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|---|---|---|
The difference between the longest and shortest diagonals of the regular n-gon equals its side. Find all possible n. | 9 | 0.25 |
Every day, Xiaoming goes to school along a flat road \(AB\), an uphill road \(BC\), and a downhill road \(CD\) (as shown in the diagram). Given that \(AB : BC : CD = 1 : 2 : 1\) and that Xiaoming's speeds on flat, uphill, and downhill roads are in the ratio 3 : 2 : 4, respectively, find the ratio of the time Xiaoming takes to go to school to the time he takes to come home. | 19:16 | 0.25 |
The rules of a card arrangement game are as follows: Arrange nine cards marked with $1,2, \cdots, 9$ randomly in a row. If the number on the first card (from the left) is $k$, then reverse the order of the first $k$ cards, which is called an operation. The game stops when no operation can be performed (i.e., the number on the first card is "1"). If an arrangement cannot be operated on and can be obtained by performing one operation on a unique other arrangement, it is called a "second stop arrangement". Find the probability of a second stop arrangement occurring among all possible arrangements. | \frac{103}{2520} | 0.125 |
Calculate the limit of the function:
$$\lim_{x \rightarrow 0} \frac{\sqrt{1-2x+3x^{2}}-(1+x)}{\sqrt[3]{x}}$$ | 0 | 0.875 |
Three distinct numbers are drawn at random from the set \(\{1, 2, \cdots, 10\}\). Find the probability that the sample variance \(s^2 \leq 1\). | \frac{1}{15} | 0.625 |
In a regular hexagon \( ABCDEF \), the diagonals \( AC \) and \( CE \) are divided by interior points \( M \) and \( N \) respectively in the ratio \(\frac{AM}{AC} = \frac{CM}{CE} = r\). If the points \( B \), \( M \), and \( N \) are collinear, find the value of \( r \). | \frac{\sqrt{3}}{3} | 0.75 |
We have 1000 solid cubes with edge lengths of 1 unit each. We want to use these small cubes to create a hollow cube with a wall thickness of 1 unit. The small cubes can be glued together but not cut. What is the maximum (external) volume of the cube we can thus create? | 2197 | 0.375 |
Find the sum of all four-digit numbers that can be formed using the digits \(1, 2, 3, 4, 5\) such that each digit appears at most once. | 399960 | 0.75 |
How many five-digit numbers are there, not divisible by 1000, in which the first, third, and last digits are even? | 9960 | 0.75 |
The weight of cows of the Red Gorbatov breed is a random variable \(X\) distributed according to the normal law with a mean \(a = 470 \, \text{kg}\) and a standard deviation \(\sigma = 30 \, \text{kg}\). What is the probability that two out of three randomly selected cows will have a weight more than 470 kg and less than 530 kg? | 0.357 | 0.125 |
How to refute the statement: "If a number is divisible by 5, then it ends with the digit 5"? | 10 | 0.625 |
Carlson and Baby have several jars of jam, each weighing an integer number of pounds.
The total weight of all Carlson's jars of jam is 13 times the total weight of all Baby's jars. Carlson gave Baby the jar with the smallest weight (among those he had), after which the total weight of his jars turned out to be 8 times the total weight of Baby's jars.
What is the maximum possible number of jars Carlson could have initially had? | 23 | 0.75 |
$O$ is the center of a circle, $C$ is the intersection point of the chord $AB$ and the radius $OD$, which is perpendicular to the chord. Given $OC = 9$ and $CD = 32$. Find the length of the chord $AB$. | 80 | 0.875 |
Given the sum \( \sum_{n=1}^{100} n=1+2+3+4+\cdots+100=5050 \), which represents the sum of 100 consecutive natural numbers starting from 1, find the units digit of \( \sum_{n=1}^{2021} n\left(n^{2}+n+1\right) \). | 3 | 0.5 |
A cyclist rode 96 km 2 hours faster than expected. At the same time, he covered 1 km more per hour than he expected to cover in 1 hour 15 minutes. What was his speed? | 16 \text{ km/h} | 0.875 |
Let T be the triangle with vertices (0, 0), (a, 0), and (0, a). Find \(\lim_{{a \to \infty}} a^4 \exp(-a^3) \int_{T} \exp(x^3 + y^3) \, dx \, dy\). | \frac{2}{9} | 0.625 |
Find the number of integers from 1 to 250 that are divisible by any of the integers 2, 3, 5, or 7. | 193 | 0.25 |
The points \(A\), \(B\), and \(C\) are the centers of three faces of a cuboid that meet at a vertex. The lengths of the sides of the triangle \(ABC\) are 4, 5, and 6. What is the volume of the cuboid? | 90\sqrt{6} | 0.875 |
A notebook sheet was colored in 23 colors, one color per cell. A pair of colors is called good if there are two adjacent cells colored in these colors. What is the minimum number of good pairs? | 22 | 0.375 |
Let the function \( f^{\prime}(x) \) be the derivative of an even function \( f(x) \) with \( x \neq 0 \), and suppose \( f(-1)=0 \). Given that \( x f^{\prime}(x) - f(x) < 0 \) for \( x > 0 \), determine the range of \( x \) such that \( f(x) > 0 \). | (-1, 0) \cup (0, 1) | 0.875 |
A line passing through the focus \( F \) of the parabola \( y^2 = 4x \) intersects the parabola at points \( M \) and \( N \). Let \( E(m, 0) \) be a point on the x-axis. The extensions of \( M E \) and \( N E \) intersect the parabola at points \( P \) and \( Q \). If the slopes \( k_1 \) of \( M N \) and \( k_2 \) of \( P Q \) satisfy \( k_1 = 3 k_2 \), then the value of the real number \( m \) is ______. | 3 | 0.5 |
Find the real polynomial \( p(x) \) of degree 4 with the largest possible coefficient of \( x^4 \) such that \( p(x) \in [0, 1] \) for all \( x \in [-1, 1] \). | 4x^4 - 4x^2 + 1 | 0.5 |
Find all real numbers \(k\) such that the inequality
\[
a^{3} + b^{3} + c^{3} + d^{3} + 1 \geq k(a + b + c + d)
\]
holds for any \(a, b, c, d \in [-1, +\infty)\). | \frac{3}{4} | 0.5 |
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 a triangle with \(\widehat{C A B}=20^{\circ}\). Let \( D \) be the midpoint of the segment \([A B]\). It is given that \(\widehat{C D B}=40^{\circ}\). What is the value of the angle \(\widehat{A B C}\)? | 70^\circ | 0.625 |
In a group consisting of $2n$ members, the following conditions hold: there exists a number $d$ such that each member of the group knows at most $d$ other members, and there are more than $d$ people in the group such that none of them know each other. Show that, no matter what $d$ is, the total number of acquaintances is less than $n^{2}$. | n^2 | 0.625 |
A metal sphere with a radius of $\sqrt[z]{16}$ is melted and recast into a cone whose lateral surface area is three times the area of its base. Find the height of the cone. | 8 | 0.875 |
The lengths of the sides of a triangle are $\sqrt{3}, \sqrt{4}(=2), \sqrt{5}$.
In what ratio does the altitude perpendicular to the middle side divide it? | 1:3 | 0.375 |
For which natural numbers \( n \) is the inequality
$$
\sin n \alpha + \sin n \beta + \sin n \gamma < 0
$$
true for any angles \( \alpha, \beta, \gamma \) that are the angles of an acute triangle? | 4 | 0.25 |
Determine all the functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) such that
\[ f\left(x^{2}+f(y)\right)=f(f(x))+f\left(y^{2}\right)+2 f(x y) \]
for all real numbers \( x \) and \( y \). | f(x) = x^2 | 0.625 |
Let \( f(x) \) be a differentiable function defined on \((- \infty, 0)\), with its derivative denoted by \( f^{\prime}(x) \). Given the inequality \( 2 f(x) + x f^{\prime}(x) > x^2 \), determine the solution set for the inequality \((x+2017)^{2} f(x+2017) - f(-1) > 0\). | (-\infty, -2018) | 0.875 |
\(r\) is the inradius of the triangle \( \triangle ABC \) and \( r' \) is the exradius for the circle touching side \( AB \). Show that \( 4rr' \leq c^2 \), where \( c \) is the length of side \( AB \). | 4rr' \leq c^2 | 0.625 |
Let \( p^{n} (p \text{ is a prime number}, n \geqslant 2) \) be a "good number". If the difference between 100 and 23 in base \( b \) is a good number, find all possible values of \( b \). | 7 | 0.5 |
The segment $A B=2 d$ is bisected perpendicularly by the line $e$. On this line $e$, the segment $R S=d$ slides. The lines $A R$ and $B S$ intersect at point $P$. What is the locus of point $P$? | x^2 + 2xy = d^2 | 0.25 |
December 31, 2013 is a Tuesday. What day of the week is June 1, 2014? (Answer using numbers: Monday is 1, Tuesday is 2, Wednesday is 3, Thursday is 4, Friday is 5, Saturday is 6, Sunday is 7.) | 7 | 0.75 |
If \( a \) is the sum of all the positive factors of 2002, find the value of \( a \).
It is given that \( x>0, y>0 \) and \( \sqrt{x}(\sqrt{x}+\sqrt{y})=3 \sqrt{y}(\sqrt{x}+5 \sqrt{y}) \). If \( b=\frac{2 x+\sqrt{x y}+3 y}{x+\sqrt{x y}-y} \), find the value of \( b \).
Given that the equation \(||x-2|-1|=c\) has only 3 integral solutions, find the value of \( c \).
If \( d \) is the positive real root of the equation \(\frac{1}{2}\left\{\frac{1}{2}\left[\frac{1}{2}\left(\frac{1}{2} x^{2}+2\right)+2\right]+2\right\}=2\), find the value of \( d \). | 2 | 0.5 |
The base of the pyramid is an isosceles right triangle, where each leg measures 8. Each of the pyramid's lateral edges is 9. Find the volume of the pyramid. | \frac{224}{3} | 0.625 |
Given a tetrahedron \( A B C D \) with side lengths \( A B = 41 \), \( A C = 7 \), \( A D = 18 \), \( B C = 36 \), \( B D = 27 \), and \( C D = 13 \), let \( d \) be the distance between the midpoints of edges \( A B \) and \( C D \). Find the value of \( d^{2} \). | 137 | 0.5 |
Given the sets \(A=\left\{1, \frac{x+y}{2}-1\right\}\) and \(B=\{-\ln(xy), x\}\), if \(A = B\) and \(0 < y < 2\), find the value of
\[
\left(x^{2} - \frac{1}{y^{2}}\right) + \left(x^{4} - \frac{1}{y^{4}}\right) + \cdots + \left(x^{2022} - \frac{1}{y^{2022}}\right).
\] | 0 | 0.75 |
Given a tetrahedron $A B C D$. All plane angles at vertex $D$ are right angles; $D A = 1, D B = 2, D C = 3$. Find the median of the tetrahedron drawn from vertex $D$. | \frac{\sqrt{14}}{3} | 0.75 |
On a board are written the numbers \(1, 2, 3, \ldots, 57\). What is the maximum number of these numbers that can be selected so that no two selected numbers differ by exactly 2.5 times? | 48 | 0.375 |
Given the set \( M = \{ (a, b) \mid a \leq -1, b \leq m \} \). If for any \((a, b) \in M\), it always holds that \(a \cdot 2^b - b - 3a \geq 0\), then the maximum value of the real number \( m \) is _____. | 1 | 0.625 |
In a class, some students study only English, some study only German, and some study both languages. What percentage of the students in the class study both languages if 80% of all students study English and 70% study German? | 50\% | 0.75 |
$a_{1}, a_{2}, a_{3}, \cdots, a_{n}$ are natural numbers satisfying $0<a_{1}<a_{2}<a_{3} \cdots<a_{n}$, and $\frac{13}{14}=\frac{1}{a_{1}} + \frac{1}{a_{2}} + \frac{1}{a_{3}}+\cdots +\frac{1}{a_{n}}$. What is the minimum value of $n$? | 4 | 0.5 |
Each hotel room can accommodate no more than 3 people. The hotel manager knows that a group of 100 football fans, who support three different teams, will soon arrive. A room can only house either men or women; and fans of different teams cannot be housed together. How many rooms need to be booked to accommodate all the fans? | 37 | 0.125 |
In triangle \(PQR\), the angle \(QRP\) is \(60^\circ\). Find the distance between the points of tangency on side \(QR\) of the inscribed circle with radius 2 and the circle with radius 3 that is tangent to the extensions of sides \(PQ\) and \(PR\). | \sqrt{3} | 0.125 |
In a tournament, each participant was supposed to play exactly one game with each of the remaining participants. However, two participants dropped out during the tournament after playing only 4 games each. As a result, the total number of games played was 62. How many participants were there in total? | 13 | 0.75 |
Find the total number of sets of positive integers \((x, y, z)\), where \(x, y\) and \(z\) are positive integers, with \(x < y < z\) such that
$$
x + y + z = 203.
$$ | 3333 | 0.75 |
Randomly select elements $m$ and $n$ (which can be the same) from the set $\{1, 2, \cdots, 100\}$. What is the probability that the unit digit of $2^{m} + 3^{n}$ is 3? | \frac{3}{16} | 0.5 |
Let \( n \) be a three-digit number such that \( 100 \leqslant n \leqslant 999 \). Find all \( n \) for which the last three digits of \( n^2 \) are equal to \( n \). | 376 \text{, } 625 | 0.375 |
Given \( x \cdot y \cdot z + y + z = 12 \), find the maximum value of \( \log_{4} x + \log_{2} y + \log_{2} z \). | 3 | 0.875 |
Let \( f(x) \) be a quadratic function such that \( 2x^{2} - 4x + 3 \leqslant f(x) \leqslant 3x^{2} - 6x + 4 \) and \( f(3) = 11 \). Find \( f(5) \). | 41 | 0.875 |
Find the maximum constant \( k \) such that for \( x, y, z \in \mathbb{R}_+ \), the following inequality holds:
$$
\sum \frac{x}{\sqrt{y+z}} \geqslant k \sqrt{\sum x},
$$
where " \( \sum \) " denotes a cyclic sum. | \sqrt{\frac{3}{2}} | 0.125 |
A math interest group in a school has 14 students. They form \( n \) different project teams. Each project team consists of 6 students, each student participates in at least 2 project teams, and any two project teams have at most 2 students in common. Find the maximum value of \( n \). | 7 | 0.25 |
A rhombus, where the shorter diagonal is equal to its side of length 1, rotates around a line passing through the end of the longer diagonal and perpendicular to this diagonal. Find the volume of the resulting solid of revolution. | \frac{3 \pi}{2} | 0.5 |
Find the smallest possible α such that if p(x) ≡ ax^2 + bx + c satisfies |p(x)| ≤ 1 on [0, 1], then |p'(0)| ≤ α. | 8 | 0.625 |
The point \( P(-1,4) \) is reflected in the \( y \)-axis to become \( Q \). The point \( Q \) is reflected in the line \( y=x \) to become \( R \). The point \( R \) is reflected in the \( x \)-axis to become \( S \). What is the area of quadrilateral \( PQRS \)? | 8 | 0.875 |
Inside an isosceles triangle \( ABC \), a point \( K \) is marked such that \( CK = AB = BC \) and \(\angle KAC = 30^\circ\). Find the angle \(\angle AKB\). | 150^{\circ} | 0.875 |
There are three simplest proper fractions with their numerators in the ratio $3: 2: 4$ and their denominators in the ratio $5: 9: 15$. When these three fractions are added together and then reduced to simplest form, the result is $\frac{28}{45}$. What is the sum of the denominators of these three fractions? | 203 | 0.5 |
Replace the asterisk $(*)$ in the expression $\left(x^{3}-2\right)^{2}+\left(x^{2}+*\right)^{2}$ with a monomial such that, after squaring and combining like terms, the resulting expression has four terms. | 2x | 0.75 |
Let \( S = \{1, 2, \cdots, 2005\} \). If any \( n \) pairwise coprime numbers in \( S \) always include at least one prime number, find the minimum value of \( n \). | 16 | 0.75 |
What is the greatest number of consecutive natural numbers, each of which has exactly four natural divisors (including 1 and the number itself)? | 3 \text{ numbers} | 0.625 |
In an infinite sequence of natural numbers, the product of any fifteen consecutive terms is equal to one million, and the sum of any ten consecutive terms is equal to \(S\). Find the maximum possible value of \(S\). | 208 | 0.375 |
Find the cosine of the angle between vectors $\overrightarrow{A B}$ and $\overrightarrow{A C}$.
$A(3, 3, -1), B(5, 5, -2), C(4, 1, 1)$ | -\frac{4}{9} | 0.875 |
The 97 numbers \( \frac{49}{1}, \frac{49}{2}, \frac{49}{3}, \ldots, \frac{49}{97} \) are written on a blackboard. We repeatedly pick two numbers \( a, b \) on the board and replace them by \( 2ab - a - b + 1 \) until only one number remains. What are the possible values of the final number? | 1 | 0.75 |
Point \( D \) lies on side \( CB \) of right triangle \( ABC \left(\angle C = 90^{\circ} \right) \), such that \( AB = 5 \), \(\angle ADC = \arccos \frac{1}{\sqrt{10}}, DB = \frac{4 \sqrt{10}}{3} \). Find the area of triangle \( ABC \). | \frac{15}{4} | 0.875 |
On the board, the number 27 is written. Every minute, the number is erased from the board and replaced with the product of its digits increased by 12. For example, after one minute, the number on the board will be $2 \cdot 7 + 12 = 26$. What number will be on the board after an hour? | 14 | 0.25 |
Is it possible to select a number \( n \geq 3 \) and fill an \( n \times (n+3) \) table (with \( n \) rows and \( n+3 \) columns) with distinct natural numbers from 1 to \( n(n+3) \) such that in each row there are three numbers, one of which is equal to the product of the other two? | \text{No} | 0.875 |
Solve the inequality \(\left(\sqrt{x^{3}-18 x-5}+2\right) \cdot\left|x^{3}-4 x^{2}-5 x+18\right| \leqslant 0\). | x = 1 - \sqrt{10} | 0.625 |
Construct regular triangles outwardly on sides $AB$ and $BC$ of parallelogram $ABCD$. The third vertices of these triangles are $E$ and $F$, respectively. Show that the sum of angles $CED$ and $AFD$ is $60^{\circ}$. | 60^\circ | 0.625 |
In the sequence \(\{a_n\}\), \(a_4=1\), \(a_{11}=9\), and the sum of any three consecutive terms is always 15. Find \(a_{2016}\). | 5 | 0.875 |
A standard deck of cards, excluding jokers, has 4 suits with 52 cards in total. Each suit has 13 cards, with face values from 1 to 13. If Feifei draws 2 hearts, 3 spades, 4 diamonds, and 5 clubs from this deck, and the sum of the face values of the spades is 11 times the sum of the face values of the hearts, while the sum of the face values of the clubs is 45 more than the sum of the face values of the diamonds, what is the sum of the face values of these 14 cards? | 101 | 0.25 |
If \( S_{n} = 1 - 2 + 3 - 4 + \ldots + (-1)^{n-1} n \), where \( n \) is a positive integer, determine the value of \( S_{17} + S_{33} + S_{50} \). | 1 | 0.875 |
How many pairs of natural numbers \(a\) and \(b\) exist such that \(a \geq b\) and the equation
\[ \frac{1}{a} + \frac{1}{b} = \frac{1}{6} \]
is satisfied? | 5 | 0.75 |
Joe has written 5 questions of different difficulties for a test, numbered 1 through 5. He wants to make sure that problem \(i\) is harder than problem \(j\) whenever \(i - j \geq 3\). In how many ways can he order the problems for his test? | 25 | 0.25 |
1. \(\lim_{x \rightarrow \infty} \frac{3 x^{2} - 1}{5 x^{2} + 2 x}\)
2. \(\lim_{n \rightarrow -\infty} \frac{n}{\sqrt{n^{2} + 1}}\)
3. \(\lim_{n \rightarrow +\infty} \frac{1 + 7^{n + 2}}{3 - 7^{n}}\)
4. \(\lim_{n \rightarrow +\infty} \frac{2 + 4 + 6 + \ldots + 2n}{1 + 3 + 5 + \ldots + (2n + 1)}\)
5. \(\lim_{x \rightarrow \frac{\pi}{4}} \frac{\operatorname{tg} 2x}{\operatorname{ctg}\left(\frac{\pi}{4} - x\right)}\)
6. \(\lim_{n \rightarrow +\infty} \frac{n^{3}}{n^{2} + 2^{2} + 3^{2} + \ldots + n^{2}}\) | 3 | 0.75 |
Given that the complex numbers \( z_{1} \) and \( z_{2} \) satisfy
\[ \left|z_{1} + z_{2}\right| = 20, \quad \left|z_{1}^{2} + z_{2}^{2}\right| = 16, \]
find the minimum value of \( \left|z_{1}^{3} + z_{2}^{3}\right| \). | 3520 | 0.875 |
Find the value of the expression \(\frac{a}{b} + \frac{b}{a}\), where \(a\) and \(b\) are the largest and smallest roots of the equation \(x^3 - 9x^2 + 9x = 1\), respectively. | 62 | 0.625 |
Let \( P \) be an arbitrary point on the ellipse \(\frac{x^{2}}{16}+\frac{y^{2}}{9}=1\) other than the endpoints of the major axis. \( F_{1} \) and \( F_{2} \) are its left and right foci respectively, and \( O \) is the center. Find the value of \( \left|P F_{1}\right| \cdot \left|P F_{2}\right| + |O P|^{2} \). | 25 | 0.75 |
Find the millionth digit after the decimal point in the decimal expansion of the fraction \(1 / 41\). | 9 | 0.875 |
Let \( \triangle ABC \) be a triangle such that \( AB = 7 \), and let the angle bisector of \( \angle BAC \) intersect line \( BC \) at \( D \). If there exist points \( E \) and \( F \) on sides \( AC \) and \( BC \), respectively, such that lines \( AD \) and \( EF \) are parallel and divide triangle \( ABC \) into three parts of equal area, determine the number of possible integral values for \( BC \). | 13 | 0.125 |
On the board, there are three quadratic equations written:
$$
\begin{gathered}
2020 x^{2}+b x+2021=0 \\
2019 x^{2}+b x+2020=0 \\
x^{2}+b x+2019=0
\end{gathered}
$$
Find the product of the roots of all the equations written on the board, given that each of them has two real roots. | 2021 | 0.875 |
Andrey, Boris, and Valentin participated in a 1 km race. (Assume each of them ran at a constant speed). Andrey finished 100 meters ahead of Boris. Boris finished 60 meters ahead of Valentin. What was the distance between Andrey and Valentin at the moment Andrey finished? | 154 \text{ meters} | 0.625 |
Given the sequence $\left\{a_{n}\right\}$ that satisfies $a_{1}=1$ and $a_{n}=2 a_{n-1}+n-2$ for $n \geqslant 2$, find the general term $a_{n}$. | a_n = 2^n - n | 0.875 |
In Sally's sequence, every term after the second is equal to the sum of the previous two terms. Also, every term is a positive integer. Her eighth term is 400. Find the minimum value of the third term in Sally's sequence. | 35 | 0.625 |
Solve the inequality:
$$
x \log _{1 / 10}\left(x^{2}+x+1\right)>0
$$ | (-\infty, -1) | 0.5 |
From May 1st to May 3rd, the provincial hospital plans to schedule 6 doctors to be on duty, with each person working 1 day and 2 people scheduled per day. Given that doctor A cannot work on the 2nd and doctor B cannot work on the 3rd, how many different scheduling arrangements are possible? | 42 | 0.5 |
A sphere is circumscribed around a regular triangular prism, the height of which is twice the side of the base. How does its volume compare to the volume of the prism? | \frac{64 \pi}{27} | 0.625 |
Let \( ABC \) be an isosceles triangle at \( A \) with \( \angle CAB = 20^\circ \). Let \( D \) be a point on the segment \( [AC] \) such that \( AD = BC \). Calculate the angle \( \angle BDC \). | 30^\circ | 0.75 |
The first term of an arithmetic sequence is 9, and the 8th term is 12. How many of the first 2015 terms of this sequence are multiples of 3? | 288 | 0.75 |
A chess player has 5 points after the 7th round (from 7 games). In how many different ways could this result have been achieved? (Winning a game gives 1 point, drawing a game gives $1 / 2$ point, and losing a game gives 0 points.) | 161 | 0.75 |
In the rectangular coordinate system \( xOy \), find the area of the graph formed by all points \( (x, y) \) that satisfy \( \lfloor x \rfloor \cdot \lfloor y \rfloor = 2013 \), where \( \lfloor x \rfloor \) represents the greatest integer less than or equal to the real number \( x \). | 16 | 0.875 |
A positive integer is called a "random number" if and only if:
(1) All digits are non-zero.
(2) The number is divisible by 11.
(3) The number is divisible by 12, and if the digits of the number are permuted in any way, the resulting number is still divisible by 12.
How many such ten-digit random numbers are there? | 50 | 0.375 |
In how many ways can we place 8 digits equal to 1 and 8 digits equal to 0 on a 4x4 board such that the sums of the numbers written in each row and column are the same? | 90 | 0.375 |
The base of a right parallelepiped is a rhombus. A plane passing through one of the sides of the lower base and the opposite side of the upper base forms an angle of $45^\circ$ with the base plane. The resulting cross-section has an area of $Q$. Determine the lateral surface area of the parallelepiped. | 2\sqrt{2}Q | 0.25 |
A construction team is building a railway tunnel. When $\frac{1}{3}$ of the task is completed, the team starts using new equipment, which increases the construction speed by $20\%$ but shortens the daily work time to $\frac{4}{5}$ of the original time for equipment maintenance. As a result, the project is completed in a total of 185 days. Based on these conditions, determine how many days it would take to complete the project without using the new equipment. | 180 | 0.5 |
Find the number of numbers \( N \) from the set \(\{1, 2, \ldots, 2018\}\) for which positive solutions \( x \) exist for the equation
$$
x^{[x]} = N
$$
(where \([x]\) is the integer part of the real number \( x \), i.e., the largest integer not exceeding \( x \)). | 412 | 0.375 |
A three-digit number is called a "concave number" if the digit in the tens place is smaller than both the digit in the hundreds place and the digit in the units place. For example, 504 and 746 are concave numbers. How many three-digit concave numbers are there if all the digits are distinct? | 240 | 0.375 |
Can \( m! + n! \) end in 1990? | \text{No} | 0.5 |
Seven old women are going to Rome. Each has seven mules, each mule carries seven bags, each bag contains seven loaves of bread, each loaf contains seven knives, each knife is in seven sheaths. How many total items are there? | 137256 | 0.625 |
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