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\( f(n) \) is defined on the set of positive integers, and satisfies the following conditions:
1. For any positive integer \( n \), \( f[f(n)] = 4n + 9 \).
2. For any non-negative integer \( k \), \( f(2^k) = 2^{k+1} + 3 \).
Determine \( f(1789) \). | 3581 | 0.875 |
In triangle \( \triangle ABC \), \( AC > AB \). Point \( P \) is the intersection of the perpendicular bisector of \( BC \) and the internal angle bisector of \( \angle A \). Draw \( PX \perp AB \), intersecting the extension of \( AB \) at point \( X \). Draw \( PY \perp AC \), intersecting \( AC \) at point \( Y \). Let \( Z \) be the intersection of \( XY \) and \( BC \). Find the value of \( \frac{BZ}{ZC} \). | 1 | 0.875 |
The base of the pyramid is an equilateral triangle with a side length of 6. One of the lateral edges is perpendicular to the plane of the base and has a length of 4. Find the radius of the sphere circumscribed around the pyramid. | 4 | 0.875 |
Let the quadratic function \( f(x) = ax^2 + bx + c \) (where \( a, b, c \in \mathbf{R} \) and \( a \neq 0 \)) satisfy the following conditions:
1. For \( x \in \mathbf{R} \), \( f(x-4) = f(2-x) \);
2. For \( x \in (0, 2) \), \( f(x) \leq \left( \frac{x+1}{2} \right)^2 \);
3. The minimum value of \( f(x) \) on \( \mathbf{R} \) is 0.
Find the maximum value of \( m \) (where \( m > 1 \)) such that there exists \( t \in \mathbf{R} \) for which \( f(x+t) \leq x \) for all \( x \in [1, m] \). | 9 | 0.625 |
Find the number of different arrangements of 8 rooks on different white squares of an 8x8 chessboard such that no rook can attack another. A chess rook attacks all squares on the row and column intersecting its position. | 576 | 0.375 |
Ezekiel has a rectangular piece of paper with an area of 40. The width of the paper is more than twice the height. He folds the bottom left and top right corners at $45^{\circ}$ and creates a parallelogram with an area of 24. What is the perimeter of the original rectangle? | 28 | 0.5 |
A clock's minute hand has length 4 and its hour hand length 3. What is the distance between the tips at the moment when it is increasing most rapidly? | \sqrt{7} | 0.375 |
Let \( x_{1} \) and \( x_{2} \) be the roots of the equation \( x^{2} + x + 1 = 0 \). Then, the value of the series \( \frac{x_{1}}{x_{2}}+\left(\frac{x_{1}}{x_{2}}\right)^{2}+\left(\frac{x_{1}}{x_{2}}\right)^{3}+\cdots+\left(\frac{x_{1}}{x_{2}}\right)^{1998} \) is equal to \(\qquad\) \(\_\_\_\_\_\). | 0 | 0.5 |
At what angle to the x-axis is the tangent to the graph of the function \( g(x) = x^2 \ln x \) inclined at the point \( x_0 = 1 \)? | \frac{\pi}{4} | 0.125 |
Anya is arranging pebbles in the sand. She first placed one pebble, then added pebbles to form a pentagon. Next, she created a larger external pentagon with pebbles, and after that another external pentagon, and so on, as shown in the picture. The numbers of pebbles she arranged in the first four pictures are: 1, 5, 12, and 22. If she continues to make such pictures, how many pebbles will there be in the 10th picture? | 145 | 0.875 |
In a tournament, each participant plays a match against every other participant. The winner of a match earns 1 point, the loser 0 points, and if the match is a draw, both players earn half a point. At the end of the tournament, participants are ranked according to their score (if several participants have the same score, their order is chosen randomly). Each participant has earned half of their points in their matches against the last ten participants (in the ranking). How many people participated in this tournament? | 25 | 0.5 |
On a drying rack, $n$ socks are hanging in a random order (as they were taken out of the washing machine). Among them are two of Absent-Minded Scientist's favorite socks. The socks are hidden by a drying sheet, so the Scientist cannot see them and picks one sock at a time by feel. Find the expected number of socks the Scientist will have taken by the time he has both of his favorite socks. | \frac{2(n+1)}{3} | 0.375 |
Given \( n > 1 \), find the maximum value of \( \sin^2 x_1 + \sin^2 x_2 + \ldots + \sin^2 x_n \), where \( x_i \) are non-negative and have sum \( \pi \). | \frac{9}{4} | 0.5 |
We repeatedly toss a coin until we get either three consecutive heads ($HHH$) or the sequence $HTH$ (where $H$ represents heads and $T$ represents tails). What is the probability that $HHH$ occurs before $HTH$? | \frac{2}{5} | 0.375 |
Write the equation of a plane passing through the point \(A\) perpendicular to the vector \(\overrightarrow{BC}\).
Given:
\[ A(5, -1, 2) \]
\[ B(2, -4, 3) \]
\[ C(4, -1, 3) \] | 2x + 3y - 7 = 0 | 0.875 |
Define the lengths of intervals $(m, n)$, $[m, n)$, $(m, n]$, and $[m, n]$ to be $n - m$ ($n, m \in \mathbf{R}$ and $n > m$). Find the sum of the lengths of the intervals for real numbers $x$ that satisfy the inequality
\[
\frac{1}{x-20}+\frac{1}{x-17} \geqslant \frac{1}{512}
\] | 1024 | 0.625 |
Let \( f(x) = x^3 - 20x^2 + x - a \) and \( g(x) = x^4 + 3x^2 + 2 \). If \( h(x) \) is the highest common factor of \( f(x) \) and \( g(x) \), find \( b = h(1) \). | 2 | 0.625 |
Is a triangle necessarily isosceles if the center of its inscribed circle is equidistant from the midpoints of two of its sides? | \text{No} | 0.375 |
Calculate the limit of the function:
$\lim _{x \rightarrow 0}\left(\operatorname{tg}\left(\frac{\pi}{4}-x\right)\right)^{\left(e^{x}-1\right) / x}$ | 1 | 0.75 |
$2017^{\ln \ln 2017}-(\ln 2017)^{\ln 2017}=$ | 0 | 0.875 |
Find all pairs of positive integers \((x, y)\) such that \(x^2 = y^2 + 7y + 6\). | (6, 3) | 0.875 |
Find the circulation of the vector field
$$
\vec{a}=\frac{y}{3} \vec{i} 3-3 x \vec{j}+x \vec{k}
$$
along the closed contour $\Gamma$
$$
\left\{\begin{array}{l}
x=2 \cos t \\
y=2 \sin t \\
z=1-2 \cos t - 2 \sin t
\end{array} \quad t \in [0,2\pi]
\right.
$$ | -\frac{52 \pi}{3} | 0.625 |
The cost of 60 copies of the first volume and 75 copies of the second volume is 2700 rubles. In reality, the total payment for all these books was only 2370 rubles because a discount was applied: 15% off the first volume and 10% off the second volume. Find the original price of these books. | 20 | 0.875 |
Let $A = \left\{a_1, a_2, a_3\right\}, B = \{-1, 0, 1\}$.
(1) How many distinct mappings are there from $A$ to $B$?
(2) Determine the number of mappings $f: A \rightarrow B$ that satisfy $f(a_1) > f(a_2) \geqslant f(a_3)$. | 4 | 0.75 |
Given the function \( f(x) = -\frac{1}{2} x^{2} + \frac{13}{2} \) on the interval \([a, b]\), if the minimum value on this interval is \(2a\) and the maximum value is \(2b\), find the interval \([a, b]\). | [1, 3] | 0.25 |
Find the maximum value of the positive real number \( k \) such that for any positive real numbers \( a \) and \( b \), the following inequality holds:
$$
\sqrt{a^{2}+k b^{2}}+\sqrt{b^{2}+k a^{2}} \geq a+b+(k-1) \sqrt{a b}
$$ | 3 | 0.875 |
Simplify the expression:
\[
\frac{p^{3}+4 p^{2}+10 p+12}{p^{3}-p^{2}+2 p+16} \cdot \frac{p^{3}-3 p^{2}+8 p}{p^{2}+2 p+6}
\] | p | 0.875 |
Vera and Anya attend a mathematics club in which more than \(91\%\) of the members are boys. Find the smallest possible number of participants in the club. | 23 | 0.875 |
On the base \(AC\) of an isosceles triangle \(ABC (AB = BC)\), point \(M\) is marked. It is known that \(AM = 7\), \(MB = 3\), \(\angle BMC = 60^\circ\). Find the length of segment \(AC\). | 17 | 0.625 |
In the plane Cartesian coordinate system \( xOy \), the focus of the parabola \( \Gamma: y^2 = 2px \) (with \( p > 0 \)) is \( F \). A tangent is drawn to \( \Gamma \) at a point \( P \) different from \( O \), and this tangent intersects the y-axis at point \( Q \). Given that \( |FP| = 2 \) and \( |FQ| = 1 \), find the dot product of the vectors \( \overrightarrow{OP} \) and \( \overrightarrow{OQ} \). | \frac{3}{2} | 0.875 |
Let \( S = \{1, 2, 3, 4\} \), and consider a sequence \( a_1, a_2, \cdots, a_n \) of \( n \) terms where \( a_i \in S \) such that for any non-empty subset \( B \) of \( S \), there exists a subsequence of \( n \) adjacent terms that forms the set \( B \). Find the minimum value of \( n \). | 8 | 0.125 |
Let \(PABC\) be a tetrahedron such that \(\angle APB = \angle APC = \angle BPC = 90^\circ\), \(\angle ABC = 30^\circ\), and \(AP^2\) equals the area of triangle \(ABC\). Compute \(\tan \angle ACB\). | 8 + 5\sqrt{3} | 0.375 |
To move a chess piece from the point $(0,0)$ to the point $(m, n)$, where $m$ and $n$ are natural numbers, and the movement must be along the coordinate axis directions, only changing direction at points with integer coordinates. How many shortest paths are there to move the chess piece? | \binom{m+n}{m} | 0.625 |
In the quadrilateral \(ABCD\), the lengths of the sides \(BC\) and \(CD\) are 2 and 6, respectively. The points of intersection of the medians of triangles \(ABC\), \(BCD\), and \(ACD\) form an equilateral triangle. What is the maximum possible area of quadrilateral \(ABCD\)? If necessary, round the answer to the nearest 0.01. | 29.32 | 0.125 |
The bisectors of the exterior angles $B$ and $C$ of triangle $ABC$ intersect at point $M$.
a) Can angle $BMC$ be obtuse?
b) Find angle $BAC$ given that $\angle BMC = \frac{\angle BAM}{2}$. | 120^\circ | 0.375 |
If integer \( x \) satisfies \( x \geq 3+\sqrt{3+\sqrt{3+\sqrt{3+\sqrt{3+\sqrt{3}}}}} \), find the minimum value of \( x \). | 6 | 0.875 |
Let $k > 0$ be an integer, and let $p = p(n)$ be a function of $n$ such that for large $n$, we have $p \geqslant (6k \ln n) n^{-1}$. Show that
$$
\lim_{n \rightarrow \infty} P\left[\alpha \geqslant \frac{1}{2} \frac{n}{k}\right] = 0.
$$ | 0 | 0.875 |
In triangle \( ABC \), let \( J \) be the center of the excircle tangent to side \( BC \) at \( A_1 \) and to the extensions of sides \( AC \) and \( AB \) at \( B_1 \) and \( C_1 \), respectively. Suppose that the lines \( A_1B_1 \) and \( AB \) are perpendicular and intersect at \( D \). Let \( E \) be the foot of the perpendicular from \( C_1 \) to line \( DJ \). Determine the angles \(\angle B E A_1\) and \(\angle A E B_1\). | 90^\circ | 0.875 |
Given that \(a\), \(b\), and \(c\) are three distinct real numbers, and the three quadratic equations
$$
\begin{array}{l}
x^{2}+a x+b=0, \\
x^{2}+b x+c=0, \\
x^{2}+c x+a=0
\end{array}
$$
each pair of equations has exactly one common root. Find the value of \(a^{2}+b^{2}+c^{2}\). | 6 | 0.75 |
What is the value of \((7 - 6 \times (-5)) - 4 \times (-3) \div (-2)\) ? | 31 | 0.625 |
Let \(ABCD\) be a rectangle with \(AB=20\) and \(AD=23\). Let \(M\) be the midpoint of \(CD\), and let \(X\) be the reflection of \(M\) across point \(A\). Compute the area of triangle \(XBD\). | 575 | 0.75 |
There are five types of gifts priced at 2 yuan, 5 yuan, 8 yuan, 11 yuan, and 14 yuan, and five types of packaging boxes priced at 3 yuan, 5 yuan, 7 yuan, 9 yuan, and 11 yuan. Each gift is paired with one packaging box. How many different total prices are there? | 19 | 0.375 |
The sum of the first thirteen terms of an arithmetic progression is $50\%$ of the sum of the last thirteen terms of this progression. The sum of all terms of this progression, excluding the first three terms, is to the sum of all terms excluding the last three terms in the ratio $5:4$. Find the number of terms in this progression. | 22 | 0.75 |
A circle is drawn inside a regular hexagon so that it touches all six sides of the hexagon. The area of the circle is \(\pi \times 64 \sqrt{3}\). What is the area of the hexagon? | 384 | 0.875 |
What prime numbers less than 17 divide the number \( 2002^{2002} - 1 \)? | 3 | 0.375 |
Find the smallest positive real number $\lambda$ such that for any three complex numbers $\mathrm{z}_{1}$, $\mathrm{z}_{2}$, $\mathrm{z}_{3} \in \{\mathrm{z} \in \mathbb{C} \mid |\mathrm{z}| < 1\}$, if $\mathrm{z}_{1} + \mathrm{z}_{2} + \mathrm{z}_{3} = 0$, then $\left| \mathrm{z}_{1} \mathrm{z}_{2} + \mathrm{z}_{2} \mathrm{z}_{3} + \mathrm{z}_{3} \mathrm{z}_{1} \right|^{2} + \left| \mathrm{z}_{1} \mathrm{z}_{2} \mathrm{z}_{3} \right|^{2} < \lambda$. | 1 | 0.625 |
Four people are sitting at four sides of a table, and they are dividing a 32-card Hungarian deck equally among themselves. If one selected player does not receive any aces, what is the probability that the player sitting opposite them also has no aces among their 8 cards? | \frac{130}{759} | 0.125 |
Given that $\sum_{i=1}^{n} a_{i} x_{i}=p$ and $\sum_{i=1}^{n} a_{i}=q$, with $a_{i} > 0 \ (i=1,2,\ldots,n)$, where $p$ and $q$ are constants, find the minimum value of $\sum_{i=1}^{n} a_{i} x_{i}^{2}$. | \frac{p^{2}}{q} | 0.125 |
If the 3-digit decimal number \( n = \overline{abc} \) satisfies that \( a \), \( b \), and \( c \) form an arithmetic sequence, then what is the maximum possible value of a prime factor of \( n \)? | 317 | 0.625 |
Five soccer teams play a match where each team plays every other team exactly once. Each match awards 3 points to the winner, 0 points to the loser, and 1 point to each team in the event of a draw. After all matches have been played, the total points of the five teams are found to be five consecutive natural numbers. Let the teams ranked 1st, 2nd, 3rd, 4th, and 5th have drawn $A$, $B$, $C$, $D$, and $E$ matches respectively. Determine the five-digit number $\overline{\mathrm{ABCDE}}$. | 13213 | 0.375 |
Inside a right triangle \(ABC\) with hypotenuse \(AC\), a point \(M\) is chosen such that the areas of triangles \(ABM\) and \(BCM\) are one-third and one-quarter of the area of triangle \(ABC\) respectively. Find \(BM\) if \(AM = 60\) and \(CM = 70\). If the answer is not an integer, round it to the nearest whole number. | 38 | 0.875 |
Find all positive integers \( x \) for which \( p(x) = x^2 - 10x - 22 \), where \( p(x) \) denotes the product of the digits of \( x \). | 12 | 0.625 |
As shown in the figure, a large rectangle is divided into 9 smaller rectangles. The areas of the three small rectangles located at the corners are 9, 15, and 12, respectively. Find the area of the small rectangle located at the fourth corner. | 20 | 0.875 |
As shown in the figure, the graph \(\Gamma\) of the function \(y = \cos x + 1\) for \(0 \leq x \leq 2\pi\) is drawn inside the rectangle \(OABC\). When \(AB\) coincides with \(OC\), forming a cylinder, find the eccentricity of the curve \(\Gamma\) on the surface of the cylinder. | \frac{\sqrt{2}}{2} | 0.875 |
Calculate the value of the product \(\left(1-\frac{1}{4}\right) \cdot\left(1-\frac{1}{9}\right) \cdot\left(1-\frac{1}{16}\right) \cdot \ldots \cdot\left(1-\frac{1}{2021^{2}}\right)\). | \frac{1011}{2021} | 0.625 |
Let \( A, B, C, \) and \( D \) be points randomly selected independently and uniformly within the unit square. What is the probability that the six lines \( \overline{AB}, \overline{AC}, \overline{AD}, \overline{BC}, \overline{BD}, \) and \( \overline{CD} \) all have positive slopes? | \frac{1}{24} | 0.5 |
The arithmetic mean of three two-digit natural numbers \( x, y, z \) is 60. What is the maximum value that the expression \( \frac{x + y}{z} \) can take? | 17 | 0.875 |
What is the minimum number of cells that need to be marked in a $7 \times 7$ grid so that in each vertical or horizontal $1 \times 4$ strip there is at least one marked cell? | 12 | 0.125 |
For normal operation, the bus depot must have at least eight buses on the line, and currently, it has ten. The probability that each bus does not go on the line is 0.1. Find the probability of normal operation of the bus depot for the coming day. | 0.9298 | 0.5 |
Given a natural number \( x = 6^n + 1 \), where \( n \) is an odd natural number. It is known that \( x \) has exactly three distinct prime divisors, one of which is 11. Find \( x \). | 7777 | 0.875 |
Determine the amount of released gas substance based on the chemical reaction equation:
$$
\mathrm{n}\left(\mathrm{CO}_{2}\right)=\mathrm{n}\left(\mathrm{CaCO}_{3}\right)=2.4 \text{ mol}
$$
Calculate the volume of the released gas:
$$
V\left(\mathrm{CO}_{2}\right)=n \cdot V_m=2.4 \cdot 22.4=53.76 \text{ L}
$$ | 53.76 \text{ L} | 0.875 |
How many three-digit natural numbers \( n \) are there for which the number \( n^3 - n^2 \) is a perfect square? | 22 | 0.625 |
Find all solutions to the following equation:
$$
\cos ^{2} x - 2 \cos x \cos y \cos (x+y) + \cos ^{2}(x+y) = a
$$
What is the condition for the equation to be solvable?
| 0 \leq a \leq 1 | 0.5 |
Nine digits: \(1, 2, 3, \ldots, 9\) are written in a certain order (forming a nine-digit number). Consider all consecutive triples of digits, and find the sum of the resulting seven three-digit numbers. What is the largest possible value of this sum? | 4648 | 0.5 |
A square fits snugly between a horizontal line and two touching circles with a radius of 1000. The line is tangent to the circles. What is the side length of the square? | 400 | 0.5 |
Into how many maximum parts can the surface of a sphere (sphere) be divided by 3, 4, and generally $n$ circles? | n^2 - n + 2 | 0.875 |
Calculate the double integral \(\iint_{D}(x + 2y) \, dx \, dy\), where the region \(D\) is bounded by the parabolas \(y = x - x^{2}\), \(y = 1 - x^{2}\), and the \(Oy\)-axis. | \frac{2}{3} | 0.5 |
Given the numbers \( x, y, z \), and \( k \) such that
\[ \frac{7}{x+y} = \frac{k}{x+z} = \frac{11}{z-y} \]
determine the value of \( k \). | 18 | 0.875 |
A large circular table has 60 chairs around it. What is the largest number of people who can sit around the table so that each person is only sitting next to exactly one other person? | 40 | 0.25 |
What is the maximum number of parts into which 5 segments can divide a plane? | 16 | 0.875 |
Find the distance from the point $M_0$ to the plane that passes through the three points $M_1$, $M_2$, and $M_3$.
$M_1(1, 0, 2)$
$M_2(1, 2, -1)$
$M_3(2, -2, 1)$
$M_0(-5, -9, 1)$ | \sqrt{77} | 0.875 |
Let $p$ be a prime number. Show that $\binom{2p}{p} \equiv 2 \pmod{p}$. | 2 | 0.125 |
Find the smallest natural number \( k \) such that for some natural number \( a \), greater than 500,000, and some natural number \( b \), the equation \(\frac{1}{a} + \frac{1}{a+k} = \frac{1}{b}\) holds. | 1001 | 0.25 |
How many five-digit numbers divisible by 3 are there that include the digit 6? | 12504 | 0.75 |
An acute angle equals to \(60^{\circ}\) and contains two circles that are tangential to each other externally. The radius of the smaller circle is \(r\). Find the radius of the larger circle. | 3r | 0.875 |
Formulate the equation of the normal to the given curve at the point with abscissa \( x_0 \).
\[ y = \frac{1 + \sqrt{x}}{1 - \sqrt{x}}, \quad x_{0} = 4 \] | y = -2x + 5 | 0.875 |
On a plane, a semicircle with a diameter \( AB = 36 \) cm was constructed; inside it, a semicircle with a diameter \( OB = 18 \) cm was constructed (with \( O \) being the center of the larger semicircle). Then, a circle was constructed, touching both semicircles and segment \( AO \). Find the radius of this circle. Justify your answer. | 8 \text{ cm} | 0.875 |
Calculate the limit of the function:
$$\lim _{x \rightarrow 0}\left(\frac{\arcsin ^{2} x}{\arcsin ^{2} 4 x}\right)^{2 x+1}$$ | \frac{1}{16} | 0.75 |
The denominators of two irreducible fractions are 600 and 700. What is the smallest possible value of the denominator of their sum when written as an irreducible fraction?
Note: We say that the fraction \( p / q \) is irreducible if the integers \( p \) and \( q \) do not have common prime factors in their factorizations. For example, \( \frac{5}{7} \) is an irreducible fraction. | 168 | 0.875 |
Find the equation of the line \( L \) such that the graph of the function
\[ y = x^4 - 4x^3 - 26x^2 \]
lies on one side of this line and has two points in common with it. | y = -60x - 225 | 0.625 |
Point \( M \) divides the side \( BC \) of the parallelogram \( ABCD \) in the ratio \( BM:MC = 1:2 \). The line \( AM \) intersects the diagonal \( BD \) at point \( K \). Find the area of the quadrilateral \( CMKD \) if the area of the parallelogram \( ABCD \) is 1. | \frac{11}{24} | 0.5 |
Two groups have an equal number of students. Each student studies at least one language: English or French. It is known that 5 people in the first group and 5 in the second group study both languages. The number of students studying French in the first group is three times less than in the second group. The number of students studying English in the second group is four times less than in the first group. What is the minimum possible number of students in one group? | 28 | 0.625 |
Given the function \( f(x) = \frac{2x^2 + \sqrt{2} \sin \left(x + \frac{\pi}{4}\right)}{2x^2 + \cos x} \), with maximum and minimum values \( a \) and \( b \) respectively, find the value of \( a + b \). | 2 | 0.875 |
Through points \(A(0 ; 14)\) and \(B(0 ; 4)\), two parallel lines are drawn. The first line, passing through point \(A\), intersects the hyperbola \(y = \frac{1}{x}\) at points \(K\) and \(L\). The second line, passing through point \(B\), intersects the hyperbola \(y = \frac{1}{x}\) at points \(M\) and \(N\).
What is the value of \(\frac{A L - A K}{B N - B M}?\) | 3.5 | 0.125 |
Calculate the volume of the solid formed by rotating around the $O Y$ axis the curvilinear trapezoid which is bounded by the hyperbola $x y=2$ and the lines $y_{1}=1, y_{2}=4$, and $y_{3}=0$. | 3\pi | 0.875 |
Find all functions \( f : [1, +\infty) \rightarrow [1, +\infty) \) that satisfy the following conditions:
1. \( f(x) \leq 2(x + 1) \);
2. \( f(x + 1) = \frac{1}{x} \left[ (f(x))^2 - 1 \right] \). | f(x) = x + 1 | 0.875 |
In triangle \(ABC\), angle \(A\) is \(40^\circ\). The triangle is randomly thrown onto a table. Find the probability that vertex \(A\) ends up east of the other two vertices. | \frac{7}{18} | 0.375 |
Find the maximum value of the parameter \(a\) for which the equation \((|x-2|+2a)^{2}-3(|x-2|+2a)+4a(3-4a)=0\) has three solutions. Specify the largest value in your answer. | 0.5 | 0.75 |
Is there a figure that has exactly two axes of symmetry but does not have a center of symmetry? | \text{No} | 0.625 |
Point \( M \) divides the side \( BC \) of the parallelogram \( ABCD \) in the ratio \( BM : MC = 2 \). Line \( AM \) intersects the diagonal \( BD \) at point \( K \). Find the area of the quadrilateral \( CMKD \) if the area of the parallelogram \( ABCD \) is 1. | \frac{11}{30} | 0.625 |
2016 students are lined up in a row and count off from left to right according to $1,2 \cdots, n$ $(n \geqslant 2)$. If the 2016th student counts to $n$, all students who count to $n$ in this round will receive a New Year's gift. How many students will definitely not receive a New Year's gift, regardless of what $n$ is? | 576 | 0.125 |
There are 29 students in a class: some are honor students who always tell the truth, and some are troublemakers who always lie.
All the students in this class sat at a round table.
- Several students said: "There is exactly one troublemaker next to me."
- All other students said: "There are exactly two troublemakers next to me."
What is the minimum number of troublemakers that can be in the class? | 10 | 0.75 |
The Small and the Big islands have a rectangular shape and are divided into rectangular counties. Each county has a road laid along one of the diagonals. On each island, these roads form a closed path that does not pass through any point more than once. Here is how the Small island is structured, which has only six counties (see the illustration).
Draw how the Big island can be structured if it has an odd number of counties. How many counties did you come up with? | 9 \text{ fiefs} | 0.875 |
Given real numbers \( x_1, x_2, \ldots, x_{2021} \) satisfy \( \sum_{i=1}^{2021} x_i^2 = 1 \), find the maximum value of \( \sum_{i=1}^{2020} x_i^3 x_{i+1}^3 \). | \frac{1}{8} | 0.875 |
For how many positive numbers less than 1000 is it true that among the numbers $2, 3, 4, 5, 6, 7, 8,$ and $9$, there is exactly one that is not its divisor? | 4 | 0.25 |
Reflect the parabola \( y = x^2 \) over the point \( (1, 1) \). What is the equation of the reflected image? | y = -x^2 + 4x - 2 | 0.875 |
In triangle \( ABC \), a point \( K \) is taken on side \( AB \), such that \( AK: BK = 1:2 \). Another point \( L \) is taken on side \( BC \), such that \( CL: BL = 2:1 \). Point \( Q \) is the intersection of lines \( AL \) and \( CK \). Find the area of triangle \( ABC \) if it is known that \( S_{BQC} = 1 \). | \frac{7}{4} | 0.25 |
Solve the following system of equations for positive numbers:
$$
\begin{cases}
x^{y} = z \\
y^{z} = x \\
z^{x} = y
\end{cases}
$$ | (1, 1, 1) | 0.25 |
Let $S$ be a set of $n$ different real numbers, and $A_{s}$ be the set of all distinct averages of any two distinct elements from $S$. For a given $n \geq 2$, what is the minimum possible number of elements in $A_{s}$? | 2n-3 | 0.875 |
There are three saline solutions with concentrations of 5%, 8%, and 9%, labeled A, B, and C, weighing 60g, 60g, and 47g respectively. We need to prepare 100g of a saline solution with a concentration of 7%. What is the maximum and minimum amount of solution A (5% concentration) that can be used? Please write down the sum of these two numbers as the answer. | 84 | 0.875 |
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