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In triangle \(ABC\), the three sides are given as \(AB = 26\), \(BC = 30\), and \(AC = 28\). Find the part of the area of this triangle that is enclosed between the altitude and the angle bisector drawn from vertex \(B\). | 36 | 0.875 |
Zeroes are written in all cells of a $5 \times 5$ board. We can take an arbitrary cell and increase by 1 the number in this cell and all cells having a common side with it. Is it possible to obtain the number 2012 in all cells simultaneously? | \text{No} | 0.625 |
Find the smallest natural number $n$ such that $\sin n^{\circ} = \sin (2016n^{\circ})$. | 72 | 0.5 |
( Fixed points and limits) Find all functions from $\mathbb{R}_{+}^{*}$ to $\mathbb{R}_{+}^{*}$ such that for all $x, y > 0$, \( f(x f(y)) = y f(x) \) and \( \lim _{x \rightarrow+\infty} f(x) = 0 \). | f(x) = \frac{1}{x} | 0.75 |
Is it possible to tile a $23 \times 23$ chessboard with $2 \times 2$ and $3 \times 3$ squares? With a $1 \times 1$ square and $2 \times 2$ and $3 \times 3$ squares? | \text{Yes} | 0.125 |
Find all positive integers \( n \) less than 1000 such that the cube of the sum of the digits of \( n \) equals \( n^2 \). | 1 \text{ and } 27 | 0.25 |
A semiprime is a number that is a product of two prime numbers. How many semiprime numbers less than 2018 can be expressed as \( x^{3}-1 \) for some natural number \( x \)? | 4 | 0.5 |
The circles OAB, OBC, and OCA have equal radius \( r \). Show that the circle ABC also has radius \( r \). | r | 0.75 |
The polynomial \( x^n + n x^{n-1} + a_2 x^{n-2} + \cdots + a_0 \) has \( n \) roots whose 16th powers have sum \( n \). Find the roots. | -1 | 0.5 |
Given that the graph of the function \( y = f(x) \) is symmetric with respect to the point \( (1,1) \) and the line \( x + y = 0 \), if \( f(x) = \log_{2}(x + 1) \) when \( x \in (0,1) \), find the value of \( f\left(\log_{2} 10\right) \). | \frac{17}{5} | 0.125 |
Find \(\sin^4 \alpha + \cos^4 \alpha\), given that \(\sin \alpha - \cos \alpha = \frac{1}{2}\). | \frac{23}{32} | 0.625 |
Determine the number of pairs of integers, \((a, b)\), with \(1 \leq a \leq 100\) so that the line with equation \(b=ax-4y\) passes through point \((r, 0)\), where \(r\) is a real number with \(0 \leq r \leq 3\), and passes through point \((s, 4)\), where \(s\) is a real number with \(2 \leq s \leq 4\). | 6595 | 0.125 |
16. Variance of the number of matches. A deck of playing cards is laid out on a table (for example, in a row). On top of each card, a card from another deck is placed. Some cards may match. Find:
a) the expected number of matches;
b) the variance of the number of matches. | 1 | 0.625 |
Let \( a, b, c \) be positive numbers such that \( a + b + c = \lambda \). If the inequality
\[ \frac{1}{a(1 + \lambda b)} + \frac{1}{b(1 + \lambda c)} + \frac{1}{c(1 + \lambda a)} \geq \frac{27}{4} \]
always holds, find the range of values for \( \lambda \). | (0, 1] | 0.875 |
Count how many 8-digit numbers there are that contain exactly four nines as digits. | 433755 | 0.375 |
Cat food is sold in large and small packages (with more food in the large package than in the small one). One large package and four small packages are enough to feed a cat for exactly two weeks. Is one large package and three small packages necessarily enough to feed the cat for 11 days? | Yes | 0.875 |
Quadrilateral \(ABCD\) is inscribed in a circle with center \(O\). Two circles \(\Omega_1\) and \(\Omega_2\) of equal radii with centers \(O_1\) and \(O_2\) are inscribed in angles \(ABC\) and \(ADC\) respectively, with the first circle touching side \(BC\) at point \(K\), and the second circle touching side \(AD\) at point \(T\).
a) Find the radius of circle \(\Omega_1\) if \(BK = 3\sqrt{3}\), \(DT = \sqrt{3}\).
b) Additionally, it is known that point \(O_1\) is the center of the circle circumscribed around triangle \(BOC\). Find the angle \(BDC\). | 30^\circ | 0.5 |
Find the functions \( f: \mathbb{Q} \rightarrow \mathbb{Q} \) such that \( f(1) = 2 \) and for all \( x, y \in \mathbb{Q} \): \( f(xy) = f(x) f(y) - f(x + y) + 1 \). | f(x) = x + 1 | 0.875 |
Show from the previous problem that the square of the hypotenuse is equal to the sum of the squares of the legs. | c^2 = a^2 + b^2 | 0.625 |
In the number \(2 * 0 * 1 * 6 * 0 * 2 *\), replace each of the 6 asterisks with any of the digits \(0, 1, 2, 3, 4, 5, 6, 7, 8\) (digits may be repeated) so that the resulting 12-digit number is divisible by 45. In how many ways can this be done? | 13122 | 0.375 |
The numbers \( x_{1}, \ldots, x_{n}, y_{1}, \ldots, y_{n} \) satisfy the condition \( x_{1}^{2}+\ldots+x_{n}^{2}+y_{1}^{2}+\ldots+y_{n}^{2} \leq 2 \). Find the maximum value of the expression
$$
A=\left(2\left(x_{1}+\ldots+x_{n}\right)-y_{1}-\ldots-y_{n}\right) \cdot \left(x_{1}+\ldots+x_{n}+2\left(y_{1}+\ldots+y_{n}\right)\right)
$$ | 5n | 0.75 |
In a right triangle, the bisector of an acute angle divides the opposite leg into segments of lengths 4 cm and 5 cm. Determine the area of the triangle. | 54 \text{ cm}^2 | 0.25 |
Angelica wants to choose a three-digit code for her suitcase lock. To make it easier to remember, Angelica wants all the digits in her code to be in non-decreasing order. How many different possible codes does Angelica have to choose from? | 220 | 0.875 |
Solve the following system of equations. It has a solution if and only if each term equals zero:
$$
\left\{\begin{array}{c}
3 x^{2}+8 x-3=0 \\
3 x^{4}+2 x^{3}-10 x^{2}+30 x-9=0
\end{array}\right.
$$ | -3 | 0.75 |
Each cell of a \(50 \times 50\) square contains a number equal to the count of \(1 \times 16\) rectangles (both vertical and horizontal) for which this cell is an endpoint. How many cells contain numbers that are greater than or equal to 3? | 1600 | 0.125 |
Anton, Vasya, Sasha, and Dima were driving from city A to city B, each taking turns at the wheel. The entire journey was made at a constant speed.
Anton drove the car for half the time Vasya did, and Sasha drove for as long as Anton and Dima together. Dima was at the wheel for only one-tenth of the distance. What fraction of the distance did Vasya drive? Provide your answer as a decimal. | 0.4 | 0.625 |
Explain a method to multiply two numbers using only addition, division by 2, and multiplication by 2, as illustrated by the following example of multiplying 97 by 23. The method involves the following steps:
1. Write the two numbers at the top of two columns.
2. In the first column, repeatedly divide the number by 2 (discarding the remainder) until you reach 1.
3. In the second column, repeatedly multiply the number by 2 the same number of times.
For example:
| 97 | 23 |
| ---: | :---: |
| 48 | 46 |
| 24 | 92 |
| 12 | 184 |
| 6 | 368 |
| 3 | 736 |
| 1 | 1472 |
| | 2231 |
Next, cross out the entries in the second column corresponding to even numbers in the first column, and sum the remaining numbers:
The numbers not crossed out are 23, 736, and 1472. Adding these together gives the product 2231.
Why does this method work? | 2231 | 0.5 |
Calculate the volume of the tetrahedron with vertices at the points $A_{1}, A_{2}, A_{3}, A_{4}$ and find its height dropped from the vertex $A_{4}$ to the face $A_{1} A_{2} A_{3}$.
$A_{1}(5, 2, 0)$
$A_{2}(2, 5, 0)$
$A_{3}(1, 2, 4)$
$A_{4}(-1, 1, 1)$ | 2\sqrt{3} | 0.625 |
Calculate: \( 2013 \div (25 \times 52 - 46 \times 15) \times 10 = \) | 33 | 0.875 |
Five athletes came to practice with their own balls, and upon leaving, each took someone else's ball. How many ways is this possible? | 44 | 0.75 |
The sequence is defined recursively:
\[ x_{0} = 0, \quad x_{n+1} = \frac{(n^2 + n + 1) x_{n} + 1}{n^2 + n + 1 - x_{n}}. \]
Find \( x_{8453} \). | 8453 | 0.875 |
Determine whether there exists an infinite number of lines \( l_{1}, l_{2}, \cdots, l_{n}, \cdots \) on the coordinate plane that satisfy the following conditions:
1. The point \( (1, 1) \) lies on \( l_{n} \) for \( n = 1, 2, \cdots \).
2. \( k_{n+1} = a_{n} - b_{n} \), where \( k_{n+1} \) is the slope of \( l_{n+1} \), and \( a_{n}, b_{n} \) are the intercepts of \( l_{n} \) on the \( x \)-axis and \( y \)-axis respectively, for \( n = 1, 2, 3, \cdots \).
3. \( k_{n} \cdot k_{n+1} \geq 0 \) for \( n = 1, 2, 3, \cdots \). | \text{No} | 0.75 |
How many ways are there to win tic-tac-toe in \(\mathbb{R}^{n}\)? That is, how many lines pass through three of the lattice points \((a_{1}, \ldots, a_{n})\) in \(\mathbb{R}^{n}\) with each coordinate \(a_{i}\) in \(\{1,2,3\}\)? Express your answer in terms of \(n\). | \frac{5^n - 3^n}{2} | 0.5 |
Given \(\left(a x^{4}+b x^{3}+c x^{2}+d x+e\right)^{5} \cdot\left(a x^{4}-b x^{3}+c x^{2}-d x+e\right)^{5}=a_{0}+a_{1} x+ a_{2} x^{2}+\cdots+a_{41} x^{10}\), find \(a_{1}+a_{3}+a_{5}+\cdots+a_{39}\). | 0 | 0.5 |
The non-zero numbers \( a, b, \) and \( c \) are such that the doubled roots of the quadratic polynomial \( x^{2}+a x+b \) are the roots of the polynomial \( x^{2}+b x+c \). What can the ratio \( a / c \) equal? | \frac{1}{8} | 0.75 |
In the plane $\alpha$, two perpendicular lines are drawn. Line $l$ forms angles of $45^{\circ}$ and $60^{\circ}$ with them. Find the angle that line $l$ makes with the plane $\alpha$. | 30^\circ | 0.875 |
Where are the points \( M(x, y) \) located if \( x = y \) (the first coordinate is equal to the second)? | y = x | 0.875 |
Let \( P \) be an arbitrary point on the graph of the function \( y = x + \frac{2}{x} \) (where \( x > 0 \)). From point \( P \), perpendiculars are drawn to the line \( y = x \) and the \( y \)-axis, with the foots of these perpendiculars being points \( A \) and \( B \) respectively. Determine the value of \( \overrightarrow{PA} \cdot \overrightarrow{PB} \). | -1 | 0.875 |
Solve the equation
$$
4^{x} - 3^{x - \frac{1}{2}} = 3^{x + \frac{1}{2}} - 2^{2x - 1}
$$ | \frac{3}{2} | 0.5 |
Let the set \( A = \{ 1, 2, \cdots, n \} \). Let \( S_n \) denote the sum of all elements in the non-empty proper subsets of \( A \), and \( B_n \) denote the number of subsets of \( A \). Find the limit \(\lim_{{n \to \infty}} \frac{S_n}{n^2 B_n}\). | \frac{1}{4} | 0.75 |
Let \( p(x) = x^2 - x + 1 \). Let \(\alpha\) be a root of \( p(p(p(p(x)))) \). Find the value of
\[
(p(\alpha) - 1) p(\alpha) p(p(\alpha)) p(p(p(\alpha)))
\] | -1 | 0.625 |
\(D\) is the point of tangency of the incircle with side \(BC\). \(N\) is an arbitrary point on segment \(ID\). The perpendicular to \(ID\) at point \(N\) intersects the circumcircle. \(O_1\) is the center of the circumcircle of \(\triangle XIY\). Find the product \(OO_1 \cdot IN\).
| Rr | 0.125 |
We take a \(6 \times 6\) chessboard, which has six rows and columns, and indicate its squares by \((i, j)\) for \(1 \leq i, j \leq 6\). The \(k\)-th northeast diagonal consists of the six squares satisfying \(i - j \equiv k \pmod{6}\); hence there are six such diagonals.
Determine if it is possible to fill the entire chessboard with the numbers \(1, 2, \ldots, 36\) (each exactly once) such that each row, each column, and each of the six northeast diagonals has the same sum. | \text{No} | 0.625 |
Given a sequence $\left\{a_{n}\right\}$ with 9 terms, where $a_{1}=a_{9}=1$, and for each $i \in \{1,2, \cdots, 8\}$, $\frac{a_{i+1}}{a_{i}} \in \left\{2,1,-\frac{1}{2}\right\}$, determine the number of such sequences. | 491 | 0.5 |
Given that $\alpha, \beta \in \mathbf{R}$, the lines $\frac{x}{\sin \alpha+\sin \beta}+\frac{y}{\sin \alpha+\cos \beta}=1$ and $\frac{x}{\cos \alpha+\sin \beta}+\frac{y}{\cos \alpha+\cos \beta}=1$ intersect on the line $y=-x$, find the value of $\sin \alpha+\cos \alpha+\sin \beta+\cos \beta$. | 0 | 0.875 |
Calculate the following expression (accurate to 8 decimal places):
$$
16\left(\frac{1}{5}-\frac{1}{3} \times \frac{1}{5^{3}}+\frac{1}{5} \times \frac{1}{5^{5}}-\frac{1}{7} \times \frac{1}{5^{7}}+\frac{1}{9} \times \frac{1}{5^{9}}-\frac{1}{11} \times \frac{1}{5^{11}}\right)-4\left(\frac{1}{239}-\frac{1}{3} \times \frac{1}{239^{3}}\right)
$$ | 3.14159265 | 0.375 |
In a right triangle \(ABC\) with a right angle at \(A\), the altitude \(AH\) is drawn. A circle passing through points \(A\) and \(H\) intersects the legs \(AB\) and \(AC\) at points \(X\) and \(Y\) respectively. Find the length of segment \(AC\), given that \(AX = 5\), \(AY = 6\), and \(AB = 9\). | 13.5 | 0.375 |
Fill the first eight positive integers in a $2 \times 4$ table, one number per cell, such that each row's four numbers increase from left to right, and each column's two numbers increase from bottom to top. How many different ways can this be done? | 14 | 0.5 |
A triangle has an angle of \(70^{\circ}\). How can you construct an angle of \(40^{\circ}\) using it? | 40^\circ | 0.625 |
Write an \( n \)-digit number using the digits 1 and 2, such that no two consecutive digits are both 1. Denote the number of such \( n \)-digit numbers as \( f(n) \). Find \( f(10) \). | 144 | 0.625 |
Given that the equation for \(x\), \(x^{2} - 4|x| + 3 = t\), has exactly three real roots. Find the value of \(t\). | 3 | 0.875 |
On the right is an equation. Nine Chinese characters represent the numbers 1 to 9. Different characters represent different numbers. What is the possible maximum value of the equation:
盼 × 望 + 树 × 翠绿 + 天空 × 湛蓝 | 8569 | 0.375 |
For \( x > 0 \), let \( f(x) = \frac{4^x}{4^x + 2} \). Determine the value of the sum
$$
\sum_{k=1}^{1290} f\left(\frac{k}{1291}\right)
$$ | 645 | 0.75 |
Let \( ABC \) be an isosceles triangle with \( A \) as the vertex angle. Let \( M \) be the midpoint of the segment \( [BC] \). Let \( D \) be the reflection of point \( M \) over the segment \( [AC] \). Let \( x \) be the angle \( \widehat{BAC} \). Determine, as a function of \( x \), the value of the angle \( \widehat{MDC} \). | \frac{x}{2} | 0.5 |
Let \( x, y, z \) be positive numbers that satisfy the following system of equations:
\[
\begin{cases}
x^2 + xy + y^2 = 108 \\
y^2 + yz + z^2 = 9 \\
z^2 + xz + x^2 = 117
\end{cases}
\]
Find the value of the expression \( xy + yz + xz \). | 36 | 0.75 |
In triangle \( ABC \), the measure of angle \( A \) is twice the measure of angle \( B \), and the lengths of the sides opposite these angles are 12 cm and 8 cm, respectively. Find the length of the third side of the triangle. | 10 \text{ cm} | 0.75 |
Solve the equation among positive numbers:
$$
\frac{x \cdot 2014^{\frac{1}{x}}+\frac{1}{x} \cdot 2014^{x}}{2}=2014
$$ | 1 | 0.5 |
The set
$$
A=\{\sqrt[n]{n} \mid n \in \mathbf{N} \text{ and } 1 \leq n \leq 2020\}
$$
has the largest element as $\qquad$ . | \sqrt[3]{3} | 0.375 |
Let \( F \) be the number of integral solutions of \( x^{2}+y^{2}+z^{2}+w^{2}=3(x+y+z+w) \). Find the value of \( F \). | 208 | 0.25 |
Find the number of different monic quadratic polynomials (i.e., with the leading coefficient equal to 1) with integer coefficients such that they have two different roots which are powers of 5 with natural exponents, and their coefficients do not exceed in absolute value $125^{48}$. | 5112 | 0.5 |
Majka examined multi-digit numbers in which odd and even digits alternate regularly. Those that start with an odd digit, she called "funny," and those that start with an even digit, she called "cheerful" (for example, the number 32387 is funny, the number 4529 is cheerful).
Majka created one three-digit funny number and one three-digit cheerful number, using six different digits without including 0. The sum of these two numbers was 1617. The product of these two numbers ended with the digits 40.
Determine Majka's numbers and calculate their product. | 635040 | 0.25 |
Let \( ABC \) be a triangle. The midpoints of the sides \( BC \), \( AC \), and \( AB \) are denoted by \( D \), \( E \), and \( F \) respectively.
The two medians \( AD \) and \( BE \) are perpendicular to each other and their lengths are \(\overline{AD} = 18\) and \(\overline{BE} = 13.5\).
Calculate the length of the third median \( CF \) of this triangle. | 22.5 | 0.75 |
Given that \( S_n \) and \( T_n \) are the sums of the first \( n \) terms of the arithmetic sequences \( \{a_n\} \) and \( \{b_n\} \), respectively, and that
\[
\frac{S_n}{T_n} = \frac{2n + 1}{4n - 2} \quad (n = 1, 2, \ldots),
\]
find the value of
\[
\frac{a_{10}}{b_3 + b_{18}} + \frac{a_{11}}{b_6 + b_{15}}.
\] | \frac{41}{78} | 0.75 |
The perimeter of a triangle is 28, and the midpoints of its sides are connected by segments. Find the perimeter of the resulting triangle. | 14 | 0.875 |
Inside triangle \( ABC \) with angles \(\angle A = 50^\circ\), \(\angle B = 60^\circ\), \(\angle C = 70^\circ\), a point \( M \) is chosen such that \(\angle AMB = 110^\circ\) and \(\angle BMC = 130^\circ\). Find \(\angle MBC\). | 20^\circ | 0.625 |
Call an integer \( n > 1 \) radical if \( 2^n - 1 \) is prime. What is the 20th smallest radical number? | 4423 | 0.625 |
Calculate the volumes of solids generated by rotating the region bounded by the graphs of the functions about the x-axis.
$$
y = \sin^2(x), x = \frac{\pi}{2}, y = 0
$$
| \frac{3\pi^2}{16} | 0.625 |
A person flips a coin, where the probability of heads up and tails up is $\frac{1}{2}$ each. Construct a sequence $\left\{a_{n}\right\}$ such that
$$
a_{n}=\left\{
\begin{array}{ll}
1, & \text{if the } n \text{th flip is heads;} \\
-1, & \text{if the } n \text{th flip is tails.}
\end{array}
\right.
$$
Let $S_{n}=a_{1}+a_{2}+\cdots+a_{n}$. Find the probability that $S_{2} \neq 0$ and $S_{8}=2$. Provide your answer in its simplest fractional form. | \frac{13}{128} | 0.5 |
How many different four-digit numbers divisible by 15 can be formed using the digits $0, 1, 3, 5, 6$ without repetition of digits? | 26 | 0.25 |
Given non-negative real numbers \( a, b, c, x, y, z \) that satisfy \( a + b + c = x + y + z = 1 \), find the minimum value of \( \left(a - x^{2}\right)\left(b - y^{2}\right)\left(c - z^{2}\right) \). | -\frac{1}{4} | 0.875 |
Vojta began writing the number of the current school year 2019202020192020... continuously. When he wrote 2020 digits, he got bored.
How many twos did he write?
Hint: How many twos would Vojta write if he wrote only 20 digits? | 757 | 0.75 |
What is the smallest positive integer that is divisible by 111 and has the last four digits as 2004? | 662004 | 0.25 |
In a small town, the police are looking for a wanderer. There is a four in five chance that he is in one of the eight bars in the town, with no preference for any particular one. Two officers visited seven bars but did not find the wanderer. What are the chances of finding him in the eighth bar? | \frac{1}{3} | 0.75 |
A bridge over a river connects two different regions of a country. At one point, one of the regions repainted its section of the bridge. If the newly painted section were 1.2 times larger, it would constitute exactly half of the entire bridge. What fraction of the bridge still needs to be painted for it to be exactly half painted? | \frac{1}{12} | 0.875 |
In a grove, there are four types of trees: birches, spruces, pines, and aspens. There are 100 trees in total. It is known that among any 85 trees, there are trees of all four types. What is the smallest number of any trees in this grove that must include trees of at least three types? | 69 | 0.625 |
On the coordinate plane, the points \(A(0, 2)\), \(B(1, 7)\), \(C(10, 7)\), and \(D(7, 1)\) are given. Find the area of the pentagon \(A B C D E\), where \(E\) is the intersection point of the lines \(A C\) and \(B D\). | 36 | 0.875 |
At the tourist base, the number of two-room cottages is twice the number of one-room cottages. The number of three-room cottages is a multiple of the number of one-room cottages. If the number of three-room cottages is tripled, it will be 25 more than the number of two-room cottages. How many cottages are there in total at the tourist base, given that there are at least 70 cottages? | 100 | 0.875 |
Let \( M \) be a finite set of numbers. It is known that among any three elements of this set, there exist two whose sum also belongs to \( M \).
What is the maximum possible number of elements in \( M \)? | 7 | 0.125 |
Given a square \(ABCD\) with side length \(a\), vertex \(A\) lies in plane \(\beta\), and the other vertices are on the same side of plane \(\beta\). The distances from points \(B\) and \(D\) to plane \(\beta\) are 1 and 2, respectively. If the dihedral angle between plane \(ABCD\) and plane \(\beta\) is 30 degrees, then \(a =\)? | 2\sqrt{5} | 0.25 |
Find the number of pairs of integers \( (x, y) \) that satisfy the condition \( x^{2} + 6xy + 5y^{2} = 10^{100} \). | 19594 | 0.375 |
Find the maximum value of the function \( f(x) = \lg 2 \cdot \lg 5 - \lg 2x \cdot \lg 5x \). | \frac{1}{4} | 0.75 |
Luna has an infinite supply of red, blue, orange, and green socks. She wants to arrange 2012 socks in a line such that no red sock is adjacent to a blue sock and no orange sock is adjacent to a green sock. How many ways can she do this? | 4 \cdot 3^{2011} | 0.25 |
When \( s \) and \( t \) range over all real numbers, the expression
$$
(s+5-3|\cos t|)^{2}+(s-2|\sin t|)^{2}
$$
achieves a minimum value of \(\qquad\). | 2 | 0.25 |
Let the sets \( A \supseteq A_{1} \supseteq A_{2} \). If \( A_{2} \sim A \), then \( A_{1} \sim A \). | A_1 \sim A | 0.5 |
All possible non-empty subsets are taken from the set of numbers $1,2,3, \ldots, n$. For each subset, the reciprocal of the product of all its numbers is taken. Find the sum of all such reciprocals. | n | 0.75 |
On an island, there are knights, liars, and followers; each person knows who is who. All 2018 island residents were lined up and each was asked to answer "Yes" or "No" to the question: "Are there more knights than liars on the island?" The residents responded one by one in such a way that the others could hear. Knights always told the truth, liars always lied. Each follower answered the same as the majority of the preceding respondents, and if the "Yes" and "No" answers were split equally, they could give either answer. It turned out that there were exactly 1009 "Yes" answers. What is the maximum number of followers that could be among the island residents? | 1009 | 0.5 |
In a meadow, ladybugs have gathered. If a ladybug has six spots on its back, it always tells the truth. If it has four spots, it always lies. There are no other types of ladybugs in the meadow. The first ladybug said, "Each of us has the same number of spots on our back." The second said, "Together we have 30 spots in total." The third disagreed, saying, "Altogether, we have 26 spots on our backs." "Of these three, exactly one told the truth," declared each of the remaining ladybugs. How many ladybugs were there in total in the meadow? | 5 | 0.5 |
Inside the triangle \(ABC\), there are points \(P\) and \(Q\) such that point \(P\) is at distances 6, 7, and 12 from lines \(AB\), \(BC\), and \(CA\) respectively, and point \(Q\) is at distances 10, 9, and 4 from lines \(AB\), \(BC\), and \(CA\) respectively. Find the radius of the inscribed circle of triangle \(ABC\). | 8 | 0.875 |
Natural numbers \( x_{1}, x_{2}, \ldots, x_{13} \) are such that \( \frac{1}{x_{1}} + \frac{1}{x_{2}} + \ldots + \frac{1}{x_{13}} = 2 \). What is the minimum value of the sum of these numbers? | 85 | 0.875 |
Find the integers \(a\), \(b\), and \(c\) such that they satisfy the condition:
\[ a^{2} + b^{2} + c^{2} + 3 < a b + 3 b + 2 c. \] | a = 1, b = 2, c = 1 | 0.625 |
Calculate the definite integral:
$$
\int_{0}^{\frac{\pi}{2}}\left(1 - 5x^{2}\right) \sin x \, dx
$$ | 11 - 5\pi | 0.875 |
On the base \( AC \) of an isosceles triangle \( ABC \) (\( AB = BC \)), a point \( M \) is marked. It is known that \( AM = 7 \), \( MB = 3 \), and \(\angle BMC = 60^{\circ}\). Find the length of the segment \( AC \). | 17 | 0.625 |
The tadpoles of the Triassic Discoglossus have five legs each, while the tadpoles of the Saber-toothed Frog have several tails (all having the same number). A Jurassic Park staff member scooped up several tadpoles along with water. It turned out that the total caught had 100 legs and 64 tails. How many tails does each Saber-toothed Frog tadpole have, if all five-legged tadpoles have one tail, and all multi-tailed tadpoles have four legs? | 3 | 0.875 |
For what maximum \( a \) is the inequality \(\frac{\sqrt[3]{\operatorname{tg} x}-\sqrt[3]{\operatorname{ctg} x}}{\sqrt[3]{\sin x}+\sqrt[3]{\cos x}}>\frac{a}{2}\) satisfied for all permissible \( x \in \left(\frac{3 \pi}{2}, 2 \pi\right) \)? If necessary, round your answer to the nearest hundredth. | 4.49 | 0.625 |
Find the smallest positive period \( T \) of the function \( f(x) = \frac{2 \sin x + 1}{3 \sin x - 5} \). | 2\pi | 0.75 |
A number \( \mathrm{A} \) is a prime number, and \( \mathrm{A}+14 \), \( \mathrm{A}+18 \), \( \mathrm{A}+32 \), \( \mathrm{A}+36 \) are also prime numbers. What is the value of \( \mathrm{A} \)? | 5 | 0.875 |
In the final of the giraffe beauty contest, two giraffes, Tall and Spotted, reached the finals. There are 135 voters divided into 5 districts, with each district divided into 9 precincts, and each precinct having 3 voters. The voters in each precinct choose the winner by majority vote; in a district, the giraffe that wins in the majority of precincts wins the district; finally, the giraffe that wins in the majority of the districts is declared the winner of the final. The giraffe Tall won. What is the minimum number of voters who could have voted for Tall? | 30 | 0.75 |
Find the sum of the first twelve terms of an arithmetic sequence if its fifth term $a_{5} = 1$ and its seventeenth term $a_{17} = 18$. | 37.5 | 0.375 |
Three points \( A \), \( B \), and \( C \) are randomly selected on the unit circle. Find the probability that the side lengths of triangle \( \triangle ABC \) do not exceed \( \sqrt{3} \). | \frac{1}{3} | 0.125 |
Given any 4-digit positive integer \( x \) not ending in '0', we can reverse the digits to obtain another 4-digit integer \( y \). For example, if \( x \) is 1234, then \( y \) is 4321. How many possible 4-digit integers \( x \) are there if \( y - x = 3177 \)? | 48 | 0.125 |
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