task_type stringclasses 4
values | problem stringlengths 14 5.23k | solution stringlengths 1 8.29k | problem_tokens int64 9 1.02k | solution_tokens int64 1 1.98k |
|---|---|---|---|---|
math | There are three pastures full of grass. The first pasture is 33 acres and can feed 22 cows for 27 days. The second pasture is 28 acres and can feed 17 cows for 42 days. How many cows can the third pasture, which is 10 acres, feed for 3 days (assuming the grass grows at a uniform rate and each acre produces the same amo... | 20 | 90 | 2 |
math | Consider the system of inequalities:
$$
\begin{cases}
x + y \leq 4 \\
3x + y \geq 3 \\
x \geq 0 \\
y \geq 0
\end{cases}
$$
Determine the length of the longest side of the quadrilateral region formed by these inequalities, expressed in simplest radical form. | 5 | 77 | 1 |
math | Determine the range of $k$ such that the function $y=(k+2)x+1$ is a decreasing function on the set of real numbers. | k < -2 | 33 | 4 |
math | Between 1000 and 9999, how many four-digit integers with all different digits have an absolute difference of 2 between the first and last digits? | 840 | 36 | 3 |
math | (Optional Elective 4-5: Inequalities)
Let $a$ and $b$ be positive real numbers such that $\frac{1}{a}+\frac{1}{b}=2\sqrt{2}$.
(1) Find the minimum value of $a^2 + b^2$.
(2) If $(a-b)^2 \geq 4(ab)^3$, find the value of $ab$. | ab = 1 | 90 | 4 |
math | Given the function $f(x)=2\sin (\omega x+ \varphi )+1$ $(\omega > 0, |\varphi|\leqslant \frac{\pi }{2})$, the distance between two adjacent intersection points of its graph and the line $y=-1$ is $\pi$. If $f(x) > 1$, it holds true for all $x\in (-\frac{\pi }{12},\frac{\pi }{3})$. Determine the range of $\varphi$. | [\frac{\pi}{6},\frac{\pi}{3}] | 112 | 14 |
math | In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. It is known that $c^{2}=a^{2}+b^{2}-4bc\cos C$, and $A-C= \frac {\pi}{2}$.
(Ⅰ) Find the value of $\cos C$;
(Ⅱ) Find the value of $\cos \left(B+ \frac {\pi}{3}\right)$. | \frac {4-3 \sqrt {3}}{10} | 109 | 15 |
math | In the diagram, $\triangle ABE$, $\triangle BCE$ and $\triangle CDE$ are right-angled, with $\angle AEB=\angle BEC = 45^\circ$, and $\angle CED = 30^\circ$, and $AE=30$. Find the length of $CE.$ | 15\sqrt{2} | 66 | 7 |
math | Determine the number of distinct odd numbers that can be formed by rearranging the digits of the number "34396". | 36 | 27 | 2 |
math | Find all pairs \((x, y)\) of integers that satisfy the equation
\[ x^{2} y + y^{2} = x^{3} \] | (0,0) \text{ and } (-4, -8) | 35 | 16 |
math | Ana, Bia, Cátia, Diana, and Elaine work as street vendors selling sandwiches. Daily, they visit Mr. Manoel's snack bar and take an identical number of sandwiches to sell. One day, Mr. Manoel was sick and left a note explaining his absence and asking each vendor to take $\frac{1}{5}$ of the sandwiches. Ana arrived first... | 75 | 212 | 2 |
math | Parallelogram $ABCD$ with $A(2,5)$, $B(4,9)$, $C(6,5)$, and $D(4,1)$ is reflected across the $x$-axis to $A'B'C'D'$ and then $A'B'C'D'$ is reflected across the line $y=x+1$ to $A''B''C''D''$. This is done such that $D'$ is the image of $D$, and $D''$ is the image of $D'$. What is the ordered pair of $D''$ in the coordi... | (-2,5) | 130 | 5 |
math | Given a linear function \( f(x) \). It is known that the distance between the points of intersection of the graphs \( y = x^2 - 1 \) and \( y = f(x) \) is \( \sqrt{30} \), and the distance between the points of intersection of the graphs \( y = x^2 \) and \( y = f(x) + 3 \) is \( \sqrt{46} \). Find the distance between... | \sqrt{38} | 129 | 6 |
math | Serge and Tanya want to show Masha a magic trick. Serge leaves the room. Masha writes down a sequence $(a_1, a_2, \ldots , a_n)$ , where all $a_k$ equal $0$ or $1$ . After that Tanya writes down a sequence $(b_1, b_2, \ldots , b_n)$ , where all $b_k$ also equal $0$ or $1$ . Then Masha either does nothing or... | n | 258 | 2 |
math | Let $a$ and $b$ be two non-coincident planes, and let $l$, $m$, $n$ be three mutually non-coincident lines. Consider the following four statements:
1. If plane $a$ is parallel to plane $b$, and line $l$ lies in plane $a$, then line $l$ is parallel to plane $b$.
2. If line $m$ lies in plane $a$, line $n$ lies in plane... | 1, 3, 4 | 255 | 7 |
math | Suppose a set 𝑆 is given as $\{-3, -1, 0, 0, 2, 5\}$. Two different numbers are randomly selected from set 𝑆 and multiplied together. Find the probability that the product is 0. | \frac{8}{15} | 56 | 8 |
math | Using the Binomial Expansion, find the first three digits to the right of the decimal point in the decimal representation of \((5^{1001} + 1)^{7/3}\). | 333 | 42 | 3 |
math | It is known that $\int_1^2x^{-1}\arctan (1+x)\ dx = q\pi\ln(2)$ for some rational number $q.$ Determine $q.$ Here, $0\leq\arctan(x)<\frac{\pi}{2}$ for $0\leq x <\infty.$ | q = \frac{3}{8} | 83 | 9 |
math | Consider integers \( \{1, 2, \ldots, 10\} \). A particle is initially at 1. It moves to an adjacent integer in the next step. What is the expected number of steps it will take to reach 10 for the first time? | 90 | 60 | 2 |
math | A given rectangle $ R$ is divided into $mn$ small rectangles by straight lines parallel to its sides. (The distances between the parallel lines may not be equal.) What is the minimum number of appropriately selected rectangles’ areas that should be known in order to determine the area of $ R$ ? | m + n - 1 | 65 | 6 |
math | Let $M, \alpha, \beta \in \mathbb{R} $ with $M > 0$ and $\alpha, \beta \in (0,1)$ . If $R>1$ is a real number, we say that a sequence of positive real numbers $\{ C_n \}_{n\geq 0}$ is $R$ -*inoceronte* if $ \sum_{i=1}^n R^{n-i}C_i \leq R^n \cdot M$ for all $n \geq 1$ . Determine the smallest real $R>1$ f... | R = \beta^{-\frac{1}{\alpha}} | 204 | 13 |
math | Given the polynomial \(1 - x + x^{2} - x^{3} + \cdots + x^{16} - x^{17}\), express it in the form \(a_{0} + a_{1} y + a_{2} y^{2} + a_{3} y^{3} + \cdots + a_{16} y^{16} + a_{17} y^{17}\), where \(y = x + 1\), and each \(a_{i}\) is a constant. Find \(a_{2}\). | 816 | 124 | 3 |
math | Two real numbers $x$ and $y$ are such that $8 y^{4}+4 x^{2} y^{2}+4 x y^{2}+2 x^{3}+2 y^{2}+2 x=x^{2}+1$. Find all possible values of $x+2 y^{2}$. | \frac{1}{2} | 72 | 7 |
math | In the rectangular coordinate system $xOy$, the rectangular coordinate equation of circle $C$ is $(x- \sqrt {3})^{2}+(y-1)^{2}=4$. Establish a polar coordinate system with the coordinate origin $O$ as the pole and the positive semi-axis of the $X$-axis as the polar axis.
1. Write the polar coordinate equation of circl... | \sqrt {3} | 139 | 5 |
math | Find all positive real solutions to: \begin{eqnarray*} (x_1^2-x_3x_5)(x_2^2-x_3x_5) &\le& 0 (x_2^2-x_4x_1)(x_3^2-x_4x_1) &\le& 0 (x_3^2-x_5x_2)(x_4^2-x_5x_2) &\le& 0 (x_4^2-x_1x_3)(x_5^2-x_1x_3) &\le & 0 (x_5^2-x_2x_4)(x_1^2-x_2x_4) &\le& 0 \end{eqnarray*} | x_1 = x_2 = x_3 = x_4 = x_5 | 183 | 19 |
math | Given a quadratic function $f(x) = ax^2 + bx + c$ ($a, b, c \in \mathbb{R}$, and $a \neq 0$), when $x \in [-3, 1]$, we have $f(x) \leq 0$; when $x \in (-\infty, -3) \cup (1, +\infty)$, we have $f(x) > 0$, and $f(2) = 5$.
(I) Find the expression of $f(x)$.
(II) If the equation $f(x) = 9m + 3$ has real solutions for ... | m \geq -\frac{7}{9} | 159 | 12 |
math | Given two arithmetic sequences \\(\{a_n\}\) and \\(\{b_n\}\) whose sums of the first \\(n\\) terms are \\(A_n\\) and \\(B_n\\) respectively, if \\( \dfrac {A_n}{B_n}= \dfrac {7n+45}{n+3}\\), then the number of positive integers that make \\( \dfrac {a_n}{b_n}\\) an integer is \_\_\_\_\_\_. | 5 | 107 | 1 |
math | Find the number of integers $n$ with $n \ge 2$ such that the remainder when $2013$ is divided by $n$ is equal to the remainder when $n$ is divided by $3$ .
*Proposed by Michael Kural* | 7 | 68 | 1 |
math | Read the following material before solving the problem: In mathematics, there are numbers with square roots that contain another square root, which can be simplified by using the complete square formula and the properties of quadratic surds. For example, $\sqrt{3+2\sqrt{2}}=\sqrt{3+2×1×\sqrt{2}}=\sqrt{{1^2}+2×1×\sqrt{2... | \sqrt{5}-2 | 166 | 6 |
math | In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $a^{2}+c^{2}-b^{2}= \sqrt {3}ac$, then the value of angle $B$ is \_\_\_\_\_\_. | \frac {\pi}{6} | 71 | 7 |
math | Let $b_1, b_2, \ldots$ be a sequence such that:
(i) $b_1 = 2$, and
(ii) for any positive integer $n$, $b_{3n}=n^2 \cdot b_n$.
What is the value of $b_{3^{100}}$? | 2 \cdot 3^{9900} | 72 | 11 |
math | Given the function $f(x)=\dfrac{b-2^{x}}{2^{x+1}+a}$ is an odd function defined on $\mathbb{R}$.
$(1)$ Find the values of real numbers $a$ and $b$;
$(2)$ Determine the monotonicity of $f(x)$ on $\mathbb{R}$;
$(3)$ If $f(k\cdot 3^{x})+f(3^{x}-9^{x}+2) > 0$ holds for any $x \geqslant 1$, find the range of the real num... | k < \dfrac{4}{3} | 133 | 10 |
math | Given $(\sqrt[3]{x} +x^{2})^{2n}$, the sum of the binomial coefficients of its expansion is greater than the sum of the binomial coefficients of the expansion of $(3x-1)^{n}$ by $992$. Find the term with the largest absolute value of the coefficient in the expansion of ${(2x- \frac{1}{x})}^{2n}$. | -15360x^{4} | 90 | 10 |
math | In triangle $\triangle ABC$, the sides $a$, $b$, $c$ opposite to angles $A$, $B$, $C$ are three consecutive even numbers, and $C=2A$. Find the value of $a$. | 8 | 49 | 1 |
math | Given an equilateral triangle $\triangle ABC$ with side length $2\sqrt{3}$, $P$ is a point in the plane of the triangle, and $|\overrightarrow{AP}-\overrightarrow{AB}-\overrightarrow{AC}|=1$. Find the maximum value of $|\overrightarrow{AP}|$. | 7 | 69 | 1 |
math | Given the function $f(x) = \ln x - x + 1$, $x \in (0, +\infty)$, and $g(x) = e^{x} - ax$. Find:
1. The maximum value of $f(x)$.
2. The range of values for $a$ such that for all $x_{1} \in (0, +\infty)$, there exists $x_{2} \in [1,2]$ satisfying $f(x_{1}) \leq g(x_{2})$. | (-\infty, \frac{e^{2}}{2}] | 116 | 15 |
math | In the tetrahedron $ABCD$, $AB=AD=CD=1$, $BD=\sqrt{2}$, and $BD \perp CD$, the plane $ABD \perp$ the plane $BCD$. If the four vertices of the tetrahedron $ABCD$ lie on the same spherical surface, calculate the volume of the sphere. | \dfrac{\sqrt{3}}{2}\pi | 78 | 11 |
math | Points $A, B, C,$ and $D$ are sequentially located on a circle. It is known that the degree measures of the smaller arcs $AB, BC, CD,$ and $AD$ have a ratio of $1: 3: 5: 6$. Find the angles of the quadrilateral $ABCD$. | 96^\circ, 132^\circ, 84^\circ, 48^\circ | 68 | 23 |
math | Let $z$ and $w$ be complex numbers such that $|3z - w| = 17$, $|z + 3w| = 4$, and $|z + w| = 6$. Find $|z|$. | 5 | 54 | 1 |
math | Let $x, y$ be positive real numbers. If \[129-x^2=195-y^2=xy,\] then $x = \frac{m}{n}$ for relatively prime positive integers $m, n$ . Find $100m+n$ .
[i]Proposed by Michael Tang | 4306 | 75 | 4 |
math | The principal of a certain school decided to take a photo of the graduating class of 2008. He arranged the students in parallel rows, all with the same number of students, but this arrangement was too wide for the field of view of his camera. To solve this problem, the principal decided to take one student from each ro... | 24 | 146 | 2 |
math | (1) Given that $x > 0$ and $y > 0$, and $\frac{2}{x} + \frac{8}{y} = 1$, find the minimum value of $xy$.
(2) Given that $x > 0$ and $y > 0$, and $x + 2y = 1$, find the minimum value of $\frac{1}{x} + \frac{1}{y}$. | 3 + 2\sqrt{2} | 97 | 9 |
math | Given that the function $f(x)=-\frac{x+a}{bx+1}$ is an odd function in the interval $[-1,1]$, what is its maximum value in this interval? | 1 | 41 | 1 |
math | Find the value of $x$ such that $\sqrt{x^2 + 16} = 12$. | 8\sqrt{2} \text{ and } -8\sqrt{2} | 24 | 18 |
math | Two years ago, the cost of producing 1 set of vaccines was 5000 yuan. With the advancement of production technology, if the annual average decrease rate of vaccine cost is $x$, then calculate the reduction in the cost of producing 1 set of vaccines now compared to producing 1 set of vaccines last year. | 5000x - 5000x^2 | 67 | 14 |
math | Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ with an angle of $\frac{\pi}{3}$ between them, $|\overrightarrow{a}|=2$, and $|\overrightarrow{b}|=1$, find the magnitude of $\overrightarrow{a}+2\overrightarrow{b}$ and the value of $|\overrightarrow{a}+\overrightarrow{b}|\cdot|\overrightarrow{a}-\overrightar... | \sqrt{21} | 100 | 6 |
math | Given the numbers $\log _{\sqrt{5x-1}}(4x+1)$, $\log _{4x+1}\left(\frac{x}{2}+2\right)^{2}$, $\log _{\frac{x}{2}+2}(5x-1)$, for which values of $x$ are two of these numbers equal and the third less than them by 1? | x = 2 | 87 | 4 |
math | During a lunch break, members of a brigade were discussing how many newspapers each person reads. It was discovered that each person subscribes to and reads exactly two newspapers, each newspaper is read by five people, and any combination of two newspapers is read by one person. How many different newspapers do the me... | 6 \text{ названий газет}, \, 15 \text{ членов} | 74 | 24 |
math | A point $(x,y)$ is randomly selected such that $0 \le x \le 4$ and $0 \le y \le 5$. What is the probability that $x+y \le 5$? Express your answer as a common fraction. | \frac{2}{5} | 54 | 7 |
math | Lynnelle and Moor love toy cars, and together, they have $27$ red cars, $27$ purple cars, and $27$ green cars. The number of red cars Lynnelle has individually is the same as the number of green cars Moor has individually. In addition, Lynnelle has $17$ more cars of any color than Moor has of any color. How man... | 22 | 97 | 2 |
math | A rectangular prism has 6 faces, each with a pair of opposite faces. Since each pair of opposite faces are unequal, then calculate the number of pairs of parallel edges. | 12 | 35 | 2 |
math | When using the method of successive subtraction to calculate the greatest common divisor (GCD) of 294 and 84, how many subtractions are needed? | 4 | 35 | 1 |
math | In a geometric sequence $\{a_n\}$, it is known that $S_6=48$ and $S_{12}=60$. Find the value of $S_{24}$. | \frac{255}{4} | 44 | 9 |
math |
A firecracker was thrown vertically upwards with a speed of 20 m/s. Three seconds after the start of its flight, it exploded into two unequal parts, the mass ratio of which is $1: 2$. The smaller fragment immediately after the explosion flew horizontally at a speed of $16 \mathrm{~m}/\mathrm{s}$. Find the magnitude of... | 17 | 119 | 2 |
math | Given a sample data $x\_1$, $x\_2$, ..., $x\_n$ with a variance of $4$, what is the standard deviation of the data $2x\_1+3$, $2x\_2+3$, ..., $2x\_n+3$? | 4 | 61 | 1 |
math | The function \( f: \mathbb{R} \rightarrow \mathbb{R} \) satisfies \( f\left(x^{2}\right) f^{\prime \prime}(x) = f^{\prime}(x) f^{\prime}\left(x^{2}\right) \) for all real \( x \). Given that \( f(1) = 1 \) and \( f^{\prime \prime \prime}(1) = 8 \), determine \( f^{\prime}(1) + f^{\prime \prime}(1) \). | 6 | 121 | 1 |
math | Calculate:<br/>$(1)-4\times 9$;<br/>$(2)10-14-\left(-5\right)$;<br/>$(3)-3×(-\frac{1}{3})^3$;<br/>$(4)-56+(-8)×(\frac{1}{8})$. | -57 | 71 | 3 |
math | Given $f(x)=ax^{3}+bx^{2}+cx+d$ intersects the x-axis at three points $(0,0)$, $(x_{1},0)$, $(x_{2},0)$, and $f(x)$ has extreme values at $x=\frac{3-\sqrt{3}}{3}$, $x=\frac{3+\sqrt{3}}{3}$, calculate the value of $x_{1} \cdot x_{2}$. | 2 | 102 | 1 |
math | Suppose that $c$ and $d$ are digits, and the repeating decimal $0.\overline{cd}$ can be expressed as a fraction in lowest terms. Assume $c$ and $d$ are not both zero, and $c \neq d$. How many different denominators are possible? | 5 | 64 | 1 |
math | Given the function $f(x)=2|x-1|+|x+2|$.
$(1)$ Find the solution set of $f(x)\leqslant 9$;
$(2)$ If the minimum value of the function $f(x)$ is $M$, and $a+b+c=M$, find the minimum value of $4a^{2}+b^{2}+c^{2}$. | 4 | 86 | 1 |
math | Using data from 1944 through 2000, the histogram shows the number of years that had a particular number of hurricanes reaching the East Coast of the U.S. For example, in 14 of those years there was exactly one hurricane each year that reached the East Coast of the U.S. What is the median number of hurricanes per year r... | 2 | 488 | 1 |
math | A sequence consists of $2020$ terms. Each term after the first is 2 larger than the previous term. The sum of the $2020$ terms is $6060$. When every second term is added up, starting with the first term and ending with the second last term, what is the sum? | 2020 | 70 | 4 |
math | Let $\{b_k\}$ be a sequence of integers such that $b_1=2$ and $b_{m+n}=b_m+b_n+m^2+n^2,$ for all positive integers $m$ and $n.$ Find $b_{12}.$ | 160 | 58 | 3 |
math | In a class, 55% of students scored at least 55% on a test. 65% of students scored at most 65% on the same test. What percentage of students scored between 55% and 65% (inclusive) on the test? | 20\% | 61 | 4 |
math | In convex quadrilateral $EFGH$, we have $\angle E \cong \angle G$ and $\angle F \cong \angle H$, $EF = GH = 200$, and $EH \neq FG$. The perimeter of $EFGH$ is $720$. Find $\lfloor 1000 \cos E \rfloor.$ | 800 | 80 | 3 |
math | Given proposition p: The real number $m$ satisfies $m^2 - 7ma + 12a^2 < 0$ ($a > 0$), and proposition q: The equation $\frac{x^2}{m-1} + \frac{y^2}{2-m} = 1$ represents an ellipse with foci on the y-axis. If $\neg p$ is a necessary but not sufficient condition for $\neg q$, find the range of values for the real number ... | \left[\frac{1}{3}, \frac{3}{8}\right] | 108 | 18 |
math | Given two numbers $P$ and $Q$, where $P > Q$, compute the percent that $P$ is above 60% of $Q$.
A) $\frac{100P - 40Q}{Q}$
B) $\frac{100P - 60Q}{Q}$
C) $\frac{100P}{Q}$
D) $\frac{60P - 40Q}{Q}$ | \frac{100P - 60Q}{Q} | 99 | 15 |
math | The 14th National People's Congress of the People's Republic of China held its first meeting on the morning of March 5, 2023, and closed on the morning of March 13. In order to understand the students' attention to major news events, a school randomly selected 100 students for a questionnaire survey. The scores of the ... | 9 | 212 | 1 |
math | In the Cartesian coordinate system $xOy$, the line $l$ is given by the parametric equations $\begin{cases} x=-\frac{1}{2}t, \\ y=3+\frac{\sqrt{3}}{2}t \end{cases}$ (where $t$ is the parameter). With the origin $O$ as the pole and the positive half-axis of $x$ as the polar axis, the polar equation of curve $C$ is $\rho ... | 3\sqrt{3} | 203 | 6 |
math | How many integers $m$ are there from 1 to 1996, such that $\frac{m^{2}+7}{m+4}$ is not a reduced fraction? | 86 | 40 | 2 |
math |
Given a positive integer \(N\) (written in base 10), define its integer substrings to be integers that are equal to strings of one or more consecutive digits from \(N\), including \(N\) itself. For example, the integer substrings of 3208 are \(3, 2, 0, 8, 32, 20, 320, 208\), and 3208. (The substring 08 is omitted from... | 88,888,888 | 162 | 10 |
math | Let points $F_{1}$ and $F_{2}$ be the two foci of the hyperbola $x^{2}- \frac{y^{2}}{3}=1$, and point $P$ be a point on the hyperbola. If $3|PF_{1}|=4|PF_{2}|$, then the area of $\triangle PF_{1}F_{2}$ is __________. | 3 \sqrt{15} | 88 | 7 |
math | Given the curve $y=\frac{\sin x}{\sin x+\cos x}-\frac{1}{2}$, calculate the slope of the tangent line at point $M\left(\frac{\pi }{4},0\right)$. | \frac{1}{2} | 52 | 7 |
math | Given sets $A=\{x|x\in\mathbb{R}, (x-2)(x-6)\leqslant 0\}$, $B=\{x|x\in\mathbb{R}, x^{2}-6x+5 \lt 0\}$, $C=\{x|x\in\mathbb{R}, m \lt x \lt m+1\}$, $U=\mathbb{R}$.
$(1)$ Find $A\cup B$, $(\complement _{U}A)\cap B$;
$(2)$ If $C\subseteq B$, find the range of values for $m$. | [1,4] | 143 | 5 |
math | Given a differentiable function \( f(x) \) defined on \( \mathbb{R} \) that satisfies: \( f′(x)+f(x) < 0 \), find the relationship between \( \frac{f(m-m^{2})}{e^{m^{2}-m+1}} \) and \( f(1) \) (where \( e \) is the base of the natural logarithm). | \dfrac {f(m-m^{2})}{e^{m^{2}-m+1}} > f(1) | 89 | 25 |
math | A rectangular box has a volume of $5400$ cubic inches and a surface area of $2352$ square inches. The sum of the lengths of its $12$ edges is $240$ inches. Find the volume of the box, in cubic inches, if its length, width, and height were each increased by one inch. | 6637 \text{ cubic inches} | 75 | 10 |
math | Let $a$ and $b$ be natural numbers, and $a = 1999b$. The sum of the greatest common divisor and the least common multiple of $a$ and $b$ is ____. | 2000b | 46 | 5 |
math | In the polar coordinate system $Ox$, the polar equation of curve $C$ is $p^{2}= \frac {144}{9+7\sin ^{2}\theta}$. Set up a Cartesian coordinate system with the pole $O$ as the origin and the polar axis as the positive $x$-axis.
$(1)$ Find the Cartesian equation of the curve $C$;
$(2)$ Suppose that the curve $C$ int... | 6(\sqrt {2}+1) | 148 | 9 |
math | Construct the parametric equations of the line of intersection of the planes \( 2x - y - 3z + 5 = 0 \) and \( x + y - 2 = 0 \). | x = t, \ y = 2 - t, \ z = t + 1 | 44 | 19 |
math | Find the minimum value of
\[2 \sqrt[4]{x} + \frac{1}{x}\] for \(x > 0.\) | 3 | 32 | 1 |
math | Determine the number of spelling errors that can occur when the English word $better$ is misspelled due to an incorrect sequence, given that each letter can be used only once. | 179 | 37 | 3 |
math | Given an arithmetic-geometric sequence $\{a\_n\}$ with the sum of its first $n$ terms denoted as $S\_n$. It is known that $S\_1=16$. A student calculated and obtained $S\_2=32$, $S\_3=76$, $S\_4=130$. Upon verification, it was found that exactly one of these numbers was incorrect. Determine the incorrect number and the... | \frac{3}{2} | 101 | 7 |
math | Given \( x > y > 0 \) and \( xy = 1 \), find the minimum value of \( \frac{3x^3 + 125y^3}{x-y} \). | 25 | 45 | 2 |
math | After two price reductions, the retail price of a certain product dropped from 800 yuan to 578 yuan. Calculate the average percentage decrease per reduction. | 15\% | 34 | 4 |
math | Mike had a bag of candies, and all candies were whole pieces that cannot be divided. Initially, Mike ate $\frac{1}{4}$ of the candies. Then, he shared $\frac{1}{3}$ of the remaining candies with his sister, Linda. Next, both Mike and his father ate 12 candies each from the remaining candies Mike had. Later, Mike’s sist... | 64 | 105 | 2 |
math | In the plane rectangular coordinate system $xOy$, the distance from the moving point $P$ to point $A(1,0)$ is twice the distance to point $B(1,3)$. Find the maximum value of the area of $\triangle PAB$. | 3 | 56 | 1 |
math | Given that function $f(x)$ is an odd function with a period of $2$, and when $x \in [0,1)$, $f(x) = \log_{10} (x+1)$, determine the value of $f(\frac{2016}{5}) + \log_{10} 18$. | 1 | 74 | 1 |
math | In triangle $ABC$, $AB=10$, $BC=12$ and $CA=14$. Point $G$ is on $\overline{AB}$, $H$ is on $\overline{BC}$, and $I$ is on $\overline{CA}$. Let $AG=s\cdot AB$, $BH=t\cdot BC$, and $CI=u\cdot CA$, where $s$, $t$, and $u$ are positive and satisfy $s+t+u=3/4$ and $s^2+t^2+u^2=3/7$. The ratio of the area of triangle $GHI$ ... | 295 | 175 | 3 |
math | Regular octagon \( CH I L D R E N \) has area 1. Determine the area of quadrilateral \( L I N E \). | 1/2 | 31 | 3 |
math | In a race, four cars each independently run timed laps around a circuit. Each car's lap time is discretely measured in seconds and can be any integer value between 150 and 155 seconds, uniformly distributed. The winner is the car with the shortest lap time. In case of a tie, the involved cars re-run the lap until a sin... | \frac{1}{3} | 151 | 7 |
math | In a classroom, $3/7$ of the students are wearing blue shirts, and $4/9$ of the students are wearing red shoes. What is the minimum number of students in the classroom wearing both a blue shirt and red shoes? | 8 | 50 | 1 |
math | The mean of the 9 data values $70, 110, x, 55, 45, 220, 85, 65, x$ is $x$. Calculate the value of $x$ given that the sum of the 9 data values is $9x$. | \frac{650}{7} | 68 | 9 |
math | Given the function $f\left(x\right)=9^{x}-2\cdot 3^{x+m}(m \gt 0)$.
$(1)$ When $m=1$, find the solution set of the inequality $f\left(x\right)\leqslant 27$;
$(2)$ If $x_{2} \gt x_{1} \gt 0$ and $x_{1}x_{2}=m^{2}$, compare the relationship between $f(x_{1})$ and $f(x_{2})$;
$(3)$ Let $g\left(x\right)=f\left(x\righ... | m = 1 | 181 | 4 |
math | Given that $F_1$ and $F_2$ are the left and right foci of the ellipse $\frac{x^{2}}{16}{+}\frac{y^{2}}{b^{2}}{=}1$, and the line $l$ passing through $F_1$ intersects the ellipse at points $A$ and $B$. If the maximum value of $|AF_2|+|BF_2|$ is $10$, find the eccentricity of the ellipse. | \frac{1}{2} | 105 | 7 |
math | Three numbers (x, y, z) are randomly selected from the interval [0,1]. If they satisfy x^2 + y^2 + z^2 > 1, then the parameter t = 1; otherwise, t = 0. Based on this, the sum of all parameters from 1000 repeated experiments is 477, determine the estimated value of the constant π. | 3.138 | 85 | 5 |
math | Given a unit right prism \( ABCD-A_1B_1C_1D_1 \), there are two moving points \( E \) and \( F \) on the edges \( BB_1 \) and \( DD_1 \) respectively, such that \( BE = D_1F \). Let the angle between line segment \( EF \) and plane \( AB \) be \(\alpha\), and the angle between line segment \( EF \) and plane \( BC_1 \)... | 90^\circ | 125 | 4 |
math | For two distinct numbers $a$ and $b$, we define $\min \{a, b\}$ as the smaller value between $a$ and $b$. For example, $\min \{2, 3\} = 2$. According to this rule, the solution to the equation $\min \left\{\frac{1}{1-x}, \frac{2}{1-x}\right\} = \frac{2}{x-1} - 3$ is ____. | \frac{7}{3} | 102 | 7 |
math | Given the function $f(x)=\sin (ωx+φ)(ω > 0,φ∈(-\dfrac{π}{2},\dfrac{π}{2}))$ with the smallest positive period of $π$, and its graph is symmetric about the line $x=\dfrac{π}{12}$. Among the following four conclusions about the function $f(x)$:
$①$ The graph is symmetric about the point $(\dfrac{π}{4},0)$;
$②$ The graph... | ②④ | 183 | 4 |
math | Given $x$ is the smallest angle in a triangle, find the range of the function $y = \sin(x + \frac{\pi}{3}) + \sin(x - \frac{\pi}{3}) + \sqrt{3}\cos(x) + 1$. | [\sqrt{3} + 1, 3] | 57 | 12 |
math | In $\triangle ABC$, $\overline{CA} = \overline{CB}$. On $CB$, equilateral triangle $BCF$ is constructed with $F$ outside $\triangle ABC$. If $y$ denotes the measure in degrees of $\angle FAB$, determine the value of $y$.
**A)** $30^\circ$
**B)** $45^\circ$
**C)** $60^\circ$
**D)** $90^\circ$
**E)** $75^\circ$ | 60^\circ | 113 | 4 |
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