On a bench of one magistrate, there are two Englishmen, two Scots, two Welshmen, one Frenchman, one Italian, one Spaniard, and one American sitting. The Englishmen do not want to sit next to each other, the Scots do not want to sit next to each other, and the Welshmen also do not want to sit next to each other.

In how many different ways can these 10 magistrate members sit on the bench so that no two people of the same nationality sit next to each other?


If \( n \) is a positive integer such that \( n^{6} + 206 \) is divisible by \( n^{2} + 2 \), find the sum of all possible values of \( n \).


For given numbers \( n \in \mathbf{N} \) and \( a \in [0; n] \), find the maximum value of the expression

\[ \left|\sum_{i=1}^{n} \sin 2 x_{i}\right| \]

under the condition that

\[ \sum_{i=1}^{n} \sin^{2} x_{i} = a. \]

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


Calculate the limit of the numerical sequence:

$$\lim _{n \rightarrow \infty} \frac{n^{2}-\sqrt{n^{3}+1}}{\sqrt[3]{n^{6}+2}-n}$$


Calculate the double integral

$$
\iint_{D} x \, dx \, dy
$$

where the region \( D \) is bounded by the lines

$$
y^{2} - 4y + x^{2} = 0, \quad y^{2} - 8y + x^{2} = 0, \quad y = \frac{x}{\sqrt{3}}, \quad x = 0
$$


Rowena has a very long, level backyard. From a particular point, she determines that the angle of elevation to a large tree in her backyard is \(15^{\circ}\). She moves \(40 \mathrm{~m}\) closer and determines that the new angle of elevation is \(30^{\circ}\). How tall, in metres, is the tree?

Solution: \boxed{20 \text{ meters}}


Given a line $f$ and two parallel lines $e$ and $g$ that are each at a unit distance from $f$ on either side. Let points $E$, $F$, and $G$ be on lines $e$, $f$, and $g$ respectively, such that the triangle formed by $E$, $F$, and $G$ has a right angle at $F$. What is the length of the altitude of the triangle from point $F$?


A group of $n$ friends takes $r$ distinct photos (no two photos have exactly the same people), where each photo contains at least one person. Find the maximum $r$ such that for every pair of photos, there is at least one person who appears in both.


Observe the equation:

$$
\begin{aligned}
(1+2+3+4)^{2} & =(1+2+3+4)(1+2+3+4) \\
& =1 \cdot 1+1 \cdot 2+1 \cdot 3+1 \cdot 4+2 \cdot 1+2 \cdot 2+2 \cdot 3+2 \cdot 4+ \\
& +3 \cdot 1+3 \cdot 2+3 \cdot 3+3 \cdot 4+4 \cdot 1+4 \cdot 2+4 \cdot 3+4 \cdot 4
\end{aligned}
$$

Note that \(4 \times 4=16\) products are formed when calculating \((1+2+3+4)^{2}\) using the distributive property.

a) How many products will be formed when calculating \((1+2+3+4)^{3}\) also using the distributive property?

b) What is the quantity of two-digit numbers that use only the digits 1, 2, 3, and 4?

c) What is the sum of the products of the digits of all four-digit numbers formed using only the digits 1, 2, 3, and 4?


The two-digit numbers $\overline{ab}$ and $\overline{\mathrm{ba}}$ are both prime numbers. How many such $\overline{ab}$ are there?


A natural number \( 1 \leq n \leq 221 \) is called lucky if, when dividing 221 by \( n \), the remainder is wholly divisible by the incomplete quotient (the remainder can be equal to 0). How many lucky numbers are there?


Investigate the convergence of the series \(\sum_{n=2}^{\infty} \frac{18}{n^{2}+n-2}\) and, if possible, find its sum.


A regular triangular prism \( ABC A_1 B_1 C_1 \) is inscribed in a sphere. The base of the prism is \(ABC \) and the lateral edges are \(AA_1, BB_1, CC_1\). Segment \(CD\) is the diameter of this sphere, and points \(K\) and \(L\) are the midpoints of edge \(AA_1\) and \(AB\) respectively. Find the volume of the prism if \(DL = \sqrt{2}\) and \(DK = \sqrt{3}\).


Given the circle \( \odot O: x^{2}+y^{2}=4 \) and the curve \( C: y=3|x-t| \), points \( A(m, n) \) and \( B(s, p) \) (where \( m, n, s, p \in \mathbf{Z}_+ \)) are on the curve \( C \). These points are such that the ratio of the distance from any point on the circle \( \odot O \) to point \( A \) and to point \( B \) is a constant \( k \) (where \( k > 1 \)). Determine the value of \( t \).


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


Let \( a, b, c \) be prime numbers such that \( a^5 \) divides \( b^2 - c \), and \( b + c \) is a perfect square. Find the minimum value of \( abc \).


The alphabet of the inhabitants of the fairy-tale planet ABV2020 consists of only three letters: A, B, and V, from which all words are formed. In any word, no two identical letters can be adjacent, and each word must contain all three letters. For example, the words AVB, VABAVA, BVBBVVA are allowed, while the words VAV, ABAAVA, AVABB are not. How many 20-letter words are there in the dictionary of this planet?


Exactly at noon, a truck left the village and headed towards the city, and at the same time, a car left the city and headed towards the village. If the truck had left 45 minutes earlier, they would have met 18 kilometers closer to the city. If the car had left 20 minutes earlier, they would have met $k$ kilometers closer to the village. Find $k$.


In how many ways can you place pieces on a chessboard so that they do not attack each other:
a) two rooks
b) two kings
c) two bishops
d) two knights
e) two queens?

All pieces are of the same color.


How many ways are there to arrange the integers from 1 to \( n \) into a sequence where each number, except for the first one on the left, differs by 1 from some number to its left (not necessarily the adjacent one)?


Given a natural number \( n \geq 3 \), the numbers \( 1, 2, 3, \ldots, n \) are written on a board. In each move, two numbers are chosen and replaced by their arithmetic mean. This process continues until only one number remains on the board. Determine the smallest integer that can be achieved at the end through an appropriate sequence of moves.


Find all functions \( f: \mathbf{Z} \rightarrow \mathbf{Z} \) such that for all \( n \in \mathbf{Z} \), \( f[f(n)] + f(n) = 2n + 3 \), and \( f(0) = 1 \).

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


In an alphabet with $n>1$ letters, a word is defined as any finite sequence of letters where any two consecutive letters are different. A word is called "good" if it is not possible to delete all but four letters from it to obtain a sequence of the form $a a b b$, where $a$ and $b$ are different letters. Find the maximum possible number of letters in a "good" word.


All the domino pieces were laid in a chain. One end had 5 points. How many points are on the other end?


In the rectangular coordinate system $xOy$, the coordinates of point $A\left(x_{1}, y_{1}\right)$ and point $B\left(x_{2}, y_{2}\right)$ are both positive integers. The angle between $OA$ and the positive direction of the x-axis is greater than $45^{\circ}$, and the angle between $OB$ and the positive direction of the x-axis is less than $45^{\circ}$. The projection of $B$ on the x-axis is $B^{\prime}$, and the projection of $A$ on the y-axis is $A^{\prime}$. The area of $\triangle OB^{\prime} B$ is 33.5 larger than the area of $\triangle OA^{\prime} A$. The four-digit number formed by $x_{1}, y_{1}, x_{2}, y_{2}$ is $\overline{x_{1} x_{2} y_{2} y_{1}}=x_{1} \cdot 10^{3}+x_{2} \cdot 10^{2}+y_{2} \cdot 10+y_{1}$. Find all such four-digit numbers and describe the solution process.


By finding a certain sixth-degree polynomial \( x^{6} + a_{1} x^{5} + \ldots + a_{5} x + a_{6} \) with integer coefficients, one of whose roots is \( \sqrt{2} + \sqrt[3]{5} \), write the sum of its coefficients \( a_{1} + a_{2} + \ldots + a_{6} \) in the answer.


Given the function \( f(x) = x^3 + 3x^2 + 6x + 14 \), where \( f(a) = 1 \) and \( f(b) = 19 \), find the value of \( a + b \).


Given positive numbers \( a, b, c, x, y, z \) such that \( cy + bz = a \), \( az + cx = b \), and \( bx + ay = c \), find the minimum value of the function \( f(x, y, z) = \frac{x^2}{1+x} + \frac{y^2}{1+y} + \frac{z^2}{1+z} \).


Calculate the integral

$$
\int_{0}^{0.1} \cos \left(100 x^{2}\right) d x
$$

with an accuracy of $\alpha=0.001$.


Let \( l \) and \( m \) be two skew lines. On \( l \), there are three points \( A, B, \) and \( C \), such that \( AB = BC \). From points \( A, B, \) and \( C \), perpendiculars \( AD, BE, \) and \( CF \) are respectively drawn to \( m \), with feet of the perpendiculars at \( D, E, \) and \( F \). Given \( AD = \sqrt{15}, BE = \frac{7}{2}, CF = \sqrt{10} \), find the distance between lines \( l \) and \( m \).


Given natural numbers \( M \) and \( N \), both greater than ten, consisting of the same number of digits, and such that \( M = 3N \). To obtain the number \( M \), one of the digits in \( N \) is increased by 2, and each of the remaining digits is increased by an odd digit. What digit could the number \( N \) end with? Find all possible answers.

(N. Agahanov)


Given that four integers \( a, b, c, d \) are all even numbers, and \( 0 < a < b < c < d \), with \( d - a = 90 \). If \( a, b, c \) form an arithmetic sequence and \( b, c, d \) form a geometric sequence, then find the value of \( a + b + c + d \).


Let \( m = 999 \ldots 99 \) be the number formed by 77 digits all equal to 9 and let \( n = 777 \ldots 77 \) be the number formed by 99 digits all equal to 7. What is the number of digits of \( m \cdot n \)?


In the expression \((x+y+z)^{2032}+(x-y-z)^{2032}\), the parentheses were expanded and like terms were combined. How many monomials \(x^{a} y^{b} z^{c}\) with a nonzero coefficient are obtained?


The probability of failure for each device during testing is 0.2. How many such devices need to be tested so that, with a probability not less than 0.9, it can be stated that at least two devices will fail?


Given a natural number \( n \), the polynomial
\[ P(x) = \sum_{h=0}^{n} \binom{n}{h} x^{n-h}(x-1)^{h} \]
can be expanded into an ascending order of \( x \) as \( a_{0} + a_{1} x + a_{2} x^{2} + \cdots + a_{n} x^{n} \). Find
\[ \left| a_{0} \right| + \left| a_{1} \right| + \left| a_{2} \right| + \cdots + \left| a_{n} \right|. \]


Given fifty distinct natural numbers, twenty-five of which do not exceed 50, and the remaining are greater than 50 but do not exceed 100. Additionally, no two of these numbers differ by exactly 50. Find the sum of these numbers.


Vanya goes to the swimming pool every Wednesday and Friday. After his latest visit, he noticed that he had already gone 10 times this month. What will be the date of his third visit next month if he continues going on Wednesdays and Fridays?


Calculate \(\sqrt{31 \times 30 \times 29 \times 28 + 1}\).


Determine the mass fraction of potassium sulfate in the original solution:

$$
\omega\left(K_{2} \mathrm{SO}_{4}\right)=\frac{m\left(K_{2} \mathrm{SO}_{4}\right) \cdot 100 \%}{m_{\mathrm{p}-\mathrm{pa}}\left(K_{2} \mathrm{SO}_{4}\right)}=\frac{2,61 \cdot 100 \%}{160}=1,63 \%
$$


Let \( A B C \) be a right triangle with \(\angle BAC = 90^{\circ}\) and \(I\) the intersection point of its angle bisectors. A line through \(I\) intersects the sides \(AB\) and \(AC\) at \(P\) and \(Q\) respectively. The distance from \(I\) to the side \(BC\) is \(1 \, \text{cm}\).

a) Find the value of \( PM \cdot NQ \).

b) Determine the minimum possible value for the area of triangle \(APQ\).

Hint: If \(x\) and \(y\) are two non-negative real numbers, then \(x + y \geq 2 \sqrt{xy}\).


Vanya thought of two natural numbers whose product is 7200. What is the greatest possible value of the greatest common divisor (GCD) of these numbers?


Lucas writes two distinct positive integers on a whiteboard. He decreases the smaller number by 20 and increases the larger number by 23, only to discover the product of the two original numbers is equal to the product of the two altered numbers. Compute the minimum possible sum of the original two numbers on the board.


An 8 by 6 grid is placed in the first quadrant with its edges along the axes. A total of 32 of the squares in the grid are shaded. A line is drawn through (0,0) and (8, c) cutting the shaded region into two equal areas. What is the value of c?

Solution: \boxed{c = 4}


Calculate the definite integral:

$$
\int_{0}^{\pi} 2^{4} \cdot \sin ^{6} x \cos ^{2} x \, dx
$$


The sum of the elements of all subsets of a set \( S \) of 4 real numbers is 2008 (here, the sum of the elements of the empty set is considered to be 0). Find the sum of all elements in \( S \).


Let \(a\) and \(b\) be strictly positive integers such that \(a^n\) divides \(b^{n+1}\) for all integers \(n \geq 1\). Show that \(a\) divides \(b\).

Solution: \boxed{a \mid b}


For each pair of real numbers \((x, y)\) with \(0 \leq x \leq y \leq 1\), consider the set
\[ A = \{ x y, x y - x - y + 1, x + y - 2 x y \}. \]

Let the maximum value of the elements in set \(A\) be \(M(x, y)\). Find the minimum value of \(M(x, y)\).


Calculate the sum of the cubes of two numbers if their sum and product are 11 and 21, respectively.


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


Points \( A, B, C \) lie on a circle \(\omega\) such that \( BC \) is a diameter. \( AB \) is extended past \( B \) to point \( B' \) and \( AC \) is extended past \( C \) to point \( C' \) such that line \( B'C' \) is parallel to \( BC \) and tangent to \(\omega\) at point \( D \). If \( B'D = 4 \) and \( C'D = 6 \), compute \( BC \).


Suppose that \((a_1, b_1), (a_2, b_2), \ldots, (a_{100}, b_{100})\) are distinct ordered pairs of nonnegative integers. Let \(N\) denote the number of pairs of integers \((i, j)\) satisfying \(1 \leq i < j \leq 100\) and \(\left|a_i b_j - a_j b_i\right| = 1\). Determine the largest possible value of \(N\) over all possible choices of the 100 ordered pairs.


The positive numbers \( x \) and \( y \) change such that \( x > y \) and \( x^{2} + y^{2} = 18xy \). Show that the following expression is constant:

\[
\log (x - y) - \frac{1}{2} (\log x + \log y)
\]

The base of the logarithms is \( \sqrt{2} \).


In a right triangle \(ABC\) with legs \(AB = 10\) and \(AC = 24\), medians \(AM\) and \(CN\) are drawn, where \(M\) and \(N\) are the midpoints of the sides \(BC\) and \(AB\), respectively. Point \(I\) lies on the side \(AC\), and \(BI\) is the angle bisector. Find the area of triangle \(MNI\).


Arrange all proper fractions in a sequence $\left\{a_{n}\right\}: \frac{1}{2}, \frac{1}{3}, \frac{2}{3}, \frac{1}{4}, \frac{2}{4}, \frac{3}{4}, \cdots$. The sorting method is: from left to right, the denominators are arranged in ascending order, and for fractions with the same denominator, the numerators are arranged in ascending order. What is the 2017th term $a_{2017}$ of this sequence?


Given a sequence \(\{a_n\}\) which satisfies \(a_n = a_{n-1} - a_{n-2}\) for \(n \geqslant 3\), if the sum of its first 1492 terms is 1985 and the sum of its first 1985 terms is 1492, find the sum of its first 2001 terms.


The decimal representation of a natural number \( N \) contains 1580 digits. Among these digits, there are threes, fives, and sevens, and there are no other digits. It is known that the number of sevens is 20 less than the number of threes. Find the remainder when \( N \) is divided by 3.


A box contains 10 identical balls. Among them, 3 balls are labeled with the number 1, 4 balls are labeled with the number 2, and 3 balls are labeled with the number 5. One ball is randomly selected from the box, recorded, and then placed back into the box. Another ball is then randomly selected. Let the sum of the numbers on the balls from the first and second selections be denoted as $\xi$.
(I) Find the distribution of the random variable $\xi$.
(II) Find the expected value $E(\xi)$ of the random variable $\xi$.


There are functions \( f(x) \) with the following properties:

- \( f(x) = a x^2 + b x + c \) for some integers \( a, b \) and \( c \) with \( a > 0 \), and
- \( f(p) = f(q) = 17 \) and \( f(p+q) = 47 \) for some prime numbers \( p \) and \( q \) with \( p < q \).

For each such function, the value of \( f(p q) \) is calculated. The sum of all possible values of \( f(p q) \) is \( S \). What are the rightmost two digits of \( S \)?


Given plane vectors $\boldsymbol{a}$ and $\boldsymbol{b}$ such that $|\boldsymbol{a}+\boldsymbol{b}|=3$, find the maximum value of $\boldsymbol{a} \cdot \boldsymbol{b}$.


Calculate the limit of the function:

$$\lim _{x \rightarrow a} \frac{\operatorname{tg} x-\operatorname{tg} a}{\ln x-\ln a}$$

Solution: \boxed{\frac{a}{\cos^2 a}}


Yaw yaw and her father are discussing the upcoming year 2016. Yaw yaw says: "I find the number 2016 interesting because the last digit 6 is exactly twice the sum of the preceding digits 2, 0, and 1." Her father says that they can define a year that meets this condition as a "lucky double year." Yaw yaw quickly calculates and says: "After 2016, the next 'lucky double year' will be when I am 17 years old." How old is Yaw yaw in 2016?


A circle passes through the intersection points of any two of the lines \( y = x \), \( y = 2x \), and \( y = 15 - 0.5x \). What is the length of the chord of the circle that lies on the line \( x + y = 16 \)?


How many triangles exist in which the measures of the angles, measured in degrees, are whole numbers?


Given the function \( f(x)=\sqrt{3} \sin 2x + 2 \cos^2 x + a \), if the minimum value of \( f(x) \) on the interval \(\left[ 0, \frac{\pi}{2} \right] \) is \(-1\), find the value of \( a \).


From the three-digit numbers \(100, 101, 102, \ldots, 699, 700\), choose \(n\) different numbers such that you can always find three of them with the same sum of digits. What is the minimum value of \(n\)?


If \( n \) is a positive integer, and \( a(n) \) is the smallest positive integer such that \( (a(n))! \) is divisible by \( n \), find all positive integers \( n \) such that \( \frac{a(n)}{n} = \frac{2}{3} \). (2003 German Math Olympiad)


If from point $P$ two perpendicular chords are drawn to a given circle, then the sum of the squares of the resulting four segments is constant.


Find the smallest two-digit number \( N \) such that the sum of digits of \( 10^N - N \) is divisible by 170.


The three different points \(A(x_1, y_1)\), \(B\left(4, \frac{9}{5}\right)\), and \(C(x_2, y_2)\) on the ellipse \(\frac{x^{2}}{25} + \frac{y^{2}}{9} = 1\), along with the focus \(F(4,0)\) have distances that form an arithmetic sequence. If the perpendicular bisector of line segment \(AC\) intersects the x-axis at point \(T\), find the slope \(k\) of the line \(BT\).


In $\triangle ABC$, the side lengths opposite to the angles $\angle A$, $\angle B$, and $\angle C$ are $a$, $b$, and $c$ respectively, and it is given that
$$
\begin{array}{l}
\sin C \cdot \cos \frac{A}{2}=(2-\cos C) \sin \frac{A}{2}, \\
\cos A=\frac{3}{5}, \quad a=4.
\end{array}
$$

Then the area of $\triangle ABC$ is ___.


A few natural numbers were multiplied together to get 224, and the smallest number was exactly half of the largest number.

How many numbers were multiplied?


What is the integer part of the following number:

$$
\sqrt{6+\sqrt{6+\ldots+\sqrt{6}}}+\sqrt[3]{6+\sqrt[3]{6+\cdots+\sqrt[3]{6}}}
$$

where both the number of square roots and cube roots is 100?


The function \( y = f(t) \) is such that the sum of the roots of the equation \( f(\sin x) = 0 \) in the interval \([3 \pi / 2, 2 \pi]\) is \( 33 \pi \), and the sum of the roots of the equation \( f(\cos x) = 0 \) in the interval \([\pi, 3 \pi / 2]\) is \( 23 \pi \). What is the sum of the roots of the second equation in the interval \([\pi / 2, \pi]?\)

Solution: \boxed{17 \pi}


How many different combinations of 4 marbles can be made from 5 indistinguishable red marbles, 4 indistinguishable blue marbles, and 2 indistinguishable black marbles?


Given the sets \(A=\{x \mid x^{2}-4x+3<0, x \in \mathbf{R}\}\) and \(B=\{x \mid 2^{1-x}+a \leq 0, x^{2}-2(a+7)x+5 \leq 0, x \in \mathbf{R}\}\). If \(A \subset B\), find the range of values for the real number \(a\).

Solution: \boxed{[-4, -1]}


Starting from 1, there are n consecutive natural numbers. If one of these numbers is removed, the average of the remaining numbers is $\frac{152}{7}$. What is the number that was removed?


The parabola \(C_{1}: x^{2}=2 py\) has its focus at \(F\). The hyperbola \(C_{2}: \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\) has its left and right foci at \(F_{1}\) and \(F_{2}\). Point \(P\) is a common point of the two curves in the first quadrant. If points \(P\), \(F\), and \(F_{1}\) are collinear and \(C_{1}\) and \(C_{2}\) share a common tangent at \(P\), find the eccentricity of \(C_{2}\).

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


A cone has a height of $12 \mathrm{~cm}$. When the lateral surface is unwrapped, we obtain a circular sector with a central angle of $100^{\circ} 48^{\prime}$. What is the surface area and the volume of the cone?

Solution: \boxed{49 \pi \text{ cm}^3}


The area of trapezoid $ABCD$ is 90. The diagonals intersect at point $O$. Segments connecting the midpoint $P$ of the base $AD$ with vertices $B$ and $C$ intersect with the diagonals of the trapezoid at points $M$ and $N$. Find the area of quadrilateral $OMPN$ if one of the bases of the trapezoid is twice as long as the other.


An equilateral triangle with side length \( n \) is divided into \( n^2 \) smaller equilateral triangles each with side length 1 by lines parallel to its sides. This creates a network of nodes connected by line segments of length 1. What is the maximum number of segments that can be chosen such that no three chosen segments form a triangle?


Three friends play the following game. They position themselves at the points with coordinates $(0,0),(1,0)$, and $(0,1)$. Every minute, one of the three friends moves somewhere along the line parallel to the line passing through the positions of the other two friends. Can they reach the coordinates $(0,0),(1,1),(0,2)$?

Solution: \boxed{\text{No}}


\(ABCD\) is a square with side length 2. \(E\) is the midpoint of \(AB\), \(F\) is the midpoint of \(BC\), \(AF\) and \(DE\) intersect at \(I\), and \(BD\) and \(AF\) intersect at \(H\). Find the area of quadrilateral \(BEIH\).


Find the largest real number \( p \) such that all three roots of the equation below are positive integers:
\[
5x^{3} - 5(p+1)x^{2} + (71p-1)x + 1 = 66p .
\]


Natural numbers \(a, b, c\) are such that \(\gcd(\operatorname{lcm}(a, b), c) \cdot \operatorname{lcm}(\gcd(a, b), c) = 200\).

What is the maximum value of \(\gcd(\operatorname{lcm}(a, b), c) ?\)


Let $G$ be a graph with $n$ vertices that does not contain any triangles (cycles of length 3). Show that $G$ has at most $n^{2} / 4$ edges.


Find the number of ways to color all natural numbers from 1 to 20 in blue and red so that both colors are used and the product of all red numbers is relatively prime to the product of all blue numbers.


Find the smallest positive integer \( n \) such that the set \( \{1, 2, 3, \cdots, 3n-1, 3n\} \) can be divided into \( n \) disjoint triples \( \{x, y, z\} \) where \( x + y = 3z \).


The set of positive even numbers $\{2, 4, 6, \cdots\}$ is grouped in increasing order such that the $n$-th group has $3n-2$ numbers:
\[
\{2\}, \{4, 6, 8, 10\}, \{12, 14, 16, 18, 20, 22, 24\}, \cdots
\]

Determine which group contains the number 2018.


In the Rhind Papyrus (Ancient Egypt), among other information, there are decompositions of fractions into a sum of fractions with numerator 1. For example,

\[
\frac{2}{73} = \frac{1}{60} + \frac{1}{219} + \frac{1}{292} + \frac{1}{x}
\]

One of the denominators is replaced by the letter \( x \). Find this denominator.


What is the exact value of the product $\sin 10^{\circ} \sin 30^{\circ} \sin 50^{\circ} \sin 70^{\circ}$?


Find the range of the function

$$
f(x)=\sqrt{g^{2}(x)-245}, \text { where } g(x)=15-2 \cos 2x-4 \sin x
$$

Solution: \boxed{[0, 14]}


Robinson had 200,000 strands of hair when he ended up on the deserted island. At that time, his hair strands were $5 \mathrm{~cm}$ long. The hair strands grew $0.5 \mathrm{~mm}$ per day, but Robinson did not cut his hair because he did not have the proper tools, and he lost 50 strands of hair each day without replacement.

How many days passed until the total length of the hair strands on Robinson's head reached its maximum?


In how many ways can the numbers from 1 to 100 be rearranged so that neighboring numbers differ by no more than 1?


What is the smallest positive integer that leaves a remainder of 2 when divided by 5, a remainder of 3 when divided by 7, and a remainder of 4 when divided by 9?


Given \(a, b > 0\), satisfy the equation for \(x\):
\[
\sqrt{|x|} + \sqrt{|x + a|} = b
\]

There are exactly three distinct real solutions \(x_1, x_2, x_3\), where \(x_1 < x_2 < x_3 = b\). Find the value of \(a + b\).


Determine all pairs of positive integers \((a, n)\) such that

$$
3^{n} = a^{2} - 16
$$

Solution: \boxed{(5, 2)}


a) Show that for any \(a \in \mathbb{N}\), the equation \(x^{2} + y^{2} + z^{2} = 8a + 7\) has no solutions in \(\mathbb{Q}^{3}\).

b) Find all natural numbers \(n\) such that \(7^{n} + 8\) is a perfect square.


Find all integers \( x, y \geq 1 \) such that \( 7^x = 3^y + 4 \).

Solution: \boxed{(1, 1)}


The area of an isosceles trapezoid circumscribed around a circle is $S$. Determine the radius of this circle if the angle at the base of the trapezoid is $30^{\circ}$.


The focus of the parabola \( y^{2} = 2px \) (with \( p > 0 \)) is \( F \), and the directrix is \( l \). Points \( A \) and \( B \) are points on the parabola such that \( \angle AFB = \frac{\pi}{3} \). Let \( M \) be the midpoint of the line segment \( AB \), and let \( N \) be the projection of \( M \) onto \( l \). What is the maximum value of \( \frac{|MN|}{|AB|} \)?


How many 100-digit natural numbers are there, each digit of which is even and the difference between any two adjacent digits is $2$?

Solution: \boxed{7 \cdot 3^{49}}


Vasya added the same natural number, which is less than both the numerator and the denominator, to the numerator and the denominator of a proper fraction. As a result, the fraction increased by more than 50%. Vasya claims that if he subtracts this number from the numerator and the denominator of the original fraction, the fraction will decrease by less than 50%. Is this possible?

Solution: \boxed{\text{No}}


Given a function \( f: \mathbf{N}^{*} \rightarrow \mathbf{N}^{*} \) satisfying \( f(f(m) + f(n)) = m + n \) for all \( m, n \in \mathbf{N}^{*} \), find \( f(2005) \).


Find the smallest positive real t such that
\[ x_1 + x_3 = 2tx_2, \]
\[ x_2 + x_4 = 2tx_3, \]
\[ x_3 + x_5 = 2tx_4 \]
has a solution \( x_1, x_2, x_3, x_4, x_5 \) in non-negative reals, not all zero.


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


In trapezoid \(A B C D\), the bases \(A D\) and \(B C\) are 8 and 18, respectively. It is known that the circumscribed circle of triangle \(A B D\) is tangent to lines \(B C\) and \(C D\). Find the perimeter of the trapezoid.


If a class of 30 students is seated in a movie theater, then in any case at least two classmates will be in the same row. If the same is done with a class of 26 students, then at least three rows will be empty. How many rows are in the theater?


What is the maximum number of \(4 \times 1\) rectangles that can be placed inside a \(6 \times 6\) square (without breaking cell boundaries)?


Let \(a, b, c\) be three distinct real numbers, and let \(p(x)\) be a polynomial with real coefficients. Given that:
1. The remainder when \(p(x)\) is divided by \(x - a\) is \(a\).
2. The remainder when \(p(x)\) is divided by \(x - b\) is \(b\).
3. The remainder when \(p(x)\) is divided by \(x - c\) is \(c\).
Find the remainder when \(p(x)\) is divided by \((x - a)(x - b)(x - c)\). (Problem from the 1990 Italian competition)

Solution: \boxed{x}


Define \( a \star = \frac{a-1}{a+1} \) for \( a \neq -1 \). Determine all real values \( N \) for which \( (N \star) \star = \tan 15^\circ \).

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


Let \( a, b \), and \( c \) be complex numbers such that \( |a|=|b|=|c|=1 \). If
\[ \frac{a^{2}}{b c}+\frac{b^{2}}{c a}+\frac{c^{2}}{a b}=1 \]
as well, determine the product of all possible values of \( |a+b+c| \).


Let $ABCD$ be a convex quadrilateral, with $E$ on $[AB]$ and $F$ on $[CD]$. Assume $AE = BE = CF = DF = EF$. The diagonals of $BCFE$ intersect at $K$ and those of $ADFE$ at $L$. The perpendicular to $[AD]$ passing through $K$ and the perpendicular to $[BC]$ passing through $L$ intersect at $T$.

Show that $TE = TF$.

Solution: \boxed{TE = TF}


Due to the reconstruction of equipment, a worker's productivity increased twice during the year by the same percentage each time. By what percentage did the worker's productivity increase each time if, in the same amount of time, the worker used to produce goods worth 2500 rubles, and now produces goods worth 2809 rubles?

Solution: \boxed{6 \%}


Given that \( n \) is a positive integer, \( P \) is a prime number, and \( pn \) has exactly 8 positive divisors, arrange them in ascending order as \( 1=d_{1}<d_{2}< \cdots <d_{8}=pn \). Additionally, let \( d_{17p-d_{3}}=\left(d_{1}+d_{2}+d_{3}\right)\left(d_{3}+d_{4}+13p\right) \). Find \( n \).


In triangle \(ABC\), lines parallel to the sides of the triangle are drawn through an arbitrary point \(O\). As a result, triangle \(ABC\) is divided into three parallelograms and three triangles. The areas of the resulting triangles are \(6 \, \text{cm}^2\), \(24 \, \text{cm}^2\), and \(54 \, \text{cm}^2\). Find the area of triangle \(ABC\).


Find all positive integers \( n \) for which all positive divisors of \( n \) can be put into the cells of a rectangular table under the following constraints:
- each cell contains a distinct divisor;
- the sums of all rows are equal; and
- the sums of all columns are equal.


Let the sequence $\left\{a_{n}\right\}$ be defined as $a_{n}=(2 n-1)(2 n+1)(2 n+3)$. Determine the greatest common divisor $d$ of the terms $a_{1}, a_{2}, \cdots, a_{2008}$.


In triangle \( ABC \), angle \( B \) is \( 80^\circ \). On side \( BC \), point \( D \) is marked such that \( AB = AD = CD \). On side \( AB \), point \( F \) is marked such that \( AF = BD \). On segment \( AC \), point \( E \) is marked such that \( AB = AE \). Find angle \( AEF \).


Vasya has three cans of paint of different colors. In how many different ways can he paint a fence of 10 boards such that any two adjacent boards are of different colors, and all three colors are used?


Find the minimum value of the function \( u(x, y) = x^{2} + \frac{81}{x^{2}} - 2xy + \frac{18}{x} \sqrt{2 - y^{2}} \) with real variables \( x \) and \( y \).


Given real numbers \( x \) and \( y \) satisfying
\[ 2^x + 3^y = 4^x + 9^y, \]
determine the range of values of \( U = 8^x + 27^y \).

Solution: \boxed{(1, 2]}


10 runners start at the same time: five in blue jerseys from one end of the running track, and five in red jerseys from the other. Their speeds are constant and different, with each runner’s speed being more than 9 km/h but less than 12 km/h. Upon reaching the end of the track, each runner immediately runs back, and upon returning to their starting point, they finish the run. The coach marks a check each time two runners in different colored jerseys meet (either face to face or one catching up with the other) (more than two runners do not meet at a point during the run). How many checks will the coach make by the time the fastest runner finishes their run?


In triangle \(ABC\), a median \(AM\) is drawn. Find the angle \(AMC\) if the angles \(BAC\) and \(BCA\) are \(45^\circ\) and \(30^\circ\) respectively.


Seryozha wrote a five-digit natural number on the board. It turned out that among any two neighboring digits, the one on the right is greater. Valera multiplied this number by 9 and calculated the sum of the digits of the resulting product. Find all possible values that this sum can take. Justify your answer.


Find the smallest constant $C$ such that for all real numbers $x, y, z$ satisfying $x + y + z = -1$, the following inequality holds:
$$
\left|x^3 + y^3 + z^3 + 1\right| \leqslant C \left|x^5 + y^5 + z^5 + 1\right|.
$$


Calculate the limit of the function:

$$
\lim _{x \rightarrow 0}\left(\frac{\sin 4 x}{x}\right)^{\frac{2}{x+2}}
$$


The hypotenuse \(AB\) of a right triangle \(ABC\) is equal to 2 and is a chord of a certain circle. The leg \(AC\) is equal to 1 and lies within the circle, and its extension intersects the circle at point \(D\) with \(CD = 3\). Find the radius of the circle.


Let \( D \) be an internal point of the acute triangle \( \triangle ABC \), such that \( \angle ADB = \angle ACB + 90^{\circ} \) and \( AC \cdot BD = AD \cdot BC \). Find the value of \( \frac{AB \cdot CD}{AC \cdot BD} \).


On a particular street in Waterloo, there are exactly 14 houses, each numbered with an integer between 500 and 599, inclusive. The 14 house numbers form an arithmetic sequence in which 7 terms are even and 7 terms are odd. One of the houses is numbered 555 and none of the remaining 13 numbers has two equal digits. What is the smallest of the 14 house numbers?

(An arithmetic sequence is a sequence in which each term after the first is obtained from the previous term by adding a constant. For example, 3, 5, 7, 9 is an arithmetic sequence with four terms.)


There are three pastures full of grass. The first pasture is 33 acres and can feed 22 cows for 27 days. The second pasture is 28 acres and can feed 17 cows for 42 days. How many cows can the third pasture, which is 10 acres, feed for 3 days (assuming the grass grows at a uniform rate and each acre produces the same amount of grass)?


Given \(a + b + c = 1\), where \(a, b, c \in \mathbf{R}^{+}\), find the maximum value of \(m\) such that \(\sqrt{4a + 1} + \sqrt{4b + 1} + \sqrt{4c + 1} \geq m\).

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


Divide a circle into \( n \) equal parts and color each point either red or blue. Starting from any point, record the colors of \( k \) points in a counterclockwise direction, which is called a " \( k \)-order color sequence." Two \( k \)-order color sequences are considered different if there is at least one position where the colors differ. If any two 3-order color sequences are different, what is the maximum value of \( n \)?


Given a function \( f(x) \) defined on the set of real numbers \(\mathbf{R}\) that satisfies:
1. \( f(-x) = f(x) \)
2. \( f(4-x) = f(x) \)

For \( x \in [0, 2] \), \( f(x) = -x^2 + 1 \). Determine \( f(x) \) for \( x \in [-6, -4] \).

Solution: \boxed{ -x^2 - 8x - 15 }


All natural numbers whose digits sum up to 5 are ordered in increasing order. What number is in the 125th place?


Given the sequence \(\{x_n\}\) defined by \(x_0 = 0\) and \(x_{n+1} = x_n + a + \sqrt{b^2 + 4ax_n}\) for \(n = 0, 1, 2, \ldots\), where \(a\) and \(b\) are given positive real numbers, find the general term of this sequence.

Solution: \boxed{x_n = an^2 + bn}


Let the circle \( O: x^{2} + y^{2} = 5 \) intersect the parabola \( C: y^{2} = 2px \) (where \( p > 0 \)) at the point \( A(x_{0}, 2) \). Let \( AB \) be a diameter of the circle \( O \), and a line passing through \( B \) intersects the parabola \( C \) at two distinct points \( D \) and \( E \). Find the product of the slopes of the lines \( AD \) and \( AE \).


Find all seven-digit numbers that contain each of the digits from 0 to 6 exactly once and satisfy the following conditions: both the first and last two digits are divisible by 2, both the first and last three digits are divisible by 3, both the first and last four digits are divisible by 4, both the first and last five digits are divisible by 5, and both the first and last six digits are divisible by 6.


Given \(a, b, c \in \mathbb{R}\), with \(a \neq 0\), consider the graph of the function \(P(x) = a x^{2} + b x + c\). Show that by using a dilation and a translation, one can obtain the graph of the function \(Q(x) = x^{2}\).

Solution: \boxed{Q(x) = x^2}


Let \( S \) be the set of rational numbers \( r \) where \( 0 < r < 1 \), and \( r \) can be expressed as a repeating decimal \( \overline{0.abcabcabc\cdots} = \overline{0.a\dot{b}\dot{c}} \), where \( a, b, \) and \( c \) are not necessarily distinct. How many distinct numerators can elements of \( S \) have when expressed in simplest fractional form?

(10th Annual American Mathematics Invitational, 1992)


In a tournament, there are 16 chess players. Determine the number of different possible schedules for the first round (schedules are considered different if they differ by the participants of at least one match; the color of the pieces and the board number are not considered).


Using matchsticks of the same length to form a $3 \times 1996$ grid (with each edge of the small squares being one matchstick, as shown in the figure). In total, $\qquad$ matchsticks are needed.


The numbers \( a \) and \( b \) are such that the polynomial \( x^4 + 3x^3 + x^2 + ax + b \) is the square of some other polynomial. Find \( b \).


In triangle \(ABC\), side \(AB\) is 21, the bisector \(BD\) is \(8 \sqrt{7}\), and \(DC\) is 8. Find the perimeter of the triangle \(ABC\).


If $f$ is a circulation, then for any set $X \subseteq V$, we have $f(X, \bar{X})=0$.

Solution: \boxed{f(X, \bar{X}) = 0}


A lattice point in the plane is a point of the form \((n, m)\), where \(n\) and \(m\) are integers. Consider a set \(S\) of lattice points. We construct the transform of \(S\), denoted by \(S^{\prime}\), by the following rule: the pair \((n, m)\) is in \(S^{\prime}\) if and only if any of \((n, m-1)\), \((n, m+1)\), \((n-1, m)\), \((n+1, m)\), or \((n, m)\) is in \(S\). How many elements are in the set obtained by successively transforming \(\{(0,0)\}\) 14 times?


Write a three-digit number after 1220 to form a seven-digit number. If this seven-digit number is a multiple of 2014, then what is this three-digit number?


Point \( P \) and line \( \ell \) are such that the distance from \( P \) to \( \ell \) is 12. Given that \( T \) is a point on \( \ell \) such that \( PT = 13 \), find the radius of the circle passing through \( P \) and tangent to \( \ell \) at \( T \).


Compute the smallest positive integer \( n \) for which
\[
\sqrt{100+\sqrt{n}} + \sqrt{100-\sqrt{n}}
\]
is an integer.


Find all 4-digit numbers that are 7182 less than the number written with the same digits in reverse order.


Determine the number of ordered 9-tuples of positive integers \((a_{1}, a_{2}, \cdots, a_{9})\) (elements in the array can be the same) that satisfy the following condition: for any \(1 \leq i < j < k \leq 9\), there exists a different \(l (1 \leq l \leq 9)\) such that
$$
a_{i} + a_{j} + a_{k} + a_{l} = 100.
$$


Let \(a\), \(b\), and \(c\) be real constants such that \(x^{2}+x+2\) is a factor of \(ax^{3}+bx^{2}+cx+5\), and \(2x-1\) is a factor of \(ax^{3}+bx^{2}+cx-\frac{25}{16}\). Find \(a+b+c\).


On a semicircle with diameter $[AD]$, and we place two points $B$ and $C$ such that $AB = BC = 1$. We assume that $AD = 3$. Calculate the length $CD$.


In the figure, in cube \(ABCD-A_{1}B_{1}C_{1}D_{1}\), what is the measure of the dihedral angle \(A-BD_{1}-A_{1}\)?


The year of the Tiger, 2022, has the following property: it is a multiple of 6 and the sum of its digits is 6. A positive integer with such properties is called a "White Tiger number". How many "White Tiger numbers" are there among the first 2022 positive integers?


Given a line \( l \) and points \( A \) and \( B \) on opposite sides of it, use a compass and straightedge to construct a point \( M \) such that the angle between \( AM \) and \( l \) is half the angle between \( BM \) and \( l \), provided these angles do not share common sides.

Solution: \boxed{M}


The wages of a worker for October and November were in the ratio $3 / 2: 4 / 3$, and for November and December, they were in the ratio $2: 8 / 3$. In December, he received 450 rubles more than in October, and for exceeding the quarterly plan, he was awarded a bonus equal to $20 \%$ of his three-month earnings. Find the size of the bonus.

Solution: \boxed{1494 \text{ rubles}}


A family of 4 people, consisting of a mom, dad, and two children, arrived in city $N$ for 5 days. They plan to make 10 trips on the subway each day. What is the minimum amount they will have to spend on tickets if the following rates are available in city $N$?

| Adult ticket for one trip | 40 rubles |
| --- | --- |
| Child ticket for one trip | 20 rubles |
| Unlimited day pass for one person | 350 rubles |
| Unlimited day pass for a group of up to 5 people | 1500 rubles |
| Unlimited three-day pass for one person | 900 rubles |
| Unlimited three-day pass for a group of up to 5 people | 3500 rubles |

Solution: \boxed{5200 \text{ rubles}}


A trapezoid is circumscribed around a circle, and its perimeter is 12. Find the median of the trapezoid.


From a single point on a circular track, a pedestrian and a cyclist start simultaneously in the same direction. The speed of the cyclist is 55% greater than the speed of the pedestrian, and therefore the cyclist periodically overtakes the pedestrian. At how many different points on the track will the overtakes occur?


A school's emergency telephone tree operates as follows: The principal calls two students. In the first level, there is only the principal. In the second level, two students are contacted. For each subsequent level, each student contacts two new students who have not yet been contacted. This process continues with each student from the previous level contacting two new students. After the 8th level, how many students in total have been contacted?


Solution: \boxed{\frac{a + 2b}{3}}


What is \( n \) if

a) the total number of different permutations of \( n \) elements is 6,227,020,800?

b) the total number of fifth-order variations without repetition of \( n \) elements is 742,560?

c) the total number of third-order variations with at least one repetition of \( n \) elements is 20,501?


The parabola \( P \) has its focus a distance \( m \) from the directrix. The chord \( AB \) is normal to \( P \) at \( A \). What is the minimum length for \( AB \)?

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


Given the real numbers \(x_1, x_2, x_3\) satisfy the equations \(x_1 + \frac{1}{2} x_2 + \frac{1}{3} x_3 = 1\) and \(x_1^2 + \frac{1}{2} x_2^2 + \frac{1}{3} x_3^2 = 3\), what is the minimum value of \(x_3\)?


\(ABC\) is a triangle such that \(BC = 10\), \(CA = 12\). Let \(M\) be the midpoint of side \(AC\). Given that \(BM\) is parallel to the external bisector of \(\angle A\), find the area of triangle \(ABC\). (Lines \(AB\) and \(AC\) form two angles, one of which is \(\angle BAC\). The external bisector of \(\angle A\) is the line that bisects the other angle.)

Solution: \boxed{8 \sqrt{14}}


The following figure shows a square \(ABCD\), and two points \(P\) and \(Q\) on the sides \(\overline{BC}\) and \(\overline{DA}\), respectively.

We now fold the square along the segment \(\overline{PQ}\), bringing vertex \(B\) to the midpoint of segment \(\overline{CD}\).

It is known that the side of the square measures 24.

a) Calculate the length of segment \(\overline{PC}\).

b) Calculate the length of segment \(\overline{AQ}\).

c) Calculate the length of segment \(\overline{PQ}\).

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


Find the number of 8-digit numbers where the product of the digits equals 9261. Present the answer as an integer.


Calculate the value of the expression \(\frac{2a - b}{3a - b} + \frac{5b - a}{3a + b}\) given that \(10a^2 - 3b^2 + 5ab = 0\) and \(9a^2 - b^2 \neq 0\).


If the product of three consecutive integers is divided by their sum, the quotient is 5. What are these numbers?

In general: when will the quotient obtained in the aforementioned manner be an integer? For instance, which group of numbers will directly follow the group identified in the previous part in magnitude?

Solution: \boxed{4, 5, 6}


Find the greatest integer \( N \) such that
\[
N \leq \sqrt{2007^{2}-20070+31}.
\]


Let \( S \) be the smallest subset of the integers with the property that \( 0 \in S \) and for any \( x \in S \), we have \( 3x \in S \) and \( 3x + 1 \in S \). Determine the number of positive integers in \( S \) less than 2008.


A regular hexagon is divided into 6 congruent equilateral triangular regions denoted as \(A, B, C, D, E, F\). Each region is to be planted with ornamental plants, with the condition that the same plant is used in any one region and different plants are used in adjacent regions. Given there are 4 different types of plants available, how many different planting schemes are possible?


In a square piece of grid paper containing an integer number of cells, a hole in the shape of a square, also consisting of an integer number of cells, was cut out. How many cells did the large square contain if, after cutting out the hole, 209 cells remained?


What is the maximum area that a rectangle can have if the coordinates of its vertices satisfy the equation \( |y-x| = (y+x+1)(5-x-y) \), and its sides are parallel to the lines \( y = x \) and \( y = -x \)? Give the square of the value of the maximum area found as the answer. (12 points)


Expanding the expression \((1+\sqrt{7})^{205}\) using the Binomial theorem, we obtain terms of the form \(C_{205}^k (\sqrt{7})^k\). Find the value of \(k\) for which such a term takes the maximum value.


In Sunny Valley, there are 10 villages. Statisticians conducted a study on the population of these villages and found the following:

1. The number of residents in any two villages differs by no more than 100 people.
2. The village of Znoinie has exactly 1000 residents, which is 90 more than the average population of the villages in the valley.

How many residents are in the village of Raduzhny, which is also located in Sunny Valley?


Find the radius of the circumcircle of an isosceles triangle with a base of 6 and a side length of 5.


The sides of a triangle are \(a, b, c\) and its area is given by \(\frac{(a+b+c)(a+b-c)}{4}\). What is the measure of the largest angle of the triangle?


Inside a sphere with radius \( R \), there are four smaller spheres each with radius \( r \). Determine the maximum possible value of \( r \). ___


Grisha: Masha, Sasha, and Natasha told the truth.

How many children actually told the truth?


Given a right-angled triangle, one of whose acute angles is $\alpha$. Find the ratio of the radii of the circumscribed and inscribed circles and determine for which value of $\alpha$ this ratio will be the smallest.

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


Find the minimum value of \(x^{2} + 4y^{2} - 2x\), where \(x\) and \(y\) are real numbers that satisfy \(2x + 8y = 3\).


Solution: \boxed{a_{4} b_{4} = -6}


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

\[
\begin{cases}
\gcd(a, b, c) = 22 \\
\mathrm{lcm}(a, b, c) = 2^{16} \cdot 11^{19}
\end{cases}
\]


Compute the sum of all positive integers \( n < 1000 \) for which \( a_{n} \) is a rational number, given the sequence \( \{a_{n}\} \) defined by \( a_{1} = 1 \) and for all integers \( n \geq 1 \):
$$
a_{n+1} = \frac{a_{n} \sqrt{n^{2} + n}}{\sqrt{n^{2} + n + 2 a_{n}^{2}}}.
$$


Allison has a coin which comes up heads \(\frac{2}{3}\) of the time. She flips it 5 times. What is the probability that she sees more heads than tails?


In the figure, the graph of the function \( y = x^2 + ax + b \) is shown. It is known that the line \( AB \) is perpendicular to the line \( y = x \). Find the length of the segment \( OC \).


Point \( P \) moves on the circle \( x^2 + (y-3)^2 = \frac{1}{4} \), and point \( Q \) moves on the ellipse \( x^2 + 4y^2 = 4 \). Find the maximum value of \( |PQ| \) and the coordinates of the corresponding point \( Q \).

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


Let \( a_{n} = 6^{n} + 8^{n} \). Find the remainder when \( a_{2018} \) is divided by 49.


An inspector, checking the quality of 400 items, determined that 20 of them belong to the second grade, and the rest to the first grade. Find the frequency of first-grade items and the frequency of second-grade items.

Solution: \boxed{0.95, 0.05}


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

This can be interpreted through either of the following equivalent conditions:
- There exist polynomials \( P(x) \) and \( Q(x) \) with integer coefficients such that \( (x+1)^{n} - 1 = (x^2 + 1) P(x) + 3 Q(x) \).
- The remainder when \( (x+1)^{n} - 1 \) is divided by \( x^2 + 1 \) is a polynomial with integer coefficients all divisible by 3.


From cities $A$ and $B$, which are 240 km apart, two cars simultaneously start driving towards each other. One car travels at 60 km/h and the other at 80 km/h. How far from the point $C$, located at the midpoint between $A$ and $B$, will the cars meet? Give the answer in kilometers, rounding to the nearest hundredth if necessary.


There are 64 small wooden cubes with an edge length of 1. Each cube has two faces painted red, and the other faces are white. Among these cubes, 20 have two adjacent faces painted red, and 44 have two opposite faces painted red. If these cubes are assembled to form one large cube, what is the maximum possible area of the red surface on the large cube?


\( \mathrm{n} \) is a positive integer not greater than 100 and not less than 10, and \( \mathrm{n} \) is a multiple of the sum of its digits. How many such \( \mathrm{n} \) are there?


Find all values of the parameter \(a\), for each of which the set of solutions to the inequality \(\frac{x^{2}+(a+1) x+a}{x^{2}+5 x+4} \geq 0\) is the union of three non-overlapping intervals. In your answer, specify the sum of the three smallest integer values of \(a\) from the obtained interval.


Two circles touch internally at point \( A \). From the center \( O \) of the larger circle, a radius \( O B \) is drawn, which touches the smaller circle at point \( C \). Find \(\angle BAC\).


The minute hand of a clock moves along the circumference of the dial. A spider is watching its tip from a certain point on the same circumference. By what angle does the tip of the minute hand rotate per minute from the spider's perspective?


Let \( x_{1}, \ldots, x_{100} \) be positive real numbers such that \( x_{i} + x_{i+1} + x_{i+2} \leq 1 \) for all \( i \leq 100 \) (with the convention that \( x_{101} = x_{1} \) and \( x_{102} = x_{2} \)). Determine the maximum value of

\[
S = \sum_{i=1}^{100} x_{i} x_{i+2}
\]


As illustrated in Figure 3.1.7, D is the incenter of triangle ABC, E is the incenter of triangle ABD, and F is the incenter of triangle BDE. If the measure of angle BFE is an integer, find the smallest possible measure of angle BFE.


Given the function \( f(x) = -\frac{1}{2} x^{2} + x \) with the domain \([m, n] (m < n)\) and the range \([k m, k n] (k > 1)\), determine the value of \( n \).


Two circles \(\omega_{1}\) and \(\omega_{2}\) with radii 10 and 13, respectively, are externally tangent at point \(P\). Another circle \(\omega_{3}\) with radius \(2 \sqrt{2}\) passes through \(P\) and is orthogonal to both \(\omega_{1}\) and \(\omega_{2}\). A fourth circle \(\omega_{4}\), orthogonal to \(\omega_{3}\), is externally tangent to \(\omega_{1}\) and \(\omega_{2}\). Compute the radius of \(\omega_{4}\).


Find the number of triples \((a, b, c)\) of positive integers such that \(a + ab + abc = 11\).


Find the limit

$$
\lim _{x \rightarrow 0}\left(\int_{0}^{\operatorname{arctg} x} e^{\sin x} \, dx \Big/ \int_{0}^{x} \cos \left(x^{2}\right) \, dx\right)
$$


Given that \(a\), \(b\), and \(c\) are three distinct real numbers, and in the quadratic equations

\[
x^{2} + ax + b = 0
\]

\[
x^{2} + bx + c = 0
\]

\[
x^{2} + cx + a = 0
\]

any two of the equations have exactly one common root, find the value of \(a^{2} + b^{2} + c^{2}\).


Two schools, A and B, each send out 5 students to participate in a long-distance race. The rules are: The $K$th student to reach the finish line scores $K$ points (no students tie for any position). The school with the lower total score wins. How many possible total scores can the winning team have?


Determine all functions \( f: \mathbb{Z} \longrightarrow \mathbb{Z} \), where \( \mathbb{Z} \) is the set of integers, such that
\[
f(m+f(f(n)))=-f(f(m+1))-n
\]
for all integers \( m \) and \( n \).

Solution: \boxed{f(n) = -n - 1}


Consider the sequence \(1, 3, 4, 9, 10, 12, 13, \ldots\) consisting of integers greater than or equal to 1, in increasing order, that are either powers of 3 or sums of distinct powers of 3 (for example: \(4=3^{1}+3^{0}, 10=3^{2}+3^{0}, 13=3^{2}+3^{1}+3^{0}\)).

Which integer is in the hundredth position?


What is the largest number of integers that can be selected from the natural numbers from 1 to 3000 such that the difference between any two of them is different from 1, 4, and 5?


Four friends, Adam, Mojmír, and twins Petr and Pavel, received a total of 52 smileys in their math class, with each receiving at least 1. The twins together have 33 smileys, but Mojmír was the most successful. How many smileys did Adam receive?


Ottó decided to assign a number to each pair \((x, y)\) and denote it as \((x \circ y)\). He wants the following relationships to hold:
a) \(x \circ y = y \circ x\)
b) \((x \circ y) \circ z = (x \circ z) \circ (y \circ z)\)
c) \((x \circ y) + z = (x + z) \circ (y + z)\).

What number should Ottó assign to the pair \((1975, 1976)\)?


Given a trapezoid $A B C D$, where the segments $A B$ and $C D$ (its non-parallel sides) are 24 and 10, respectively. Points $X$ and $Y$ are marked on side $A B$ such that $A X=6$, $X Y=8$, and $Y B=10$. The distances from points $X$ and $Y$ to the line $C D$ are 23 and 27, respectively.

(a) (1 point) Find the area of triangle $A C D$.

(b) (3 points) Find the area of trapezoid $A B C D$.


The real numbers \(a, b, c\) satisfy \(5^{a}=2^{b}=\sqrt{10^{c}}\), and \(ab \neq 0\). Find the value of \(\frac{c}{a}+\frac{c}{b}\).


A notebook sheet is painted in 23 colors, with each cell in the sheet painted in one of these colors. A pair of colors is called "good" if there exist two adjacent cells painted in these colors. What is the minimum number of good pairs?


Three circles of different radii are pairwise tangent to each other. The segments connecting their centers form a right triangle. Find the radius of the smallest circle if the radii of the largest and the medium circles are 6 cm and 4 cm.

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


The opposite sides of a quadrilateral inscribed in a circle intersect at points \(P\) and \(Q\). Find \(PQ\), if the tangents to the circle from points \(P\) and \(Q\) are equal to \(a\) and \(b\).

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


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


Let \( n \) be a positive integer. By removing the last three digits of \( n \), one gets the cube root of \( n \). Find a possible value of \( n \).


How many trials must we conduct at a minimum so that the probability of:

a) Rolling a sum of 15 exactly once with 3 dice,

b) Rolling a sum of at least 15 exactly once with 3 dice,

is greater than $1/2$?


Let \( a \) be a prime number and \( b \) be a positive integer. Given the equation \( 9(2a + b)^2 = 509(4a + 511b) \), find the values of \( a \) and \( b \).

Solution: \boxed{a = 251, \, b = 7}


A number $a$ is randomly selected from $1, 2, 3, \cdots, 10$, and a number $b$ is randomly selected from $-1, -2, -3, \cdots, -10$. What is the probability that $a^{2} + b$ is divisible by 3?


If $n$ is a natural number, we denote by $n!$ the product of all integers from 1 to $n$. For example: $5! = 1 \times 2 \times 3 \times 4 \times 5$ and $13! = 1 \times 2 \times 3 \times 4 \times 5 \times \ldots \times 12 \times 13$. By convention, $0! = 1$. Find three different integers $a, b$, and $c$ between 0 and 9 such that the three-digit number $abc$ is equal to $a! + b! + c!$.


Each New Year, starting from the first year AD, Methuselah, who is still alive today, sends a greeting to his best friend. The greeting formula remains unchanged for almost two millennia: "Happy New Year 1", "Happy New Year 2", "Happy New Year 3", and so on, "Happy New Year 1978", and finally, "Happy New Year 1979".

Which digit has Methuselah used the least frequently so far?


Two lines enclose an angle of $36^{\circ}$. A grasshopper jumps back and forth from one line to the other such that the length of each of its jumps is always the same. Show that it can reach at most ten different points.


Let \( Q \) be the set of \( n \)-term sequences, each element of which is 0 or 1. Let \( A \) be a subset of \( Q \) with \( 2^{n-1} \) elements. Show that there are at least \( 2^{n-1} \) pairs \( (a, b) \), where \( a \in A \), \( b \in Q \backslash A \), and the sequences \( a \) and \( b \) differ in exactly one position.

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


If a certain number of cats ate a total of 999,919 mice, and all cats ate the same number of mice, how many cats were there in total? Additionally, each cat ate more mice than there were cats.


In a pocket, there are six small balls that are identical in size and shape, each labeled with one of the numbers $-5, -4, -3, -2, 2, 1$. We randomly draw one ball from the pocket and denote the number on it as $a$. Determine the probability that the vertex of the parabola given by \( y = x^2 + 2x + a + 2 \) falls in the third quadrant, and the equation
\[
\frac{ax}{x-2} = \frac{3x+2}{2-x} + 2
\]
has an integer solution.


Given the real number \( x \) satisfies the equation \(\sin \left(x + 20^\circ \right) = \cos \left( x + 10^\circ \right) + \cos \left( x - 10^\circ \right)\), find the value of \(\tan x\).


Two real numbers \(x\) and \(y\) are such that \(8y^4 + 4x^2y^2 + 4xy^2 + 2x^3 + 2y^2 + 2x = x^2 + 1\). Find all possible values of \(x + 2y^2\).


Given point \( A(4,0) \) and \( B(2,2) \), while \( M \) is a moving point on the ellipse \(\frac{x^{2}}{25} + \frac{y^{2}}{9} = 1\), the maximum value of \( |MA| + |MB| \) is ______.

Solution: \boxed{10 + 2 \sqrt{10}}


On one side of a street, six neighboring houses will be built. The houses can be made of brick or wood, but as a fire safety measure, two wooden houses cannot be adjacent to each other. In how many ways can the construction of these houses be planned?


Formulate the equation of a line passing through the intersection point of the lines \( x + y - 2 = 0 \) and \( 3x + 2y - 5 = 0 \), and perpendicular to the line \( 3x + 4y - 12 = 0 \).

Solution: \boxed{4x - 3y - 1 = 0}


Find the largest \( \mathrm{C} \) such that for all \( \mathrm{y} \geq 4 \mathrm{x}>0 \), the inequality \( x^{2}+y^{2} \geq \mathrm{C} x y \) holds.


The solution to the equation \(\arcsin x + \arcsin 2x = \arccos x + \arccos 2x\) is


What is the ratio of the areas of a regular hexagon inscribed in a circle to a regular hexagon circumscribed about the same circle?


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


Determine all distinct positive integers \( x \) and \( y \) such that

\[ \frac{1}{x} + \frac{1}{y} = \frac{2}{7} \]

Solution: \boxed{(4, 28)}


Svetlana takes a triplet of numbers and transforms it by the rule: at each step, each number is replaced by the sum of the other two. What is the difference between the largest and smallest numbers in the triplet on the 1580th step of applying this rule, if the initial triplet of numbers was $\{80, 71, 20\}$? If the problem allows for multiple answers, list them all as a set.


On the lateral side \(CD\) of trapezoid \(ABCD (AD \parallel BC)\), point \(M\) is marked. A perpendicular \(AH\) is dropped from vertex \(A\) to segment \(BM\). It turns out that \(AD = HD\). Find the length of segment \(AD\), given that \(BC = 16\), \(CM = 8\), and \(MD = 9\).


On a rectangular sheet of graph paper of size \( m \times n \) cells, several squares are placed such that their sides align with the vertical and horizontal lines of the paper. It is known that no two squares coincide, and no square contains another square within itself. What is the maximum number of such squares?

Solution: \boxed{m \times n}


If \( R = 1 \times 2 + 2 \times 2^2 + 3 \times 2^3 + \ldots + 10 \times 2^{10} \), find the value of \( R \).


What is the minimum value of the function \( y = \sin^4 x + \cos^4 x + \sec^4 x + \csc^4 x \)?


The hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ has a right focus at $F$ and an eccentricity of $e$. A line passing through point $F$ with a slope angle of $\frac{\pi}{3}$ intersects this hyperbola at points $A$ and $B$. If the midpoint of $A B$ is $M$ and the distance $|FM|$ equals the semi-focal distance, find $e$.


Find the sum of the integers that belong to the set of values of the function \(f(x) = \log_{2}(5 \cos 2x + 11)\) for \(x \in \left[1.25 \left(\arctan\left(\frac{1}{3}\right)\right) \cos\left(\pi + \arcsin(-0.6)\right), \arctan 2\right]\).


All angles of an $n$-gon $A_{1} A_{2} A_{3} \ldots A_{n}$ are equal; and

$$
A_{1} A_{2} \leqslant A_{2} A_{3} \leqslant A_{3} A_{4} \leqslant \ldots \leqslant A_{n-1} A_{n} \leqslant A_{n} A_{1} .
$$

What values can the ratio $\frac{A_{1} A_{n}}{A_{1} A_{2}}$ take?


Given that \( x \) and \( y \) are greater than 0, and \( x^{2}+y \geq x^{3}+y^{2} \), find the maximum value of \( x^{2}+y^{2} \).


As shown, \(U\) and \(C\) are points on the sides of triangle \(MN H\) such that \(MU = s\), \(UN = 6\), \(NC = 20\), \(CH = s\), and \(HM = 25\). If triangle \(UNC\) and quadrilateral \(MUCH\) have equal areas, what is \(s\)?

Solution: \boxed{s = 4}


Given that \( n \in \mathbf{N}^{+} \), and subsets \( A_{1}, A_{2}, \cdots, A_{2n+1} \) of a set \( B \) satisfy:
1. \( \left|A_{i}\right| = 2n \) for \( i = 1, 2, \cdots, 2n+1 \);
2. \( \left|A_{i} \cap A_{j}\right| = 1 \) for \( 1 \leqslant i < j \leqslant 2n+1 \);
3. Each element in \( B \) belongs to at least two of the \( A_{i} \).

For which values of \( n \) is it possible to label the elements of \( B \) with 0 or 1 such that each \( A_{i} \) contains exactly \( n \) elements labeled 0?

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


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


Several schoolchildren went mushroom picking. The schoolchild who gathered the most mushrooms collected \( \frac{1}{5} \) of the total amount of mushrooms, while the one who gathered the least collected \( \frac{1}{7} \) of the total amount. How many schoolchildren were there?


There are 11 members in total from two units, A and B, with unit A having 7 members and unit B having 4 members. A group of 5 people is to be formed from these members.

1. How many ways can the group be formed if it must include 2 people from unit B?
2. How many ways can the group be formed if it must include at least 2 people from unit B?
3. How many ways can the group be formed if a specific person from unit B and a specific person from unit A cannot be in the group simultaneously?


Let \( k \) be a given positive integer, and let \( P \) be a point on plane \( \alpha \). In plane \( \alpha \), if it is possible to draw \( n \) lines that do not pass through point \( P \) such that any ray starting from \( P \) and lying in this plane intersects with at least \( k \) of these \( n \) lines, find the minimum value of \( n \).

Solution: \boxed{2k + 1}


1. How many different ways are there to choose 4 letters from $a, b, b, c, c, c, d, d, d, d$?
2. How many different four-digit numbers can be formed by choosing 4 digits from $1, 2, 2, 3, 3, 3, 4, 4, 4, 4$?


Given an isosceles triangle \(ABC\) with base \(AC\). A circle with radius \(R\) and center at point \(O\) passes through points \(A\) and \(B\) and intersects the line \(BC\) at point \(M\), distinct from \(B\) and \(C\). Find the distance from point \(O\) to the center of the circumcircle of triangle \(ACM\).

Solution: \boxed{R}


For any positive integer \( n \), connect the origin \( O \) and the point \( A_{n}(n, n+3) \). Let \( f(n) \) denote the number of lattice points on the line segment \( O A_{n} \) excluding the endpoints. Find \( f(1) + f(2) + \cdots + f(2006) \).


Find all pairs of integers \( x \) and \( y \) such that

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

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


Calculate \(\cos (\alpha + \beta)\) if \(\cos \alpha - \cos \beta = -\frac{3}{5}\) and \(\sin \alpha + \sin \beta = \frac{7}{4}\).


Mark has a cursed six-sided die that never rolls the same number twice in a row, and all other outcomes are equally likely. Compute the expected number of rolls it takes for Mark to roll every number at least once.


Given a regular hexagon \( A B C D E F \) with side length \( 10 \sqrt[4]{27} \), find the area of the union of triangles \( ACE \) and \( BDF \).


Find all values of \( n \in \mathbf{N} \) possessing the following property: if you write the numbers \( n^3 \) and \( n^4 \) next to each other (in decimal system), then in the resulting sequence, each of the 10 digits \( 0, 1, \ldots, 9 \) appears exactly once.


Let \( a, b, \) and \( c \) be strictly positive real numbers such that \( a^2 + b^2 + c^2 = \frac{1}{2} \). Show that

\[
\frac{1 - a^2 + c^2}{c(a + 2b)} + \frac{1 - b^2 + a^2}{a(b + 2c)} + \frac{1 - c^2 + b^2}{b(c + 2a)} \geq 6
\]


There are 200 different cards on the table with numbers $201, 203, 205, \ldots, 597, 599$ (each card has exactly one number, and each number appears exactly once). In how many ways can you choose 3 cards such that the sum of the numbers on the chosen cards is divisible by 3?


From the first 2005 natural numbers, \( k \) of them are arbitrarily chosen. What is the least value of \( k \) to ensure that there is at least one pair of numbers such that one of them is divisible by the other?


In the Martian calendar, a year consists of 5882 days, with each month having either 100 or 77 days. How many months are in the Martian calendar?


Given positive integers \( n \) and \( m \), let \( A = \{1, 2, \cdots, n\} \) and define \( B_{n}^{m} = \left\{\left(a_{1}, a_{2}, \cdots, a_{m}\right) \mid a_{i} \in A, i=1,2, \cdots, m\} \right. \) satisfying:
1. \( \left|a_{i} - a_{i+1}\right| \neq n-1 \), for \( i = 1, 2, \cdots, m-1 \);
2. Among \( a_{1}, a_{2}, \cdots, a_{m} \) (with \( m \geqslant 3 \)), at least three of them are distinct.

Find the number of elements in \( B_{n}^{m} \) and in \( B_{6}^{3} \).


The sequence \( 1, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 1, \ldots \) is formed as follows: write down infinitely many '1's, insert a '2' between the first and the second '1's, insert two '2's between the second and the third '1's, insert three '2's between the third and the fourth '1's, and so on. If \( a_{n} \) denotes the \( n \)-th term of the sequence, find the value of \( a_{1} a_{2} + a_{2} a_{3} + \cdots + a_{2013} a_{2014} \).


A circle with radius \( r \) is inscribed in a circle with radius \( R \). Point \( A \) is the point of tangency. A line perpendicular to the line connecting the centers intersects the smaller circle at point \( B \) and the larger circle at point \( C \). Find the radius of the circumcircle of triangle \( A B C \).

Solution: \boxed{\sqrt{rR}}


Given for any \( x_{1}, x_{2}, \cdots, x_{2020} \in [0, 4] \),

the equation
$$
\left|x - x_{1}\right| + \left|x - x_{2}\right| + \cdots + \left|x - x_{2020}\right| = 2020a
$$

has at least one root in the interval \([0, 4]\), find the value of \( a \).


Arrange all powers of 3 and any finite sums of distinct powers of 3 into an increasing sequence:
$$
1, 3, 4, 9, 10, 12, 13, \cdots
$$

Find the 100th term of this sequence.


Determinants of nine digits. Nine positive digits can be arranged in the form of a 3rd-order determinant in 9! ways. Find the sum of all such determinants.


Find the number of first kind circular permutations formed by selecting 6 elements (with repetition allowed) from the 3-element set \( A = \{a, b, c\} \).


Points \( A_1 \), \( B_1 \), and \( C_1 \) are located on the sides \( BC \), \( AC \), and \( AB \) of triangle \( ABC \) respectively, such that \( BA_1 : A_1C = CB_1 : B_1A = AC_1 : C_1B = 1 : 3 \). Find the area of the triangle formed by the intersections of the lines \( AA_1 \), \( BB_1 \), and \( CC_1 \), given that the area of triangle \( ABC \) is 1.


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

\[ f\left(x^{2}-y^{2}\right)=x f(x)-y f(y) \]

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


In the trapezoid \(ABCD (BC \parallel AD)\), the diagonals intersect at point \(M\). Given \(BC = b\) and \(AD = a\), find the ratio of the area of triangle \(ABM\) to the area of trapezoid \(ABCD\).


How many solutions does the equation

\[ x^{2}+y^{2}+2xy-1988x-1988y=1989 \]

have in the set of positive integers?


Find the maximum value of the natural number \( a \) such that the inequality \(\frac{1}{n+1}+\frac{1}{n+2}+\cdots+\frac{1}{3n+1}>2a-5\) holds for all natural numbers \( n \).


Given \( n \) real numbers, satisfying \( \left|x_{i}\right| < 1 \) for \( i = 1, \ldots, n \), and \( \left| x_1 \right| + \left| x_2 \right| + \cdots + \left| x_n \right| \geq 19 + \left| x_1 + \cdots + x_n \right| \), find the smallest possible value of \( n \).


A regular 2017-gon \( A_1 A_2 \cdots A_{2017} \) is inscribed in a unit circle \( O \). If two different vertices \( A_i \) and \( A_j \) are chosen randomly, what is the probability that \( \overrightarrow{O A_i} \cdot \overrightarrow{O A_j} > \frac{1}{2} \)?


Given points \( A(1,1) \), \( B\left(\frac{1}{2},0\right) \), and \( C\left(\frac{3}{2},0\right) \), the plane region \( G \) is enclosed by the lines passing through points \( A \) and \( B \), through points \( A \) and \( C \), and the line \( y=a \) where \( 0 < a < 1 \). If the probability of any point in the rectangular region \(\{(x, y) \mid 0<x<2, 0<y<1\}\) falling into the region \( G \) is \(\frac{1}{16}\), find the value of \( a \).


Teacher Li plans to buy 25 souvenirs for students from a store that has four types of souvenirs: bookmarks, postcards, notebooks, and pens, with 10 pieces available for each type (souvenirs of the same type are identical). Teacher Li intends to buy at least one piece of each type. How many different purchasing plans are possible? (Answer in numeric form.).


Given a sequence of integers $\left\{a_{n}\right\}$ satisfying $a_{n}=a_{n-1}-a_{n-2}$ for $n \geq 3$, if the sum of the first 1492 terms is 1985 and the sum of the first 1985 terms is 1492, find the sum of the first 2001 terms.


Let \( I \) be the universal set, and let non-empty sets \( P \) and \( Q \) satisfy \( P \subsetneq Q \subsetneq I \). If there exists a set operation expression involving \( P \) and \( Q \) such that the result of the operation is the empty set \( \varnothing \), then one such expression could be _____ (just write one expression).

Solution: \boxed{P \cap Q^c}


Given that $\sin ^{10} x+\cos ^{10} x=\frac{11}{36}$, find the value of $\sin ^{14} x+\cos ^{14} x$.


It is sufficient to check that $DF_{1}=BC$. Similarly, $DF_{2}=BC$ and the problem is solved.

Solution: \boxed{D F_{1} = B C}


Daniel and Scott are playing a game where a player wins as soon as he has two points more than his opponent. Both players start at zero points, and points are earned one at a time. If Daniel has a 60% chance of winning each point, what is the probability that Daniel will win the game?


Two snowplows are working to clear snow. The first can clear the entire street in 1 hour, while the second can do it in 75% of that time. They started working together and after 20 minutes, the first machine stopped working. How much more time is needed for the second machine to finish the job?

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


Find the number of all Young diagrams with weight $s$, if
a) $s=4$; b) $s=5$;
c) $s=6$; d) $s=7$.

Refer to the definition of Young diagrams in the handbook.


Two circles are constructed on a plane in such a way that each passes through the center of the other. Points $P$ and $Q$ are the points of their intersection. A line passing through point $P$ intersects the first circle at point $A$ and the second circle at point $B$ such that $P$ is between $A$ and $B$. This line makes an angle of $15^{\circ}$ with the line connecting the centers. Find the area of triangle $ABQ$ if it is known that $PQ = 2\sqrt{3}$.

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


Given a pyramid \(S-ABC\) with height \(SO = 3\) and a base whose side length is 6, a perpendicular is drawn from point A to its opposite face \(SBC\) with the foot of the perpendicular being \(O'\). On the line segment \(AO'\), a point \(P\) is selected such that \(\frac{AP}{PO'} = 8\). Find the area of the cross-section through point \(P\) and parallel to the base.


How many four-digit numbers are there that are divisible by 17 and end in 17?


Given a regular 200-sided polygon \( A_{1} A_{2} \cdots A_{200} \), connect the diagonals \( A_{i} A_{i+9} \) for \( i=1, 2, \cdots, 200 \), where \( A_{i+200}=A_{i} \) for \( i=1,2, \cdots, 9 \). How many different intersection points are there inside the regular 200-sided polygon formed by these 200 diagonals?


If the inequality $(x - 2) e^x < a x + 1$ has exactly three distinct integer solutions, find the minimum value of the integer $a$.


Isosceles trapezoid \(ABCD\) with bases \(AB\) and \(CD\) has a point \(P\) on \(AB\) with \(AP = 11\), \(BP = 27\), \(CD = 34\), and \(\angle CPD = 90^\circ\). Compute the height of isosceles trapezoid \(ABCD\).


Does \(xy = z\), \(yz = x\), \(zx = y\) have any solutions in positive reals apart from \(x = y = z = 1\)?

Solution: \boxed{x = y = z = 1}


Find the solutions of the system of equations

$$
\begin{aligned}
a^{3}+3 a b^{2}+3 a c^{2}-6 a b c & =1 \\
b^{3}+3 b a^{2}+3 b c^{2}-6 a b c & =1 \\
c^{3}+3 c a^{2}+3 c b^{2}-6 a b c & =1
\end{aligned}
$$

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


A positive integer is written on each corner of a square such that numbers on opposite vertices are relatively prime while numbers on adjacent vertices are not relatively prime. What is the smallest possible value of the sum of these 4 numbers?


We form a number from three consecutive digits. If we write these digits in reverse order and subtract the resulting number from the original one, the difference is always 198.


Maruska wrote one of ten consecutive natural numbers on each of ten cards. However, she lost one card. The sum of the numbers on the remaining nine cards was 2012. What number was written on the lost card?


Given that \(\alpha\) and \(\beta\) are real numbers, for any real numbers \(x\), \(y\), and \(z\), it holds that
$$
\alpha(x y + y z + z x) \leqslant M \leqslant \beta\left(x^{2} + y^{2} + z^{2}\right),
$$
where \(M = \sum \sqrt{x^{2} + x y + y^{2}} \sqrt{y^{2} + y z + z^{2}}\,\) and \(\sum\) represents cyclic sums. Find the maximum value of \(\alpha\) and the minimum value of \(\beta\).


Find the maximum length of a horizontal segment with endpoints on the graph of the function \( y = x^3 - x \).


The total length of the highway from Lishan Town to the provincial capital is 189 kilometers, passing through the county town. The county town is 54 kilometers away from Lishan Town. In the morning at 8:30, a bus departs from Lishan Town to the county town and arrives at 9:15. After a 15-minute stop, it heads to the provincial capital, arriving by 11:00 AM. Another bus departs from the provincial capital directly to Lishan Town at 9:00 AM on the same day, traveling at 60 kilometers per hour. When the two buses meet, the one traveling from the provincial capital to Lishan Town has been traveling for how many minutes?


Five people play several games of dominoes (2 vs 2) in such a way that each player partners with each other player exactly once, and opposes each other player exactly twice. Find the number of games played and all possible ways to arrange the players.

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


An archipelago consists of $N \geqslant 7$ islands. Any two islands are connected by at most one bridge. It is known that from each island no more than 5 bridges lead out, and among any 7 islands there must be two connected by a bridge. What is the maximum value that $N$ can take?


The height \( AH \) of triangle \( ABC \) is equal to its median \( BM \). On the extension of side \( AB \) beyond point \( B \), point \( D \) is marked such that \( BD = AB \). Find the angle \( BCD \).


Pile up 2019 stones into one pile. First, person A splits this pile into two piles and writes the product of the number of stones in each pile on the blackboard. Then, person A selects one pile from the two and splits it into two more piles, again writing the product of the number of stones in each pile on the blackboard. Person A continues this process until all piles have exactly 1 stone. At this point, what is the total sum of the numbers on the blackboard?


Consider pairs of sequences of positive real numbers \( a_1 \geq a_2 \geq a_3 \geq \cdots \) and \( b_1 \geq b_2 \geq b_3 \geq \cdots \), and the sums \( A_n = a_1 + \cdots + a_n \), \( B_n = b_1 + \cdots + b_n \) for \( n = 1, 2, \ldots \). For any pair, define \( c_i = \min \{a_i, b_i\} \) and \( C_n = c_1 + \cdots + c_n \) for \( n = 1, 2, \ldots \).
(a) Does there exist a pair \( \left(a_i\right)_{i \geq 1},\left(b_i\right)_{i \geq 1} \) such that the sequences \( \left(A_n\right)_{n \geq 1} \) and \( \left(B_n\right)_{n \geq 1} \) are unbounded while the sequence \( \left(C_n\right)_{n \geq 1} \) is bounded?
(b) Does the answer to question (a) change by assuming additionally that \( b_i = \frac{1}{i} \) for \( i = 1, 2, \ldots \)?
Justify your answer.

Solution: \boxed{No}


In the right triangle \(ABC\), the hypotenuse \(AB\) is 10. \(AD\) is the angle bisector of \(\angle A\). Segment \(DC\) is 3. Find \(DB\).


Find all positive integers \( n \) such that \( n \) divides \( 2^{n} - 1 \).


The sequence \(\left(x_{n}\right)\) is defined recursively by \(x_{0}=1, x_{1}=1\), and:

\[ x_{n+2}=\frac{1+x_{n+1}}{x_{n}} \]

for all \(n \geq 0\). Calculate \(x_{2007}\).


Arrange all positive odd numbers in ascending order. Take the first number as \(a_1\), take the sum of the subsequent two numbers as \(a_2\), then take the sum of the subsequent three numbers as \(a_3\), and so on. This generates the sequence \(\{a_n\}\), i.e.
\[ a_1 = 1, \quad a_2 = 3 + 5, \quad a_3 = 7 + 9 + 11, \quad \cdots \cdots \]

Find the value of \(a_1 + a_2 + \cdots + a_{20}\).


Solution: \boxed{e - 2}


Can a finite set of points in space, not lying in the same plane, have the following property: for any two points \(A\) and \(B\) from this set, there exist two other points \(C\) and \(D\) from this set such that \(AB \parallel CD\) and these lines do not coincide?

Solution: \boxed{\text{Yes}}


In quadrilateral \(ABCD\), the diagonals \(AC\) and \(BD\) intersect at point \(O\). The areas of \(\triangle AOB\), \(\triangle AOD\), and \(\triangle BOC\) are \(12\), \(16\), and \(18\), respectively. What is the area of \(\triangle COD\)?


In the city where the Absent-Minded Scientist lives, phone numbers consist of 7 digits. The Scientist easily remembers a phone number if it is a palindrome, meaning it reads the same forwards and backwards. For example, the number 4435344 is easy for the Scientist to remember because it is a palindrome. However, the number 3723627 is not a palindrome, making it difficult for the Scientist to remember. Find the probability that the phone number of a new random acquaintance is easy for the Scientist to remember.


Given that \(b\) and \(c\) are both integers and \(c < 2000\). If the quadratic equation \(x^{2} - bx + c = 0\) has roots whose real parts are both greater than 1, how many pairs \((b, c)\) satisfy this condition?


Find the smallest positive integer \( b \) for which \( 7 + 7b + 7b^2 \) is a fourth power.


In a parallelogram with sides 2 and 4, a diagonal equal to 3 is drawn. A circle is inscribed in each of the resulting triangles. Find the distance between the centers of the circles.


The line \( K M_{1} \) intersects the extension of \( A B \) at point \( N \). Find the measure of angle \( DNA \).


Does there exist a natural number that, when divided by the sum of its digits, gives 2014 as both the quotient and the remainder? If there is more than one such number, write their sum as the answer. If no such number exists, write 0 as the answer.


Let \( f:[0,1) \rightarrow \mathbb{R} \) be a function that satisfies the following condition: if
\[ x = \sum_{n=1}^{\infty} \frac{a_n}{10^n} = .a_1 a_2 a_3 \ldots \]
is the decimal expansion of \( x \) and there does not exist a positive integer \( k \) such that \( a_n = 9 \) for all \( n \geq k \), then
\[ f(x) = \sum_{n=1}^{\infty} \frac{a_n}{10^{2n}} . \]

Determine \( f'\left(\frac{1}{3}\right) \).


Find all odd integers \( n \geq 1 \) such that \( n \) divides \( 3^{n} + 1 \).

Solution: \boxed{n = 1}


A bag of rice takes Liu Bei 5 days to finish eating alone, and Guan Yu 3 days to finish eating alone. A bag of wheat takes Guan Yu 5 days to finish eating alone, and Zhang Fei 4 days to finish eating alone. Liu Bei's daily food consumption is $\qquad$% less than Zhang Fei's daily food consumption.


Let

$$
A = \sqrt{5} + \sqrt{22+2 \sqrt{5}}, \quad B = \sqrt{11+2 \sqrt{29}} + \sqrt{16-2 \sqrt{29} + 2 \sqrt{55-10 \sqrt{29}}}
$$

What is the relative magnitude of the numbers $A$ and $B$?

Solution: \boxed{A = B}


Four boys and three girls went to the forest to collect mushrooms. Each of them found several mushrooms, and in total they collected 70 mushrooms. No two girls gathered the same number of mushrooms, and any three boys together brought at least 43 mushrooms. The number of mushrooms collected by any two children differed by no more than 5 times. Masha collected the most mushrooms among the girls. How many mushrooms did she gather?


On a straight line, there are \(2n\) points, and the distance between any two adjacent points is 1. A person starts from the 1st point and jumps to other points, jumping \(2n\) times and returning to the 1st point. During these \(2n\) jumps, the person must visit all \(2n\) points. How should they jump to maximize the total distance jumped?


Let \( a, b, c, d \) be strictly positive real numbers satisfying \( (a+c)(b+d)=ac+bd \). Determine the smallest value that

\[
\frac{a}{b} + \frac{b}{c} + \frac{c}{d} + \frac{d}{a}
\]

can take.


Find the number of 7-tuples \(\left(n_{1}, \ldots, n_{7}\right)\) of integers such that
\[
\sum_{i=1}^{7} n_{i}^{6} = 96957
\]


In the quadrilateral \(ABCD\), it is known that \(AB = BD\) and \(\angle ABD = \angle DBC\), with \(\angle BCD = 90^\circ\). A point \(E\) on segment \(BC\) is such that \(AD = DE\). What is the length of segment \(BD\), given that \(BE = 7\) and \(EC = 5\)?


Two cars, Car A and Car B, travel towards each other from cities A and B, which are 330 kilometers apart. Car A starts from city A first. After some time, Car B starts from city B. The speed of Car A is $\frac{5}{6}$ of the speed of Car B. When the two cars meet, Car A has traveled 30 kilometers more than Car B. Determine how many kilometers Car A had traveled before Car B started.


In triangle \( \triangle ABC \), what is the maximum value of \( \sin \frac{A}{2} \sin \frac{B}{2} \sin \frac{C}{2} \)?


Among the positive integers not greater than 2017, how many are divisible by 12 but not divisible by 20?


The diagonal \(AC\) of a convex quadrilateral \(ABCD\) is the diameter of the circumcircle. Find the ratio of the areas of triangles \(ABC\) and \(ACD\) if it is known that the diagonal \(BD\) divides \(AC\) in the ratio 2:1 (counting from point \(A\)), and \(\angle BAC = 30^{\circ}\).


A chocolate bar has a size of $4 \times 10$ tiles. In one move, it is allowed to break one of the existing pieces into two along a straight break line. What is the minimum number of moves required to break the entire chocolate bar into pieces of size one tile each?


Given a triangle with area \( A \) and perimeter \( p \), let \( S \) be the set of all points that are a distance 5 or less from a point of the triangle. Find the area of \( S \).

Solution: \boxed{A + 5p + 25\pi}


If the real numbers $\alpha, \beta, \gamma$ form a geometric sequence with a common ratio of 2, and $\sin \alpha, \sin \beta, \sin \gamma$ form a geometric sequence, find the value of $\cos \alpha$.


In triangle \( ABC \), \( AC = 3 AB \). Let \( AD \) bisect angle \( A \) with \( D \) lying on \( BC \), and let \( E \) be the foot of the perpendicular from \( C \) to \( AD \). Find \( \frac{[ABD]}{[CDE]} \). (Here, \([XYZ]\) denotes the area of triangle \( XYZ \)).


Petya showed Vasya 37 identical-looking cards laid out in a row. He said that the hidden sides of the cards contain all the numbers from 1 to 37 (each exactly once) in such a way that the number on any card starting from the second one is a divisor of the sum of the numbers written on all the preceding cards. Then Petya revealed to Vasya that the number 37 is written on the first card and the number 1 is written on the second card. Vasya said that he then knows what number is written on the third card. What number is written on the third card?


The expression \( x + \frac{1}{x} \) has a maximum for \( x < 0 \) and a minimum for \( x > 0 \). Find the area of the rectangle whose sides are parallel to the axes and two of whose vertices are the maximum and minimum values of \( x + \frac{1}{x} \).


If $x+y-2$ is a factor of the polynomial $x^{2}+a x y+b y^{2}-5 x + y + 6$ with respect to $x$ and $y$, then what is the value of $a-b$?


Inside an angle of $30^{\circ}$ with vertex $A$, a point $K$ is chosen such that its distances to the sides of the angle are 1 and 2. Through point $K$, all possible lines intersecting the sides of the angle are drawn. Find the minimum area of a triangle that is cut off by one of these lines from the angle.


Given that \( F_{1} \) and \( F_{2} \) are the left and right foci of the hyperbola \( C: \frac{x^{2}}{4}-\frac{y^{2}}{12}=1 \), and point \( P \) lies on the hyperbola \( C \). Let \( G \) and \( I \) be the centroid and incenter of \( \triangle F_{1} P F_{2} \) respectively. If \( G I \) is parallel to the \( x \)-axis, find the circumradius \( R \) of \( \triangle F_{1} P F_{2} \).


Let \( x = \cos \theta \). Express \( \cos 3\theta \) in terms of \( x \).

Solution: \boxed{4x^3 - 3x}


1) It is known that the sequence \( u_{1}=\sqrt{2}, u_{2}=\sqrt{2+\sqrt{2}}, u_{3}=\sqrt{2+\sqrt{2+\sqrt{2}}}, \ldots, u_{n}=\sqrt{2+u_{n-1}} \) converges. Find its limit.
2) Find the limit of the sequence \( u_{n}=\frac{2^{n}}{n!} \).


Place 48 chess pieces into 9 boxes, with each box containing at least 1 piece and every box containing a different number of pieces. What is the maximum number of pieces that can be placed in the box with the most pieces?


Find the odd prime number \( p \) that satisfies the following condition: There exists a permutation \( b_1, b_2, \cdots, b_{p-1} \) of \( 1, 2, \cdots, p-1 \) such that \( 1^{b_1}, 2^{b_2}, \cdots, (p-1)^{b_{p-1}} \) forms a reduced residue system modulo \( p \).


We have a two-pan balance and want to be able to weigh any object with an integer mass between 1 and 40 kg. What is the minimum number of weights with integer masses needed to achieve this:

1. if the weights can only be placed on one pan of the balance?
2. if the weights can be placed on both pans of the balance?

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


Find the derivative of the scalar field \( u = x z^{2} + 2 y z \) at the point \( M_{0}(1,0,2) \) along the curve described by

\[
\left\{
\begin{array}{l}
x = 1 + \cos t \\
y = \sin t - 1 \\
z = 2
\end{array}
\right.
\]


1. In what base is 16324 the square of 125?
2. In what base does \( 4 \cdot 13 = 100 \)?


Solution: \boxed{\sqrt{3 + \sqrt{6}}}


Given an odd function \( f(x) \) defined on \( \mathbf{R} \) whose graph is symmetric about the line \( x=2 \), and when \( 0 < x \leq 2 \), \( f(x) = x + 1 \). Find the value of \( f(-100) + f(-101) \).


An engineer arrives at the train station every day at 8 AM. At exactly 8 AM, a car arrives at the station and takes the engineer to the factory. One day, the engineer arrived at the station at 7 AM and started walking towards the car. Upon meeting the car, he got in and arrived at the factory 20 minutes earlier than usual. How long did the engineer walk? The speeds of the car and the engineer are constant.


In a football tournament, each team is supposed to play one match against each of the other teams. However, during the tournament, half of the teams were disqualified and did not participate further. As a result, a total of 77 matches were played, and the disqualified teams managed to play all their matches against each other, with each disqualified team having played the same number of matches. How many teams were there at the beginning of the tournament?


An infinite sequence of positive real numbers is defined by \(a_{0}=1\) and \(a_{n+2}=6 a_{n}-a_{n+1}\) for \(n=0,1,2, \cdots\). Find the possible value(s) of \(a_{2007}\).


As shown in the diagram, through point \( D \) draw \( D O \) bisecting \( \angle A D C \). Through point \( B \) draw \( B O \) bisecting \( \angle A B C \), with \( B O \) and \( D O \) intersecting at point \( O \). Given that \( \angle A = 35^\circ \) and \( \angle O = 42^\circ \), find \( \angle C \).


Let \( F \) be a finite nonempty set of integers and let \( n \) be a positive integer. Suppose that:
- Any \( x \in F \) may be written as \( x = y + z \) for some \( y, z \in F \);
- If \( 1 \leq k \leq n \) and \( x_{1}, \ldots, x_{k} \in F \), then \( x_{1}+ \cdots + x_{k} \neq 0 \).

Show that \( F \) has at least \( 2n+2 \) distinct elements.

Solution: \boxed{2n + 2}


Given \(x, y, z \in (0, 2]\), find the maximum value of the expression

\[
A = \frac{\left(x^{3} - 6\right) \sqrt[3]{x + 6} + \left(y^{3} - 6\right) \sqrt[3]{y + 6} + \left(z^{3} - 6\right) \sqrt[3]{z + 6}}{x^{2} + y^{2} + z^{2}}
\]


Given the real numbers \( x \) and \( y \) satisfy the equation \( 2x^2 + 3xy + 2y^2 = 1 \), find the minimum value of \( x + y + xy \).


Define the sequence \(\left\{a_n\right\}\) as follows:
\[
\begin{aligned}
& a_1 = 1, \\
& a_2 = 3, \\
& a_3 = 5, \\
& a_n = a_{n-1} - a_{n-2} + a_{n-3} \quad \text{for} \; n = 4, 5, \ldots
\end{aligned}
\]

Determine the sum of the first 2015 terms of this sequence, \( S_{2015} \).


Find the number of positive integers less than 10000 that contain the digit 1.


Suppose \( f \) is a function satisfying \( f\left(x+x^{-1}\right)=x^{6}+x^{-6} \), for all \( x \neq 0 \). Determine \( f(3) \).


The medians drawn from vertices $A$ and $B$ of triangle $ABC$ are perpendicular to each other. Find the area of the square with side $AB$, given that $BC = 28$ and $AC = 44$.


An isosceles trapezoid is circumscribed around a circle. The lateral side of the trapezoid is equal to $a$, and the segment connecting the points of tangency of the lateral sides with the circle is equal to $b$. Find the diameter of the circle.

Solution: \boxed{D = \sqrt{ab}}


In how many ways can all ten digits be arranged in a row such that the digits $2, 5, 8$ are placed next to each other?


In a $3 \times 4$ grid, you need to place 4 crosses so that there is at least one cross in each row and each column. How many ways are there to do this?


There were 7 boxes in a warehouse. Some of them were filled with 7 more boxes, and this was done several times. In the end, there were 10 non-empty boxes. How many boxes are there in total?


As shown in the figure, a large square is divided into six smaller squares. If two smaller squares share more than one common point, they are called adjacent. Fill the numbers $1, 2, 3, 4, 5, 6$ into the smaller squares in the right figure, with each smaller square containing one number such that the difference between the numbers in any two adjacent smaller squares is never 3. How many different ways can this be done?


Reimu and Sanae play a game using 4 fair coins. Initially both sides of each coin are white. Starting with Reimu, they take turns to color one of the white sides either red or green. After all sides are colored, the 4 coins are tossed. If there are more red sides showing up, then Reimu wins, and if there are more green sides showing up, then Sanae wins. However, if there is an equal number of red sides and green sides, then neither of them wins. Given that both of them play optimally to maximize the probability of winning, what is the probability that Reimu wins?


Determine the value of \(\lambda\) for which the equation

\[
\lambda x^{2}+4xy+y^{2}-4x-2y-3=0
\]

represents a pair of straight lines.


Given \( w=\sqrt{2p-q}+\sqrt{3q-2p}+\sqrt{6-2q} \), where \( p \) and \( q \) are real numbers that make \( w \) meaningful, determine the maximum value of \( w \).


The vulgar fraction \(\frac{1}{221}\) is represented as a repeating decimal. Find the length of the repeating period. (For example, the length of the repeating period for the fraction \(\frac{25687}{99900}=0.25712712712\ldots=0.25(712)\) is 3.)


In triangle \( \triangle ABC \), the sides opposite to angles \( \angle A \), \( \angle B \), and \( \angle C \) are \( a \), \( b \), and \( c \) respectively. Given that \( \angle ABC = 120^\circ \), the bisector of \( \angle ABC \) intersects \( AC \) at point \( D \) and \( BD = 1 \). Find the minimum value of \( 4a + c \).


A quadrilateral $ABCD$ is such that $AC = DB$. Let $M$ be any point in the plane. Show that $MA < MB + MC + MD$.

Solution: \boxed{MA < MB + MC + MD}


Points \(E\) and \(F\) are on sides \(AD\) and \(BC\), respectively, of square \(ABCD\). Given that \(BE = EF = FD = 30\), determine the area of the square.


The sequence \(\left\{a_{n}\right\}\) satisfies: \(a_{1}=1\), and for each \(n \in \mathbf{N}^{*}\), \(a_{n}\) and \(a_{n+1}\) are the two roots of the equation \(x^{2}+3nx+b_{n}=0\). Find \(\sum_{k=1}^{20} b_{k}\).


The numbers \(a\) and \(b\) are such that the polynomial \(x^{4} + x^{3} - x^{2} + ax + b\) is the square of some other polynomial. Find \(b\).


The team members' numbers are uniquely selected from the positive integers 1 to 100. If the number of any team member is neither the sum of the numbers of any two other team members nor twice the number of another team member, what is the maximum number of members in this sports team?


Among the applicants who passed the entrance exams to a technical university, the grades of "excellent" were received as follows: 48 people in mathematics, 37 people in physics, 42 people in literature, 75 people in mathematics or physics, 76 people in mathematics or literature, 66 people in physics or literature, and 4 people in all three subjects. How many applicants received only one "excellent" grade? Exactly two "excellent" grades? At least one "excellent" grade?


Given that the radius of the inscribed circle of triangle \( \triangle ABC \) is 2 and \(\tan A = -\frac{4}{3}\), find the minimum value of the area of triangle \( \triangle ABC \).

Solution: \boxed{18 + 8 \sqrt{5}}


Distribute 12 different objects among 3 people so that each person receives 4 objects. In how many ways is this possible?


Two cars start simultaneously towards each other from cities $A$ and $Б$, which are 220 km apart. Their speeds are 60 km/h and 80 km/h. At what distance from the point $C$, located halfway between $A$ and $Б$, will the cars meet? Provide the answer in kilometers, rounding to the nearest hundredth if necessary.


If you get 58 out of 84 questions correct on a test, what is your accuracy percentage?


Calculate: $5 \times 13 \times 31 \times 73 \times 137$


For an acute triangle \(ABC\), find the values of \(\alpha\) such that \(A\) is equidistant from the circumcenter \(O\) and the orthocenter \(H\).


How many digits are in the number \(2^{1000}\)?


Points \( E \) and \( F \) lie on sides \( AB \) and \( BC \) respectively of rhombus \( ABCD \), such that \( AE = 5BE \) and \( BF = 5CF \). It is known that triangle \( DEF \) is equilateral. Find the angle \( BAD \).


A circle is tangent to both branches of the hyperbola \( x^{2} - 20y^{2} = 24 \) as well as the \( x \)-axis. Compute the area of this circle.

Solution: \boxed{504 \pi}


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


Let \( d = \overline{xyz} \) be a three-digit number that cannot be divisible by 10. If the sum of \( \overline{xyz} \) and \( \overline{zyx} \) is divisible by \( c \), find the largest possible value of this integer \( d \).


Find the sum of squares of all distinct complex numbers \(x\) satisfying the equation
$$
0 = 4x^{10} - 7x^{9} + 5x^{8} - 8x^{7} + 12x^{6} - 12x^{5} + 12x^{4} - 8x^{3} + 5x^{2} - 7x + 4.
$$


Find the largest natural number in which all digits are different and each pair of adjacent digits differs by 6 or 7.


Given an $n$-degree polynomial $P(x)$ such that $P(j)=2^{j-1}$ for $j=1, 2, \ldots, n, n+1$, what is the value of $P(n+2)$?

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


Let \(a\), \(b\), \(c\) be nonzero real numbers such that \(a+b+c=0\) and \(a^3 + b^3 + c^3 = a^5 + b^5 + c^5\). Find the value of \(a^2 + b^2 + c^2\).


Find a positive integer whose last digit is 2, and if this digit is moved from the end to the beginning, the new number is twice the original number.


What is the smallest natural number $n$ such that the decimal representation of $n$! ends with ten zeros?


Let \( x \) and \( y \) be real numbers such that \( x^2 + y^2 = 2 \) and \( |x| \neq |y| \). Find the minimum value of \( \frac{1}{(x+y)^2} + \frac{1}{(x-y)^2} \).


A contest has six problems worth seven points each. On any given problem, a contestant can score either 0, 1, or 7 points. How many possible total scores can a contestant achieve over all six problems?


Calculate

$$
\frac{6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 - 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}{4 \cdot 3 \cdot 2 \cdot 1}
$$


A root of unity is a complex number that is a solution to \( z^{n}=1 \) for some positive integer \( n \). Determine the number of roots of unity that are also roots of \( z^{2}+a z+b=0 \) for some integers \( a \) and \( b \).


Let it be known that for a random variable \(\xi\), its expectation \(\mathrm{E} \xi\) is negative, and \(\mathrm{E} e^{\theta \xi} = 1\) for some \(\theta \neq 0\). Show that \(\theta > 0\).

Solution: \boxed{\theta > 0}


At what speed does the distance traveled at uniform motion ($s = v t$) equal the same number as the time?


The general solution of the equation \(\cos \frac{x}{4}=\cos x\) is \((\quad)\). In the interval \((0, 24 \pi)\), there are ( ) distinct solutions.