problem stringlengths 14 4.09k | solution stringlengths 1 802k | task_type stringclasses 3
values | problem_tokens int64 5 895 |
|---|---|---|---|
Let \( p \) be an odd prime number. Find all natural numbers \( k \) such that
$$
\sqrt{k^{2} - pk}
$$
is a positive integer. | k = \left( \frac{p + 1}{2} \right)^2 | math | 40 |
Successive discounts of $10\%$ and $20\%$ are equivalent to a single discount of: | 28\% | math | 25 |
The following relationships are given:
① The relationship between apple production and climate;
② The relationship between a student and his/her student ID;
③ The relationship between the diameter at breast height and the height of the same species of trees in a forest;
④ The relationship between a point on a curve and... | ①③ | math | 89 |
Given the area of a sector is 1 and its perimeter is 4, calculate the radian measure of the central angle of the sector. | 2 | math | 29 |
Let \(\overline{ab}\) denote a two-digit number with tens digit \(a\) and unit digit \(b\). Find a two-digit number \(\overline{xy}\) satisfying \(\overline{xy} = (x - y)!(\overline{yx} - 3)\). | 42 | math | 66 |
Given a sequence $\{a\_n\}$ where all terms are positive numbers, it satisfies the equation $\log\_2 a\_n = 1 + \log\_2 a\_{n-1}, n \in \mathbb{N}^*, n \geq 2$, and $a\_1 = 2$.
(I) Find the general formula for the sequence $\{a\_n\}$;
(II) Let $c\_n = (3n - 1) \cdot a\_n$, find the sum of the first $n$ terms of the seq... | m = 2, n = 12 | math | 243 |
Given that $f(x+2)$ is an even function on $\mathbb{R}$, and when $x>2$, $f(x)=x^2+1$, determine the expression for $f(x)$ when $x<2$. | x^2 - 8x + 17 | math | 51 |
Given a real number $a < 0$, the function $f(x)= \begin{cases} x^{2}+2a, & x < 1 \\ -x, & x\geqslant 1 \end{cases}$, determine the range of the real number $a$ such that $f(1-a)\geqslant f(1+a)$. | [-2, -1] | math | 81 |
For what values of the real number $m$ is the complex number $z=m^2-1+(m^2-m-2)i$ respectively:
$(1)$ a real number;
$(2)$ a complex number with a non-zero real part;
$(3)$ a purely imaginary number. | m = 1 | math | 60 |
When $m$ is such that the function $y = (m+2)x + 4x - 5$ is a linear function. | m \neq -6 | math | 30 |
Given that triangle ABC is a triangle with altitude AD and median BE, and the lengths of BC, AD, and AC are known, determine the length of EC. | \frac{1}{2}AC | math | 33 |
A company is planning to increase the annual production of a product by implementing technical reforms in 2013. According to the survey, the product's annual production volume $x$ (in ten thousand units) and the technical reform investment $m$ (in million yuan, where $m \ge 0$) satisfy the equation $x = 3 - \frac{k}{m ... | 21 | math | 266 |
Given that the function $y=f(x)$ is the inverse function of $y=a^x$ ($a > 0$ and $a \ne 1$), and its graph passes through the point $\left(\begin{matrix} a^2, & a \end{matrix}\right)$, find $f(x)=$ . | \log_2 x | math | 70 |
Given $\triangle BAD$ is right-angled at $B$, on $AD$ there is a point $C$ for which $AC=CD$ and $AB=BC$. Determine the magnitude of $\angle DAB$. | 60^\circ | math | 46 |
In the Cartesian coordinate system $xOy$, the parametric equations of curve $C_1$ are
$$
\begin{cases}
x = 2 + 2\cos \varphi, \\
y = 2\sin \varphi
\end{cases}
(\varphi \text{ as the parameter}).
$$
With the origin $O$ as the pole and the positive x-axis as the polar axis, establish a polar coordinate system. The polar ... | \alpha = \dfrac{3\pi}{4} | math | 275 |
Given that point A lies on the circle C: x² + y² = 1, a line is drawn from point A perpendicular to the y-axis, intersecting the y-axis at point B. Point P satisfies the equation $\overrightarrow {BP}=2 \overrightarrow {BA}$.
1. Find the trajectory equation of point P.
2. Let Q be a point on the line l: x = 3, and O t... | \frac{3}{2} | math | 116 |
At the Pawsitive Pup Training Center, dogs can learn to do three tricks: jump, fetch, and shake. Of the dogs at the center: \begin{tabular}{l@{\qquad}l}
70 dogs can jump & 30 dogs can jump and fetch \\
40 dogs can fetch & 20 dogs can fetch and shake \\
50 dogs can shake & 25 dogs can jump and shake \\
15 dogs can do al... | 115 | math | 121 |
Given the sequence $a_{n}= \frac {1}{4n^{2}-1}$ for all natural numbers $n$, calculate the sum of the first 10 terms of the sequence $\{a_{n}\}$. | \frac{10}{21} | math | 48 |
Let \( a \) and \( b \) be non-zero real numbers and \( x \in \mathbb{R} \). If
\[
\frac{\sin^{4} x}{a^{2}} + \frac{\cos^{4} x}{b^{2}} = \frac{1}{a^{2} + b^{2}},
\]
find the value of \(\frac{\sin^{2008} x}{a^{2006}} + \frac{\cos^{2008} x}{b^{2006}}\). | \frac{1}{(a^2 + b^2)^{1003}} | math | 122 |
If
\[\frac{\cos^4 \alpha}{\cos^2 \beta} + \frac{\sin^4 \alpha}{\sin^2 \beta} = 1,\]then find the sum of all possible values of
\[\frac{\sin^4 \beta}{\sin^2 \alpha} + \frac{\cos^4 \beta}{\cos^2 \alpha}.\] | 1 | math | 87 |
Define $\mathbf{A} = \begin{pmatrix} 0 & 1 \\ 4 & 0 \end{pmatrix}.$ Find the vector $\mathbf{v}$ such that
\[(\mathbf{A}^7 + \mathbf{A}^5 + \mathbf{A}^3 + \mathbf{A} + \mathbf{I}) \mathbf{v} = \begin{pmatrix} 5 \\ 0 \end{pmatrix}.\] | \begin{pmatrix} -\frac{5}{28899} \\ \frac{1700}{28899} \end{pmatrix} | math | 112 |
Given the set of numbers $\{-8,-6,-4,0,3,5,7\}$, find the minimum possible product of three different numbers from this set. | -280 | math | 36 |
A block of iron solidifies from molten iron, and its volume reduces by $\frac{1}{34}$. Then, if this block of iron melts back into molten iron (with no loss in volume), by how much does its volume increase? | \frac{1}{33} | math | 53 |
Let the random variable $X \sim B(2,p)$ and the random variable $Y \sim B(3,p)$. If $P(X \geqslant 1) = \frac{5}{9}$, then calculate the value of $D(\sqrt{3}Y+1)$. | 2 | math | 65 |
Let $1 \leq n \leq 2021$ be a positive integer. Jack has $2021$ coins arranged in a line where each coin has an $H$ on one side and a $T$ on the other. At the beginning, all coins show $H$ except the nth coin. Jack can repeatedly perform the following operation: he chooses a coin showing $T$ , and turns over... | n = 1011 | math | 143 |
The graph represented by the equation $(x^2-9)^2(x^2-y^2)^2=0$ has how many solutions in the Cartesian plane. | 4 | math | 34 |
Solve the following equations:
$(1) (x+1)^{2} = 4$;
$(2) 3x^{2} - 1 = 2x$. | x_1 = 1, x_2 = -\frac{1}{3} | math | 41 |
Given three fixed points on the plane \\(A(-1,0)\\), \\(B(3,0)\\), \\(C(1,4)\\). Find the equation of the circle that passes through points \\(A\\), \\(B\\), and \\(C\\). | {x}^{2}+{y}^{2}-2x-3y-3=0 | math | 63 |
Sophia is 68 inches tall. Using the conversion 1 inch = 2.54 cm, how tall is Sophia in centimeters? Additionally, convert Sophia's height from centimeters to meters. Express both answers as decimals to the nearest tenth. | 1.7 | math | 53 |
Given six positive consecutive integers start with $c$, the average of these integers is $d$. Find the average of $7$ consecutive integers that start with $d$. | c+5.5 | math | 34 |
Find the direction cosines for the direction vector of the line given by the equations:
$$
\left\{\begin{array}{l}
2x - 3y - 3z + 4 = 0 \\
x + 2y + z - 5 = 0
\end{array}\right.
$$ | \cos \alpha = \frac{3}{\sqrt{83}}, \quad \cos \beta = -\frac{5}{\sqrt{83}}, \quad \cos \gamma = \frac{7}{\sqrt{83}} | math | 68 |
Let $f(x)$ be an even function defined on $\mathbb{R}$, which is monotonically increasing in the interval $(-\infty, 0)$, and satisfies $f(-a^2 + 2a - 5) < f(2a^2 + a + 1)$. Find the range of real numbers $a$. | (-4, 1) | math | 76 |
Find the equation of the line that passes through the intersection point of lines $l_1: 2x-3y+10=0$ and $l_2: 3x+4y-2=0$, and that is perpendicular to the line $3x-2y+5=0$. | 2x + 3y - 2 = 0 | math | 66 |
Let \( S = \{x + iy : -2 \leq x \leq 2, -2 \leq y \leq 2\} \). A complex number \( z = x + iy \) is chosen uniformly at random from \( S \). Compute the probability that the transformation \( \left(\frac{1}{2} + \frac{i}{2}\right)z \) results in a number that remains within \( S \). | 1 | math | 97 |
The line $l$ with slope 1 intersects the ellipse $\frac{x^2}{4} + y^2 = 1$ at points A and B. Find the maximum value of $|AB|$. | \frac{4\sqrt{10}}{5} | math | 44 |
The sum of the first n terms of the sequence {a_n} is denoted by S_n, and it is given that for all n∈ℕ, 2S_n = 3a_n + 4. Find the expression for S_n. | 2-2\times3^{n} | math | 52 |
A line with a slope of $-1$ is drawn through the right vertex $A$ of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ ($a > 0, b > 0$). This line intersects the two asymptotes of the hyperbola at points $B$ and $C$. If $2\overrightarrow{AB} = \overrightarrow{BC}$, then the eccentricity of the hyperbola is \_\_\_\_... | \sqrt{5} | math | 117 |
Given the sequence $$\sqrt{3}, \sqrt{7}, \sqrt{11}, \sqrt{15}, \ldots$$, find the term number in which $$5\sqrt{3}$$ appears. | 19 | math | 47 |
A line passes through point A (3, 0) and is perpendicular to the line $2x+y-5=0$. | x-2y-3=0 | math | 27 |
Given set A={1, 2, 3} and set B={x|x^2-x-2≤0}, find A∩B. | \{1, 2\} | math | 31 |
Given the set of numbers $2$, $0$, $1$, $5$, calculate the probability that the number $2$ is the median of the three different numbers selected. | \dfrac{1}{2} | math | 36 |
Elective 4-4: Coordinate System and Parametric Equations
In the Cartesian coordinate system, the parametric equations of curve $C$ are $\begin{cases}x= \sqrt{5}\cos \alpha \\ y=\sin \alpha\end{cases}$ (where $\alpha$ is the parameter). Taking the origin $O$ as the pole and the positive half-axis of $x$ as the polar ax... | \frac{10 \sqrt{2}}{3} | math | 196 |
The base of a triangular piece of paper $ABC$ is $15\text{ cm}$ long. The paper is folded down over the base, with the crease $DE$ parallel to the base of the paper. The area of the triangle that projects below the base is $25\%$ that of the area of triangle $ABC$. Calculate the length of $DE$, in cm. | 7.5\text{ cm} | math | 82 |
In the complex plane, the corresponding points of the complex numbers \( z_1, z_2, z_3 \) are \( Z_1, Z_2, Z_3 \) respectively. Given that:
\[ \left|z_1\right| = \left|z_2\right| = \sqrt{2}, \overrightarrow{O Z_1} \cdot \overrightarrow{O Z_2} = 0, \text{ and } \left|z_1 + z_2 - z_3\right| = 2, \]
find the range of val... | [0, 4] | math | 142 |
Let $\triangle ABC$ have sides $a$, $b$, and $c$ opposite to angles $A$, $B$, and $C$ respectively. Given that $A = \frac{\pi}{3}$ and $\frac{b+c}{\sin B + \sin C} = 2$. Find:<br/>
$(1)$ the length of side $a$;<br/>
$(2)$ if the area of $\triangle ABC$ is $\frac{\sqrt{3}}{2}$, find the perimeter of $\triangle ABC$. | 3 + \sqrt{3} | math | 111 |
In triangle \( T_{0} \), a triangle was formed using its midlines and named \( T_{1} \). In triangle \( T_{1} \), a triangle was formed using its midlines and named \( T_{2} \). Continuing in this manner, triangle \( T_{10} \) was obtained. Find the ratio of the sum of the areas of all these eleven triangles to the are... | 1398101 | math | 123 |
$ABCD$ is a square with centre $O$ . Two congruent isosceles triangle $BCJ$ and $CDK$ with base $BC$ and $CD$ respectively are constructed outside the square. let $M$ be the midpoint of $CJ$ . Show that $OM$ and $BK$ are perpendicular to each other. | OM \perp BK | math | 91 |
In a circle with center $O$ and radius $r$, chord $AB$ is drawn with length equal to $\sqrt{2}r$ units. From $O$, a perpendicular to $AB$ meets $AB$ at point $M$. From $M$, a perpendicular to $OA$ meets $OA$ at point $D$. What is the area of triangle $MDA$ expressed in terms of $r$?
A) $\frac{3r^2}{16}$
B) $\frac{\pi r... | \frac{r^2}{4\sqrt{3}} | math | 166 |
Given the function \( f(n) = k \), where \( k \) is the \( n \)-th digit after the decimal point of the repeating decimal \( 0.\dot{9}1827364\dot{5} \), determine the value of \( \underbrace{f\{f \cdots f[f}_{1996 \uparrow f}(1)]\} \). | 4 | math | 88 |
Arrange the natural numbers from 1 to 1982 in a certain order in a row. A computer reads two adjacent numbers from left to right (the 1st and 2nd, the 2nd and 3rd, etc.). If the larger number is on the left side, the computer swaps their positions, then continues to read the next pair, until it reaches the end. After t... | 100 | math | 132 |
If \( P \) is the circumcenter of \( \triangle ABC \), and
\[
\overrightarrow{P A}+\overrightarrow{P B}+\lambda \overrightarrow{P C}=\mathbf{0}, \quad \text{with} \quad \angle C=120^{\circ},
\]
determine the value of the real number \( \lambda \). | -1 | math | 84 |
Given $\cos (75^{\circ}+\alpha)= \frac {5}{13}$, where $\alpha$ is an angle in the third quadrant,
(1) Find the value of $\sin (75^{\circ}+\alpha)$.
(2) Find the value of $\cos (\alpha-15^{\circ})$.
(3) Find the value of $\sin (195^{\circ}-\alpha)+\cos (105^{\circ}-\alpha)$. | -\frac{10}{13} | math | 110 |
Given the proposition "If $a$, $b$, $c$ are in geometric progression, then $b^2 = ac$", determine the number of true propositions among its converse, inverse, and contrapositive. | 1 | math | 44 |
Given that the domain of $y=f(2^{x})$ is $(-\infty,1]$, then the domain of $y=f[\log_{3}(2-x)]$ is ______. | [-7,1) | math | 43 |
Given functions $f(x)=\ln x$ and $g(x)=\frac{1}{2}ax+b$, where $f(x)$ and $g(x)$ have the same tangent line at $x=1$.
$(1)$ Find the value of $a+2b$;
$(2)$ Solve the inequality $f(x) \lt g(x)$. | (0,1) \cup (1,+\infty) | math | 78 |
Given $\sin \left( \frac{\pi}{4}+\alpha\right)\sin \left( \frac{\pi}{4}-\alpha\right)= \frac{1}{6}$, and $\alpha\in\left( \frac{\pi}{2},\pi\right)$, find the value of $\tan 4\alpha$. | \frac{4 \sqrt{2}}{7} | math | 74 |
Consider the geometric sequence \( \left(a+\log _{2} 3\right),\left(a+\log _{4} 3\right),\left(a+\log _{8} 3\right) \). What is the common ratio of this sequence? | \frac{1}{3} | math | 58 |
Given four propositions, determine the number of true propositions. | 0 | math | 11 |
Given that the function $f(x) = \log\,_{\frac{1}{2}} \frac{a}{x^{2}+1}$ defined on $\mathbb{R}$ has a range of $[-1,+\infty)$, determine the value of the real number $a$. | 2 | math | 64 |
Point \((x,y)\) is randomly picked from the rectangular region with vertices at \((0,0), (3000,0), (3000,4500),\) and \((0,4500)\). What is the probability that \(x < 3y\)? Express your answer as a common fraction. | \frac{11}{18} | math | 75 |
Given that Jack earns 25 dollars per hour, of which 2.3% is deducted for a specific healthcare contribution, calculate the number of cents contributed towards healthcare per hour. | 57.5 | math | 37 |
Let $x_1=97$ , and for $n>1$ let $x_n=\frac{n}{x_{n-1}}$ . Calculate the product $x_1x_2 \ldots x_8$ . | 384 | math | 51 |
Given that the terminal side of angle α intersects with a circle centered at the origin of the coordinate plane and having a radius of 1 at point P( sin 2π/3, cos 2π/3), find the smallest positive value of angle α. | \frac {11π}{6} | math | 54 |
Find the standard equation of the ellipse that satisfies the following conditions:
1. The coordinates of the two foci are $(-2,0)$ and $(2,0)$, respectively. The sum of the distances from a point $P$ on the ellipse to the two foci equals $6$. Find the equation of the ellipse.
2. The foci of the ellipse are $F_{1}(0,-5)... | \dfrac{y^2}{40} + \dfrac{x^2}{15} = 1 | math | 119 |
A cylindrical tank with radius $4$ feet and height $9$ feet is lying on its side. The tank is filled with water to a depth of $2$ feet. Calculate the volume of water in the tank, in cubic feet. | 48\pi - 36\sqrt{3} | math | 49 |
Given the hyperbola $$\frac {x^{2}}{9} - \frac {y^{2}}{16} = 1$$ with its left and right foci marked as F<sub>1</sub> and F<sub>2</sub> respectively. If there is a point P on the hyperbola such that the angle ∠F<sub>1</sub>PF<sub>2</sub> is 90°, find the area of triangle ΔF<sub>1</sub>PF<sub>2</sub>, denoted as $$S_{△F... | 16 | math | 138 |
The minimum value of the function $y= \frac {4x^{2}+2x+5}{x^{2}+x+1}(x > 1)$ is $\boxed{\text{answer}}$. | \frac {16-2 \sqrt {7}}{3} | math | 46 |
$F$ is the focus of the parabola $y^{2}=2x$, $A$ and $B$ are two points on the parabola, ($|AF|+|BF|=8$), then calculate the distance from the midpoint of line segment $AB$ to the $y$ axis. | \frac {7}{2} | math | 66 |
Parallelogram $PQRS$ has vertices $P(4,4)$, $Q(-2,-2)$, $R(-8,-2)$, and $S(-2,4)$. If a point is selected at random from the region determined by the parallelogram, what is the probability that the point lies below the $x$-axis? | \frac{1}{2} | math | 76 |
Given the function $f(x) = \log_a(a^{2x} - 2a^x - 2)$ where $a > 1$, determine the range of $x$ for which $f(x) > 0$. | (\log_a 3, +\infty) | math | 50 |
In triangle $XYZ$, $\angle Y = 90^\circ$, $YZ = 4$, and $XY = \sqrt{34}$. What is $\tan X$? | \frac{2\sqrt{2}}{3} | math | 39 |
In what numeral system is the number 11111 a perfect square? | B = 3 | math | 17 |
Two identical polygons were cut out of cardboard, overlaid, and pinned together at a certain point. When one polygon is rotated around this "axis" by $25^{\circ} 30^{\prime}$, it coincides again with the second polygon. What is the minimum possible number of sides of such polygons? | 240 | math | 67 |
(1) Given the function $f(x)=(\frac{1}{3})^{x}$
(1) If $y=f(x)$ and $y=f^{-1}(x)$ are inverse functions, find the monotonic intervals of $g(x)=f^{-1}(x^{2}+2x-3)$.
(2) When $x \in [-1,1]$, find the maximum and minimum values of $y=[f(x)]^{2}-2f(x)+3$. | 6 | math | 104 |
Two radii OA and OB of a circle c with midpoint O are perpendicular. Another circle touches c in point Q and the radii in points C and D, respectively. Determine $ \angle{AQC}$ . | 45^\circ | math | 45 |
Two vertical poles stand on sloped ground. The bottoms of the poles are 20 feet apart. One pole is 12 feet tall, and the other is 30 feet tall. The ground slopes upward between the poles, starting from the shorter pole and rising linearly at a rate of 1 foot height increase every 4 feet horizontal. How long, in feet, i... | \sqrt{569} | math | 97 |
Given the sequence 2008, 2009, 1, -2008, -2009, ..., starting from the second term, each term is equal to the sum of the term before it and the term after it. Calculate the sum of the first 2016 terms of this sequence, $S_{2016}$. | 0 | math | 80 |
Find the derivative.
$$
y=\frac{5^{x}(2 \sin 2 x+\cos 2 x \cdot \ln 5)}{4+\ln ^{2} 5}
$$ | 5^x \cos 2x | math | 44 |
Find the equation of the line that passes through point A(2, -3) and is parallel to the line y = x. | x - y = 5 | math | 27 |
In the elective course 4-4: Coordinate System and Parametric Equations
Given curves $C_{1}$ with parametric equations
$$
\begin{cases}
x = -4 + \cos t \\
y = -3 + \sin t
\end{cases}
$$
where $t$ is the parameter, and $C_{2}$ with parametric equations
$$
\begin{cases}
x = 8 \cos \theta \\
y = -3 \sin \theta
\end{cases}
... | \frac{2\sqrt{5}}{5} | math | 244 |
A certain school established a soccer club to enrich students' extracurricular activities. In order to determine if students' preference for soccer is related to gender, $100$ male and $100$ female students were randomly selected for a survey. Some of the data is shown in the table below:
| | Likes Soccer | Do... | E(\xi) = \frac{11}{6} | math | 407 |
Given the polynomial $f(x) = 2x^7 + x^6 + x^4 + x^2 + 1$, calculate the value of $V_2$ using Horner's method when $x=2$. | 10 | math | 49 |
Given the expression $3^{3^{3^{3}}}$, determine the number of distinct values obtained under different evaluations of exponentiation. | 1 | math | 28 |
The hyperbola $x^2-\frac{y^2}{m}=1$ has one of its foci at $(-3,0)$. Determine the value of $m$. | 8 | math | 40 |
Given 5 people stand in a row, and there is exactly 1 person between person A and person B, determine the total number of possible arrangements. | 36 | math | 31 |
Find the real roots for the polynomial equation
\[ x^n - x^{n-1} + x^{n-2} - \dots + (-1)^{n-1}x + (-1)^n = 0, \]
where $n$ is a positive integer. | 1 | math | 59 |
Given $133a$ is a positive integer, $a < 100$, and $a^{3} + 23$ is divisible by 24, determine the number of values of $a$. | 5 | math | 47 |
Determine the remainder when $1 + 3 + 3^2 + \cdots + 3^{1000}$ is divided by $500$. | 1 | math | 36 |
Solve the following system of equations in integers:
$$
\left\{\begin{array}{l}
x^{2}-y^{2}-z^{2}=1 \\
y+z-x=3
\end{array}\right.
$$ | \{ (9, 8, 4), (-3, -2, 2), (9, 4, 8), (-3, 2, -2) \} | math | 50 |
Let $(x,y,z)$ be an ordered triplet of real numbers that satisfies the following system of equations: \begin{align*}x+y^2+z^4&=0,y+z^2+x^4&=0,z+x^2+y^4&=0.\end{align*} If $m$ is the minimum possible value of $\lfloor x^3+y^3+z^3\rfloor$ , find the modulo $2007$ residue of $m$ . | 2004 | math | 113 |
Let $x_1, x_2, \dots, x_{50}$ be real numbers such that $x_1 + x_2 + \dots + x_{50} = 0$ and
\[
\frac{x_1}{1+x_1} + \frac{x_2}{1+x_2} + \dots + \frac{x_{50}}{1+x_{50}} = 1.
\]
Find the value of
\[
\frac{x_1^2}{1+x_1} + \frac{x_2^2}{1+x_2} + \dots + \frac{x_{50}^2}{1+x_{50}}.
\] | 1 | math | 152 |
A certain city has the following water usage fee standards: for each household, if the monthly water usage does not exceed $6m^{3}$, the water fee is charged at a rate of $a$ yuan per cubic meter; if it exceeds $6m^{3}$, the part that does not exceed is still charged at $a$ yuan per cubic meter, and the part that excee... | 11 \text{ cubic meters} | math | 236 |
Given the universal set $U=\{-2,-1,0,1,2\}$, and the set $A=\{x\in Z|x^{2}+x-2 < 0\}$, determine the complement of $A$ in $U$. | \{-2,1,2\} | math | 56 |
What is the tenth number in the row of Pascal's triangle that has 100 numbers? | \binom{99}{9} | math | 20 |
A point $(x, y)$ is randomly picked from inside the rectangle with vertices $(0,0)$, $(6,0)$, $(6,3)$, and $(0,3)$. What is the probability that $2x < y$? | \frac{1}{8} | math | 53 |
Given the proposition "There exists $x \in \mathbb{R}$, such that $|x - a| + |x + 1| \leq 2$" is false, then the range of values for the real number $a$ is ______. | (-\infty, -3) \cup (1, +\infty) | math | 56 |
Find the distance from point \( M_{0} \) to the plane passing through three points \( M_{1}, M_{2}, M_{3} \).
\( M_{1}(0, -1, -1) \)
\( M_{2}(-2, 3, 5) \)
\( M_{3}(1, -5, -9) \)
\( M_{0}(-4, -13, 6) \) | 2 \sqrt{45} | math | 97 |
Find the minimum value of the distance AB, where A and B are the points of intersection of the line $y=m$ with $y=2x-3$ and the curve $y=x+e^x$, respectively. | 2 | math | 47 |
The sum of the maximum and minimum values of the function $y=2\sin \left( \frac{\pi x}{6}- \frac{\pi}{3}\right)$ where $(0\leqslant x\leqslant 9)$ is to be determined. | 2-\sqrt{3} | math | 59 |
Given that $a$ and $b$ are positive real numbers, and the line $(a+1)x+2y-1=0$ is perpendicular to the line $3x+(b-2)y+2=0$, find the minimum value of $\dfrac{3}{a} + \dfrac{2}{b}$. | 25 | math | 71 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.