problem stringlengths 10 5.15k | answer dict |
|---|---|
Given the line $l: \lambda x-y-\lambda +1=0$ and the circle $C: x^{2}+y^{2}-4y=0$, calculate the minimum value of $|AB|$. | {
"answer": "2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Using the property that in finding the limit of the ratio of two infinitesimals, they can be replaced with their equivalent infinitesimals (property II), find the following limits:
1) \(\lim _{x \rightarrow 0} \frac{\sin 4 x}{\sin 3 x}\)
2) \(\lim _{x \rightarrow 0} \frac{\tan^{2} 2 x}{\sin ^{2} \frac{x}{3}}\)
3) \(\lim _{x \rightarrow 0} \frac{x \sin 2 x}{(\arctan 5 x)^{2}}\)
4) \(\lim _{n \rightarrow+\infty} \frac{\tan^{3} \frac{1}{n} \cdot \arctan \frac{3}{n \sqrt{n}}}{\frac{2}{n^{3}} \cdot \tan \frac{1}{\sqrt{n}} \cdot \arcsin \frac{5}{n}}\) | {
"answer": "\\frac{3}{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Mia and Jake ordered a pizza cut into 12 equally-sized slices. Mia wanted a plain pizza but Jake wanted pepperoni on one-third of the pizza. The cost of a plain pizza was $12, and the additional cost for pepperoni on part of the pizza was $3. Jake ate all the pepperoni slices and three plain slices. Mia ate the rest. Each paid for what they ate. How much more did Jake pay than Mia? | {
"answer": "2.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $3 \in \{a, a^2 - 2a\}$, then the value of the real number $a$ is ______. | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Point $P$ is on the circle $C_{1}: x^{2}+y^{2}-8x-4y+11=0$, and point $Q$ is on the circle $C_{2}: x^{2}+y^{2}+4x+2y+1=0$. What is the minimum value of $|PQ|$? | {
"answer": "3\\sqrt{5} - 5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the real numbers \( x_1, x_2, \ldots, x_{2001} \) satisfy \( \sum_{k=1}^{2000} \left|x_k - x_{k+1}\right| = 2001 \). Let \( y_k = \frac{1}{k} \left( x_1 + x_2 + \cdots + x_k \right) \) for \( k = 1, 2, \ldots, 2001 \). Find the maximum possible value of \( \sum_{k=1}^{2000} \left| y_k - y_{k+1} \right| \). | {
"answer": "2000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What percent of the positive integers less than or equal to $120$ have no remainders when divided by $6$? | {
"answer": "16.67\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A cube with an edge length of 1 and its circumscribed sphere intersect with a plane to form a cross section that is a circle and an inscribed equilateral triangle. What is the distance from the center of the sphere to the plane of the cross section? | {
"answer": "$\\frac{\\sqrt{3}}{6}$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Simplify first, then evaluate: $(1-\frac{2}{{m+1}})\div \frac{{{m^2}-2m+1}}{{{m^2}-m}}$, where $m=\tan 60^{\circ}-1$. | {
"answer": "\\frac{3-\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two workers were assigned to produce a batch of identical parts; after the first worked for \(a\) hours and the second for \(0.6a\) hours, it turned out that they had completed \(\frac{5}{n}\) of the entire job. After working together for another \(0.6a\) hours, they found that they still had \(\frac{1}{n}\) of the batch left to produce. How many hours will it take for each of them, working separately, to complete the whole job? The number \(n\) is a natural number; find it. | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $O$ be the origin, and $F$ be the right focus of the ellipse $C: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ where $a > b > 0$. The line $l$ passing through $F$ intersects the ellipse $C$ at points $A$ and $B$. Two points $P$ and $Q$ on the ellipse satisfy
$$
\overrightarrow{O P}+\overrightarrow{O A}+\overrightarrow{O B}=\overrightarrow{O P}+\overrightarrow{O Q}=0,
$$
and the points $P, A, Q,$ and $B$ are concyclic. Find the eccentricity of the ellipse $C$. | {
"answer": "\\frac{\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In how many ways can you arrange the digits of 11250 to get a five-digit number that is a multiple of 2? | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a math interest class, the teacher gave a problem for everyone to discuss: "Given real numbers $a$, $b$, $c$ not all equal to zero satisfying $a+b+c=0$, find the maximum value of $\frac{|a+2b+3c|}{\sqrt{{a}^{2}+{b}^{2}+{c}^{2}}}."$ Jia quickly offered his opinion: Isn't this just the Cauchy inequality? We can directly solve it; Yi: I am not very clear about the Cauchy inequality, but I think we can solve the problem by constructing the dot product of vectors; Bing: I am willing to try elimination, to see if it will be easier with fewer variables; Ding: This is similar to the distance formula in analytic geometry, can we try to generalize it to space. Smart you can try to use their methods, or design your own approach to find the correct maximum value as ______. | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given triangle $\triangle ABC$ with sides $a$, $b$, $c$ opposite to angles $A$, $B$, $C$ respectively, satisfying $\frac{a}{{2\cos A}}=\frac{b}{{3\cos B}}=\frac{c}{{6\cos C}}$, then $\sin 2A=$____. | {
"answer": "\\frac{3\\sqrt{11}}{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the polar equation of curve $C$ is $\rho\sin^2\theta-8\cos\theta=0$, with the pole as the origin of the Cartesian coordinate system $xOy$, and the polar axis as the positive half-axis of $x$. In the Cartesian coordinate system, a line $l$ with an inclination angle $\alpha$ passes through point $P(2,0)$.
$(1)$ Write the Cartesian equation of curve $C$ and the parametric equation of line $l$;
$(2)$ Suppose the polar coordinates of points $Q$ and $G$ are $(2, \frac{3\pi}{2})$ and $(2,\pi)$, respectively. If line $l$ passes through point $Q$ and intersects curve $C$ at points $A$ and $B$, find the area of $\triangle GAB$. | {
"answer": "16\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $ABC$ be a triangle such that midpoints of three altitudes are collinear. If the largest side of triangle is $10$ , what is the largest possible area of the triangle? | {
"answer": "25",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A cube with a side length of 10 is divided into 1000 smaller cubes with a side length of 1. A number is written in each small cube such that the sum of the numbers in each column of 10 cubes (in any of the three directions) is zero. In one of the small cubes (denoted as \( A \)), the number one is written. Three layers pass through cube \( A \), each parallel to the faces of the larger cube (with each layer having a thickness of 1). Find the sum of all the numbers in the cubes that are not in these layers. | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given there are ten steps from the first floor to the second floor, calculate the total number of ways Xiao Ming can go from the first floor to the second floor. | {
"answer": "89",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $y=\cos(2x+\frac{\pi}{4})$, determine the $x$-coordinate of one of the symmetric centers of the translated graph after translating it to the left by $\frac{\pi}{6}$ units. | {
"answer": "\\frac{11\\pi}{24}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate plane, a polar coordinate system is established with the origin as the pole and the non-negative half of the x-axis as the polar axis. It is known that point A has polar coordinates $$( \sqrt{2}, \frac{\pi}{4})$$, and the parametric equation of line $l$ is:
$$\begin{cases} x= \frac{3}{2} - \frac{\sqrt{2}}{2}t \\ y= \frac{1}{2} + \frac{\sqrt{2}}{2}t \end{cases}$$ (where $t$ is the parameter), and point A lies on line $l$.
(Ⅰ) Find the corresponding parameter $t$ of point A;
(Ⅱ) If the parametric equation of curve C is:
$$\begin{cases} x=2\cos\theta \\ y=\sin\theta \end{cases}$$ (where $\theta$ is the parameter), and line $l$ intersects curve C at points M and N, find the length of line segment |MN|. | {
"answer": "\\frac{4\\sqrt{2}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The positive integer \( N \) is the smallest one whose digits add to 41. What is the sum of the digits of \( N + 2021 \)?
A) 10
B) 12
C) 16
D) 2021
E) 4042 | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a circle $O$ with radius $1$, $PA$ and $PB$ are two tangents to the circle, and $A$ and $B$ are the points of tangency. The minimum value of $\overrightarrow{PA} \cdot \overrightarrow{PB}$ is \_\_\_\_\_\_. | {
"answer": "-3+2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\cos (α- \frac {π}{6})+ \sin α= \frac {4}{5} \sqrt {3}$, and $α \in (0, \frac {π}{3})$, find the value of $\sin (α+ \frac {5}{12}π)$. | {
"answer": "\\frac{7 \\sqrt{2}}{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $$\frac {1}{3} \leq a \leq 1$$, if the function $f(x) = ax^2 - 2x + 1$ has a domain of $[1, 3]$.
(1) Find the minimum value of $f(x)$ in its domain (expressed in terms of $a$);
(2) Let the maximum value of $f(x)$ in its domain be $M(a)$, and the minimum value be $N(a)$. Find the minimum value of $M(a) - N(a)$. | {
"answer": "\\frac {1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many ways are there to distribute 7 balls into 4 boxes if the balls are not distinguishable and neither are the boxes? | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a cube with a side length of \(4\), if a solid cube of side length \(1\) is removed from each corner, calculate the total number of edges of the resulting structure. | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the plane rectangular coordinate system $xOy$, the parametric equations of the line $l$ are $\left\{\begin{array}{l}x=1+\frac{1}{2}t\\ y=\frac{\sqrt{3}}{2}t\end{array}\right.$ (where $t$ is a parameter). Taking $O$ as the pole and the positive half-axis of the $x$-axis as the polar axis, the polar coordinate equation of the curve $C$ is $ρsin^{2}\frac{θ}{2}=1$.
$(1)$ Find the rectangular coordinate equation of $C$ and the polar coordinates of the intersection points of $C$ with the $y$-axis.
$(2)$ If the line $l$ intersects $C$ at points $A$ and $B$, and intersects the $x$-axis at point $P$, find the value of $\frac{1}{|PA|}+\frac{1}{|PB|}$. | {
"answer": "\\frac{\\sqrt{7}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given vectors $\vec{m}=(2a\cos x,\sin x)$ and $\vec{n}=(\cos x,b\cos x)$, the function $f(x)=\vec{m}\cdot \vec{n}-\frac{\sqrt{3}}{2}$, and $f(x)$ has a y-intercept of $\frac{\sqrt{3}}{2}$, and the closest highest point to the y-axis has coordinates $\left(\frac{\pi}{12},1\right)$.
$(1)$ Find the values of $a$ and $b$;
$(2)$ Move the graph of the function $f(x)$ to the left by $\varphi (\varphi > 0)$ units, and then stretch the x-coordinates of the points on the graph by a factor of $2$ without changing the y-coordinates, to obtain the graph of the function $y=\sin x$. Find the minimum value of $\varphi$. | {
"answer": "\\frac{5\\pi}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $g : \mathbb{R} \to \mathbb{R}$ be a function such that
\[g(g(x - y)) = g(x) g(y) - g(x) + g(y) - 2xy\]for all $x,$ $y.$ Find the sum of all possible values of $g(1).$ | {
"answer": "-\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In △ABC, B = $$\frac{\pi}{3}$$, AB = 8, BC = 5, find the area of the circumcircle of △ABC. | {
"answer": "\\frac{49\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider the function $f(x) = x^3 + 3\sqrt{x}$. Evaluate $3f(3) - 2f(9)$. | {
"answer": "-1395 + 9\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $g(x, y)$ be the function for the set of ordered pairs of positive coprime integers such that:
\begin{align*}
g(x, x) &= x, \\
g(x, y) &= g(y, x), \quad \text{and} \\
(x + y) g(x, y) &= y g(x, x + y).
\end{align*}
Calculate $g(15, 33)$. | {
"answer": "165",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, the lengths of the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $a^2 = b(b + c)$. Find the value of $\frac{B}{A}$. | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The decimal representation of \(\dfrac{1}{25^{10}}\) consists of a string of zeros after the decimal point, followed by non-zero digits. Find the number of zeros in that initial string of zeros after the decimal point. | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are \( n \) distinct lines in the plane. One of these lines intersects exactly 5 of the \( n \) lines, another one intersects exactly 9 of the \( n \) lines, and yet another one intersects exactly 11 of them. Which of the following is the smallest possible value of \( n \)? | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the smallest positive real number \(x\) such that
\[\lfloor x^2 \rfloor - x \lfloor x \rfloor = 7.\] | {
"answer": "\\frac{71}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Rectangle $ABCD$ has area $4032$. An ellipse with area $4032\pi$ passes through points $A$ and $C$ and has foci at points $B$ and $D$. Determine the perimeter of the rectangle. | {
"answer": "8\\sqrt{2016}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ $(a > b > 0)$, the symmetric point $Q$ of the right focus $F(c, 0)$ with respect to the line $y = \dfrac{b}{c}x$ is on the ellipse. Find the eccentricity of the ellipse. | {
"answer": "\\dfrac{\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=e^{x}$, $g(x)=-x^{2}+2x+b(b\in\mathbb{R})$, denote $h(x)=f(x)- \frac {1}{f(x)}$
(I) Determine the parity of $h(x)$ and write down the monotonic interval of $h(x)$, no proof required;
(II) For any $x\in[1,2]$, there exist $x_{1}$, $x_{2}\in[1,2]$ such that $f(x)\leqslant f(x_{1})$, $g(x)\leqslant g(x_{2})$. If $f(x_{1})=g(x_{2})$, find the value of the real number $b$. | {
"answer": "e^{2}-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Rotate a square around a line that lies on one of its sides to form a cylinder. If the volume of the cylinder is $27\pi \text{cm}^3$, then the lateral surface area of the cylinder is _________ $\text{cm}^2$. | {
"answer": "18\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $O$ is the coordinate origin, and vectors $\overrightarrow{OA}=(\sin α,1)$, $\overrightarrow{OB}=(\cos α,0)$, $\overrightarrow{OC}=(-\sin α,2)$, and point $P$ satisfies $\overrightarrow{AB}=\overrightarrow{BP}$.
(I) Denote function $f(α)=\overrightarrow{PB} \cdot \overrightarrow{CA}$, find the minimum positive period of function $f(α)$;
(II) If points $O$, $P$, and $C$ are collinear, find the value of $| \overrightarrow{OA}+ \overrightarrow{OB}|$. | {
"answer": "\\frac{\\sqrt{74}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In rectangle $ABCD,$ $P$ is a point on side $\overline{BC}$ such that $BP = 9$ and $CP = 27.$ If $\tan \angle APD = 2,$ then find $AB.$ | {
"answer": "27",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\sin(x+\pi) + \cos(x-\pi) = \frac{1}{2}, x \in (0, \pi)$.
(1) Find the value of $\sin x \cos x$;
(2) Find the value of $\sin x - \cos x$. | {
"answer": "\\frac{\\sqrt{7}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $g$ be a function satisfying $g(x^2y) = g(x)/y^2$ for all positive real numbers $x$ and $y$. If $g(800) = 4$, what is the value of $g(7200)$? | {
"answer": "\\frac{4}{81}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What would the 25th number be in a numeric system where the base is five? | {
"answer": "100",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that a group of students is sitting evenly spaced around a circular table and a bag containing 120 pieces of candy is circulated among them, determine the possible number of students if Sam picks both the first and a final piece after the bag has completed exactly two full rounds. | {
"answer": "60",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Triangles $ABC$ and $AFG$ have areas $3012$ and $10004$, respectively, with $B=(0,0),$ $C=(335,0),$ $F=(1020, 570),$ and $G=(1030, 580).$ Find the sum of all possible $x$-coordinates of $A$. | {
"answer": "1800",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The ten smallest positive odd numbers \( 1, 3, \cdots, 19 \) are arranged in a circle. Let \( m \) be the maximum value of the sum of any one of the numbers and its two adjacent numbers. Find the minimum value of \( m \). | {
"answer": "33",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The Elvish language consists of 4 words: "elara", "quen", "silva", and "nore". In a sentence, "elara" cannot come directly before "quen", and "silva" cannot come directly before "nore"; all other word combinations are grammatically correct (including sentences with repeated words). How many valid 3-word sentences are there in Elvish? | {
"answer": "48",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a frustum $ABCD-A_{1}B_{1}C_{1}D_{1}$ with a rectangular lower base, where $AB=2A_{1}B_{1}$, the height is $3$, and the volume of the frustum is $63$, find the minimum value of the perimeter of the upper base $A_{1}B_{1}C_{1}D_{1}$. | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=2\sqrt{3}\cos^2\left(\frac{\pi}{2}+x\right)-2\sin(\pi+x)\cos x-\sqrt{3}$.
$(1)$ Find the extreme values of $f(x)$ on the interval $\left[\frac{\pi}{4}, \frac{\pi}{2}\right]$.
$(2)$ If $f(x_0-\frac{\pi}{6})=\frac{10}{13}$, where $x_0\in\left[\frac{3\pi}{4}, \pi\right]$, find the value of $\sin 2x_0$. | {
"answer": "-\\frac{5+12\\sqrt{3}}{26}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( x_{1}, x_{2}, \cdots, x_{n} \) and \( a_{1}, a_{2}, \cdots, a_{n} \) be two sets of arbitrary real numbers (where \( n \geqslant 2 \)) that satisfy the following conditions:
1. \( x_{1} + x_{2} + \cdots + x_{n} = 0 \)
2. \( \left| x_{1} \right| + \left| x_{2} \right| + \cdots + \left| x_{n} \right| = 1 \)
3. \( a_{1} \geqslant a_{2} \geqslant \cdots \geqslant a_{n} \)
Determine the minimum value of the real number \( A \) such that the inequality \( \left| a_{1} x_{1} + a_{2} x_{2} + \cdots + a_{n} x_{n} \right| \leqslant A ( a_{1} - a_{n} ) \) holds, and provide a justification for this value. | {
"answer": "1/2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system $xOy$, the parametric equation of line $l$ is $\begin{cases} x=1+\dfrac{\sqrt{2}}{2}t \\ y=2+\dfrac{\sqrt{2}}{2}t \end{cases}$ ($t$ is the parameter), in the polar coordinate system (with the same unit length as the Cartesian coordinate system $xOy$, and the origin $O$ as the pole, and the non-negative half-axis of $x$ as the polar axis), the equation of circle $C$ is $\rho=6\sin\theta$.
- (I) Find the standard equation of circle $C$ in Cartesian coordinates;
- (II) If point $P(l,2)$, suppose circle $C$ intersects line $l$ at points $A$ and $B$, find the value of $|PA| + |PB|$. | {
"answer": "2\\sqrt{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the expression $\frac{810 \times 811 \times 812 \times \cdots \times 2010}{810^{n}}$ is an integer, find the maximum value of $n$. | {
"answer": "149",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $P$ be a moving point on the line $3x+4y+3=0$, and through point $P$, draw two tangents to the circle $C$: $x^{2}+y^{2}-2x-2y+1=0$, with the points of tangency being $A$ and $B$, respectively. Find the minimum value of the area of quadrilateral $PACB$. | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
12 students are standing in two rows, with 4 in the front row and 8 in the back row. Now, 2 students from the back row are to be selected to stand in the front row. If the relative order of the other students remains unchanged, the number of different rearrangement methods is ______. | {
"answer": "28",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A square has a 6x6 grid, where every third square in each row following a checkerboard pattern is shaded. What percent of the six-by-six square is shaded? | {
"answer": "33.33\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $S$ be the set of positive real numbers. Let $f : S \to \mathbb{R}$ be a function such that
\[ f(x)f(y) = f(xy) + 2023 \left( \frac{1}{x} + \frac{1}{y} + 2022 \right) \]
for all $x, y > 0.$
Let $n$ be the number of possible values of $f(2)$, and let $s$ be the sum of all possible values of $f(2)$. Find $n \times s$. | {
"answer": "\\frac{4047}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=\sin(\omega x+\varphi)$ is monotonically increasing on the interval ($\frac{π}{6}$,$\frac{{2π}}{3}$), and the lines $x=\frac{π}{6}$ and $x=\frac{{2π}}{3}$ are the two symmetric axes of the graph of the function $y=f(x)$, evaluate the value of $f(-\frac{{5π}}{{12}})$. | {
"answer": "\\frac{\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the rectangular coordinate system, a polar coordinate system is established with the origin as the pole and the positive semi-axis of the x-axis as the polar axis. Given circle C: ρ = 2cosθ - 2sinθ, and the parametric equation of line l is x = t, y = -1 + 2√2t (t is the parameter). Line l intersects with circle C at points M and N, and point P is any point on circle C that is different from M and N.
1. Write the rectangular coordinate equation of C and the general equation of l.
2. Find the maximum area of triangle PMN. | {
"answer": "\\frac{10\\sqrt{5}}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a sector with a central angle of 120° and an area of $3\pi$, which is used as the lateral surface area of a cone, find the surface area and volume of the cone. | {
"answer": "\\frac{2\\sqrt{2}\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the largest possible distance between two points, one on the sphere of radius 15 with center $(3, -5, 7),$ and the other on the sphere of radius 95 with center $(-10, 20, -25)$? | {
"answer": "110 + \\sqrt{1818}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that point $P$ is a moving point on circle $C_{1}$: $\left(x-1\right)^{2}+y^{2}=1$, point $Q$ is a moving point on circle $C_{2}$: $\left(x-4\right)^{2}+\left(y-1\right)^{2}=4$, and point $R$ moves on the line $l: x-y+1=0$, find the minimum value of $|PR|+|QR|$. | {
"answer": "\\sqrt{26}-3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Alex sent 150 text messages and talked for 28 hours, given a cell phone plan that costs $25 each month, $0.10 per text message, $0.15 per minute used over 25 hours, and $0.05 per minute within the first 25 hours. Calculate the total amount Alex had to pay in February. | {
"answer": "142.00",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Solve the following equations:
(1) $\frac {1}{2}x - 2 = 4 + \frac {1}{3}x$
(2) $\frac {x-1}{4} - 2 = \frac {2x-3}{6}$
(3) $\frac {1}{3}[x - \frac {1}{2}(x-1)] = \frac {2}{3}(x - \frac {1}{2})$
(4) $\frac {x}{0.7} - \frac {0.17-0.2x}{0.03} = 1$ | {
"answer": "\\frac {14}{17}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $$f(x)= \begin{cases} f(x+1), x < 4 \\ ( \frac {1}{2})^{x}, x\geq4\end{cases}$$, find $f(\log_{2}3)$. | {
"answer": "\\frac {1}{24}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The slope angle of the tangent line to the curve y=\frac{1}{3}x^{3} at x=1 is what value? | {
"answer": "\\frac{\\pi}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The area enclosed by the curves $y=e^{x}$, $y=e^{-x}$, and the line $x=1$ is $e^{1}-e^{-1}$. | {
"answer": "e+e^{-1}-2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A factory packs its products in cubic boxes. In one store, they put $512$ of these cubic boxes together to make a large $8\times 8 \times 8$ cube. When the temperature goes higher than a limit in the store, it is necessary to separate the $512$ set of boxes using horizontal and vertical plates so that each box has at least one face which is not touching other boxes. What is the least number of plates needed for this purpose? | {
"answer": "21",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
You are standing at the edge of a river which is $1$ km wide. You have to go to your camp on the opposite bank . The distance to the camp from the point on the opposite bank directly across you is $1$ km . You can swim at $2$ km/hr and walk at $3$ km-hr . What is the shortest time you will take to reach your camp?(Ignore the speed of the river and assume that the river banks are straight and parallel). | {
"answer": "\\frac{2 + \\sqrt{5}}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On a standard dice, the sum of the numbers of pips on opposite faces is always 7. Four standard dice are glued together as shown. What is the minimum number of pips that could lie on the whole surface?
A) 52
B) 54
C) 56
D) 58
E) 60 | {
"answer": "58",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In right triangle $ABC$, $\angle C=90^{\circ}$, $AB=9$, $\cos B=\frac{2}{3}$, calculate the length of $AC$. | {
"answer": "3\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A box contains 4 cards, each with one of the following functions defined on \\(R\\): \\(f_{1}(x)={x}^{3}\\), \\(f_{2}(x)=|x|\\), \\(f_{3}(x)=\sin x\\), \\(f_{4}(x)=\cos x\\). Now, if we randomly pick 2 cards from the box and multiply the functions on the cards to get a new function, the probability that the resulting function is an odd function is \_\_\_\_\_. | {
"answer": "\\dfrac{2}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A plane parallel to the base of a cone divides the height of the cone into two equal segments. What is the ratio of the lateral surface areas of the two parts of the cone? | {
"answer": "\\frac{1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the integer $n$, $-180 \le n \le 180,$ such that $\sin n^\circ = \cos 682^\circ.$ | {
"answer": "128",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let set $A=\{x \mid |x-2| \leq 2\}$, and $B=\{y \mid y=-x^2, -1 \leq x \leq 2\}$, then $A \cap B=$ ? | {
"answer": "\\{0\\}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $ABC$, $AB = 18$ and $BC = 12$. Find the largest possible value of $\tan A$. | {
"answer": "\\frac{2\\sqrt{5}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two tigers, Alice and Betty, run in the same direction around a circular track with a circumference of 400 meters. Alice runs at a speed of \(10 \, \text{m/s}\) and Betty runs at \(15 \, \text{m/s}\). Betty gives Alice a 40 meter head start before they both start running. After 15 minutes, how many times will they have passed each other?
(a) 9
(b) 10
(c) 11
(d) 12 | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $0 < α < \dfrac {π}{2}$, and $\cos ( \dfrac {π}{3}+α)= \dfrac {1}{3}$, find the value of $\cos α$. | {
"answer": "\\dfrac {2 \\sqrt {6}+1}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, let the sides opposite to angles $A$, $B$, and $C$ be $a$, $b$, and $c$, respectively, and $\frac{\cos C}{\cos B} = \frac{3a-c}{b}$.
(1) Find the value of $\sin B$;
(2) If $b = 4\sqrt{2}$ and $a = c$, find the area of $\triangle ABC$. | {
"answer": "8\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Three non-collinear lattice points $A,B,C$ lie on the plane $1+3x+5y+7z=0$ . The minimal possible area of triangle $ABC$ can be expressed as $\frac{\sqrt{m}}{n}$ where $m,n$ are positive integers such that there does not exists a prime $p$ dividing $n$ with $p^2$ dividing $m$ . Compute $100m+n$ .
*Proposed by Yannick Yao* | {
"answer": "8302",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the least possible value of \((x+1)(x+2)(x+3)(x+4)+2023\) where \(x\) is a real number? | {
"answer": "2022",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A circle $U$ has a circumference of $18\pi$ inches, and segment $AB$ is a diameter. If the measure of angle $UAV$ is $45^{\circ}$, what is the length, in inches, of segment $AV$? | {
"answer": "9\\sqrt{2 - \\sqrt{2}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given points P(0, -3) and Q(5, 3) in the xy-plane; point R(x, m) is taken so that PR + RQ is a minimum where x is fixed to 3, determine the value of m. | {
"answer": "\\frac{3}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For how many positive integers $n$ less than or equal to 500 is $$(\sin (t+\frac{\pi}{4})+i\cos (t+\frac{\pi}{4}))^n=\sin (nt+\frac{n\pi}{4})+i\cos (nt+\frac{n\pi}{4})$$ true for all real $t$? | {
"answer": "125",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Circles with centers at $(5,5)$ and $(20,15)$ are both tangent to the $x$-axis. What is the distance between the closest points of the two circles? | {
"answer": "5 \\sqrt{13} - 20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A bug starts at a vertex of a square. On each move, it randomly selects one of the three vertices where it is not currently located and crawls along a side of the square to that vertex. Given that the probability that the bug moves to its starting vertex on its eighth move is \( \frac{p}{q} \), where \( p \) and \( q \) are relatively prime positive integers, find \( p + q \). | {
"answer": "2734",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and $\frac{\cos C}{\cos B}= \frac{2a-c}{b}$.
(1) Find $B$;
(2) If $\tan \left(A+ \frac{π}{4}\right) =7$, find the value of $\cos C$. | {
"answer": "\\frac{-4+3\\sqrt{3}}{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the sum of all positive integers such that their expression in base $5$ digits is the reverse of their expression in base $11$ digits. Express your answer in base $10$. | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A three-digit number has different digits in each position. By writing a 2 to the left of this three-digit number, we get a four-digit number; and by writing a 2 to the right of this three-digit number, we get another four-digit number. The difference between these two four-digit numbers is 945. What is this three-digit number? | {
"answer": "327",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Triangle $ABC$ is a right triangle with $AB = 6$, $BC = 8$, and $AC = 10$. Point $D$ is on line $\overline{BC}$ such that $\overline{AD}$ bisects angle $BAC$. The inscribed circles of $\triangle ADB$ and $\triangle ADC$ have radii $r_1$ and $r_2$, respectively. What is $r_1/r_2$?
A) $\frac{24}{35}$
B) $\frac{35}{24}$
C) $\frac{15}{28}$
D) $\frac{7}{15}$ | {
"answer": "\\frac{24}{35}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the smallest positive real number $x$ such that
\[\lfloor x^2 \rfloor - x \lfloor x \rfloor = 8.\] | {
"answer": "\\frac{89}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $y=\sin 3x$, determine the horizontal shift required to obtain the graph of the function $y=\sin \left(3x+\frac{\pi }{4}\right)$. | {
"answer": "\\frac{\\pi}{12}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Rectangle $EFGH$ has an area of $4032$. An ellipse with area $4032\pi$ passes through points $E$ and $G$ and has foci at $F$ and $H$. Determine the perimeter of rectangle $EFGH$. | {
"answer": "8\\sqrt{2016}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A stock investment increased by 30% in 2006. Starting at this new value, what percentage decrease is needed in 2007 to return the stock to its original price at the beginning of 2006? | {
"answer": "23.077\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $C= \dfrac {\pi}{6}$, $a=1$, $b= \sqrt {3}$, find the measure of $B$. | {
"answer": "\\dfrac {2\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $ABC$, $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively. Given vectors $\overrightarrow{m} = (\cos A, \sin A)$ and $\overrightarrow{n} = (\cos B, -\sin B)$, and $|\overrightarrow{m} - \overrightarrow{n}| = 1$.
(1) Find the degree measure of angle $C$;
(2) If $c=3$, find the maximum area of triangle $ABC$. | {
"answer": "\\frac{3\\sqrt{3}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the smallest solution to the equation \[\frac{1}{x-3} + \frac{1}{x-5} = \frac{5}{x-4}.\] | {
"answer": "4 - \\frac{\\sqrt{15}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the radius of a hemisphere is 2, calculate the maximum lateral area of the inscribed cylinder. | {
"answer": "4\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x) = \sin 2x + \sqrt{3}\cos 2x$, stretch the x-coordinates of all points on the graph to twice their original length, and then shift all points on the graph to the right by $\frac{\pi}{6}$ units, and find the equation of one of the axes of symmetry for the resulting function $g(x)$. | {
"answer": "\\frac{\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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