, and ray $QD$ intersects circle $\omega_{B}$ again at point $S$. Find the ratio of the areas of triangles $ABR$ and $ACS$.


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


Define the efficiency of a natural number \( n \) as the proportion of all natural numbers from 1 to \( n \) (inclusive) that share a common divisor with \( n \) greater than 1. For example, the efficiency of the number 6 is \( \frac{2}{3} \).

a) Does a number exist with efficiency greater than 80%? If so, find the smallest such number.

b) Is there a number whose efficiency is the highest possible (i.e., not less than that of any other number)? If so, find the smallest such number.


Let the function \( f(x)=\frac{x^{2}+x+16}{x} \) where \( 2 \leqslant x \leqslant a \) and the real number \( a > 2 \). If the range of \( f(x) \) is \([9, 11]\), then find the range of values for \( a \).

Solution: \boxed{[4, 8]}


The function \( f \) is defined on the set of integers and satisfies:
\[
f(n)=\left\{
\begin{array}{ll}
n-3 & \text{if } n \geq 1000 \\
f[f(n+5)] & \text{if } n < 1000
\end{array}
\right.
\]
Find \( f(84) \).


A section of a rectangular parallelepiped $A B C D A_{1} B_{1} C_{1} D_{1}$ by a plane passing through its vertex $D$ intersects the lateral edges $A A_{1}$, $B B_{1}$, and $C C_{1}$ at points $K$, $M$, and $N$, respectively. Find the ratio of the volume of the pyramid with vertex at point $P$ and base $D K M N$ to the volume of the given parallelepiped, if it is known that point $P$ divides the edge $D D_{1}$ in the ratio $D P : P D_{1} = m : n$.


For which values of \( n \) is the expression \( 2^{n} + 1 \) a nontrivial power of a natural number?


Four points are chosen independently and at random on the surface of a sphere (using the uniform distribution). What is the probability that the center of the sphere lies inside the resulting tetrahedron?


Pigsy picked a bag of wild fruits from the mountain to bring back to Tang Monk. As he walked, he got hungry and took out half of them, hesitated, put back two, and then ate the rest. This happened four times: taking out half, putting back two, and then eating. By the time he reached Tang Monk, there were 5 wild fruits left in the bag. How many wild fruits did Pigsy originally pick?


The polynomial \( P(x) \) is such that \( P\left(x^{2}\right) \) has \( 2n+1 \) roots. What is the minimum number of roots that the derivative of the polynomial \( P(x) \) can have?

(In both cases, distinct roots are considered, without considering multiplicity.)

Solution: \boxed{n}


There is a box containing 3 red balls and 3 white balls, which are identical in size and shape. A fair die is rolled, and the number rolled determines the number of balls drawn from the box. What is the probability that the number of red balls drawn is greater than the number of white balls drawn?


In an isosceles triangle \( ABC \) (\( AB = BC \)), a circle is inscribed. A line parallel to side \( BC \) and tangent to the circle intersects side \( AB \) at a point \( N \) such that \( AN = \frac{3}{8} AB \). Find the radius of the circle if the area of triangle \( ABC \) is 12.


In a \(3 \times 3\) table, 9 numbers are arranged such that all six products of these numbers in the rows and columns of the table are different. What is the maximum number of numbers in this table that can be equal to one?


Divide an equilateral triangle into $n^{2}$ equally sized smaller equilateral triangles. Number some of these smaller triangles with the numbers $1, 2, 3, \cdots, m$ such that adjacent triangles have adjacent numbers. Show that $m \leq n^{2} - n + 1$.

Solution: \boxed{m \leq n^2 - n + 1}


Find all triplets of positive integers \((m, n, p)\) with \(p\) being a prime number, such that \(2^{m} p^{2} + 1 = n^{5}\).

Solution: \boxed{(1, 3, 11)}


The triangle ABC has area 1. Take points X on AB and Y on AC so that the centroid G is on the opposite side of XY to B and C. Show that the area of BXGY plus the area of CYGX is greater than or equal to 4/9. When do we have equality?


In the right triangle \(ABC\) with the right angle at \(C\), points \(P\) and \(Q\) are the midpoints of the angle bisectors drawn from vertices \(A\) and \(B\). The circle inscribed in the triangle touches the hypotenuse at point \(H\). Find the angle \(PHQ\).


Let \( a_1, a_2, \ldots, a_{24} \) be integers with sum 0 and satisfying \( |a_i| \leq i \) for all \( i \). Find the greatest possible value of \( a_1 + 2a_2 + 3a_3 + \cdots + 24a_{24} \).


Find all strictly increasing functions \( f: \mathbb{N} \rightarrow \mathbb{N} \) such that for all \( n \geq 0 \), \( f(f(n)) < n + 1 \).

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


Triangle \(ABC\) has side lengths \(AB=2\sqrt{5}\), \(BC=1\), and \(CA=5\). Point \(D\) is on side \(AC\) such that \(CD=1\), and \(F\) is a point such that \(BF=2\) and \(CF=3\). Let \(E\) be the intersection of lines \(AB\) and \(DF\). Find the area of \(CDEB\).


Given that $\left\{a_{n}\right\}$ is a geometric sequence and $a_{1} a_{2017}=1$, and if $f(x)=\frac{2}{1+x^{2}}$, find $\sum_{i=1}^{2017} f\left(a_{i}\right)$.


Triangle \(A B C\) has side lengths \(A B = 65\), \(B C = 33\), and \(A C = 56\). Find the radius of the circle tangent to sides \(A C\) and \(B C\) and to the circumcircle of triangle \(A B C\).


Let \( k>1 \) be a real number. Calculate:

(a) \( L = \lim _{n \rightarrow \infty} \int_{0}^{1}\left(\frac{k}{\sqrt[n]{x} + k - 1}\right)^{n} \, \mathrm{d}x \).

(b) \( \lim _{n \rightarrow \infty} n\left[L - \int_{0}^{1}\left(\frac{k}{\sqrt[n]{x} + k - 1}\right)^{n} \, \mathrm{d}x\right] \).

Solution: \boxed{ \frac{k}{(k-1)^2} }


In a plane, 100 points are marked. It turns out that 40 marked points lie on each of two different lines \( a \) and \( b \). What is the maximum number of marked points that can lie on a line that does not coincide with \( a \) or \( b \)?


Let $S$ be a set of $n$ distinct real numbers, and let $A_{s}$ be the set of averages of all distinct pairs of elements from $S$. For a given $n \geq 2$, what is the minimum possible number of elements in $A_{s}$?

Solution: \boxed{2n - 3}


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

$$
r=\frac{5}{2} \sin \phi, \quad r=\frac{3}{2} \sin \phi
$$

Solution: \boxed{\pi}


In the right triangle \(ABC\) with a right angle at \(A\), the altitude \(AH\) is drawn. A circle passing through points \(A\) and \(H\) intersects the legs \(AB\) and \(AC\) at points \(X\) and \(Y\) respectively. Find the length of segment \(AC\) given that \(AX = 5\), \(AY = 6\), and \(AB = 9\).


Find the maximum number of white dominoes that can be cut from the board shown on the left. A domino is a $1 \times 2$ rectangle.


If \( P(x, y) \) is a point on the hyperbola \(\frac{x^{2}}{8}-\frac{y^{2}}{4}=1\), then the minimum value of \(|x-y|\) is ________.


Two boxes together contain 65 balls of various sizes. Each ball is white, black, red, or yellow. Every time we take out five balls of the same color, at least two are of the same size.

(a) What is the maximum number of types of balls that exist in the boxes? Two balls are considered different types when they have different colors or sizes.

(b) Show that there are at least three balls, in the same box, which are of the same type.


Find the smallest positive integer \( n \) such that, if there are initially \( 2n \) townspeople and 1 goon, then the probability that the townspeople win is greater than 50%.


The sum of \( m \) distinct positive even numbers and \( n \) distinct positive odd numbers is 1987. Determine the maximum value of \( 3m + 4n \) for all such \( m \) and \( n \). Provide a proof for your conclusion.


Given the sequence $\left\{a_{n}\right\}$ that satisfies $a_{1}=p, a_{2}=p+1, a_{n+2}-2 a_{n+1}+a_{n}=n-20$, where $p$ is a given real number and $n$ is a positive integer, find the value of $n$ that makes $a_{n}$ minimal.


How many functions \( f: \{1, 2, \ldots, 10\} \rightarrow \{1, 2, \ldots, 10\} \) satisfy the property that \( f(i) + f(j) = 11 \) for all values of \( i \) and \( j \) such that \( i + j = 11 \)?


Let \( A(2,0) \) be a fixed point on the plane, \( P\left(\sin \left(2 t-60^{\circ}\right), \cos \left(2 t-60^{\circ}\right)\right) \) be a moving point. When \( t \) changes from \( 15^{\circ} \) to \( 45^{\circ} \), the area swept by the line segment \( AP \) is ______.


In preparation for an exam, three students solved 100 problems. Each student solved 60 problems, and every problem was solved by at least one student. A problem is considered difficult if it was solved by only one student. A problem is considered easy if it was solved by all three students. Are there more easy problems or difficult problems, and by how many?


A $10 \times 10$ table is filled with numbers from 1 to 100: in the first row, the numbers from 1 to 10 are listed in ascending order from left to right; in the second row, the numbers from 11 to 20 are listed in the same way, and so on; in the last row, the numbers from 91 to 100 are listed from left to right. Can a segment of 7 cells of the form $\quad$ be found in this table, where the sum of the numbers is $455?$ (The segment can be rotated.)

Solution: \boxed{Yes}


There are 70 points on a circle. Choose one point and label it 1. Move clockwise, skipping one point, and label the next point 2. Skip two points and label the next point 3. Continue this pattern, until all numbers from 1 to 2014 have been labeled on the points. Each point may have more than one number labeled on it. What is the smallest integer labeled on the point that is also labeled with 2014?


How many pairs of two-digit positive integers have a difference of 50?


The famous Fibonacci sequence is defined as follows:

$$
\begin{gathered}
F_{0}=0, F_{1}=1 \\
F_{n+2}=F_{n+1}+F_{n}
\end{gathered}
$$

1. Let $\varphi$ and $\varphi'$ be the two roots of the equation $x^2 - x - 1 = 0$.

$$
\varphi=\frac{1+\sqrt{5}}{2}, \varphi^{\prime}=\frac{1-\sqrt{5}}{2}
$$

Show that there exist two real numbers $\lambda$ and $\mu$ such that

$$
F_{n}=\lambda \cdot \varphi^{n}+\mu \cdot \varphi^{\prime n}
$$

Can you generalize the result to any sequences of the type $u_{n+2} = a u_{n+1} + b u_{n}$?

2. Show that

$$
\lim_{n \rightarrow \infty} \frac{F_{n+1}}{F_{n}} = \varphi
$$

3. Show the following formulas:

$$
\begin{gathered}
\sum_{i=1}^{n} F_{i}^{2}=F_{n} F_{n+1} \\
F_{2n-1}=F_{n}^{2} + F_{n+1}^{2} \\
m \mid n \Rightarrow F_{m} \mid F_{n}
\end{gathered}
$$

Solution: \boxed{\varphi}


A polynomial \( P \) has four roots: \( \frac{1}{4}, \frac{1}{2}, 2, 4 \). The product of the roots is 1, and \( P(1) = 1 \). Find \( P(0) \).


Find the number of all five-digit numbers \( \overline{abcde} \) where all digits are different, and \( a < b < c > d > e \).


In a \( 29 \times 29 \) table, the numbers \( 1, 2, 3, \ldots, 29 \) are each written 29 times. It is found that the sum of the numbers above the main diagonal is three times the sum of the numbers below this diagonal. Find the number written in the central cell of the table.


The great commander, Marshal of the Soviet Union Georgy Konstantinovich Zhukov, was born in the village of Strelkovka in the Kaluga province. He lived for 78 years. In the 20th century, he lived 70 years more than in the 19th century. In what year was G.K. Zhukov born?


On the minor axis of an ellipse, at a distance $e=O F$ (where $O$ is the center of the ellipse and $F$ is one of the foci) from the center, two points are taken, and from these points, perpendiculars to an arbitrary tangent are drawn. Show that the sum of the squares of these perpendiculars is $2 a^{2}$.

Solution: \boxed{2 a^2}


In triangle \( A B C \), the angle bisector \( A M \) is perpendicular to the median \( B K \). Find the ratios \( B P : P K \) and \( A P : P M \), where \( P \) is the intersection point of the angle bisector and the median.

Solution: \boxed{AP : PM = 3:1}


In a magical country, there are only two types of people: type A people who always tell the truth and type B people who always lie. One day, 2014 citizens of this country lined up in a row, and each person said, "There are more type B people behind me than type A people in front of me." How many type A people are there among these 2014 citizens?


A steamboat travels from point $A$ to point $B$ in $a$ hours and back in $b$ hours. How many hours will a raft take to drift from $A$ to $B$?


Let \( A \) and \( B \) be two positive four-digit integers such that \( A \times B = 16^5 + 2^{10} \). Find the value of \( A + B \).


The transgalactic ship encountered an amazing meteor shower. One part of the meteors flies along a straight line with equal speeds, one after another, at equal distances from each other. Another part flies similarly but along another straight line, parallel to the first one, with the same speeds but in the opposite direction, at the same distance from each other. The ship flies parallel to these lines. Astronaut Gavrila recorded that the ship encounters meteors flying towards it every 7 seconds, and those flying in the same direction as the ship every 13 seconds. He wondered how often the meteors would pass by if the ship were stationary. He thought it necessary to take the arithmetic mean of the two given times. Is Gavrila right? If so, write this arithmetic mean as the answer. If not, indicate the correct time in seconds, rounded to one decimal place.


Find the smallest number \( k \) such that \(\frac{t_{a} + t_{b}}{a + b} < k\), where \( a \) and \( b \) are two side lengths of a triangle, and \( t_{a} \) and \( t_{b} \) are the lengths of the angle bisectors corresponding to these sides, respectively.


Suppose \( S = \{1,2, \cdots, 2005\} \). Find the minimum value of \( n \) such that every subset of \( S \) consisting of \( n \) pairwise coprime numbers contains at least one prime number.


Petra had natural numbers from 1 to 9 written down. She added two of these numbers, erased them, and wrote the resulting sum in place of the erased addends. She then had eight numbers left, which she was able to divide into two groups with the same product.

Determine the largest possible product of these groups.


A circle is tangent to the extensions of two sides \( AB \) and \( AD \) of the square \( ABCD \), and the point of tangency cuts off a segment of 2 cm from vertex \( A \). Two tangents are drawn from point \( C \) to this circle. Find the side of the square if the angle between the tangents is \( 30^\circ \), and it is known that \( \sin 15^\circ = \frac{\sqrt{3}-1}{2 \sqrt{2}} \).

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


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


A group consisting of 7 young men and 7 young women is randomly paired up. Find the probability that at least one pair consists of two young women. Round your answer to the nearest hundredth.


Through the midpoint of the altitude $BB_1$ of triangle $ABC$, a line $DE$ is drawn parallel to the base. How does the area of triangle $BDE$ compare to the area of trapezoid $ADEC$?


There are 207 different cards with numbers $1, 2, 3, 2^{2}, 3^{2}, \ldots, 2^{103}, 3^{103}$ (each card has one number, and each number appears exactly once). How many ways can you choose 3 cards such that the product of the numbers on the selected cards is a square of an integer that is divisible by 6?


Determine the value of \( c \) such that the equation

$$
5x^{2} - 2x + c = 0
$$

has roots with a ratio of \(-\frac{3}{5}\).


The diagonals of a trapezoid are 12 and 6, and the sum of the bases is 14. Find the area of the trapezoid.

Solution: \boxed{16 \sqrt{5}}


There are 100 pieces, and two people take turns taking pieces. Each time, one is allowed to take either 1 or 2 pieces. The person who takes the last piece wins. If you go first, how many pieces should you take in the first turn to guarantee a win?


In triangle \( \triangle ABC \), the sides opposite to the angles \( A \), \( B \), and \( C \) are of lengths \( a \), \( b \), and \( c \) respectively. Point \( G \) satisfies
$$
\overrightarrow{GA} + \overrightarrow{GB} + \overrightarrow{GC} = \mathbf{0}, \quad \overrightarrow{GA} \cdot \overrightarrow{GB} = 0.
$$
If \((\tan A + \tan B) \tan C = m \tan A \cdot \tan B\), then \( m = \).


Given a constant \( a \in (0,1) \), and \( |x| + |y| \leqslant 1 \), what is the maximum value of the function \( f(x, y) = a x + y \)?


Find the value of the expression \( 1! \cdot 3 - 2! \cdot 4 + 3! \cdot 5 - 4! \cdot 6 + \ldots - 2000! \cdot 2002 + 2001! \).


By how much did the dollar exchange rate change during the year 2014 (from January 1, 2014, to December 31, 2014)? Give your answer in rubles, rounded to the nearest whole number.


In an $n$-throw game of dice, the sum of the points from all throws must be greater than $2^n$ to pass the round.

1. How many rounds can a person pass at most?
2. What is the probability that the person passes the first three rounds in succession?

(Note: The dice is a fair six-sided cube with faces numbered 1 through 6. When the dice come to rest after being thrown, the number on the upper face is the point scored.)


The radii of the excircles of a triangle are 3, 10, and 15 respectively. What is the perimeter of the triangle?


8 people decide to hold daily meetings subject to the following rules. At least one person must attend each day. A different set of people must attend on different days. On day N, for each 1 ≤ k < N, at least one person must attend who was present on day k. How many days can the meetings be held?


Solution: \boxed{\lambda = 1}


Let $\Gamma_{1}$ and $\Gamma_{2}$ be two circles intersecting at points $\mathrm{P}$ and $\mathrm{Q}$. Let $A$ and $B$ be two points on $\Gamma_{1}$ such that $A, P, Q,$ and $B$ are concyclic in this order, and let $C$ and $D$ be the second intersections with $\Gamma_{2}$ of lines (AP) and (BQ) respectively. Show that $(AB) \parallel (CD)$.

Solution: \boxed{AB \parallel CD}


Solution: \boxed{(-\infty, \infty)}


Given \(0 < x < 1, a > 0, a \neq 1\), compare the magnitudes of \(p = \left|\log_a(1 - x)\right|\) and \(q = \left|\log_a(1 + x)\right| \).

Solution: \boxed{p > q}


Given that \( f(x) \) is an \( n \)-degree polynomial of \( x \) where \( n > 0 \), and for any real number \( x \), the equation \( 8 f\left(x^{3}\right) - x^{6} f(2x) - 2 f\left(x^{2}\right) + 12 = 0 \) holds.
(1) Find \( f(x) \).

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


A client of a brokerage company deposited 12,000 rubles into a brokerage account at a rate of 60 rubles per dollar with instructions to the broker to invest the amount in bonds of foreign banks, which have a guaranteed return of 12% per annum in dollars.

(a) Determine the amount in rubles that the client withdrew from their account after a year if the ruble exchange rate was 80 rubles per dollar, the currency conversion fee was 4%, and the broker's commission was 25% of the profit in the currency.

(b) Determine the effective (actual) annual rate of return on investments in rubles.

(c) Explain why the actual annual rate of return may differ from the one you found in point (b). In which direction will it differ from the above value?


In quadrilateral \(ABCD\), side \(AB\) is equal to diagonal \(AC\) and is perpendicular to side \(AD\), while diagonal \(AC\) is perpendicular to side \(CD\). On side \(AD\), a point \(K\) is taken such that \(AC = AK\). The angle bisector of \(\angle ADC\) intersects \(BK\) at point \(M\). Find the angle \(\angle ACM\).


Solve the systems of equations:
a)
\[
\begin{cases}
x + y + xy = 5, \\
xy(x + y) = 6;
\end{cases}
\]

b)
\[
\begin{cases}
x^3 + y^3 + 2xy = 4, \\
x^2 - xy + y^2 = 1.
\end{cases}
\]

Solution: \boxed{(1, 1)}


Let \( ABC \) be a triangle such that \(\angle CAB = 20^\circ\). Let \( D \) be the midpoint of segment \([AB]\). It is given that \(\angle CDB = 40^\circ\). What is the measure of \(\angle ABC\)?


Oleg has 550 rubles and wants to buy an odd number of tulips, making sure that no color is repeated. In the store where Oleg goes, one tulip costs 49 rubles, and there are eleven different shades available. How many different ways are there for Oleg to give flowers to his mother? (The answer should be a compact expression that does not contain summation signs, ellipses, etc.)


Given \(\frac{ab}{a+b}=3\), \(\frac{bc}{b+c}=6\), and \(\frac{ac}{a+c}=9\), determine the value of \(\frac{c}{ab}\).


Suppose there are 128 ones written on a blackboard. At each step, you can erase any two numbers \(a\) and \(b\) from the blackboard and write \(ab + 1\). After performing this operation 127 times, only one number is left. Let \(A\) be the maximum possible value of this remaining number. Find the last digit of \(A\).


In a rhombus with an acute angle of $30^{\circ}$, a circle is inscribed, and a square is inscribed in the circle. Find the ratio of the area of the rhombus to the area of the square.


Let \( n \) be a fixed integer, \( n \geq 2 \).
(I) Find the minimum constant \( c \) such that the inequality
\[ \sum_{1 \leq i < j \leq n} x_i x_j \left( x_i^2 + x_j^2 \right) \leq c \left( \sum_{i=1}^{n} x_i \right)^4 \]
holds.
(II) For this constant \( c \), determine the necessary and sufficient conditions for which equality holds.


2002 is a palindromic year, meaning it reads the same backward and forward. The previous palindromic year was 11 years ago (1991). What is the maximum number of non-palindromic years that can occur consecutively (between the years 1000 and 9999)?


As shown in the figure, an equilateral triangle $ABC$ with a side length of 4 is folded along a line $EF$ parallel to side $BC$ in such a way that plane $AEF$ is perpendicular to plane $BCFE$. Let $O$ be the midpoint of $EF$.
(1) Find the cosine value of the dihedral angle $F-AE-B$.
(2) If $BE$ is perpendicular to plane $AOC$, find the length of the fold $EF$.


In triangle \(ABC\), the sides \(AC = 14\) and \(AB = 6\) are given. A circle with center \(O\), constructed on side \(AC\) as the diameter, intersects side \(BC\) at point \(K\). It is given that \(\angle BAK = \angle ACB\). Find the area of triangle \(BOC\).


Let \( P(x) \) be the polynomial of minimal degree such that \( P(k) = \frac{720k}{k^2 - 1} \) for \( k \in \{2, 3, 4, 5\} \). Find the value of \( P(6) \).


Points \( A, B, C, D \) are on the same circle, and \( BC = CD = 4 \). Let \( E \) be the intersection of \( AC \) and \( BD \), with \( AE = 6 \). If the lengths of segments \( BE \) and \( DE \) are both integers, what is the length of \( BD \)?


Calculate the limit of the function:

\[
\lim _{x \rightarrow a}\left(2-\frac{x}{a}\right)^{\tan\left(\frac{\pi x}{2a}\right)}
\]


Two prime numbers that differ by exactly 2 are called twin primes. For example, 3 and 5 are twin primes, as are 29 and 31. In number theory research, twin primes are one of the most popular topics. If a pair of twin primes both do not exceed 200, what is the maximum sum of these two prime numbers?


Given three points \( A, B, C \) forming a triangle with angles \( 30^{\circ}, 45^{\circ}, 105^{\circ} \). Two of these points are selected, and the perpendicular bisector of the segment connecting them is drawn, after which the third point is reflected relative to this perpendicular bisector, resulting in a fourth point \( D \). This procedure continues with the set of the four resulting points: two points are chosen, a perpendicular bisector is drawn, and all points are reflected relative to it. What is the maximum number of distinct points that can be obtained as a result of repeatedly applying this procedure?


In a Cartesian coordinate system, \( A(1,2) \), \( B(3,0) \), and \( P \) are points on the circle \( (x-3)^{2}+(y-2)^{2}=1 \). Suppose
$$
\overrightarrow{O P}=\lambda \overrightarrow{O A}+\mu \overrightarrow{O B} \quad (\lambda, \mu \in \mathbf{R}).
$$

Find the minimum value of \( 11\lambda + 9\mu \).


Admiral Ackbar needs to send a 5-character message through hyperspace to the Rebels. Each character is a lowercase letter, and the same letter may appear more than once in a message. When the message is beamed through hyperspace, the characters come out in a random order. Ackbar chooses his message so that the Rebels have at least a \(\frac{1}{2}\) chance of getting the same message he sent. How many distinct messages could he send?


Given the function \( f(x)=\frac{m-2 \sin x}{\cos x} \) is monotonically decreasing on the interval \(\left(0, \frac{\pi}{2}\right)\), find the range of the real number \( m \).

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


Teacher Kadrnožková bought tickets at the zoo ticket office for her students and herself. An adult ticket was more expensive than a student ticket but not more than twice the price. Teacher Kadrnožková paid a total of 994 CZK. Teacher Hnízdo had three more students than his colleague, and therefore, he paid 1120 CZK for his students and himself.

1. How many students did teacher Hnízdo have with him?
2. How much did an adult ticket cost?


At vertex $A$ of a unit square $ABCD$, an ant begins its journey. It needs to reach point $C$, where the entrance to an anthill is located. Points $A$ and $C$ are separated by a vertical wall in the shape of an isosceles right triangle with hypotenuse $BD$. Find the length of the shortest path the ant must take to reach the anthill.


Given that \( y = f(x) \) is an increasing function on \(\mathbb{R}\) and \( f(x) = f^{-1}(x) \), show that \( f(x) = x \).

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


Two ants crawled along their own closed routes on a $7 \times 7$ board. Each ant crawled only along the sides of the cells of the board and visited each of the 64 vertices of the cells exactly once. What is the minimum possible number of such sides that both the first and the second ant crawled along?


Along the shore of a circular lake, there are apple trees. Petya and Vasya start walking from point $A$ on the shore in opposite directions along the shore, counting all the apple trees they encounter and all the apples growing on the trees. When they meet at some point $B$, they compare their results. It turns out that Petya counted twice as many apple trees as Vasya, and seven times more apples than Vasya. Surprised by this result, they decided to repeat the experiment. They started from point $B$ in the same directions and met again at point $C$. It turned out that on the way from $B$ to $C$, Petya again counted twice as many apple trees as Vasya, and seven times more apples than Vasya. Their surprise grew, and they decided to repeat the experiment once more. Setting out from $C$ in the same directions, they met at point $D$. It turned out that Petya again counted twice as many apple trees as Vasya. Who counted more apples on the way from $C$ to $D$, and by how many times?

Answer: On the way from $C$ to $D$, Vasya counted 3 times more apples than Petya.


The side length of the base of a regular triangular pyramid is \(a\). The lateral edge forms an angle of \(60^{\circ}\) with the plane of the base. Find the radius of the sphere circumscribed around the pyramid.


Solve the equation \(x^{3} - \lfloor x \rfloor = 3\), where \(\lfloor x \rfloor\) represents the greatest integer less than or equal to \(x\).


In triangle \(ABC\), the angle bisectors \(AD\) and \(BE\) are drawn. Find the measure of angle \(C\), given that \(AD \cdot BC = BE \cdot AC\) and \(AC \neq BC\).


\(A\) and \(B\) are positive integers less than 10 such that \(21A104 \times 11 = 2B8016 \times 9\).
Find \(A\).
Find \(B\).


Calculate the definite integral:

$$
\int_{\pi / 4}^{\arccos (1 / \sqrt{26})} \frac{36 \, dx}{(6 - \tan x) \sin 2x}
$$

Solution: \boxed{6 \ln 5}


Let \( \triangle DEF \) be a triangle and \( H \) the foot of the altitude from \( D \) to \( EF \). If \( DE = 60 \), \( DF = 35 \), and \( DH = 21 \), what is the difference between the minimum and the maximum possible values for the area of \( \triangle DEF \)?


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


There are 10 different balls that need to be placed into 8 different empty boxes, with each box containing at least one ball. How many ways are there to do this?


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


In the isosceles triangle \(ABC\) (\(AB = BC\)), a point \(D\) is taken on the side \(BC\) such that \(BD : DC = 1 : 4\). In what ratio does the line \(AD\) divide the altitude \(BE\) of triangle \(ABC\), counting from vertex \(B\)?


At the school reunion, 45 people attended. It turned out that any two of them who have the same number of acquaintances among the attendees are not acquainted with each other. What is the maximum number of pairs of acquaintances that could be among the attendees?


For positive integers \( n \) and \( k \), let \( \mho(n, k) \) be the number of distinct prime divisors of \( n \) that are at least \( k \). For example, \( \mho(90,3) = 2 \), since the only prime factors of 90 that are at least 3 are 3 and 5. Find the closest integer to
\[
\sum_{n=1}^{\infty} \sum_{k=1}^{\infty} \frac{\mho(n, k)}{3^{n+k-7}}
\]


Find the smallest positive integer \( n \) that satisfies the following condition: For any \( n \) points \( A_1, A_2, \dots, A_n \) chosen on the circumference of a circle \( O \), among the \(\binom{n}{2}\) angles \(\angle A_i O A_j \) (where \( 1 \leq i < j \leq n \)), there are at least 2007 angles that do not exceed \( 120^\circ \).


A digit was crossed out from a six-digit number, resulting in a five-digit number. When this five-digit number was subtracted from the original six-digit number, the result was 654321. Find the original six-digit number.


In an equilateral triangular prism \( S-ABC \), the lateral edges are equal in length to the edges of the base. If \( E \) and \( F \) are the midpoints of \( SC \) and \( AB \) respectively, what is the angle between the skew lines \( EF \) and \( SA \)?


In triangle \( \triangle ABC \), \(\cos A\), \(\sin A\), and \(\tan B\) form a geometric sequence with a common ratio of \(\frac{3}{4}\). What is \(\cot C\)?


Rectangle \( R_{0} \) has sides of lengths 3 and 4. Rectangles \( R_{1}, R_{2} \), and \( R_{3} \) are formed such that:
- all four rectangles share a common vertex \( P \),
- for each \( n=1,2,3 \), one side of \( R_{n} \) is a diagonal of \( R_{n-1} \),
- for each \( n=1,2,3 \), the opposite side of \( R_{n} \) passes through a vertex of \( R_{n-1} \) such that the center of \( R_{n} \) is located counterclockwise of the center of \( R_{n-1} \) with respect to \( P \).

Compute the total area covered by the union of the four rectangles.


A cube is dissected into 6 pyramids by connecting a given point in the interior of the cube with each vertex of the cube, so that each face of the cube forms the base of a pyramid. The volumes of five of these pyramids are 200, 500, 1000, 1100, and 1400. What is the volume of the sixth pyramid?


ABC is an acute-angled triangle with area 1. A rectangle R has its vertices on the sides of the triangle, with two vertices on BC, one on AC, and one on AB. Another rectangle S has its vertices on the sides of the triangle formed by the points A, R3, and R4, with two vertices on R3R4 and one on each of the other two sides. What is the maximum total area of rectangles R and S over all possible choices of triangle and rectangles?


Horizontal parallel segments \( AB = 10 \) and \( CD = 15 \) are the bases of trapezoid \( ABCD \). Circle \(\gamma\) of radius 6 has its center within the trapezoid and is tangent to sides \( AB \), \( BC \), and \( DA \). If side \( CD \) cuts out an arc of \(\gamma\) measuring \(120^\circ\), find the area of \(ABCD\).


Calculate the limit of the function:

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


Let $x$ and $y$ be non-zero real numbers such that $\frac{x \sin \frac{\pi}{5} + y \cos \frac{\pi}{5}}{x \cos \frac{\pi}{5} - y \sin \frac{\pi}{5}} = \tan \frac{9\pi}{20}$.
(1) Find the value of $\frac{y}{x}$;
(2) In triangle $ABC$, if $\tan C=\frac{y}{x}$, find the maximum value of $\sin 2A + 2 \cos B$.


A coin is tossed 10 times. Find the probability that two heads do not appear consecutively at any time.


Given that \(\theta_{1}, \theta_{2}, \cdots, \theta_{n}\) are all non-negative real numbers and satisfy the equation
\[
\theta_{1} + \theta_{2} + \cdots + \theta_{n} = \pi.
\]

Find the maximum value of \(\sin^2 \theta_{1} + \sin^2 \theta_{2} + \cdots + \sin^2 \theta_{n}\).


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

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


Given the sequence $\left\{a_{n}\right\}$ which satisfies $a_{n} = a_{n-1} - a_{n-2}$ for $n \geq 3$. If the sum of its first 1492 terms is 1985 and the sum of its first 1985 terms is 1492, find the sum of its first 2001 terms.


The numbers \(a, b, c, d\) belong to the interval \([-6, 6]\). Find the maximum value of the expression \(a + 2b + c + 2d - ab - bc - cd - da\).


The integers from 1 to \( n \) are written in increasing order from left to right on a blackboard. David and Goliath play the following game: starting with David, the two players alternate erasing any two consecutive numbers and replacing them with their sum or product. Play continues until only one number on the board remains. If it is odd, David wins, but if it is even, Goliath wins. Find the 2011th smallest positive integer greater than 1 for which David can guarantee victory.


In the figure below, triangle \(ABC\) and rectangle \(PQRS\) have the same area and the same height of 1. For each value of \(x\) between 0 and 1, trapezoid \(ABED\) of height \(x\) is drawn, and then rectangle \(PQNM\) of the same area as the trapezoid is drawn, as shown in the figure. Let \(f\) be the function that assigns to each \(x\) the height of rectangle \(PQNM\).

a) What is the ratio between \(AB\) and \(PQ\)?

b) What is the value of \(f\left(\frac{1}{2}\right)\)?

c) Find the expression of \(f(x)\) and draw the graph of \(f\).

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


How many children did my mother have?

If you asked me this question, I would only tell you that my mother dreamed of having at least 19 children, but she couldn't make this dream come true; however, I had three times more sisters than cousins, and twice as many brothers as sisters. How many children did my mother have?


A part of a book has fallen out. The number of the first fallen page is 387, and the number of the last page consists of the same digits but in a different order. How many sheets fell out of the book?


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


If \( a, b, c, \) and \( d \) satisfy \( \frac{a}{b} = \frac{2}{3} \) and \( \frac{c}{b} = \frac{1}{5} \) and \( \frac{c}{d} = \frac{7}{15} \), what is the value of \( \frac{a b}{c d} \)?


Is there a positive integer \( m \) such that the equation
\[ \frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{abc} = \frac{m}{a+b+c} \]
has infinitely many solutions in positive integers \( a, b, c \)?


Let $a$ be an integer such that $x^2 - x + a$ divides $x^{13} + x + 90$. Find the value of $a$.


What is the smallest number of regular hexagons with a side length of 1 needed to completely cover a disc with a radius of 1?


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


Given an isosceles trapezoid $\mathrm{ABCE}$ with bases $\mathrm{BC}$ and $\mathrm{AE}$, where $\mathrm{BC}$ is smaller than $\mathrm{AE}$ with lengths 3 and 4 respectively. The smaller lateral side $\mathrm{AB}$ is equal to $\mathrm{BC}$. Point $\mathrm{D}$ lies on $\mathrm{AE}$ such that $\mathrm{AD}$ : $\mathrm{DE}=3:1$. Point $\mathrm{F}$ lies on $\mathrm{AD}$ such that $\mathrm{AF}$ : $\mathrm{FD}=2:1$. Point $\mathrm{G}$ lies on $\mathrm{BD}$ such that $\mathrm{BG}$ : $\mathrm{GD}=1:2$. Determine the angle measure $\angle \mathrm{CFG}$.


We need to calculate the expression with three significant digits' accuracy. To simplify the division, find an expression of the form (2) - with coefficients $A, B, C$ such that when multiplying it by (1), we obtain a rational denominator (a formula table can be used).

$$
\begin{gathered}
529 \\
\hline 12 \sqrt[3]{9}+52 \sqrt[3]{3}+49 \\
A \sqrt[3]{9}+B \sqrt[3]{3}+C
\end{gathered}
$$


120 identical spheres are tightly packed in the shape of a regular triangular pyramid. How many spheres are in the base?


If \( n \in \mathbf{N}^{*} \), then \( \lim_{n \rightarrow \infty} \sin^{2}\left(\pi \sqrt{n^{2}+n}\right) = \) ?


Define a subset $A$ of $\mathbf{Z}_{+}$ (the set of positive integers) as a "sum-free set" if and only if for any two (possibly identical) elements $x, y$ in $A$, their sum $x + y$ is not in $A$. Find all surjective functions $f: \mathbf{Z}_{+} \rightarrow \mathbf{Z}_{+}$ such that the image $\{f(a) \mid a \in A\}$ is also a sum-free set for any sum-free set $A \subseteq \mathbf{Z}_{+}$.

Note: A function $f: \mathbf{Z}_{+} \rightarrow \mathbf{Z}_{+}$ is called surjective if for any positive integer $n$, there exists a positive integer $m$ such that $f(m) = n$.

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


A semicircle with a radius of $3 \text{ cm}$ contains a $1 \text{ cm}$ radius semicircle and a $2 \text{ cm}$ radius semicircle within it, as shown in the figure. Calculate the radius of the circle that is tangent to all three semicircular arcs.


Usually, I go up the escalator in the subway. I have calculated that when I walk up the moving escalator, I ascend 20 steps, and it takes me exactly 60 seconds. My wife walks up the stairs more slowly and only ascends 16 steps; therefore, her total time to ascend the escalator is longer - it is 72 seconds.

How many steps would I have to climb if the escalator suddenly breaks down?


In triangle \(ABC\), angle \(A\) is \(40^\circ\). The triangle is randomly thrown onto a table. Find the probability that vertex \(A\) ends up east of the other two vertices.


Calculate the limit of the function:

$\lim _{x \rightarrow \frac{\pi}{2}} \frac{1-\sin ^{3} x}{\cos ^{2} x}$


Given 1997 points on a plane such that from any four points, three can be chosen to be collinear (on the same line). Show that at least 1996 of the points are on the same line.


Draw a rectangle. Connect the midpoints of the opposite sides to get 4 congruent rectangles. Connect the midpoints of the lower right rectangle for a total of 7 rectangles. Repeat this process infinitely. Let \( n \) be the minimum number of colors we can assign to the rectangles so that no two rectangles sharing an edge have the same color and \( m \) be the minimum number of colors we can assign to the rectangles so that no two rectangles sharing a corner have the same color. Find the ordered pair \((n, m)\).

Solution: \boxed{(3, 4)}


Let \([x]\) be the largest integer not greater than \(x\), for example, \([2.5] = 2\). If \(a = 1 + \frac{1}{2^{2}} + \frac{1}{3^{2}} + \cdots + \frac{1}{2004^{2}}\) and \(S = [a]\), find the value of \(a\).


The equation \( x^{4} - 7x - 3 = 0 \) has exactly two real roots \( a \) and \( b \), where \( a > b \). Find the value of the expression \( \frac{a - b}{a^{4} - b^{4}} \).


The sum of the digits of the birth years of Jean and Jack are equal, and the age of each of them starts with the same digit. Can you determine the difference in their ages?


Four students from Harvard, one of them named Jack, and five students from MIT, one of them named Jill, are going to see a Boston Celtics game. However, they found out that only 5 tickets remain, so 4 of them must go back. Suppose that at least one student from each school must go see the game, and at least one of Jack and Jill must go see the game. How many ways are there of choosing which 5 people can see the game?


A rectangle has the same perimeter and area as a rhombus, one of whose angles measures $30^{\circ}$. What is the ratio of the sides of the rectangle?

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


A small village has \( n \) people. During their yearly elections, groups of three people come up to a stage and vote for someone in the village to be the new leader. After every possible group of three people has voted for someone, the person with the most votes wins. This year, it turned out that everyone in the village had the exact same number of votes! If \( 10 \leq n \leq 100 \), what is the number of possible values of \( n \)?


In the figure, \( AB \) is tangent to the circle at point \( A \), \( BC \) passes through the center of the circle, and \( CD \) is a chord of the circle that is parallel to \( AB \). If \( AB = 6 \) and \( BC = 12 \), what is the length of \( CD \)?

Solution: \boxed{7.2 \; \text{units}}


Describe all positive integer solutions $(m, n)$ of the equation $8m - 7 = n^2$ and provide the first value of $m$ (if it exists) greater than 1959.


Calculate the limit of the numerical sequence:

$$
\lim _{n \rightarrow \infty} \frac{\sqrt{3 n-1}-\sqrt[3]{125 n^{3}+n}}{\sqrt[3]{n}-n}
$$


A pentagon is inscribed around a circle, with the lengths of its sides being whole numbers, and the lengths of the first and third sides equal to 1. Into what segments does the point of tangency divide the second side?


Given the function \( f(x) = \frac{m - 2 \sin x}{\cos x} \) is monotonically decreasing on the interval \(\left(0, \frac{\pi}{2} \right)\), find the range of the real number \( m \).

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


Calculate the area of a pentagon whose vertices have the coordinates: \((-1, 2)\), \((1, 1)\), \((2, 3)\), \((3, -5)\), and \((-2, -4)\).


The sum of the positive numbers \(a, b, c,\) and \(d\) is 4. Find the minimum value of the expression

$$
\frac{a^{8}}{\left(a^{2}+b\right)\left(a^{2}+c\right)\left(a^{2}+d\right)}+\frac{b^{8}}{\left(b^{2}+c\right)\left(b^{2}+d\right)\left(b^{2}+a\right)}+\frac{c^{8}}{\left(c^{2}+d\right)\left(c^{2}+a\right)\left(c^{2}+b\right)}+\frac{d^{8}}{\left(d^{2}+a\right)\left(d^{2}+b\right)\left(d^{2}+c\right)}
$$


A positive number \( x \) was increased by 69%. By what percentage did the number \( \sqrt{\frac{x}{5}} \) increase?


Circles \(\omega_{1}\) and \(\omega_{2}\) intersect at points \(A\) and \(B\), and a circle centered at point \(O\) encompasses circles \(\omega_{1}\) and \(\omega_{2}\), touching them at points \(C\) and \(D\) respectively. It turns out that points \(A, C\), and \(D\) are collinear. Find the angle \(\angle ABO\).


Find the minimum number of planes that divide a cube into at least 300 parts.


For any positive integer \( n \), if
$$
1^{n} + 2^{n} + \cdots + (n-1)^{n} < C n^{n},
$$
then the minimum value of \( C \) is ______.

Solution: \boxed{ \frac{1}{e-1} }


Find all odd natural numbers greater than 500 but less than 1000, for which the sum of the last digits of all divisors (including 1 and the number itself) is equal to 33.


In an eight-digit number, each digit (except the last one) is greater than the following digit. How many such numbers are there?


Given positive integers \( m \), \( n \), and \( r \), where \( 1 \leq r \leq m \leq n \). An \( m \times n \) grid has a set of \( m \) squares that form a generalized diagonal (or simply diagonal) if each pair of squares in the set are in different rows and different columns.

Consider coloring some squares red on an \( m \times n \) grid such that each row and each column contains at most \( r \) red squares. Determine the minimum value of the positive integer \( a \) such that for any coloring scheme, it is possible to find \( a \) diagonals on the grid such that all the red squares are on these diagonals.

Solution: \boxed{r}


What is the smallest natural number $n$ for which there exist natural numbers $x$ and $y$ satisfying the equation a) $x \cdot (x+n) = y^{2}$; b) $x \cdot (x+n) = y^{3}$?

Solution: \boxed{n = 2}


In the bottom-left corner of a chessboard, there is a checker piece. It can be moved one square up, one square to the right, or one square diagonally down-left. Is it possible to move the checker in such a way that it visits all the squares on the board exactly once? (7 points)

Solution: \boxed{\text{No}}


Point \( M \) is the midpoint of side \( BC \) of the triangle \( ABC \), where \( AB = 17 \), \( AC = 30 \), and \( BC = 19 \). A circle is constructed with diameter \( AB \). A point \( X \) is chosen arbitrarily on this circle. What is the minimum possible length of the segment \( MX \)?