Let \(\alpha\), \(\beta\), and \(\gamma\) be real numbers satisfying \(\alpha + \beta + \gamma = 2\) and \(\alpha \beta \gamma = 4\). Let \(v\) be the minimum value of \(|\alpha| + |\beta| + |\gamma|\). Find the value of \(v\).


Two positive integers are written on the blackboard, one being 2002 and the other being a number less than 2002. If the arithmetic mean of the two numbers is an integer $m$, then the following operation can be performed: one of the numbers is erased and replaced by $m$. What is the maximum number of times this operation can be performed?


Find the pairs of positive integers \((x, y)\) that satisfy \(x > y\) and \((x - y)^{xy} = x^y y^x\).

Solution: \boxed{(4, 2)}


Let \( a, b \), and \( c \) be positive real numbers. Determine the largest total number of real roots that the following three polynomials may have among them: \( ax^2 + bx + c \), \( bx^2 + cx + a \), and \( cx^2 + ax + b \).


In city $\mathrm{N}$, there are exactly three monuments. One day, a group of 42 tourists arrived in this city. Each tourist took no more than one photograph of each of the three monuments. It turned out that any two tourists together had photographs of all three monuments. What is the minimum number of photographs that all the tourists together could have taken?


Let \(ABCD\) be a quadrilateral such that \(AD = BC\), \( (AB) \) and \( (CD) \) are parallel, and \(AB > CD\). Let \(E\) be the midpoint of \([AC]\), and \(F\) the point of intersection of the diagonals \((AC)\) and \((BD)\). The line parallel to \((BD)\) passing through \(E\) intersects the line \((CD)\) at point \(G\).

1. Show that triangle \(CGA\) is a right triangle at \(G\).
2. Let \(CD = b\) and \(AB = a\). Calculate the ratio \(\frac{EG}{CF}\) as a function of \(a\) and \(b\).

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


Let \( R \) be the circumradius of \( \triangle ABC \), \( r \) be the inradius of \( \triangle ABC \), and \( d \) be the distance between the circumcenter \( O \) and the incenter \( I \). Then \( d^{2} = R^{2} - 2Rr \).

Solution: \boxed{d^2 = R^2 - 2 R r}


Given \(\theta = \arctan \frac{5}{12}\), find the principal value of the argument of the complex number \(z = \frac{\cos 2\theta + i \sin 2\theta}{239 + i}\).


Using the digits 1 to 5 and a multiplication sign $\times$, Clara forms the product of two numbers with the $\times$ sign between them. How should Clara arrange the digits to obtain the largest possible product?


Count the paths composed of \( n \) rises and \( n \) descents of the same amplitude.


Find all three-digit numbers that are five times the product of their digits.


Alberto, Bernardo, and Carlos are collectively listening to three different songs. Each is simultaneously listening to exactly two songs, and each song is being listened to by exactly two people. In how many ways can this occur?


16 progamers are playing in a single elimination tournament. Each round, each of the remaining progamers plays against another and the loser is eliminated. Each time a progamer wins, he will have a ceremony to celebrate. A player's first ceremony is ten seconds long, and each subsequent ceremony is ten seconds longer than the previous one. What is the total length in seconds of all the ceremonies over the entire tournament?


The plane figure $W$ is the set of all points whose coordinates $(x, y)$ satisfy the inequality: $(|x| + |4 - |y|| - 4)^{2} \leqslant 4$. Draw the figure $W$ and find its area.


Given the set \( A = \{0, 1, 2, \cdots, 9\} \), and a collection of non-empty subsets \( B_1, B_2, \cdots, B_k \) of \( A \) such that for \( i \neq j \), \( B_i \cap B_j \) has at most two elements, determine the maximum value of \( k \).


In a finite sequence of real numbers, the sum of any 7 consecutive terms is negative, while the sum of any 11 consecutive terms is positive. What is the maximum number of terms this sequence can have?


Let \( f_{n+1} = \left\{ \begin{array}{ll} f_n + 3 & \text{if } n \text{ is even} \\ f_n - 2 & \text{if } n \text{ is odd} \end{array} \right. \).
If \( f_1 = 60 \), determine the smallest possible value of \( n \) satisfying \( f_m \geq 63 \) for all \( m \geq n \).


A positive integer \( n \) is picante if \( n! \) (n factorial) ends in the same number of zeroes whether written in base 7 or in base 8. How many of the numbers \( 1, 2, \ldots, 2004 \) are picante?


A square is inscribed in another square such that its vertices lie on the sides of the first square, and its sides form angles of $60^{\circ}$ with the sides of the first square. What fraction of the area of the given square is the area of the inscribed square?

Solution: \boxed{4 - 2\sqrt{3}}


100 bulbs were hung in a row on the New Year's tree. Then the bulbs began to switch according to the following algorithm: all bulbs were lit initially, then after one second, every second bulb was turned off, and after another second, every third bulb switched: if it was on, it was turned off, and if it was off, it was turned on. Again, after one second, every fourth bulb switched, after another second - every fifth bulb, and so on. This procedure continued for 100 seconds. Determine the probability that a randomly chosen bulb is lit after all this (bulbs do not burn out or break).


Determine the maximum value of the sum
$$
\sum_{i<j} x_{i} x_{j}\left(x_{i}+x_{j}\right)
$$
over all \( n \)-tuples \(\left(x_{1}, \ldots, x_{n}\right)\), satisfying \( x_{i} \geq 0 \) and \(\sum_{i=1}^{n} x_{i} = 1 \).


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


There are a few integers \( n \) such that \( n^{2}+n+1 \) divides \( n^{2013}+61 \). Find the sum of the squares of these integers.


Egor wrote a number on the board and encoded it according to the rules of letter puzzles (different letters correspond to different digits, the same letters to the same digits). The result was the word "ГВАТЕМАЛА". How many different numbers could Egor have originally written if his number was divisible by 30?


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


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


Solution: \boxed{(6, 1)}


$V, W, X, Y, Z$ are 5 digits in base-5. The three base-5 three-digit numbers $(V Y Z)_{5}, (V Y X)_{5}, (V V W)_{5}$ form an arithmetic sequence with a common difference of 1. In decimal, what is the value of the three-digit number $(X Y Z)_{5}$?


Let \( F \) be the number of integral solutions of \( x^2 + y^2 + z^2 + w^2 = 3(x + y + z + w) \). Find the value of \( F \).


A sphere of radius 1 is covered in ink and rolling around between concentric spheres of radii 3 and 5. If this process traces a region of area 1 on the larger sphere, what is the area of the region traced on the smaller sphere?


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


Determine the highest natural power of 2007 that divides 2007!.


How many solutions does the equation \(\left|\left| |x-1| - 1 \right| - 1 \right| = 1\) have? The modulus function \( |x| \) evaluates the absolute value of a number; for example \( |6| = |-6| = 6 \).


Each of the 33 warriors either always lies or always tells the truth. It is known that each warrior has exactly one favorite weapon: a sword, a spear, an axe, or a bow.

One day, Uncle Chernomor asked each warrior four questions:
- Is your favorite weapon a sword?
- Is your favorite weapon a spear?
- Is your favorite weapon an axe?
- Is your favorite weapon a bow?

13 warriors answered "yes" to the first question, 15 warriors answered "yes" to the second question, 20 warriors answered "yes" to the third question, and 27 warriors answered "yes" to the fourth question.

How many of the warriors always tell the truth?


Four cars, \( A, B, C, \) and \( D \) start simultaneously from the same point on a circular track. \( A \) and \( B \) drive clockwise, while \( C \) and \( D \) drive counterclockwise. All cars move at constant (but pairwise different) speeds. Exactly 7 minutes after the race begins, \( A \) meets \( C \) for the first time, and at the same moment, \( B \) meets \( D \) for the first time. After another 46 minutes, \( A \) and \( B \) meet for the first time. After how much time from the start of the race will \( C \) and \( D \) meet for the first time?


The teacher invented a puzzle by replacing the example \(a + b = c\) with the addition of two natural numbers with their digits replaced by letters: identical digits are replaced by identical letters, and different digits by different letters. (For example, if \(a = 23\), \(b = 528\), then \(c = 551\), resulting in the puzzle \(A B + V A \Gamma = VVD\) precisely up to the choice of letters). It turned out that the original example can be uniquely restored from the given puzzle. Find the smallest possible value of the sum \(c\).


There are coins with values of 1, 2, 3, and 5 cruzeiros, each valued at their respective weights in grams (1, 2, 3, and 5 grams). It has been found that one of them is fake and differs in weight from the normal (but it is not known whether it is heavier or lighter than the real one). How can the fake coin be identified with the fewest weighings on a balance scale without weights?


\( f(n) \) is defined on the set of positive integers, and satisfies the following conditions:
1. For any positive integer \( n \), \( f[f(n)] = 4n + 9 \).
2. For any non-negative integer \( k \), \( f(2^k) = 2^{k+1} + 3 \).

Determine \( f(1789) \).


Four points, \( A, B, C, \) and \( D \), are chosen randomly on the circumference of a circle with independent uniform probability. What is the expected number of sides of triangle \( ABC \) for which the projection of \( D \) onto the line containing the side lies between the two vertices?


Determine, with proof, the value of \(\log _{2} 3 \log _{3} 4 \log _{4} 5 \ldots \log _{255} 256\).


How many necklaces can be made from five white beads and two black beads?


Solution: \boxed{a = -10}


There are knights, liars, and followers living on an island; each knows who is who among them. All 2018 islanders were arranged in a row and asked to answer "Yes" or "No" to the question: "Are there more knights on the island than liars?". They answered in turn such that everyone else could hear. Knights told the truth, liars lied. Each follower gave the same answer as the majority of those who answered before them, and if "Yes" and "No" answers were equal, they gave either answer. It turned out that the number of "Yes" answers was exactly 1009. What is the maximum number of followers that could have been among the islanders?


Given a sequence $\left\{a_{n}\right\}$ where all terms are positive and the sum of the first $n$ terms $S_{n}$ satisfies $6 S_{n} = a_{n}^{2} + 3 a_{n} + 2$. If $a_{2}, a_{4}, a_{9}$ form a geometric sequence, find the general formula for the sequence.

Solution: \boxed{a_n = 3n - 2}


How many ordered quadruples \((a, b, c, d)\) of four distinct numbers chosen from the set \(\{1, 2, 3, \ldots, 9\}\) satisfy \(b<a\), \(b<c\), and \(d<c\)?


There are 2005 points inside $\triangle ABC$ that are not collinear, plus the 3 vertices of $\triangle ABC$, making a total of 2008 points. How many non-overlapping small triangles can be formed by connecting these points? $\qquad$


Given the sequence \(\{a_n\}\) which satisfies \(a_1 = 1, a_2 = 2, a_{n+1} - 3a_n + 2a_{n-1} = 1\) (for \(n \geq 2\), \(n \in \mathbf{N}^*\)), find the general formula for \(\{a_n\}\).

Solution: \boxed{2^n - n}


The probability that a purchased light bulb will work is 0.95. How many light bulbs need to be purchased so that with a probability of 0.99 there will be at least 5 working ones?


The numbers 1, 2, 3, 4, 5, 6, 7 are written in a circle in some order. A number is called "good" if it is equal to the sum of the two numbers written next to it. What is the maximum possible number of "good" numbers among those written?


A positive integer cannot be divisible by 2 or 3, and there do not exist non-negative integers \(a\) and \(b\) such that \(|2^a - 3^b| = n\). Find the smallest value of \(n\).


Rachel does jumps, each of which is \(168 \mathrm{~cm}\) long. Joel does jumps, each of which is \(120 \mathrm{~cm}\) long. Mark does jumps, each of which is \(72 \mathrm{~cm}\) long.

(a) On Monday, Rachel completes 5 jumps and Joel completes \(n\) jumps. Rachel and Joel jump the same total distance. Determine the value of \(n\).

(b) On Tuesday, Joel completes \(r\) jumps and Mark completes \(t\) jumps. Joel and Mark jump the same total distance, and \(r\) and \(t\) are integers. If \(11 \leq t \leq 19\), determine the values of \(r\) and \(t\).

(c) On Wednesday, Rachel completes \(a\) jumps, Joel completes \(b\) jumps, and Mark completes \(c\) jumps. Each of \(a, b\) and \(c\) is a positive integer, and Rachel, Joel and Mark each jump the same total distance. Determine the minimum possible value of \(c\) and explain why this value is indeed the minimum.

Solution: \boxed{c = 35}


Solution: \boxed{\text{No}}


The teacher fills some numbers into the circles in the diagram (each circle can and must contain only one number). The sum of the three numbers in each of the left and right closed loops is 30, and the sum of the four numbers in each of the top and bottom closed loops is 40. If the number in circle \(X\) is 9, what is the number in circle \(Y\)?


In a theater performance of King Lear, the locations of Acts II-V are drawn by lot before each act. The auditorium is divided into four sections, and the audience moves to another section with their chairs if their current section is chosen as the next location. Assume that all four sections are large enough to accommodate all chairs if selected, and each section is chosen with equal probability. What is the probability that the audience will have to move twice compared to the probability that they will have to move only once?


Find the maximum value of the expression

$$
A=\frac{1}{\sin ^{6} \alpha+\cos ^{6} \alpha} \text { for } 0 \leq \alpha \leq \frac{\pi}{2}
$$


Does there exist an irrational number \(\alpha > 1\) such that
$$
\left\lfloor \alpha^n \right\rfloor \equiv 0 \pmod{2017}
$$
for all integers \(n \geq 1\)?

Solution: \boxed{\text{Yes}}


Given real numbers \( x \) and \( y \) that satisfy \[ \frac{x^{2}}{9}+\frac{y^{2}}{4}=1 \], find the maximum value of the function \( U = x + y \).


$$
\frac{\cos ^{2}\left(\frac{5 \pi}{4}-2 \alpha\right)-\sin ^{2}\left(\frac{5 \pi}{4}-2 \alpha\right)}{\left(\cos \frac{\alpha}{2}+\sin \frac{\alpha}{2}\right)\left(\cos \left(2 \pi-\frac{\alpha}{2}\right)+\cos \left(\frac{\pi}{2}+\frac{\alpha}{2}\right)\right) \sin \alpha}
$$

Solution: \boxed{4 \cos 2 \alpha}


When a two-digit number is multiplied by a three-digit number, a four-digit number of the form \( A = \overline{abab} \) is obtained. Find the largest \( A \), given that \( A \) is divisible by 14.


Natural numbers \(a, x\), and \(y\), each greater than 100, are such that \(y^2 - 1 = a^2 (x^2 - 1)\). What is the smallest possible value of the fraction \(a / x\)?


Two swimmers are training in a rectangular quarry. The first swimmer prefers to start at the corner of the quarry, so he swims diagonally to the opposite corner and back. The second swimmer prefers to start at a point that divides one of the sides of the quarry in the ratio \(2018: 2019\). He swims along a quadrilateral, visiting one point on each side, and returns to the starting point. Can the second swimmer choose points on the other three sides in such a way that his path is shorter than the first swimmer's? What is the minimum value that the ratio of the length of the longer path to the shorter one can have?


The perimeter of triangle \( ABC \) is 1. Circle \( \omega \) is tangent to side \( BC \) and the extensions of side \( AB \) at point \( P \) and side \( AC \) at point \( Q \). The line passing through the midpoints of \( AB \) and \( AC \) intersects the circumcircle of triangle \( APQ \) at points \( X \) and \( Y \). Find the length of segment \( XY \).


Hooligan Vasily tore out an entire chapter from a book, with the first page numbered 231, and the number of the last page consisted of the same digits. How many sheets did Vasily tear out of the book?


At a ball, there are \(n\) married couples. In each couple, the husband and wife are of the exact same height, but no two couples share the same height. When the waltz begins, all attendees randomly pair up: each man dances with a randomly chosen woman. Find the expected value of the random variable \(X\), which represents the number of men who are shorter than their partners.


Let \( P \) be any point on the ellipse \(\frac{x^{2}}{16}+\frac{y^{2}}{9}=1\) other than the endpoints of the major axis, \(F_{1}\) and \(F_{2}\) be the left and right foci respectively, and \(O\) be the center. Then \(\left|P F_{1}\right| \cdot \left|P F_{2}\right| + |O P|^{2} = \, \underline{\hspace{2cm}}\).


Let the line \( y = a \) intersect the curve \( y = \sin x \) (for \( 0 \leqslant x \leqslant \pi \)) at points \( A \) and \( B \). If \( |AB| = \frac{\pi}{5} \), find the value of \( a \) (accurate to 0.0001).


Rózsa, Ibolya, and Viola have decided to solve all the problems in their problem collection. Rózsa solves \(a\), Ibolya \(b\), and Viola \(c\) problems daily. (Only one of them works on a problem.) If Rózsa solved 11 times, Ibolya 7 times, and Viola 9 times more problems daily, they would finish in 5 days; whereas if Rózsa solved 4 times, Ibolya 2 times, and Viola 3 times more problems daily, they would finish in 16 days. How many days will it take for them to complete the solutions?


For how many one-digit positive integers \( k \) is the product \( k \cdot 234 \) divisible by 12?


Sergey wrote down a certain five-digit number and multiplied it by 9. To his surprise, he obtained a number consisting of the same digits, but in reverse order. What number did Sergey write down?


What is the maximum length of the segment intercepted by the sides of a triangle on the tangent to the inscribed circle, drawn parallel to the base, if the perimeter of the triangle is $2p$?


A square with a side length of 36 cm was cut into three rectangles in such a way that the areas of all three rectangles are equal and any two rectangles have a common section of the boundary. What is the total length (in cm) of the made cuts?

Solution: \boxed{60 \text{ cm}}


Show that for any sequence \( a_0, a_1, \ldots, a_n, \ldots \) consisting of ones and minus ones, there exist such \( n \) and \( k \) such that

\[ |a_0 a_1 \cdots a_k + a_1 a_2 \cdots a_{k+1} + \cdots + a_n a_{n+1} \cdots a_{n+k}| = 2017. \]

Solution: \boxed{S = 2017}


Let \( A \), \( B \), and \( C \) be pairwise independent events with equal probabilities, and \( A \cap B \cap C = \varnothing \). Find the maximum possible value for the probability \( \mathrm{P}(A) \).


In triangle \( \triangle ABC \), \( M \) is the midpoint of side \( BC \), and \( N \) is the midpoint of segment \( BM \). Given \( \angle A = \frac{\pi}{3} \) and the area of \( \triangle ABC \) is \( \sqrt{3} \), find the minimum value of \( \overrightarrow{AM} \cdot \overrightarrow{AN} \).

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


Two spheres of one radius and two spheres of another radius are arranged so that each sphere touches three other spheres and a given plane. Find the ratio of the radii of the spheres.

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


Find the point \( M' \) symmetric to the point \( M \) with respect to the plane.

\( M(-1, 0, 1) \)

\(2x + 4y - 3 = 0\)

Solution: \boxed{(0, 2, 1)}


The brother left home 6 minutes later than his sister, following her, and caught up with her in 12 minutes. How many minutes would it take him to catch up with her if he walked twice as fast? Both the brother and sister walk at a constant speed.

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


Let \( F_{1} \) and \( F_{2} \) be the left and right foci of the hyperbola \( C: x^{2} - \frac{y^{2}}{24} = 1 \) respectively, and let \( P \) be a point on the hyperbola \( C \) in the first quadrant. If \( \frac{\left|P F_{1}\right|}{\left|P F_{2}\right|} = \frac{4}{3} \), find the radius of the inscribed circle of the triangle \( P F_{1} F_{2} \).


Let \( S \) be the set of points whose coordinates \( x \), \( y \), and \( z \) are integers that satisfy \( 0 \leq x \leq 2 \), \( 0 \leq y \leq 3 \), and \( 0 \leq z \leq 4 \). Two distinct points are randomly chosen from \( S \). Find the probability that the midpoint of the two chosen points also belongs to \( S \).


On side \(AB\) of an equilateral triangle \(ABC\), a point \(K\) is chosen, and on side \(BC\), points \(L\) and \(M\) are chosen such that \(KL = KM\) and point \(L\) is closer to \(B\) than \(M\).

a) Find the angle \(MKA\) if it is known that \(\angle BKL = 10^\circ\).

b) Find \(MC\) if \(BL = 2\) and \(KA = 3\).

Provide justification for each part of the answers.


Calculate the limit of the function:

$\lim _{x \rightarrow 0}\left(\operatorname{tg}\left(\frac{\pi}{4}-x\right)\right)^{\left(e^{x}-1\right) / x}$


A train passes a post in 5 seconds. In how many seconds will the train and an electric train pass by each other, if the speed of the train is 2 times the speed of the electric train, and the length of the train is 3 times the length of the electric train?


Find all integer values of the parameter \(a\) for which the system has at least one solution:

$$
\left\{\begin{array}{l}
y - 2 = x(x + 2) \\
x^{2} + a^{2} + 2x = y(2a - y)
\end{array}\right.
$$

In the answer, indicate the sum of the found values of parameter \(a\).


In a regular 1976-gon, the midpoints of all sides and the midpoints of all diagonals are marked. What is the maximum number of marked points that lie on one circle?


Given the function $f(x)=\frac{1}{3} x^{3}-(a-1) x^{2}+b^{2} x$, where $a \in\{1,2,3,4\}$ and $b \in\{1,2,3\}$, find the probability that the function $f(x)$ is an increasing function on $\mathbf{R}$.


Calculate the area enclosed by the cardioid $\rho = 1 + \cos \varphi$ for $0 \leqslant \varphi \leqslant 2\pi$.


Let $\{a_n\}$ be a sequence of positive numbers with $b_n$ representing the sum of the first $n$ terms of $\{a_n\}$. The product of the first $n$ terms of the sequence $\{b_n\}$ is denoted by $c_n$, and $b_n + c_n = 1$. What is the number in the sequence $\left\{\frac{1}{a_n}\right\}$ that is closest to 2009?


Two circles of radius 12 have their centers on each other. \( A \) is the center of the left circle, and \( AB \) is a diameter of the right circle. A smaller circle is constructed tangent to \( AB \) and the two given circles, internally to the right circle and externally to the left circle. Find the radius of the smaller circle.


Let $a_{1}, a_{2}, a_{3}, a_{4}, a_{5}$ be a permutation of the numbers $1, 2, 3, 4, 5$. If there does not exist $1 \leq i < j < k \leq 5$ such that $a_{i} < a_{j} < a_{k}$, then the number of such permutations is ______.


A quadrilateral is inscribed in a circle with a radius of 13. The diagonals of the quadrilateral are perpendicular to each other. One of the diagonals is 18, and the distance from the center of the circle to the point where the diagonals intersect is \( 4 \sqrt{6} \).

Find the area of the quadrilateral.

Solution: \boxed{18 \sqrt{161}}


Find the greatest common divisor (GCD) of all the numbers of the form \( n^{13} - n \).


Vasya has three cans of paint of different colors. In how many different ways can he paint a fence consisting of 10 planks so that any two adjacent planks are different colors and he uses all three colors? Provide a justification for your answer.


A weight supported by a helical spring is lifted by a distance $b$ and then released. It begins to fall, with its acceleration described by the equation $g = -p^2 s$, where $p$ is a constant and $s$ is the distance from the equilibrium position. Find the equation of motion.

Solution: \boxed{s(t) = b \cos(pt)}


Suppose \( f \) is a function that satisfies \( f(2) = 20 \) and \( f(2n) + n f(2) = f(2n+2) \) for all positive integers \( n \). What is the value of \( f(10) \)?


Given \( S = \frac{1}{9} + \frac{1}{99} + \frac{1}{999} + \cdots + \frac{1}{\text{1000 nines}} \), what is the 2016th digit after the decimal point in the value of \( S \)?


On the median \(A A_{1}\) of triangle \(A B C\), a point \(M\) is taken such that \(A M : M A_{1} = 1 : 3\). In what ratio does the line \(B M\) divide the side \(A C\)?


In rectangle \( ABCD \), a circle \(\omega\) is constructed using side \( AB \) as its diameter. Let \( P \) be the second intersection point of segment \( AC \) with circle \(\omega\). The tangent to \(\omega\) at point \( P \) intersects segment \( BC \) at point \( K \) and passes through point \( D \). Find \( AD \), given that \( KD = 36 \).


Given some triangles with side lengths \(a \,\text{cm}, 2 \,\text{cm}\) and \(b \,\text{cm}\), where \(a\) and \(b\) are integers and \(a \leq 2 \leq b\). If there are \(q\) non-congruent classes of triangles satisfying the above conditions, find the value of \(q\).


Given a regular tetrahedron \(P-ABC\) with edge length 1, let \(E\), \(F\), \(G\), and \(H\) be points on edges \(PA\), \(PB\), \(CA\), and \(CB\) respectively. If \(\overrightarrow{EF}+\overrightarrow{GH}=\overrightarrow{AB}\) and \(\overrightarrow{EH} \cdot \overrightarrow{FG}=\frac{1}{18}\), then find \(\overrightarrow{EG} \cdot \overrightarrow{FH}\).


$\bigcirc \bigcirc \div \square=14 \cdots 2$, how many ways are there to fill the square?


Grandmother has apples of three varieties in her garden: Antonovka, Grushovka, and Bely Naliv. If the Antonovka apples were tripled, the total number of apples would increase by 70%. If the Grushovka apples were tripled, the total number of apples would increase by 50%. By what percentage would the total number of apples change if the Bely Naliv apples were tripled?


The sequence \(a_{0}, a_{1}, a_{2}, \cdots, a_{n}\) satisfies \(a_{0}=\sqrt{3}\), \(a_{n+1}=\left\lfloor a_{n}\right\rfloor + \frac{1}{\left\{a_{n}\right\}}\), where \(\left\lfloor a_{n}\right\rfloor\) and \(\left\{a_{n}\right\}\) represent the integer part and fractional part of \(a_{n}\), respectively. Find \(a_{2016}\).


Compute the number of quadruples \((a, b, c, d)\) of positive integers satisfying
\[ 12a + 21b + 28c + 84d = 2024. \]


Positive integers \( a, b \), and \( c \) have the property that \( a^b \), \( b^c \), and \( c^a \) end in 4, 2, and 9, respectively. Compute the minimum possible value of \( a + b + c \).


Find the sum of the first twelve terms of an arithmetic sequence if its fifth term $a_{5} = 1$ and its seventeenth term $a_{17} = 18$.


There are 700 cards in a box, in six colors: red, orange, yellow, green, blue, and white. The ratio of the number of red, orange, and yellow cards is $1: 3: 4$, and the ratio of the number of green, blue, and white cards is $3:1:6$. Given that there are 50 more yellow cards than blue cards, determine the minimum number of cards that must be drawn to ensure that there are at least 60 cards of the same color among the drawn cards.


It is known that \( A \) is the largest number that is the product of several natural numbers whose sum is 2011.

What is the largest power of three that divides \( A \)?


A \(10 \times 1\) rectangular pavement is to be covered by tiles which are either green or yellow, each of width 1 and of varying integer lengths from 1 to 10. Suppose you have an unlimited supply of tiles for each color and for each of the varying lengths. How many distinct tilings of the rectangle are there, if at least one green and one yellow tile should be used, and adjacent tiles should have different colors?


Let \( a, b, c \) be pairwise distinct positive integers such that \( a+b, b+c \) and \( c+a \) are all square numbers. Find the smallest possible value of \( a+b+c \).


In the tetrahedron $ABCD$, the angles $ABD$ and $ACD$ are obtuse. Show that $AD > BC$.

Solution: \boxed{AD > BC}


In triangle \(ABC\), if \(\cos^{2} A + \cos^{2} B + \cos^{2} C = \sin^{2} B\), then \(\tan A \tan C =\)


What is the smallest possible perimeter of a triangle whose side lengths are all squares of distinct positive integers?


As shown in the diagram, the abacus has three sections, each with 10 beads. By dividing the beads in each section into top and bottom parts, two three-digit numbers are formed. The requirement is that the three-digit number in the top part must have distinct digits and be a multiple of the three-digit number in the bottom part. Determine the three-digit number in the top part.


The sum of four numbers in a geometric progression is -40, and the sum of their squares is 3280. Find this progression.

Solution: \boxed{2, -6, 18, -54}


The number of positive integers \( k \) such that there exists a pair of positive integers \( (a, b) \) satisfying
$$
\frac{k}{2015}(a+b)=\text{lcm}(a, b)
$$
where \(\text{lcm}(a, b)\) denotes the least common multiple of \(a\) and \(b\), is ______.


Let a three-digit number \( n = \overline{abc} \), where \( a \), \( b \), and \( c \) can form an isosceles (including equilateral) triangle as the lengths of its sides. How many such three-digit numbers \( n \) are there?


A shopkeeper wants to sell his existing stock of a popular item at one and a half times its current price. To achieve this, he plans to implement a phased price increase. He starts with a certain price increase, and when two-thirds of the stock has been sold, he raises the price again by the same percentage. This new price remains in effect until the entire stock is sold. What percentage price increases are needed to achieve the desired goal?


For a positive integer \( n \), let the sum of its digits be denoted as \( s(n) \), and the product of its digits be denoted as \( p(n) \). If \( s(n) + p(n) = n \) holds true, then \( n \) is called a "magic number." Find the sum of all magic numbers.


A circle is tangent to the extensions of two sides \(AB\) and \(AD\) of the square \(ABCD\), and the points of tangency cut off segments of length 4 cm from vertex \(A\). From point \(C\), two tangents are drawn to this circle. Find the side length of the square if the angle between the tangents is \(60^\circ\).


Knowing that segment \( CD \) has a length of 6 and its midpoint is \( M \), two triangles \( \triangle ACD \) and \( \triangle BCD \) are constructed on the same side with \( CD \) as a common side and both having a perimeter of 16, such that \( \angle AMB = 90^{\circ} \). Find the minimum area of \( \triangle AMB \).


Given that for any \( x \in [0,1] \), the inequality
\[ f(x) = k \left( x^2 - x + 1 \right) - x^4 (1 - x)^4 \geqslant 0, \]
holds, find the minimum value of \( k \).


In the diagram, there is a map consisting of 8 cities connected by 12 roads. How many ways are there to close 5 roads simultaneously for repairs such that it is still possible to travel from any city to any other city?


In an isosceles trapezoid, the bases are 40 and 24, and its diagonals are mutually perpendicular. Find the area of the trapezoid.


Compute the limit of the function:

$$
\lim _{x \rightarrow 0} \frac{\cos (1+x)}{\left(2+\sin \left(\frac{1}{x}\right)\right) \ln (1+x)+2}
$$

Solution: \boxed{\frac{\cos 1}{2}}


There are four individuals: Jia, Yi, Bing, and Ding. It is known that the average age of Jia, Yi, and Bing is 1 year more than the average age of all four people. The average age of Jia and Yi is 1 year more than the average age of Jia, Yi, and Bing. Jia is 4 years older than Yi, and Ding is 17 years old. How old is Jia?


Let \( x \) be a non-zero real number such that \( x + \frac{1}{x} = \sqrt{2019} \). What is the value of \( x^{2} + \frac{1}{x^{2}} \)?


Aunt Martha bought some nuts. She gave Tommy one nut and a quarter of the remaining nuts, Bessie received one nut and a quarter of the remaining nuts, Bob also got one nut and a quarter of the remaining nuts, and finally, Jessie received one nut and a quarter of the remaining nuts. It turned out that the boys received 100 more nuts than the girls.

How many nuts did Aunt Martha keep for herself?


What is the smallest number of triangular pyramids (tetrahedrons) into which a cube can be divided?


Egor borrowed 28 rubles from Nikita and then repaid them in four payments. It turned out that Egor always returned a whole number of rubles, and each payment amount always increased and was exactly divisible by the previous one. What amount did Egor pay back the last time?


Determine the number of integers \( n \) such that \( 1 \leq n \leq 10^{10} \), and for all \( k = 1, 2, \ldots, 10 \), the integer \( n \) is divisible by \( k \).


If Xiao Zhang's daily sleep time is uniformly distributed between 6 hours and 9 hours, what is the probability that Xiao Zhang's average sleep time over two consecutive days is at least 7 hours?


How many ordered pairs $(A, B)$ are there where $A$ and $B$ are subsets of a fixed $n$-element set and $A \subseteq B$?


Compute the limit of the numerical sequence:

$$\lim _{n \rightarrow \infty} \frac{2-5+4-7+\ldots+2 n-(2 n+3)}{n+3}$$


Some of the 20 metal cubes, which are identical in size and appearance, are made of aluminum, and the rest are made of duralumin (which is heavier). How can you determine the number of duralumin cubes using 11 weighings on a balance scale without weights?


Let \( f(m) \) be the product of the digits of the positive integer \( m \). Find the positive integer solutions to the equation \( f(m) = m^2 - 10m - 36 \).


The apex of a regular pyramid with a square base $ABCD$ of unit side length is $E$. Point $P$ lies on the base edge $AB$ and point $Q$ lies on the lateral edge $EC$ such that $PQ$ is perpendicular to both $AB$ and $EC$. Additionally, we know that $AP : PB = 6 : 1$. What are the lengths of the lateral edges?


What is the condition for two medians of a triangle to be perpendicular to each other?

Solution: \boxed{a^2 + b^2 = 5c^2}


Let \( S = \{1, 2, \ldots, 2013\} \). Find the number of ordered triples \((A, B, C)\) of subsets of \(S\) such that \(A \subseteq B\) and \(A \cup B \cup C = S\).


50 students with blond hair, brown hair, and red hair are sitting around a round table. It is known that in any group of students sitting consecutively, there is at least one student with brown hair between any two students with blond hair, and at least one student with red hair between any two students with brown hair. What is the minimum number of students with red hair that can be sitting at this table?


In triangle \(ABC\), the side \(AC\) is longer than \(AB\). The line \(l\) is the bisector of the external angle at \(C\). A line passing through the midpoint of \(AB\) and parallel to \(l\) intersects \(AC\) at point \(E\). Find \(CE\) if \(AC = 7\) and \(CB = 4\). (The external angle of a triangle is the angle adjacent to the internal angle at a given vertex.)


What is the highest power of 3 that divides the number whose decimal representation consists of $3^n$ ones?


The numbers \( a, b, c, d \) belong to the interval \([-13.5, 13.5]\). Find the maximum value of the expression \( a + 2b + c + 2d - ab - bc - cd - da \).


Given the sequence \(\left\{a_{n}\right\}\), satisfy the equation \(\frac{a_{n+1}+a_{n}-1}{a_{n+1}-a_{n}+1}=n\) for \(n \in \mathbf{N}^{*}\), and \(a_{2}=6\).


We have \( n \geq 3 \) points in the plane. We suppose that the area of any triangle formed by 3 of these points does not exceed 1.

Show that all these points can be placed inside a triangle with an area of at most 4.


Let \( ABC \) be a triangle with \(\angle A = 60^\circ\). Line \(\ell\) intersects segments \( AB \) and \( AC \) and splits triangle \( ABC \) into an equilateral triangle and a quadrilateral. Let \( X \) and \( Y \) be on \(\ell\) such that lines \( BX \) and \( CY \) are perpendicular to \(\ell\). Given that \( AB = 20 \) and \( AC = 22 \), compute \( XY \).


After adding the coefficients \( p \) and \( q \) with the roots of the quadratic polynomial \( x^2 + px + q \), the result is 2, and after multiplying them, the result is 12. Find all such quadratic polynomials.

Solution: \boxed{x^2 + 3x + 2}


One vertex of an equilateral triangle lies on a circle, and the other two vertices divide a certain chord into three equal parts.

At what angle is the chord seen from the center of the circle?


Find the smallest positive integer $n$ such that the polynomial $(x+1)^{n}-1$ is divisible by $x^{2}+1$ modulo 3.


Find the largest constant \( K \) such that for all positive real numbers \( a, b \), and \( c \), the following inequality holds

$$
\sqrt{\frac{ab}{c}} + \sqrt{\frac{bc}{a}} + \sqrt{\frac{ac}{b}} \geqslant K \sqrt{a+b+c}
$$


Let \(ABC\) be a triangle such that \(AB=7\), and let the angle bisector of \(\angle BAC\) intersect line \(BC\) at \(D\). If there exist points \(E\) and \(F\) on sides \(AC\) and \(BC\), respectively, such that lines \(AD\) and \(EF\) are parallel and divide triangle \(ABC\) into three parts of equal area, determine the number of possible integral values for \(BC\).


Given that the complex number \( z \) satisfies \( |z+\sqrt{3}i| + |z-\sqrt{3}i| = 4 \), find the minimum value of \( |z - \mathrm{i}| \).


Find the number of triples of natural numbers \((a, b, c)\) that satisfy the system of equations:

\[
\left\{
\begin{array}{l}
\gcd(a, b, c) = 6 \\
\operatorname{lcm}(a, b, c) = 2^{15} \cdot 3^{16}
\end{array}
\right.
\]


The numbers from 1 to 8 are arranged at the vertices of a cube in such a way that the sum of the numbers at any three vertices on the same face is at least 10. What is the minimum possible sum of the numbers on the vertices of one face?


Draw a convex pentagon \(ABCDE\) where \(AB = CD\), \(BC = ED\), \(AC = BD = EC\), and the diagonal \(BE\) intersects diagonal \(AC\) at a \(90^\circ\) angle, while \(AD\) intersects \(AC\) at a \(75^\circ\) angle.

Solution: \boxed{ABCDE}


Point \( A \) lies on the line \( y = \frac{5}{12} x - 11 \), and point \( B \) lies on the parabola \( y = x^{2} \). What is the minimum length of segment \( AB \)?


Among all integer solutions of the equation \(20x + 19y = 2019\), find the one for which the value \(|x - y|\) is minimized. Write down the product \(xy\) as the answer.


Camp Koeller offers exactly three water activities: canoeing, swimming, and fishing. None of the campers is able to do all three of the activities. In total, 15 of the campers go canoeing, 22 go swimming, 12 go fishing, and 9 do not take part in any of these activities. Determine the smallest possible number of campers at Camp Koeller.


For non-negative real numbers $x_{0}$, $x_{1}, \cdots, x_{n}$ such that their sum is 1 and $x_{n+1}=x_{1}$, find the maximum value of
$$
S=\sqrt{x_{0}+\sum_{i=1}^{n} \frac{\left(x_{i}-x_{i+1}\right)^{2}}{2n}} + \sum_{i=1}^{n} \sqrt{x_{i}} \quad (n \geqslant 3).
$$


In triangle \(ABC\), angle \(C\) is \(75^\circ\) and angle \(B\) is \(60^\circ\). The vertex \(M\) of an isosceles right triangle \(BCM\) with hypotenuse \(BC\) is located inside triangle \(ABC\). Find angle \(MAC\).


The sum of the ages of five people \( A, B, C, D, \) and \( E \) is 256 years. The age difference between any two people is not less than 2 years and not more than 10 years. What is the minimum age of the youngest person among them?


Name three whole numbers whose product equals their sum.

Find 12.5% of the number 44.


Consider a function \( f: \mathbb{Z} \rightarrow \mathbb{Z} \) such that for every integer \( n \geq 0 \), there are at most \( 0.001 n^{2} \) pairs of integers \( (x, y) \) for which \( f(x + y) \neq f(x) + f(y) \) and \( \max \{|x|, |y|\} \leq n \). Is it possible that for some integer \( n \geq 0 \), there are more than \( n \) integers \( a \) such that \( f(a) \neq a \cdot f(1) \) and \( |a| \leq n \)?

Solution: \boxed{\text{No}}


There are 16 students who form a $4 \times 4$ square matrix. In an examination, their scores are all different. After the scores are published, each student compares their score with the scores of their adjacent classmates (adjacent refers to those directly in front, behind, left, or right; for example, a student sitting in a corner has only 2 adjacent classmates). A student considers themselves "happy" if at most one classmate has a higher score than them. What is the maximum number of students who will consider themselves "happy"?


Two pedestrians started walking at dawn. Each was walking at a constant speed. One walked from point $A$ to point $B$, and the other from point $B$ to point $A$. They met at noon and continued walking without stopping. One arrived at $B$ at 4 PM, and the other arrived at $A$ at 9 PM. At what hour was dawn on that day?


1. What is the units digit of \(3^{1789}\)?
2. What is the units digit of \(1777^{1777^{1777}}\)?


The diagonals of a quadrilateral are equal, and the lengths of its midlines are p and q. Find the area of the quadrilateral.

Solution: \boxed{pq}


There are four cups, each with the mouth facing up. Each time, three cups are flipped, and cups that have been flipped before are allowed to be flipped again. After $n$ flips, all the cups have the mouth facing down. What is the smallest value of the positive integer $n$?

Note: Flipping a cup means turning it from mouth up to mouth down or from mouth down to mouth up.


Find a six-digit number which, when multiplied by 2, 3, 4, 5, and 6, results in six-digit numbers that use the same digits in a different order.


There are 30 volumes of an encyclopedia arranged in some order on a bookshelf. In one operation, it is allowed to swap any two adjacent volumes. What is the minimum number of operations required to guarantee arranging all the volumes in the correct order (from the first to the thirtieth from left to right) regardless of the initial arrangement?


Given a circle, two points \( P \) and \( Q \) on this circle, and a line. Find a point \( M \) on the circle such that the lines \( MP \) and \( MQ \) intercept a segment \( AB \) of a given length on the given line.

Solution: \boxed{M}


Given a four-digit number that satisfies the following conditions: (1) If the units digit and the hundreds digit are swapped, and the tens digit and the thousands digit are swapped, the number increases by 5940; (2) When divided by 9, the remainder is 8. Find the smallest odd number that meets these conditions.


In triangle \( \triangle ABC \), \( BC = a, AC = b, AB = c \), and \( D \) is the midpoint of \( AC \). If \( a^2 + b^2 + c^2 = ab + bc + ca \), then \( \angle CBD = \quad \).


Solve the following equation in the set of integer pairs:

$$
(x+2)^{4}-x^{4}=y^{3} \text {. }
$$

Solution: \boxed{(-1, 0)}


In the plane Cartesian coordinate system, the area of the region corresponding to the set of points $\{(x, y) \mid(|x|+|3 y|-6)(|3 x|+|y|-6) \leq 0\}$ is ________.


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


The sum of the distances from a point inside an equilateral triangle of perimeter length \( p \) to the sides of the triangle is \( s \). Show that \( s \sqrt{12} = p \).

Solution: \boxed{s \sqrt{12} = p}


As shown in the figure, $A$ and $B$ are endpoints of a diameter of a circular track. Three micro-robots, Alpha, Beta, and Gamma, start simultaneously on the circular track and move uniformly in a circular motion. Alpha and Beta start from point $A$, and Gamma starts from point $B$. Beta moves clockwise, while Alpha and Gamma move counterclockwise. After 12 seconds, Alpha reaches point $B$. Nine seconds later, Alpha catches up with Gamma for the first time and also coincides with Beta for the first time. When Gamma reaches point $A$ for the first time, how many seconds will it take for Beta to reach point $B$ for the first time?


What values can the expression \((x-y)(y-z)(z-x)\) take, if it is known that \(\sqrt{x-y+z}=\sqrt{x}-\sqrt{y}+\sqrt{z}\)?


Students from grades 9A, 9B, and 9C gathered for a ceremony. Mary Ivanovna counted 27 students from grade 9A, 29 students from grade 9B, and 30 students from grade 9C. Ilia Grigorievich decided to count the total number of students present from all three grades and ended up with 96 students.

It turned out that Mary Ivanovna's count for each grade could be off by no more than 2 students. Additionally, Ilia Grigorievich's total count could be off by no more than 4 students.

How many students from grade 9A were present at the ceremony?


Find the two-digit number whose digits are distinct and whose square is equal to the cube of the sum of its digits.


A tree is 24 meters tall. A snail at the bottom of the tree wants to climb to the top. During the day, it climbs 6 meters up, and at night, it slides down 4 meters. After how many days can the snail reach the top of the tree?


Find the largest natural number \( n \) for which the system of inequalities

\[ 1 < x < 2, \]
\[ 2 < x^2 < 3, \]
\[ \vdots \]
\[ n < x^n < n+1 \]

has a solution.

Solution: \boxed{n = 4}


Find all values of \( a \) for which the system

$$
\left\{\begin{array}{l}
2^{b x}+(a+1) b y^{2}=a^{2} \\
(a-1) x^{3}+y^{3}=1
\end{array}\right.
$$

has at least one solution for any value of \( b \) (\(a, b, x, y \in \mathbf{R}\)).


The base of a quadrangular pyramid is a rhombus \(ABCD\) in which \(\angle BAD = 60^\circ\). It is known that \(SD = SB\) and \(SA = SC = AB\). Point \(E\) is taken on edge \(DC\) such that the area of triangle \(BSE\) is the smallest among all sections of the pyramid containing segment \(BS\) and intersecting segment \(DC\). Find the ratio \(DE : EC\).


The length of the diagonal of a rectangular parallelepiped is 3. What is the maximum possible surface area of such a parallelepiped?


Several children purchase two types of items priced at 3 yuan and 5 yuan each. Each child buys at least one item, but the total amount spent by each child must not exceed 15 yuan. Xiaomin stated that among the children, there are definitely at least three children who buy the same quantity of each type of item. What is the minimum number of children?


Let \( A B C D \) be a convex quadrilateral inscribed in a circle \( \Gamma \). The line parallel to \( (B C) \) passing through \( D \) intersects \( (C A) \) at \( P \), \( (A B) \) at \( Q \), and intersects the circle \( \Gamma \) again at \( R \). The line parallel to \( (A B) \) passing through \( D \) intersects \( (C A) \) at \( S \), \( (B C) \) at \( T \), and intersects the circle \( \Gamma \) again at \( U \).

Show that if \( P Q = Q R \), then \( S T = T U \).

Solution: \boxed{TS = TU}


Given the plane vectors $a, b,$ and $c$ such that:
$$
\begin{array}{l}
|\boldsymbol{a}|=|\boldsymbol{b}|=|\boldsymbol{c}|=2, \boldsymbol{a}+\boldsymbol{b}+\boldsymbol{c}=\mathbf{0}.
\end{array}
$$

If $0 \leqslant x \leqslant \frac{1}{2} \leqslant y \leqslant 1$, then find the minimum value of
$$
|x(\boldsymbol{a}-\boldsymbol{c})+y(\boldsymbol{b}-\boldsymbol{c})+\boldsymbol{c}|.
$$


As shown in the right image, a frog stands at position 1. On the first jump, it jumps 1 step to position 2; on the second jump, it jumps 2 steps to position 4; on the third jump, it jumps 3 steps to position 1; ...; on the nth jump, it jumps n steps. After the frog has jumped 20 times in a clockwise direction, it will be at position ( ).


Calculate the sum

$$
S=\frac{2014}{2 \cdot 5}+\frac{2014}{5 \cdot 8}+\frac{2014}{8 \cdot 11}+\ldots+\frac{2014}{2012 \cdot 2015}
$$

In the answer, indicate the remainder when the even number closest to the obtained value of $S$ is divided by 5.


An apple, pear, orange, and banana were placed in four boxes (one fruit in each box). Labels were made on the boxes:

1. An orange is here.
2. A pear is here.
3. If a banana is in the first box, then either an apple or a pear is here.
4. An apple is here.

It is known that none of the labels correspond to reality.

Determine which fruit is in which box. In the answer, record the box numbers sequentially, without spaces, where the apple, pear, orange, and banana are located, respectively (the answer should be a 4-digit number).


Every day, from Monday to Friday, an old man went to the blue sea and cast his net into the sea. Each day, the number of fish caught in the net was no greater than the number caught on the previous day. In total, over the five days, the old man caught exactly 100 fish. What is the minimum total number of fish he could have caught over three specific days: Monday, Wednesday, and Friday?


What is the smallest area a right triangle can have if its hypotenuse contains the point \(M(1, 0)\) and its legs lie on the lines \(y = -2\) and \(x = 0\)?

Solution: \boxed{S_{min} = 4}


From 1999 to 2010, over 12 years, the price increase was $150\%$ (i.e., in 1999, an item costing 100 yuan would cost 150 yuan more in 2010 to purchase the same item). If the wages of a company's frontline employees have not changed in these 12 years, calculate the percentage decrease in their wages in terms of purchasing power.

Solution: \boxed{60 \%}


There are 16 people in Nastya's room, each of whom either is friends with or hostile to every other person. Upon entering the room, each person writes down the number of friends who are already there, and upon leaving, writes down the number of enemies still remaining in the room. What can the sum of all the recorded numbers be after everyone has first entered and then left the room?


The diagram shows a right-angled triangle \( ACD \) with a point \( B \) on the side \( AC \). The sides of triangle \( ABD \) have lengths 3, 7, and 8. What is the area of triangle \( BCD \)?


Given a natural number \( n \), find the number of distinct triplets of natural numbers whose sum is \( 6n \).


Calculate $\Delta f\left(P_{0}\right)$ and $d f\left(P_{0}\right)$ for the function $f(x, y) = x^{2} y$ at the point $P_{0} = (5, 4)$ with $\Delta x = 0.1$ and $\Delta y = -0.2$.


A subset \( H \) of the set of numbers \(\{1, 2, \ldots, 100\}\) has the property that if an element is in \( H \), then ten times that element is not in \( H \). What is the maximum number of elements that \( H \) can have?


Let \( a \) and \( b \) be positive numbers. Find the minimum value of the fraction \(\frac{(a+b)(a+2)(b+2)}{16ab}\). Justify your answer.


Find the sum of all factors \( d \) of \( N = 19^{88} - 1 \) that are of the form \( d = 2^a \cdot 3^b \) (where \( a \) and \( b \) are natural numbers).


Given set \( S \) such that \( |S| = 10 \). Let \( A_1, A_2, \cdots, A_k \) be non-empty subsets of \( S \), and the intersection of any two subsets contains at most two elements. Find the maximum value of \( k \).


Point \( F \) is the midpoint of side \( BC \) of square \( ABCD \). A perpendicular \( AE \) is drawn to segment \( DF \). Find the angle \( CEF \).


In a convex quadrilateral, the lengths of the diagonals are 2 cm and 4 cm. Find the area of the quadrilateral, given that the lengths of the segments connecting the midpoints of the opposite sides are equal.

Solution: \boxed{4 \text{ cm}^2}


Points \( P \) and \( Q \) are located on the sides \( AB \) and \( AC \) of triangle \( ABC \) such that \( AP:PB = 1:4 \) and \( AQ:QC = 3:1 \). Point \( M \) is chosen randomly on side \( BC \). Find the probability that the area of triangle \( ABC \) exceeds the area of triangle \( PQM \) by no more than two times. Find the mathematical expectation of the random variable - the ratio of the areas of triangles \( PQM \) and \( ABC \).


Let \( A_{1}, A_{2}, A_{3} \) be three points in the plane, and for convenience, let \( A_{4} = A_{1}, A_{5} = A_{2} \). For \( n = 1, 2, \) and 3, suppose that \( B_{n} \) is the midpoint of \( A_{n} A_{n+1} \), and suppose that \( C_{n} \) is the midpoint of \( A_{n} B_{n} \). Suppose that \( A_{n} C_{n+1} \) and \( B_{n} A_{n+2} \) meet at \( D_{n} \), and that \( A_{n} B_{n+1} \) and \( C_{n} A_{n+2} \) meet at \( E_{n} \). Calculate the ratio of the area of triangle \( D_{1} D_{2} D_{3} \) to the area of triangle \( E_{1} E_{2} E_{3} \).


A regular dodecagon \( P_1, P_2, \ldots, P_{12} \) is inscribed in a unit circle with center \( O \). Let \( X \) be the intersection of \( P_1 P_5 \) and \( OP_2 \), and let \( Y \) be the intersection of \( P_1 P_5 \) and \( OP_4 \). Let \( A \) be the area of the region bounded by \( XY \), \( XP_2 \), \( YP_4 \), and the minor arc \( \widehat{P_2 P_4} \). Compute \( \lfloor 120A \rfloor \).


A randomly selected phone number consists of 5 digits. What is the probability that: 1) all digits are different; 2) all digits are odd?


Inside square \(ABCD\), a point \(M\) is chosen such that \(\angle MAB = 60^\circ\) and \(\angle MCD = 15^\circ\). Find \(\angle MBC\).


What is the sum of all three-digit numbers \( n \) such that \( \frac{3n+2}{5n+1} \) is not in simplest form?


Two concentric circles have radii 2006 and 2007. $ABC$ is an equilateral triangle inscribed in the smaller circle and $P$ is a point on the circumference of the larger circle. Given that a triangle with side lengths equal to $PA, PB$, and $PC$ has area $\frac{a \sqrt{b}}{c}$, where $a, b, c$ are positive integers such that $a$ and $c$ are relatively prime and $b$ is not divisible by the square of any prime, find $a+b+c$.


Inflation over two years will be:

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

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


If the three-digit number \( m \) satisfies the following conditions: (1) The sum of its digits is 12; (2) \( 2m \) is also a three-digit number, and the sum of its digits is 6. How many such three-digit numbers \( m \) are there?


Initially, all the squares of an $8 \times 8$ grid are white. You start by choosing one of the squares and coloring it gray. After that, you may color additional squares gray one at a time, but you may only color a square gray if it has exactly 1 or 3 gray neighbors at that moment (where a neighbor is a square sharing an edge).

Is it possible to color all the squares gray? Justify your answer.

Solution: \boxed{\text{No}}


At a certain middle school, there are 110 teachers who know both English and Russian. According to the statistics, there are 75 teachers who know English and 55 teachers who know Russian. How many teachers at this school know English but not Russian?


Write a system of equations for a line that passes through the origin and forms equal angles with the three coordinate axes. Determine the measure of these angles. How many solutions does the problem have?


Given a convex quadrilateral \(ABCD\) in which \(\angle BAC=20^\circ\), \(\angle CAD=60^\circ\), \(\angle ADB=50^\circ\), and \(\angle BDC=10^\circ\). Find \(\angle ACB\).


Determine all strictly positive integers that are coprime with all numbers of the form \(2^{n} + 3^{n} + 6^{n} - 1\), for \(n\) being a natural number.


Let \( s(n) \) denote the number of 1's in the binary representation of \( n \). Compute
\[
\frac{1}{255} \sum_{0 \leq n<16} 2^{n}(-1)^{s(n)} .
\]


Given \( f(u) = u^{2} + au + (b-2) \), where \( u = x + \frac{1}{x} \) (with \( x \in \mathbb{R} \) and \( x \neq 0 \)). If \( a \) and \( b \) are real numbers such that the equation \( f(u) = 0 \) has at least one real root, find the minimum value of \( a^{2} + b^{2} \).


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


The natural numbers $1,2,3, \ldots, 10$ are written on a board. It is allowed to write the number $a^{2}$ if the number $a$ is already on the board, or write the least common multiple (LCM) of the numbers $a$ and $b$ if the numbers $a$ and $b$ are already on the board. Is it possible to obtain the number 1,000,000 using these operations?

Solution: \boxed{\text{No}}


During a school recess, \( n \) children are sitting in a circle, playing a game. A teacher walks clockwise around the circle, distributing candies to the children according to the following rule:

1. Pick a child and give them and the next child (clockwise) one candy each.
2. Skip the next child, and give the following child one candy.
3. Skip the next two children, and give the following child one candy.
4. Continue this pattern (skip three children, give one candy, and so on).

Determine the value of \( n \) such that all children eventually (possibly after the teacher makes many rounds) receive at least one candy.

(Note: This problem was part of the 31st International Mathematical Olympiad shortlist and the 1990 Asia-Pacific Mathematical Olympiad.)


In a certain triangle, the difference of two sides: $b-c$ is twice the distance of the angle bisector $f_{\alpha}$, originating from the common endpoint $A$ of the two sides, from the altitude foot $M$. What is the angle $\alpha$ enclosed by the two sides?


Given the function
$$
f(x)=x^{4}+a x^{3}+b x^{2}+a x+1 \quad (a, b \in \mathbf{R})
$$

which has at least one root, find the minimum value of \( a^2 - b \).


There are 5 integers written on the board. The sums of these integers taken in pairs resulted in the following set of 10 numbers: $6, 9, 10, 13, 13, 14, 17, 17, 20, 21$. Determine which numbers are written on the board. Provide their product as the answer.


Find the area of a right-angled triangle if the altitude drawn from the right angle divides it into two triangles with the radii of the inscribed circles equal to 3 and 4.


Let $\gamma$ and $\gamma^{\prime}$ be two circles that are externally tangent at point $A$. Let $t$ be a common tangent to both circles, touching $\gamma$ at $T$ and $\gamma^{\prime}$ at $T^{\prime}$. Let $M$ be the midpoint of $\left[T T^{\prime}\right]$.

Show that $M T=M A=M T^{\prime}$.

Solution: \boxed{MT = MA = MT'}


In an exam with 3 questions, four friends checked their answers after the test and found that they got 3, 2, 1, and 0 questions right, respectively. When the teacher asked how they performed, each of them made 3 statements as follows:

Friend A: I got two questions correct, and I did better than B, C scored less than D.
Friend B: I got all questions right, C got them all wrong, and A did worse than D.
Friend C: I got one question correct, D got two questions right, B did worse than A.
Friend D: I got all questions right, C did worse than me, A did worse than B.

If each person tells as many true statements as the number of questions they got right, let \(A, B, C, D\) represent the number of questions each of A, B, C, and D got right, respectively. Find the four-digit number \(\overline{\mathrm{ABCD}}\).


Determine the odd prime number \( p \) such that the sum of digits of the number \( p^{4} - 5p^{2} + 13 \) is the smallest possible.


Given that 3 is a solution of the inequality $(x-a)(x+2a-1)^{2}(x-3a) \leq 0$, determine the range of the real number $a$.

Solution: \boxed{\{-1\} \cup [1,3]}


Out of 8 circular disks with radius \( r \), 7 are fixed on a table such that their centers are at the vertices and center of a regular hexagon with side length \( 2r \), and the 8th disk touches one of the 6 outer disks. The 8th disk is rolled around the 7 fixed disks once without slipping, until it returns to its starting position. How many times does the 8th disk rotate about its own center during this process? (Provide an explanation for the answer.)


A bag contains 6 red balls and 8 white balls. If 5 balls are randomly placed into Box $A$ and the remaining 9 balls are placed into Box $B$, what is the probability that the sum of the number of white balls in Box $A$ and the number of red balls in Box $B$ is not a prime number? (Answer with a number)


If the real number \(x\) satisfies \(\arcsin x > \arccos x\), then what is the range of the function \( f(x) = \sqrt{2 x^{2} - x + 3} + 2^{\sqrt{x^{2} - x}} \)?


At an oil refinery, a tank was filled with crude oil with a sulfur concentration of $2 \%$. A portion of this oil was then sent for production, and the same amount of oil with a sulfur concentration of $3 \%$ was added to the tank. Again, the same amount of oil was sent for production, but this time oil with a sulfur concentration of $1.5 \%$ was added. As a result, the sulfur concentration in the oil in the tank became the same as it was initially. Determine what fraction of the oil in the tank was sent to production each time.


Given the set
$$
A=\{n|n \in \mathbf{N}, 11| S(n), 11 \mid S(n+1)\} \text {, }
$$
where \(S(m)\) represents the sum of the digits of a natural number \(m\). Find the smallest number in the set \(A\).


There are three identical red balls, three identical yellow balls, and three identical green balls. In how many different ways can they be split into three groups of three balls each?


Find the most probable number of hits in the ring in five throws if the probability of hitting the ring with the ball in one throw is $p=0.6$.


Let \( M \) be a set composed of a finite number of positive integers,
\[
M = \bigcup_{i=1}^{20} A_i = \bigcup_{i=1}^{20} B_i, \text{ where}
\]
\[
A_i \neq \varnothing, B_i \neq \varnothing \ (i=1,2, \cdots, 20)
\]
satisfying the following conditions:
1. For any \( 1 \leqslant i < j \leqslant 20 \),
\[
A_i \cap A_j = \varnothing, \ B_i \cap B_j = \varnothing;
\]
2. For any \( 1 \leqslant i \leqslant 20, \ 1 \leqslant j \leqslant 20 \), if \( A_i \cap B_j = \varnothing \), then \( \left|A_i \cup B_j\right| \geqslant 18 \).

Find the minimum number of elements in the set \( M \) (denoted as \( |X| \) representing the number of elements in set \( X \)).


In a regular quadrilateral pyramid, the dihedral angle at the lateral edge is $120^{\circ}$. Find the lateral surface area of the pyramid if the area of its diagonal section is $S$.


Given vectors \(\overrightarrow{O P}=\left(2 \cos \left(\frac{\pi}{2}+x\right),-1\right)\) and \(\overrightarrow{O Q}=\left(-\sin \left(\frac{\pi}{2}- x\right), \cos 2 x\right)\), and the function \(f(x)=\overrightarrow{O P} \cdot \overrightarrow{O Q}\). If \(a, b, c\) are the sides opposite angles \(A, B, C\) respectively in an acute triangle \( \triangle ABC \), and it is given that \( f(A) = 1 \), \( b + c = 5 + 3 \sqrt{2} \), and \( a = \sqrt{13} \), find the area \( S \) of \( \triangle ABC \).


Is it true that in space, angles with respectively perpendicular sides are equal or sum up to $180^{\circ}$?

Solution: \boxed{\text{No}}


Calculate the definite integral:

$$
\int_{0}^{2 \pi} \sin ^{2}\left(\frac{x}{4}\right) \cos ^{6}\left(\frac{x}{4}\right) d x
$$


Let \( n \) be a natural number. What digit is immediately after the decimal point in the decimal representation of \( \sqrt{n^{2} + n} \)?


Kolya, an excellent student in the 7th-8th grade, found the sum of the digits of all the numbers from 0 to 2012 and added them all together. What number did he get?


Let \( a_{1}, a_{2}, a_{3}, a_{4} \) be four distinct numbers from the set \(\{1, 2, \ldots, 100\}\) such that they satisfy the following equation:
\[
\left(a_{1}^{2}+a_{2}^{2}+a_{3}^{2}\right)\left(a_{2}^{2}+a_{3}^{2}+a_{4}^{2}\right) = \left(a_{1} a_{2}+a_{2} a_{3}+a_{3} a_{4}\right)^{2}.
\]

How many such ordered quadruples \(\left(a_{1}, a_{2}, a_{3}, a_{4}\right)\) are there?


Given the set $T$ of all positive divisors of $2004^{100}$, determine the maximum possible number of elements in a subset $S$ of $T$ such that no element of $S$ is a multiple of any other element in $S$.


A ball thrown vertically upwards has a height above the ground that is a quadratic function of its travel time. Xiaohong throws two balls vertically upwards at intervals of 1 second. Assuming the height above the ground is the same at the moment of release for both balls, and both balls reach the same maximum height 1.1 seconds after being thrown, find the time $t$ seconds after the first ball is thrown such that the height above the ground of the first ball is equal to the height of the second ball. Determine $t = \qquad$ .

Solution: \boxed{1.6 \text{ seconds}}


Find the set of positive integers \( k \) such that \(\left(x^{2}-1\right)^{2k}+\left(x^{2}+2x\right)^{2k}+(2x+1)^{2k}=2\left(1+x+x^{2}\right)^{2k}\) is valid for all real numbers \( x \).

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


The square was cut into 25 smaller squares, of which exactly one has a side length different from 1 (each of the others has a side length of 1).

Find the area of the original square.


Given a real number $\alpha$, determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ depending on $\alpha$ such that

$$
f(f(x+y) f(x-y))=x^{2}+\alpha y f(y)
$$

for all $x, y \in \mathbb{R}$.

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


In how many different ways can 5 numbers be chosen from $\{1,2,\ldots, 18\}$ such that no two of these numbers are consecutive?


For \( a, b, c > 0 \), find the maximum value of the expression

\[
A = \frac{a^{4} + b^{4} + c^{4}}{(a + b + c)^{4} - 80(a b c)^{4/3}}
\]


In how many ways can the set of integers \(\{1,2, \ldots, 1995\}\) be partitioned into three nonempty sets so that none of these sets contains two consecutive integers?

Solution: \boxed{2^{1993} - 1}


A point is randomly thrown onto the segment [6, 11], and let \( k \) be the resulting value. Find the probability that the roots of the equation \( \left(k^{2}-2k-15\right)x^{2}+(3k-7)x+2=0 \) satisfy the condition \( x_{1} \leq 2x_{2} \).


How many five-digit numbers are there that end with six and are divisible by three?


If the inequality \(\frac{y}{4} - \cos^2 x \geq a \sin x - \frac{9}{y}\) holds for all positive real numbers \(x\) and \(y\), then what is the range of the real number \(a\)?

Solution: \boxed{[-3, 3]}


Let \( f(n) \) be a function defined on the set of positive integers with non-negative integer values. Given the conditions:
- \( f(2) = 0 \)
- \( f(3) > 0 \)
- \( f(9999) = 3333 \)
- For all \( m \) and \( n \), \( f(m+n) - f(m) - f(n) = 0 \) or 1

Determine the value of \( f(1982) \).


The diameter \( AB \) and the chord \( CD \) of a circle intersect at point \( E \), such that \( CE = DE \). Tangents to the circle at points \( B \) and \( C \) intersect at point \( K \). Segments \( AK \) and \( CE \) intersect at point \( M \). Find the area of triangle \( CKM \) if \( AB = 10 \) and \( AE = 1 \).


Find all 4-digit integers of the form \( aabb \) (when written in base 10) that are perfect squares.


A bug is located at the point $(0,0)$ on a coordinate grid. How many ways are there for it to reach the cell (7,3) if the bug can move only to the right and upwards, and if it is located at a point with matching coordinates, it cannot stop there?


The 3-digit number \( AAA \), where \( A \neq 0 \), and the 6-digit number \( AAABBB \) satisfy the following equality:

\[ AAA \times AAA + AAA = AAABBB \]

Find \( A \) and \( B \).

Solution: \boxed{B = 0}


Anaëlle has \( 2n \) pieces, which are labeled \( 1, 2, 3, \ldots, 2n \), as well as a red and a blue box. She now wants to distribute all \( 2n \) pieces into the two boxes so that the pieces \( k \) and \( 2k \) end up in different boxes for each \( k = 1, 2, \ldots, n \). How many ways does Anaëlle have to do this?


The sequence of numbers \( a_{1}, a_{2}, \ldots, a_{2022} \) is such that \( a_{n} - a_{k} \geq n^{3} - k^{3} \) for any \( n \) and \( k \) where \( 1 \leq n \leq 2022 \) and \( 1 \leq k \leq 2022 \). Furthermore, \( a_{1011} = 0 \). What values can \( a_{2022} \) take?

Solution: \boxed{2022^3 - 1011^3}


The numbers from 1 to 10 are divided into two groups such that the product of the numbers in the first group is divisible by the product of the numbers in the second group.

What is the smallest possible value of the quotient of the division of the first product by the second product?


A metal bar with a temperature of $20{ }^{\circ} \mathrm{C}$ is placed into water that is initially at $80{ }^{\circ} \mathrm{C}$. After thermal equilibrium is reached, the temperature is $60{ }^{\circ} \mathrm{C}$. Without removing the first bar from the water, another metal bar with a temperature of $20{ }^{\circ} \mathrm{C}$ is placed into the water. What will the temperature of the water be after the new thermal equilibrium is reached?

Solution: \boxed{50^\circ \mathrm{C}}


As shown in Figure 14-13, AC and CE are two diagonals of the regular hexagon ABCDEF. Points M and N divide AC and CE internally, respectively, such that AM:AC = CN:CE = r. If points B, M, and N are collinear, determine the value of r.


For which $n \in \mathbf{N}^{*}$ is $n \times 2^{n+1} + 1$ a square?


What is the minimum number of points that need to be marked inside a convex $n$-gon so that each triangle with vertices at the vertices of this $n$-gon contains at least one marked point inside?


Calculate the value of:
$$
\cos^2 \frac{\pi}{17} + \cos^2 \frac{2\pi}{17} + \cdots + \cos^2 \frac{16\pi}{17}
$$


Sheep Thievery. Robbers stole $\frac{1}{3}$ of the flock of sheep and $\frac{1}{3}$ of a sheep. Another gang stole $\frac{1}{4}$ of the remaining sheep and $\frac{1}{4}$ of a sheep. Then a third gang of robbers stole $\frac{1}{5}$ of the remaining sheep and an additional $\frac{3}{5}$ of a sheep, after which 409 sheep remained in the flock.

How many sheep were initially in the flock?


A nine-digit number is odd. The sum of its digits is 10. The product of the digits of the number is non-zero. The number is divisible by seven. When rounded to three significant figures, how many millions is the number equal to?


(1) Given that \( a > 0 \) and \( b > 0 \), and \( a + b = 1 \), find the minimum value of \(\left(a + \frac{1}{a^{2}}\right)^{2} + \left(b + \frac{1}{b^{2}}\right)^{2}\). Also, find the maximum value of \(\left(a + \frac{1}{a}\right)\left(b + \frac{1}{b}\right)\).

(2) Given that \( a > 0 \), \( b > 0 \), and \( c > 0 \), and \( a + b + c = 1 \), find the minimum value of \( u = \left(a + \frac{1}{a}\right)^{3} + \left(b + \frac{1}{b}\right)^{3} + \left(c + \frac{1}{c}\right)^{3} \).


On the sides \( AB, BC \), and \( AC \) of triangle \( ABC \), points \( M, N, \) and \( K \) are taken respectively so that \( AM:MB = 2:3 \), \( AK:KC = 2:1 \), and \( BN:NC = 1:2 \). In what ratio does the line \( MK \) divide the segment \( AN \)?


Let the function \( f: \mathbf{R} \rightarrow \mathbf{R} \) satisfy \( f(0) = 1 \) and for any \( x, y \in \mathbf{R} \), it holds that
\[ f(xy + 1) = f(x)f(y) - f(y) - x + 2. \]
Determine \( f(x) \).

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


When \( N \) takes all values from 1, 2, 3, ..., to 2015, how many numbers of the form \( 3^n + n^3 \) are divisible by 7?


Find the smallest integer that needs to be added to the expression \((a+2)(a+5)(a+8)(a+11)\) so that the resulting sum is positive for any value of \(a\).


Misha calculated the products \(1 \times 2, 2 \times 3, 3 \times 4, \ldots, 2017 \times 2018\). How many of these products have their last digit as zero?


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


There are \( n \) girls \( G_{1}, \ldots, G_{n} \) and \( n \) boys \( B_{1}, \ldots, B_{n} \). A pair \( \left(G_{i}, B_{j}\right) \) is called suitable if and only if girl \( G_{i} \) is willing to marry boy \( B_{j} \). Given that there is exactly one way to pair each girl with a distinct boy that she is willing to marry, what is the maximal possible number of suitable pairs?


If a positive integer \( n \) makes the equation \( x^{3} + y^{3} = z^{n} \) have a positive integer solution \( (x, y, z) \), then \( n \) is called a "good number." How many good numbers are there that do not exceed 2,019?


Using 4 different colors to paint the 4 faces of a regular tetrahedron (each face is an identical equilateral triangle) so that different faces have different colors, how many different ways are there to paint it? (Coloring methods that remain different even after any rotation of the tetrahedron are considered different.)


Let \( n \) be the smallest positive integer such that the sum of its digits is 2011. How many digits does \( n \) have?


How many distinct permutations of the letters of the word REDDER are there that do not contain a palindromic substring of length at least two? (A substring is a contiguous block of letters that is part of the string. A string is palindromic if it is the same when read backwards.)


Given that \( n! \), in decimal notation, has exactly 57 ending zeros, find the sum of all possible values of \( n \).


In a rectangular parallelepiped \(ABCDEFGHA'B'C'D'\), which has \(AB = BC = a\) and \(AA' = b\), the shape was orthogonally projected onto some plane containing the edge \(CD\). Find the maximum area of the projection.

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


The players Alfred and Bertrand jointly determine a polynomial \( x^n + a_{n-1} x^{n-1} + \cdots + a_0 \) with the given degree \( n \geq 2 \). They alternately choose the value of one coefficient in \( n \) moves, with all coefficients being integers and \( a_0 \neq 0 \). Alfred takes the first move. Alfred wins if the polynomial has an integer root in the end.

(a) For which \( n \) can Alfred force a win if the coefficients \( a_j \) are chosen from right to left, i.e., for \( j = 0, 1, \ldots, n-1 \)?

(b) For which \( n \) can Alfred force a win if the coefficients \( a_j \) are chosen from left to right, i.e., for \( j = n-1, n-2, \ldots, 0 \)?

(Theresia Eisenkölbl, Clemens Heuberger)

Answer: In both cases, Alfred can force a win if and only if \( n \) is odd.

Solution: \boxed{n \text{ is odd}}


For each value \( n \in \mathbb{N} \), determine how many solutions the equation \( x^{2} - \left\lfloor x^{2} \right\rfloor = \{x\}^{2} \) has on the interval \([1, n]\).

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


Two athletes run around an oval track at constant speeds. The first athlete completes the track 5 seconds faster than the second athlete. If they start running from the same point on the track in the same direction, they will meet again for the first time after 30 seconds. How many seconds will it take for them to meet again for the first time if they start running from the same point on the track in opposite directions?


There are three piles of stones: one with 51 stones, another with 49 stones, and a third with 5 stones. You are allowed to either combine any piles into one or divide a pile with an even number of stones into two equal piles. Is it possible to end up with 105 piles, each containing one stone?

Solution: \boxed{\text{No}}


Let $T$ be the set of all positive divisors of $2004^{100}$, and let $S$ be a subset of $T$ such that no element in $S$ is an integer multiple of any other element in $S$. Find the maximum value of $|S|$.


Each artist in the creative collective "Patience and Labor" has their own working schedule. Six of them paint one picture every two days, another eight of them paint one picture every three days, and the rest never paint pictures. From September 22 to September 26, they painted a total of 30 pictures. How many pictures will they paint on September 27?


From home to work and back. If a person walks to work and takes transport back, it takes him a total of one and a half hours. If he takes transport both ways, the entire journey takes him 30 minutes. How much time will it take for the person to walk both to work and back?

Solution: \boxed{2.5 \text{ hours}}


Find the minimum value of the function \( f(x) = x^{2} + 2x + \frac{6}{x} + \frac{9}{x^{2}} + 4 \) for \( x > 0 \).

Solution: \boxed{10 + 4 \sqrt{3}}


\[
\left(\left(\sqrt{mn} - \frac{mn}{m+\sqrt{mn}}\right) \div \frac{\sqrt[4]{mn} - \sqrt{n}}{m-n} - m \sqrt{n}\right)^{2} \div \sqrt[3]{mn \sqrt{mn}} - \left(\frac{m}{\sqrt{m^{4}-1}}\right)^{-2}.
\]


Find all natural numbers $n$ for which the number $n^2 + 3n$ is a perfect square.

Solution: \boxed{n = 1}


When buying 2 shirts and 1 tie, 2600 rubles are paid. The second purchase will have a 25% discount. The price of the third shirt is \(1200 \cdot 0.75 = 900\) rubles. The total price of the entire purchase is \(2600 + 900 = 3500\) rubles.

Solution: \boxed{3500 \text{ rubles}}


What is the largest number of integers that we can choose from the set $\{1, 2, 3, \ldots, 2017\}$ such that the difference between any two of them is not a prime number?


Initially, there are 33 ones written on the board. Every minute, Karlson erases any two numbers and writes their sum on the board, then eats a number of candies equal to the product of the two erased numbers. What is the maximum number of candies he can eat in 33 minutes?


Given a movable point $P(x, y)$ that satisfies the following conditions:
\[
\left\{
\begin{array}{l}
2x + y \leq 2, \\
x \geq 0, \\
\left( x + \sqrt{x^2 + 1} \right) \left( y + \sqrt{y^2 + 1} \right) \geq 1,
\end{array}
\right.
\]
find the area of the region formed by the point $P(x, y)$.


Suppose \( n \) is a given integer greater than 2. There are \( n \) indistinguishable bags, and the \( k \)-th bag contains \( k \) red balls and \( n-k \) white balls \((k=1,2, \cdots, n)\). From each bag, three balls are drawn consecutively without replacement, and the third ball drawn from each bag is placed into a new bag. What is the expected number of white balls in the new bag?


Determine the simplest polynomial equation with integer coefficients, one of whose roots is $\sqrt{2} + \sqrt{3}$.

Solution: \boxed{x^4 - 10x^2 + 1 = 0}


For what maximum \( a \) is the inequality \(\frac{\sqrt[3]{\operatorname{tg} x}-\sqrt[3]{\operatorname{ctg} x}}{\sqrt[3]{\sin x}+\sqrt[3]{\cos x}}>\frac{a}{2}\) satisfied for all permissible \( x \in \left(\frac{3 \pi}{2}, 2 \pi\right) \)? If necessary, round your answer to the nearest hundredth.


A cube with a side of 10 is divided into 1000 smaller cubes each with an edge of 1. A number is written in each small cube such that the sum of the numbers in each column of 10 cubes (along any of the three directions) equals 0. In one of the cubes (denoted as A), the number 1 is written. There are three layers passing through cube A, and these layers are parallel to the faces of the cube (each layer has a thickness of 1). Find the sum of all the numbers in the cubes that do not lie in these layers.


Let's call a set of four numbers chosen from the set {1,2,3,4,5,6,7} good, if no two numbers in this set add up to 8. How many good sets of four numbers are there?


ABC is an equilateral triangle with a side length of 10. A point D is taken on side AB, a point E on side AC, and points F and G on side BC such that triangles ADE, BDG, and CEF are also equilateral. AD = 3. Find FG.


In triangle \( \triangle ABC \), \( AB = 8 \), \( BC = 11 \), and \( AC = 6 \). The points \( P \) and \( Q \) are on \( BC \) such that \( \triangle PBA \) and \( \triangle QAC \) are each similar to \( \triangle ABC \). What is the length of \( PQ \)?


In the figure, \(ABCD\) is a square with an area that is \(\frac{7}{32}\) of the area of triangle \(XYZ\). What is the ratio between \(XA\) and \(XY\)?


Ivan Semenovich leaves for work at the same time every day, travels at the same speed, and arrives exactly at 9:00 AM. One day, he overslept and left 40 minutes later than usual. To avoid being late, Ivan Semenovich increased his speed by 60% and arrived at 8:35 AM. By what percentage should he have increased his usual speed to arrive exactly at 9:00 AM?

Solution: \boxed{30 \%}


A function \( f \) is defined on the set of positive integers and satisfies the condition \( f(1) = 2002 \) and \( f(1) + f(2) + \cdots + f(n) = n^{2} f(n) \) for \( n > 1 \). What is the value of \( f(2002) \) ?


On a table, 28 coins of the same size but possibly different masses are arranged in the shape of a triangle. It is known that the total mass of any trio of coins that touch each other pairwise is 10 grams. Find the total mass of all 18 coins on the border of the triangle.

Solution: \boxed{60 \text{ grams}}


Let \( S = \left\{ A = \left(a_{1}, a_{2}, \cdots, a_{8}\right) \mid a_{i} = 0 \text{ or } 1, i = 1,2, \cdots, 8 \right\} \). For any two elements \( A = \left(a_{1}, a_{2}, \cdots, a_{8}\right) \) and \( B = \left(b_{1}, b_{2}, \cdots, b_{8}\right) \) in \( S \), define \( d(A, B) = \sum_{i=1}^{8} \left| a_{i} - b_{i} \right| \), which is called the distance between \( A \) and \( B \). What is the maximum number of elements that can be chosen from \( S \) such that the distance between any two of them is at least \( 5 \)?


Is it possible to arrange all natural numbers from 1 to 2018 in a circle so that the sum of any three consecutive numbers is an odd number?

Solution: \boxed{\text{No}}


If a positive integer cannot be written as the difference of two square numbers, then the integer is called a "cute" integer. For example, 1, 2, and 4 are the first three "cute" integers. Find the \(2010^{\text{th}}\) "cute" integer.


A car rental company owns 100 cars. When the monthly rental fee per car is 3000 Yuan, all the cars are rented out. For every 50 Yuan increase in the monthly rental fee per car, one more car remains unrented. The maintenance cost for each rented car is 150 Yuan per month, and the maintenance cost for each unrented car is 50 Yuan per month. At what monthly rental fee per car does the car rental company achieve maximum monthly revenue? What is the maximum monthly revenue?

Solution: \boxed{307050 \text{ yuan}}


Suppose the edge length of a regular tetrahedron $ABC D$ is 1 meter. A bug starts at point $A$ and moves according to the following rule: at each vertex, it chooses one of the three edges connected to this vertex with equal probability and crawls along this edge to the next vertex. What is the probability that the bug will be back at point $A$ after crawling for 4 meters?


As shown in the figure, the cross-section of the reflector on the movie projector lamp is a part of an ellipse. The filament is located at the focus \( F_{2} \), and the distance between the filament and the vertex of the reflector \( A \) is \( |F_{2} A| = 1.5 \) cm. The length of the latus rectum \( |BC| = 5.4 \) cm. In order to ensure that the film gate of the projector receives the strongest light, how far should the lamp be placed from the film gate?

Solution: \boxed{12 \text{ meters}}


Arrange 3 identical black balls and 3 identical white balls in a row from left to right. If any position (including that position) to the left always has at least as many black balls as white balls, we call this arrangement a "valid arrangement". What is the probability of having a "valid arrangement"?


The sum of 100 numbers is 1000. The largest of these numbers was doubled, while another number was decreased by 10. After these actions, the sum of all numbers remained unchanged. Find the smallest of the original numbers.


Given positive real numbers \( a, b, c, d \) that satisfy the equalities

\[ a^{2}+d^{2}-ad = b^{2}+c^{2}+bc \quad \text{and} \quad a^{2}+b^{2} = c^{2}+d^{2}, \]

find all possible values of the expression \( \frac{ab+cd}{ad+bc} \).


Represent in the rectangular coordinate system those pairs of real numbers \((a ; b)\) for which the two-variable polynomial

$$
x(x+4) + a\left(y^2 - 1\right) + 2by
$$

can be factored into the product of two first-degree polynomials.

Solution: \boxed{(a + 2)^2 + b^2 = 4}


Among all natural numbers that are multiples of 20, the sum of those that do not exceed 3000 and are also multiples of 14 is $\qquad$


Baptiste wants to grab a pair of socks in the dark. He has 6 red socks, 13 blue socks, and 8 white socks. How many socks does he need to take to be sure to have 2 of the same color?


Point $A$ lies on the line $y=\frac{8}{15} x-6$, and point $B$ on the parabola $y=x^{2}$. What is the minimum length of the segment $AB$?


A great-grandfather banker left an inheritance to his newborn great-grandson. According to the bank agreement, the amount in the great-grandson's account increases. Each year, on the day after his birthday, the current amount increases by one million rubles more than the previous year. Thus, if the initial amount was zero rubles, after one year it will be +1 million rubles; after 2 years, $1+2$ million rubles; after 3 years, $1+2+3$; and so on. According to the agreement, the process will stop, and the great-grandson will receive the money when the amount in the account is a three-digit number consisting of three identical digits.

How old will the great-grandson be when the conditions of the agreement are fulfilled?


Two circles with radii 3 and 4, and a distance of 5 between their centers, intersect at points \(A\) and \(B\). A line through point \(B\) intersects the circles at points \(C\) and \(D\), such that \(CD = 8\) and point \(B\) lies between points \(C\) and \(D\). Find the area of triangle \(ACD\).


Vitya found the smallest possible natural number which, when multiplied by 2, results in a perfect square, and when multiplied by 3, results in a perfect cube. What number did Vitya find?


Let $ABC$ be an isosceles triangle with the apex at $A$. Let $M$ be the midpoint of segment $[BC]$. Let $D$ be the symmetric point of $M$ with respect to the segment $[AC]$. Let $x$ be the angle $\widehat{BAC}$. Determine, as a function of $x$, the value of the angle $\widehat{MDC}$.


A subset \( X \) of the set of "two-digit" numbers \( 00, 01, \ldots, 98, 99 \) is such that in any infinite sequence of digits there are two adjacent digits forming a number from \( X \). What is the smallest number of elements that can be contained in \( X \)?


In a circle with radius $R$, two mutually perpendicular diameters are given. An arbitrary point on the circumference is projected onto these diameters. Find the distance between the projections of the point.

Solution: \boxed{R}


The tangent line drawn to the point \( A \) on the cubic parabola \( y = ax^3 \) intersects the \( Y \)-axis at point \( B \) and intersects the curve again at another point \( C \). Determine the geometric locus of the midpoints of the distances \( AB \) and \( BC \), considering that the location of point \( A \) varies.

Solution: \boxed{y = 5 a x^3}


As shown in the diagram, the area of parallelogram \(ABCD\) is 60. The ratio of the areas of \(\triangle ADE\) and \(\triangle AEB\) is 2:3. Find the area of \(\triangle BEF\).


Solution: \boxed{x = 1}


The polynomial \( G(x) \) with real coefficients takes the value 2022 at exactly five distinct points \( x_{1}<x_{2}<x_{3}<x_{4}<x_{5} \). It is known that the graph of the function \( y=G(x) \) is symmetric with respect to the line \( x=-7 \).

(a) Find \( x_{1}+x_{3}+x_{5} \).

(b) What is the minimum degree that \( G(x) \) can have?


The inequality \( \left|f^{\prime}(0)\right| \leq A \) holds for all quadratic functions \( f(x) \) satisfying \( |f(x)| \leq 1 \) for \( 0 \leq x \leq 1 \). Find the minimum value of the real number \( A \).


\(ABCD\) is a convex quadrilateral in which \(AC\) and \(BD\) meet at \(P\). Given \(PA = 1\), \(PB = 2\), \(PC = 6\), and \(PD = 3\). Let \(O\) be the circumcenter of \(\triangle PBC\). If \(OA\) is perpendicular to \(AD\), find the circumradius of \(\triangle PBC\).


Find the smallest value of the expression \(\left|36^{m} - 5^{n}\right|\), where \(m\) and \(n\) are natural numbers.


A swimmer goes downstream in a river from point \( P \) to point \( Q \), and then crosses a still lake to point \( R \), taking a total of 3 hours. When the swimmer travels from \( R \) to \( Q \) and then back to \( P \), it takes 6 hours. If the lake water is also flowing with the same speed as the river, the journey from \( P \) to \( Q \) and then to \( R \) takes \(\frac{5}{2}\) hours. Under these conditions, how many hours does it take for the swimmer to travel from \( R \) to \( Q \) and then to \( P \)?


Does there exist a six-digit natural number which, when multiplied by 9, results in the same digits but in reverse order?


As shown in the figure, quadrilateral \(ABCD\) is a square, \(ABGF\) and \(FGCD\) are rectangles, point \(E\) is on \(AB\), and \(EC\) intersects \(FG\) at point \(M\). If \(AB = 6\) and the area of \(\triangle ECF\) is 12, find the area of \(\triangle BCM\).


Given positive real numbers \( x \) and \( y \) satisfy:
\[
\left(2 x+\sqrt{4 x^{2}+1}\right)\left(\sqrt{y^{2}+4}-2\right) \geqslant y
\]
then the minimum value of \( x + y \) is ______.


The height of an isosceles trapezoid is $h$. The upper base of the trapezoid is seen from the midpoint of the lower base at an angle of $2 \alpha$, and the lower base is seen from the midpoint of the upper base at an angle of $2 \beta$. Find the area of the trapezoid in the general case and calculate it without tables if $h=2, \alpha=15^{\circ}, \beta=75^{\circ}$.


In the trapezoid \(ABCD\), which is circumscribed about a circle, \(BC \parallel AD\), \(AB = CD\), and \(\angle BAD = 45^\circ\). The area of the trapezoid is 10.

Find \(AB\).


A line parallel to the bases of a trapezoid divides it into two similar trapezoids.

Find the segment of this line that is enclosed within the trapezoid, given that the lengths of the bases are \( a \) and \( b \).

Solution: \boxed{\sqrt{ab}}


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


Given vectors \( u_1, u_2, \ldots, u_n \) in the plane, each with a length of at most 1, and with a sum of zero, show that these vectors can be rearranged into \( v_1, v_2, \ldots, v_n \) such that all partial sums \( v_1, v_1 + v_2, v_1 + v_2 + v_3, \ldots, v_1 + v_2 + \cdots + v_n \) have lengths at most \(\sqrt{5}\).


Petya and Vasya simultaneously set off on scooters towards each other. There is a bridge exactly in the middle between them. The road from Petya to the bridge is paved, while the road from Vasya to the bridge is dirt. It is known that they travel at the same speeds on the dirt road, but Petya moves 3 times faster on the paved road than on the dirt road. Petya reached the bridge in one hour and continued moving without stopping. How much time after starting will Petya meet Vasya?


Find all values of \( a \) for which the equation \( a^{2}(x-2) + a(39-20x) + 20 = 0 \) has at least two distinct roots.


For the quadratic polynomial \( f(x) = ax^2 - ax + 1 \), it is known that \( |f(x)| \leq 1 \) for \( 0 \leq x \leq 1 \). Find the maximum possible value of \( a \).


Let \( f(x) = x^4 + ax^3 + bx^2 + cx + d \) be a polynomial whose roots are all negative integers. If \( a + b + c + d = 2009 \), find \( d \).


Mr. Pipkins said, "I was walking along the road at a speed of $3 \frac{1}{2}$ km/h when suddenly a car sped past me, almost knocking me off my feet."

"What was its speed?" his friend asked.

"I can tell you now. From the moment it sped past me until it disappeared around the bend, I took 27 steps. Then I continued walking without stopping and covered another 135 steps until I reached the bend."

"We can easily determine the speed of the car if we assume that both your speeds were constant," his friend replied.

Solution: \boxed{21 \text{ km/h}}


The Small and Large islands have a rectangular shape and are divided into rectangular counties. In each county, a road is laid along one of the diagonals. On each island, these roads form a closed path that does not pass through any point twice. Here's how the Small island is arranged, where there are only six counties (see the figure).

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


There are 10 cups on a table, 5 of them with the opening facing up and 5 with the opening facing down. Each move involves flipping 3 cups simultaneously. What is the minimum number of moves required to make all cup openings face the same direction?


Let \( f(x) = x^3 - x^2 \). For a given value of \( c \), the graph of \( f(x) \), together with the graph of the line \( c + x \), split the plane up into regions. Suppose that \( c \) is such that exactly two of these regions have finite area. Find the value of \( c \) that minimizes the sum of the areas of these two regions.


Find all natural numbers \( n \) for which the fractions \( \frac{1}{n} \) and \( \frac{1}{n+1} \) can both be expressed as finite decimal numbers.

Solution: \boxed{1 \text{ and } 4}


The right triangular prism \(ABC-A_1B_1C_1\) has a base \(\triangle ABC\) which is an equilateral triangle. Points \(P\) and \(E\) are movable points (including endpoints) on \(BB_1\) and \(CC_1\) respectively. \(D\) is the midpoint of side \(BC\), and \(PD \perp PE\). Find the angle between lines \(AP\) and \(PE\).


Let's call two positive integers almost neighbors if each of them is divisible (without remainder) by their difference. In a math lesson, Vova was asked to write down in his notebook all the numbers that are almost neighbors with \(2^{10}\). How many numbers will he have to write down?


In an isosceles trapezoid, one of the acute angles at the base is $45^{\circ}$. Show that the area of this trapezoid is obtained by dividing the difference of the squares of the parallel sides by 4.

Solution: \boxed{\frac{a^2 - b^2}{4}}


Find the perimeter of a rectangle if the sum of the lengths of its three different sides can be equal to 6 or 9.


For how many integer values of \( b \) does there exist a polynomial function with integer coefficients such that \( f(2) = 2010 \) and \( f(b) = 8 \)?


Given the parabola \( y^2 = 4x \) with focus \( F \) and vertex \( O \), let \( M \) be a variable point on the parabola. What is the maximum value of \( \frac{|MO|}{|MF|} \)?

A. \( \frac{\sqrt{3}}{3} \)

B. \( \frac{2\sqrt{3}}{3} \)

C. \( \frac{4}{3} \)

D. \( \sqrt{3} \)

Solution: \boxed{ B}


Alex has a $20 \times 16$ grid of lightbulbs, initially all off. He has 36 switches, one for each row and column. Flipping the switch for the $i$th row will toggle the state of each lightbulb in the $i$th row (so that if it were on before, it would be off, and vice versa). Similarly, the switch for the $j$th column will toggle the state of each bulb in the $j$th column. Alex makes some (possibly empty) sequence of switch flips, resulting in some configuration of the lightbulbs and their states. How many distinct possible configurations of lightbulbs can Alex achieve with such a sequence? Two configurations are distinct if there exists a lightbulb that is on in one configuration and off in another.


Find the sum of all even numbers from 10 to 31. Calculate using different methods.


The pairwise products \( ab, bc, cd \), and \( da \) of positive integers \( a, b, c, \) and \( d \) are 64, 88, 120, and 165 in some order. Find \( a + b + c + d \).


Solution: \boxed{x = 1}


Let the set
\[ S=\{1, 2, \cdots, 12\}, \quad A=\{a_{1}, a_{2}, a_{3}\} \]
where \( a_{1} < a_{2} < a_{3}, \quad a_{3} - a_{2} \leq 5, \quad A \subseteq S \). Find the number of sets \( A \) that satisfy these conditions.


The cube $A B C D A^{\prime} B^{\prime} C^{\prime} D^{\prime}$ shown in Figure 2 is cut into two parts by a plane passing through point $A$, the trisection point $H$ on the edge $C C^{\prime}$ closer to $C$, and the midpoint $F$ on the edge $D D^{\prime}$. Determine the volume ratio of the two parts.


Three friends are sitting in front of the TV. It is known that each of them is either always right about everything or always wrong about everything. The first said: "None of us have seen this movie." The second said: "I have seen this movie, but both of you have not." The third said: "I have seen this movie." Determine how many of these friends are always right, given that at least one of them said everything correctly and at least one of them made a mistake.


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


Given point $O$ is the origin, $\overrightarrow{O M}=(-1,1)$, $\overrightarrow{N M}=(-5,-5)$, set $A=\{\overrightarrow{O R} | R N \mid=2\}$, $\overrightarrow{O P}, \overrightarrow{O Q} \in A, \overrightarrow{M P}=\lambda \overrightarrow{M Q}(\lambda \in \mathbf{R}, \lambda \neq 0)$, find $\overrightarrow{M P} \cdot \overrightarrow{M Q}$.


Given a sequence \( \left\{a_n\right\} \) that satisfies \( a_n=\left[(2+\sqrt{5})^n+\frac{1}{2^n}\right] \) for \( n \in \mathbf{N}^* \), where \([x]\) represents the greatest integer less than or equal to the real number \( x \). Let \( C \) be a real number such that for any positive integer \( n \), the following holds:
\[ \sum_{k=1}^{n} \frac{1}{a_k a_{k+2}} \leqslant C \]
Find the minimum value of \( C \).


Find the minimum and maximum values of the function \( f(x) \) on the interval \([0, 2]\). Provide the sum of these values.

\[
f(x) = (x + 1)^5 + (x - 1)^5
\]


For any finite sequence of positive integers \(\pi\), let \(S(\pi)\) be the number of strictly increasing subsequences in \(\pi\) with length 2 or more. For example, in the sequence \(\pi=\{3,1,2,4\}\), there are five increasing subsequences: \(\{3,4\}, \{1,2\}, \{1,4\}, \{2,4\}\), and \(\{1,2,4\}\), so \(S(\pi)=5\). In an eight-player game of Fish, Joy is dealt six cards of distinct values, which she puts in a random order \(\pi\) from left to right in her hand. Determine
$$
\sum_{\pi} S(\pi)
$$
where the sum is taken over all possible orders \(\pi\) of the card values.


For which $n$ can an $n \times n$ grid be divided into one $2 \times 2$ square and some number of strips consisting of five cells, in such a way that the square touches the side of the board?

Solution: \boxed{n \equiv 2 \pmod{5}}


Using 12 different animal patterns to make some animal cards, with each card containing 4 different animal patterns, such that any two cards have exactly one animal pattern in common. What is the maximum number of cards that can be made?


In $\triangle ABC$, $AB=\sqrt{5}$, $BC=1$, and $AC=2$. $I$ is the incenter of $\triangle ABC$ and the circumcircle of $\triangle IBC$ intersects $AB$ at $P$. Find $BP$.

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


On the surface of a sphere with a radius of 1, there are three points \( A \), \( B \), and \( C \). Given that the spherical distances from \( A \) to \( B \) and \( C \) are both \(\frac{\pi}{2}\), and the spherical distance between \( B \) and \( C \) is \(\frac{\pi}{3}\), find the distance from the center of the sphere to the plane \( ABC \).