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Simplify the following expression:
\[
\frac{\sin ^{2} 4 \alpha + 4 \sin ^{4} 2 \alpha - 4 \sin ^{2} 2 \alpha \cos ^{2} 2 \alpha}{4 - \sin ^{2} 4 \alpha - 4 \sin ^{2} 2 \alpha}
\] | \tan^4 2\alpha | 0.25 |
Solve the inequality:
$$
\log _{3+\sin x-\cos x}\left(3-\frac{\cos 2 x}{\cos x+\sin x}\right) \geq e^{\sqrt{x}}
$$ | 0 | 0.875 |
Let the set \( M = \{1, 2, \cdots, 10\} \),
\[ A = \{(x, y, z) \mid x, y, z \in M, \text{ and } 9 \mid (x^3 + y^3 + z^3) \} . \]
The number of elements in the set \( A \) is \(\quad\). | 243 | 0.375 |
From a sequence of natural numbers, all numbers that are squares or cubes of integers have been removed. Which number remains in the 100th position? | 112 | 0.5 |
Identical coins are arranged on a table in the shape of a hexagon. If they are arranged such that the side of the hexagon is made up of 2 coins, 7 coins are needed, and if the side consists of 3 coins, a total of 19 coins is required. How many coins are needed to construct a hexagon with a side made up of 10 coins? | 271 | 0.875 |
Given a triangular pyramid \( S-ABC \) whose base \( ABC \) is an isosceles right triangle with \( AB \) as the hypotenuse, and satisfying \( SA = SB = SC = 2 \) and \( AB = 2 \), assume that the four points \( S, A, B, C \) are all on the surface of a sphere centered at point \( O \). Find the distance from point \( O \) to the plane \( ABC \). | \frac{\sqrt{3}}{3} | 0.875 |
Find the sum of all three-digit natural numbers that do not contain the digits 0 or 5. | 284160 | 0.75 |
Given \(a\) is a digit from 1 to 9, if the repeating decimal \(0.1\mathrm{a} = \frac{1}{\mathrm{a}}\), find \(a\). | 6 | 0.875 |
Find all positive integers $n > 3$ such that there exist $n$ points $A_{1}, A_{2}, \cdots, A_{n}$ in the plane and real numbers $r_{1}, r_{2}, \cdots, r_{n}$ satisfying the following conditions:
(1) Any 3 points among $A_{1}, A_{2}, \cdots, A_{n}$ are not collinear;
(2) For each triplet of points $\left\{A_{i}, A_{j}, A_{k}\right\}$ (where $1 \leq i < j < k \leq n$), the area of triangle $\triangle A_{i} A_{j} A_{k}$, denoted by $S_{ijk}$, is equal to $r_{i} + r_{j} + r_{k}$.
(Note: This is a problem from the 36th IMO in 1995) | 4 | 0.875 |
Determine the largest value of \( a \) such that \( a \) satisfies the equations \( a^2 - bc - 8a + 7 = 0 \) and \( b^2 + c^2 + bc - 6a + 6 = 0 \) for some real numbers \( b \) and \( c \). | 9 | 0.875 |
Find the mass of the plate $D$ with surface density $\mu = 16 x + \frac{9 y^2}{2}$, bounded by the curves
$$x = \frac{1}{4}, \quad y = 0, \quad y^2 = 16 x \quad (y \geq 0)$$ | 2 | 0.75 |
In a bag, there are 10 white balls, 5 black balls, and 4 blue balls. All the balls are identical except for color. To ensure that there are at least 2 balls of each color among the balls drawn, what is the minimum value of $n$ for the number of balls drawn? | 17 | 0.875 |
Given that \( x, y, z \) are positive real numbers such that \( x + y + z = 1 \), find the minimum value of the function \( f(x, y, z) = \frac{3x^{2} - x}{1 + x^{2}} + \frac{3y^{2} - y}{1 + y^{2}} + \frac{3z^{2} - z}{1 + z^{2}} \), and provide a proof. | 0 | 0.875 |
In a regular hexagon $A B C D E F$, the diagonals $A C$ and $C E$ are divided by points $M$ and $N$ respectively in the following ratios: $\frac{A M}{A C} = \frac{C N}{C E} = r$. If points $B$, $M$, and $N$ are collinear, determine the ratio $r$. | \frac{\sqrt{3}}{3} | 0.75 |
Determine the number of pairs of integers \((a, b)\) such that \(1 \leq a \leq 30\), \(3 \leq b \leq 30\) and \(a\) is divisible by both \(b\) and \(b-2\). | 22 | 0.125 |
In a certain year, a certain date was never a Sunday in any month. Determine that date. | 31 | 0.25 |
A tetrahedron. There are 6 rods of different lengths, and it is known that no matter how they are ordered, they can form a tetrahedron (triangular pyramid). How many different tetrahedrons can be formed with these rods? | 30 | 0.625 |
Given that \( P \) is a moving point on side \( AF \) of a regular hexagon \( ABCDEF \) with side length 1, find the minimum value of \(\overrightarrow{PD} \cdot \overrightarrow{PE}\). | \frac{3}{2} | 0.375 |
The integers from \(1\) to \(n\) are written in increasing order from left to right on a blackboard. David and Goliath play the following game: starting with David, the two players alternate erasing any two consecutive numbers and replacing them with their sum or product. Play continues until only one number on the board remains. If it is odd, David wins, but if it is even, Goliath wins. Find the 2011th smallest positive integer greater than \(1\) for which David can guarantee victory. | 4022 | 0.125 |
In the royal dining hall, there are three tables with three identical pies. For lunch, the king invites six princes to his table. At the second table, there can be 12 to 18 courtiers, and at the third table, there can be 10 to 20 knights. Each pie is divided into equal pieces according to the number of people seated at the table. There is a rule at the court: the amount of food a knight receives combined with the amount a courtier receives equals the amount received by the king. Determine the maximum possible number of knights the king can invite for lunch that day. How many courtiers will be seated at their table in this case? | 14 | 0.5 |
\(ABCD\) is a square with side length 9. Let \(P\) be a point on \(AB\) such that \(AP: PB = 7:2\). Using \(C\) as the center and \(CB\) as the radius, a quarter circle is drawn inside the square. The tangent from \(P\) meets the circle at \(E\) and \(AD\) at \(Q\). The segments \(CE\) and \(DB\) meet at \(K\), while \(AK\) and \(PQ\) meet at \(M\). Find the length of \(AM\). | \frac{85}{22} | 0.5 |
The children went to the forest for mushrooms. If Anya gives half of her mushrooms to Vita, all the children will have an equal number of mushrooms. But if instead, Anya gives all her mushrooms to Sasha, Sasha will have as many mushrooms as all the others combined. How many children went for mushrooms? | 6 | 0.875 |
When the natural number \( n \) is divided by 8, the remainder is 6. What is the remainder when \( n^2 \) is divided by 32? | 4 | 0.875 |
Vasya thought of three natural numbers with a sum of 1003. After calculating their product, Vasya noticed that it ends in $N$ zeros. What is the maximum possible value of $N$? | 7 | 0.625 |
Along a straight alley, 100 lanterns are placed at equal intervals, numbered sequentially from 1 to 100. Simultaneously, Petya and Vasya start walking towards each other from opposite ends of the alley with different constant speeds (Petya starts from the first lantern, and Vasya starts from the hundredth lantern). When Petya reached the 22nd lantern, Vasya was at the 88th lantern. At which lantern will they meet? If the meeting occurs between two lanterns, indicate the smaller number of the two. | 64 | 0.875 |
Find the Cauchy principal value of the integral \( I = \int_{-\infty}^{\infty} \sin(2x) \, dx \). | 0 | 0.875 |
A positive integer \( N \) and \( N^2 \) end with the same sequence of digits \(\overline{abcd}\), where \( a \) is a non-zero digit. Find \(\overline{abc}\). | 937 | 0.75 |
Among all polynomials \( P(x) \) with integer coefficients for which \( P(-10)=145 \) and \( P(9)=164 \), compute the smallest possible value of \( |P(0)| \). | 25 | 0.875 |
Given \((1-2x)^7 = \sum_{k=0}^{7} a_k x^k\), what is \(2a_2 + 3a_3 + 4a_4 + 5a_5 + 6a_6 + 7a_7\)? | 0 | 0.5 |
In the Cartesian coordinate plane \( xOy \), circle \(\Omega\) passes through the points \((0,0)\), \((2,4)\), and \((3,3)\). The maximum distance from a point on circle \(\Omega\) to the origin is \(\underline{\quad}\). | 2\sqrt{5} | 0.875 |
Given \(0 < a < b\), draw two lines \(l\) and \(m\) through two fixed points \(A(a, 0)\) and \(B(b, 0)\), respectively, such that they intersect the parabola \(y^2 = x\) at four distinct points. When these four points lie on a common circle, find the locus of the intersection point \(P\) of the two lines \(l\) and \(m\). | x = \frac{a + b}{2} | 0.125 |
In the given triangle \( \triangle ABC \) with an area of 36, point \( D \) is on \( AB \) such that \( BD = 2AD \), and point \( E \) is on \( DC \) such that \( DE = 2EC \). What is the area of \( \triangle BEC \)? | 8 | 0.875 |
Express 33 as the sum of \(n\) consecutive natural numbers. When \(n\) is maximized, replace all "+" signs in the resulting sum with "×". What is the product of the terms in this expression? | 20160 | 0.75 |
Given points \( A(4,0) \) and \( B(2,2) \) are inside the ellipse \( \frac{x^{2}}{25}+\frac{y^{2}}{9}=1 \), and \( M \) is a point on the ellipse, find the maximum value of \( |MA| + |MB| \). | 10 + 2 \sqrt{10} | 0.75 |
Let \( n \) be the second smallest integer that can be written as the sum of two positive cubes in two different ways. Compute \( n \). | 4104 | 0.375 |
We have a lighter and two cords (not necessarily identical). When either cord is lit, it takes one hour to burn completely. How can you measure forty-five minutes? | 45 \text{ minutes} | 0.875 |
Real numbers \( x \) and \( y \) satisfy \( 4x^2 - 5xy + 4y^2 = 5 \). Let \( s = x^2 + y^2 \). Find the value of \( \frac{1}{s_{\max}} \). | \frac{3}{10} | 0.875 |
In the number A, the digits are in ascending order (from left to right). What is the sum of the digits of the number \( 9 \cdot A \) ? | 9 | 0.25 |
In triangle \( \triangle ABC \), if \( \frac{\cos A}{\sin B} + \frac{\cos B}{\sin A} = 2 \), and the perimeter of \( \triangle ABC \) is 12, find the maximum possible value of its area. | 36(3 - 2\sqrt{2}) | 0.125 |
In a tetrahedron \(ABCD\), the angle between faces \(ABC\) and \(BCD\) is \(30^{\circ}\), the area of \(\triangle ABC\) is \(120\), the area of \(\triangle BCD\) is \(80\), and \(BC = 10\). Find the volume of this tetrahedron. | 320 | 0.375 |
In Lhota, there was an election for the mayor. Two candidates ran: Mr. Schopný and his wife, Dr. Schopná. The village had three polling stations. In the first and second stations, Dr. Schopná received more votes. The vote ratios were $7:5$ in the first station and $5:3$ in the second station. In the third polling station, the ratio was $3:7$ in favor of Mr. Schopný. The election ended in a tie, with both candidates receiving the same number of votes. In what ratio were the valid votes cast in each polling station if we know that the same number of people cast valid votes in the first and second polling stations? | 24 : 24 : 25 | 0.75 |
Each of the 10 people is either a knight, who always tells the truth, or a liar, who always lies. Each of them has thought of a natural number. Then the first said, "My number is greater than 1," the second said, "My number is greater than 2," and so on until the tenth said, "My number is greater than 10." Afterwards, in a different order, they each said one sentence: "My number is less than 1," "My number is less than 2," and so on until "My number is less than 10." What is the maximum number of knights that could be among these 10 people? | 8 | 0.5 |
After walking so much that his feet get really tired, the beaver staggers so that, at each step, his coordinates change by either $(+1,+1)$ or $(+1,-1)$. Now he walks from $(0,0)$ to $(8,0)$ without ever going below the $x$-axis. How many such paths are there? | 14 | 0.75 |
4n points are arranged around a circle and are colored alternately yellow and blue. The yellow points are divided into pairs, with each pair connected by a yellow line segment. Similarly, the blue points are divided into pairs and each pair is connected by a blue line segment. At most two segments meet at any point inside the circle. Show that there are at least n points of intersection between a yellow segment and a blue segment. | n | 0.125 |
Vanya decided to give Masha a bouquet of an odd number of flowers, consisting of yellow and red tulips, so that the count of one color differs from the count of the other by exactly one. Yellow tulips cost 50 rubles each, and red tulips cost 31 rubles each. What is the maximum number of tulips he can buy for Masha's birthday bouquet without spending more than 600 rubles? | 15 | 0.75 |
Evaluate the expression:
\[
\frac{1-\frac{1}{3}}{1 \times \frac{1}{2} \times \frac{1}{3}}+\frac{\frac{1}{3}-\frac{1}{5}}{\frac{1}{3} \times \frac{1}{4} \times \frac{1}{5}}+\cdots+\frac{\frac{1}{2015}-\frac{1}{2017}}{\frac{1}{2015} \times \frac{1}{2016} \times \frac{1}{2017}}
\] | 2034144 | 0.625 |
On a circle, points $A, B, C, D, E, F, G$ are located clockwise as shown in the diagram. It is known that $AE$ is the diameter of the circle. Additionally, it is known that $\angle ABF = 81^\circ$ and $\angle EDG = 76^\circ$. How many degrees is the angle $FCG$? | 67^\circ | 0.375 |
A traveler visited a village where each person either always tells the truth or always lies. The villagers stood in a circle, and each told the traveler whether the neighbor to their right was truthful. Based on these reports, the traveler was able to determine exactly what proportion of the villagers are liars. Determine what this proportion is. | \frac{1}{2} | 0.75 |
Let the function \( f(x) = \frac{1}{2} + \log_2 \frac{x}{1-x} \) and \( S_n = \sum_{i=1}^{n-1} f\left(\frac{i}{n}\right) \), where \( n \in \mathbf{N}^{*} \) and \( n \geq 2 \). Find \( S_n \). | \frac{n-1}{2} | 0.625 |
Calculate the areas of the regions bounded by the curves given in polar coordinates.
$$
r=\cos 2 \phi
$$ | \frac{\pi}{2} | 0.75 |
Find the solution of the equation \( y^{\prime} - y = \cos x - \sin x \), which satisfies the condition that \( y \) is bounded as \( x \rightarrow +\infty \). | y = \sin x | 0.875 |
In the isosceles triangle \(ABC\), \(\angle ACB = \angle ABC = 40^\circ\). On the ray \(AC\), mark off the segment \(AD\) such that \(AD = BC\). What is the measure of \(\angle BDC\)? | 30^\circ | 0.125 |
Consider a list of six numbers. When the largest number is removed from the list, the average is decreased by 1. When the smallest number is removed, the average is increased by 1. When both the largest and the smallest numbers are removed, the average of the remaining four numbers is 20. Find the product of the largest and the smallest numbers. | 375 | 0.75 |
In a table tennis match between players A and B, they follow the rules that: the winner of each game gets 1 point and the loser gets 0 points. The match stops when one player has 2 more points than the other or after a maximum of 6 games. Let the probability that A wins each game be $\frac{3}{4}$ and the probability that B wins each game be $\frac{1}{4}$, and assume the outcomes of each game are independent. Find the expected number of games $\xi$ played when the match stops, denoted as $\mathrm{E} \xi$. | \frac{97}{32} | 0.375 |
A number was multiplied by the first digit and resulted in 494, by the second digit and resulted in 988, and by the third digit and resulted in 1729. Find this number. | 247 | 0.875 |
Find the largest natural number \( n \) with the following property: for any odd prime number \( p \) less than \( n \), the difference \( n - p \) is also a prime number. | 10 | 0.25 |
The absolute value of a number \( x \) is equal to the distance from 0 to \( x \) along a number line and is written as \( |x| \). For example, \( |8|=8, |-3|=3 \), and \( |0|=0 \). For how many pairs \( (a, b) \) of integers is \( |a|+|b| \leq 10 \)? | 221 | 0.375 |
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 |
Find all positive integer solutions \((x, y, z, t)\) to the equation \(2^{y} + 2^{z} \times 5^{t} - 5^{x} = 1\). | (2, 4, 1, 1) | 0.75 |
Determine the radius of the sphere that touches the faces of the unit cube passing through vertex $A$ and the edges passing through vertex $B$. | 2 - \sqrt{2} | 0.625 |
Four boys, \( A, B, C, \) and \( D \) made three statements each about the same number \( x \). We know that each of them has at least one statement that is true, but also at least one statement that is false. Determine whether \( x \) can be identified. The statements are:
\( A \):
1. The reciprocal of \( x \) is not less than 1.
2. The decimal representation of \( x \) does not contain the digit 6.
3. The cube of \( x \) is less than 221.
\( B \):
4. \( x \) is an even number.
5. \( x \) is a prime number.
6. \( x \) is an integer multiple of 5.
\( C \):
7. \( x \) cannot be expressed as a ratio of two integers.
8. \( x \) is less than 6.
9. \( x \) is a square of a natural number.
\( D \):
10. \( x \) is greater than 20.
11. \( x \) is positive, and its base-10 logarithm is at least 2.
12. \( x \) is not less than 10. | 25 | 0.375 |
Find the smallest natural number \( n \) for which:
a) \( n! \) is divisible by 2016;
b) \( n! \) is divisible by \( 2016^{10} \).
(Note that \( n! = 1 \cdot 2 \cdot 3 \cdot \ldots \cdot n \)). | n=63 | 0.5 |
Given $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are three natural numbers, and the least common multiple (LCM) of $\mathrm{a}$ and $\mathrm{b}$ is $60$, the LCM of $\mathrm{a}$ and $\mathrm{c}$ is $270$, find the LCM of $\mathrm{b}$ and $\mathrm{c}$. | 540 | 0.5 |
In decimal notation, the square of a number with more than one digit has its tens digit equal to 7. What is the units digit of this square number? | 6 | 0.5 |
The lateral faces of a pentagonal pyramid \( S A B C D E \) are acute-angled triangles. We will call a lateral edge of the pyramid good if it is equal to the height of the opposite lateral face, drawn from the apex of the pyramid. For example, edge \( S A \) is good if it is equal to the height of triangle \( S C D \), drawn from vertex \( S \). What is the maximum number of good edges that the pyramid can have? | 2 | 0.375 |
Given the sequence: $\frac{2}{3}, \frac{2}{9}, \frac{4}{9}, \frac{6}{9}, \frac{8}{9}, \frac{2}{27}, \frac{4}{27}, \cdots$, $\frac{26}{27}, \cdots, \frac{2}{3^{n}}, \frac{4}{3^{n}}, \cdots, \frac{3^{n}-1}{3^{n}}, \cdots$. Find the position of $\frac{2018}{2187}$ in the sequence. | 1552 | 0.75 |
Masha has 2 kg of "Lastochka" candies, 3 kg of "Truffle" candies, 4 kg of "Ptichye moloko" candies, and 5 kg of "Citron" candies. What is the maximum number of New Year gifts she can make if each gift must contain 3 different types of candies, 100 grams of each type? | 45 | 0.25 |
Let \( A = \cos^2 10^\circ + \cos^2 50^\circ - \sin 40^\circ \sin 80^\circ \). Determine the value of \( 100A \). | 75 | 0.875 |
As shown in the figure, the side length of square $ABCD$ is 8 centimeters. Point $E$ is on the extension of $AB$, and quadrilateral $BEFG$ is a square. Find the area of triangle $AFC$. | 32 | 0.75 |
\[
\frac{\sin 7^{\circ}+\sin 8^{\circ} \cos 15^{\circ}}{\cos 7^{\circ}-\sin 8^{\circ} \sin 15^{\circ}} =
\] | 2 - \sqrt{3} | 0.875 |
A number appears on a computer screen. It is known that if \( x \) appears on the screen, the number \( x^{2}-2x+1 \) appears immediately after. If the first number to appear is 2, what is the 2020th number to appear? | 1 | 0.75 |
Standa and Jana received two three-digit numbers. Standa placed a decimal point after the first digit of the first number and after the second digit of the second number, then added the resulting decimal numbers and got a result of 50.13. Jana placed a decimal point after the second digit of the first number and after the first digit of the second number, then added the resulting decimal numbers and got a result of 34.02.
Determine the sum of the original three-digit numbers. | 765 | 0.375 |
Given a rectangle with a length of 8 and a width of 4, it is folded along a diagonal and pressed flat. Find the area of the overlapping part (grey triangle). | 10 | 0.125 |
A regular hexagon and an equilateral triangle have the same perimeter. What is the ratio of their areas? | \frac{3}{2} | 0.75 |
A person is waiting at the $A$ HÉV station. They get bored of waiting and start moving towards the next $B$ HÉV station. When they have traveled $1 / 3$ of the distance between $A$ and $B$, they see a train approaching $A$ station at a speed of $30 \mathrm{~km/h}$. If they run at full speed either towards $A$ or $B$ station, they can just catch the train. What is the maximum speed at which they can run? | 10 \text{ km/h} | 0.375 |
Suppose positive real numbers \( x, y, z \) satisfy \( x y z = 1 \). Find the maximum value of \( f(x, y, z) = (1 - yz + z)(1 - zx + x)(1 - xy + y) \) and the corresponding values of \( x, y, z \). | 1 | 0.75 |
A wolf saw a roe deer a few meters away and started chasing it along a straight forest path. The wolf's jump is $22\%$ shorter than the roe deer's jump. Both animals jump at a constant speed. All jumps of the roe deer are of the same length, and the wolf's jumps are also equal to each other. There is an interval of time during which both the wolf and the roe deer make some integer number of jumps. Each time it turns out that the wolf has made $t\%$ more jumps than the roe deer. Find the largest integer value of $t$ for which the wolf cannot catch up with the roe deer. | 28 | 0.625 |
Removing the jokers from a standard deck of cards, we have 52 cards remaining. If five cards are drawn at random from this deck, what is the probability that at least two of the cards have the same rank (number or letter $K$, $Q$, $J$, $A$)? Calculate this probability to two decimal places. | 0.49 | 0.5 |
A farmer presented 6 types of sour cream in containers of \(9, 13, 17, 19, 20, \text{ and } 38\) liters at the market. On the first day, he sold the sour cream from three containers entirely, and on the second day, he sold the contents of two more containers completely. The volume of sour cream sold on the first day was twice the volume sold on the second day. Which containers were emptied on the first day? Indicate the maximum possible total volume of sour cream sold on the first day in the answer. | 66 | 0.625 |
Given a triangle ABC with sides \(a\), \(b\), and \(c\) satisfying \(a^2 + b^2 + 3c^2 = 7\), what is the maximum area of triangle ABC? | \frac{\sqrt{7}}{4} | 0.5 |
Calculate the definite integral:
$$
\int_{-\pi / 2}^{0} 2^{8} \cdot \sin^2(x) \cos^6(x) \, dx
$$ | 5 \pi | 0.375 |
Find all natural numbers \(a\) for which the number
$$
\frac{a+1+\sqrt{a^{5}+2 a^{2}+1}}{a^{2}+1}
$$
is also a natural number. | a = 1 | 0.875 |
Given 99 positive numbers arranged in a circle, it is found that for any four consecutive numbers, the sum of the first two numbers in the clockwise direction is equal to the product of the last two numbers in the clockwise direction. What can be the sum of all 99 numbers placed in a circle? | 198 | 0.5 |
In a country with 20 cities, an airline wants to organize bidirectional flights between them such that it is possible to reach any city from any other city with no more than $\mathrm{k}$ layovers. The number of flight routes should not exceed four. What is the minimum value of $\mathrm{k}$ for which this is possible? | 2 | 0.25 |
Solve the inequality \(\frac{\sqrt{\frac{x}{\gamma}+(\alpha+2)}-\frac{x}{\gamma}-\alpha}{x^{2}+a x+b} \geqslant 0\).
Indicate the number of integer roots of this inequality in the answer. If there are no integer roots or there are infinitely many roots, indicate the number 0 on the answer sheet.
Given:
\[
\alpha = 3, \gamma = 1, a = -15, b = 54.
\] | 7 | 0.625 |
Let $p$ be a prime number and $q$ be a prime divisor of $p^{p-1}+\ldots+ p+1$. Show that $q \equiv 1 (\bmod p)$. | q \equiv 1 \pmod{p} | 0.625 |
How many three-digit positive integers are there such that the three digits of every integer, taken from left to right, form an arithmetic sequence? | 45 | 0.375 |
A right triangle is circumscribed around a circle with a radius of 4, and its hypotenuse is 26. Find the perimeter of the triangle. | 60 | 0.875 |
In a right triangle $ABC$, the leg $AC = 15$ and the leg $BC = 20$. On the hypotenuse $AB$, the segment $AD$ is 4 units long, and the point $D$ is connected to $C$. Find $CD$. | 13 | 0.875 |
Find the area of a triangle given that two of its sides are 1 and $\sqrt{13}$, and the median to the third side is 2. | \sqrt{3} | 0.875 |
What is the product
\[
\log_{3} 2 \cdot \log_{4} 3 \cdot \log_{5} 4 \ldots \log_{10} 9
\]
if it is known that \(\lg 2=0,3010\)? | 0.3010 | 0.875 |
a) $\sqrt{8}$
b) $\sqrt[3]{a^{8}}$
c) $\sqrt[3]{16 x^{4}}$ | 2 x \sqrt[3]{2 x} | 0.5 |
In a carriage, any group of $m \ (m \geq 3)$ passengers have only one common friend (where if A is a friend of B, then B is also a friend of A; no one is their own friend). What is the maximum number of friends that the person with the most friends in this carriage can have? | m | 0.375 |
The mid-term exam results of Chinese, Mathematics, and English for a certain class are as follows: There are 18 students who have achieved excellence in at least one course; 9 students have achieved excellence in Chinese, 11 in Mathematics, and 8 in English; 5 students have achieved excellence in both Chinese and Mathematics, 3 in both Mathematics and English, and 4 in both Chinese and English.
(1) How many students have achieved excellence in at least one of the courses, Chinese or Mathematics?
(2) How many students have achieved excellence in all three courses, Chinese, Mathematics, and English? | 2 | 0.875 |
The numbers \(2^{2021}\) and \(5^{2021}\) are written out one after the other. How many digits were written in total? | 2022 | 0.375 |
Let the set \( T = \{0, 1, \dots, 6\} \),
$$
M = \left\{\left.\frac{a_1}{7}+\frac{a_2}{7^2}+\frac{a_3}{7^3}+\frac{a_4}{7^4} \right\rvert\, a_i \in T, i=1,2,3,4\right\}.
$$
If the elements of the set \( M \) are arranged in decreasing order, what is the 2015th number? | \frac{386}{2401} | 0.625 |
In triangle \(ABC\), \(AB = 20\), \(BC = 21\), and \(CA = 29\). Point \(M\) is on side \(AB\) with \(\frac{AM}{MB}=\frac{3}{2}\), while point \(N\) is on side \(BC\) with \(\frac{CN}{NB}=2\). Points \(P\) and \(Q\) are on side \(AC\) such that line \(MP\) is parallel to \(BC\) and line \(NQ\) is parallel to \(AB\). Suppose \(MP\) and \(NQ\) intersect at point \(R\). Find the area of triangle \(PQR\). | \frac{224}{15} | 0.25 |
A circle with radius \( R \) is inscribed in an isosceles trapezoid. The upper base of the trapezoid is half of its height. Find the area of the trapezoid. | 5R^2 | 0.625 |
In the city where Absent-Minded Scientist lives, phone numbers consist of 7 digits. The Scientist easily remembers a phone number if it is a palindrome, which means it reads the same forwards and backwards. For example, the number 4435344 is easily remembered by the Scientist because it is a palindrome. However, the number 3723627 is not a palindrome, so the Scientist finds it hard to remember. Find the probability that a new random acquaintance's phone number will be easily remembered by the Scientist. | 0.001 | 0.125 |
30 tigers and 30 foxes are divided into 20 groups, each with 3 animals. Tigers always tell the truth, and foxes always lie. When asked, "Is there a fox in your group?", 39 out of the 60 animals answered "No." How many groups consist entirely of tigers? | 3 | 0.875 |
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