problem stringlengths 10 5.15k | answer dict |
|---|---|
In triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. It is known that $b\sin(C+\frac{π}{3})-c\sin B=0$.
$(1)$ Find the value of angle $C$.
$(2)$ If the area of $\triangle ABC$ is $10\sqrt{3}$ and $D$ is the midpoint of $AC$, find the minimum value of $BD$. | {
"answer": "2\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The robot vacuum cleaner is programmed to move on the floor according to the law:
$$\left\{\begin{array}{l} x = t(t-6)^2 \\ y = 0, \quad 0 \leq t \leq 7 \\ y = (t-7)^2, \quad t \geq 7 \end{array}\right.$$
where the axes are chosen parallel to the walls and the movement starts from the origin. Time $t$ is measured in minutes, and coordinates are measured in meters. Find the distance traveled by the robot in the first 7 minutes and the absolute change in the velocity vector during the eighth minute. | {
"answer": "\\sqrt{445}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If \( e^{i \theta} = \frac{3 + i \sqrt{2}}{4}, \) then find \( \cos 3\theta. \) | {
"answer": "\\frac{9}{64}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given lines $l_{1}$: $\rho\sin(\theta-\frac{\pi}{3})=\sqrt{3}$ and $l_{2}$: $\begin{cases} x=-t \\ y=\sqrt{3}t \end{cases}$ (where $t$ is a parameter), find the polar coordinates of the intersection point $P$ of $l_{1}$ and $l_{2}$. Additionally, three points $A$, $B$, and $C$ lie on the ellipse $\frac{x^{2}}{4}+y^{2}=1$, with $O$ being the coordinate origin. If $\angle{AOB}=\angle{BOC}=\angle{COA}=120^{\circ}$, find the value of $\frac{1}{|OA|^{2}}+\frac{1}{|OB|^{2}}+\frac{1}{|OC|^{2}}$. | {
"answer": "\\frac{15}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two people are playing a game. One person thinks of a ten-digit number, and the other can ask questions about which digits are in specific sets of positions in the number's sequence. The first person answers the questions without indicating which digits are in which exact positions. What is the minimum number of questions needed to reliably guess the number? | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=2\sin ωx\cos ωx-2\sqrt{3} \cos ^{2}ωx+\sqrt{3} (ω > 0)$, and the distance between two adjacent symmetry axes of the graph of $y=f(x)$ is $\frac{π}{2}$.
(I) Find the period of the function $f(x)$;
(II) In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Angle $C$ is acute, and $f(C)=\sqrt{3}$, $c=3\sqrt{2}$, $\sin B=2\sin A$. Find the area of $\triangle ABC$. | {
"answer": "3\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a triangle $ABC$ with internal angles $A$, $B$, and $C$ opposite to the sides $a$, $b$, and $c$ respectively. If $(2a-c)\cos B=b\cos C$, and the dot product $\vec{AB}\cdot \vec{BC} = -3$,
1. Find the area of $\triangle ABC$;
2. Find the minimum value of side $AC$. | {
"answer": "\\sqrt{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a function $f(x)$ defined on $R$ such that $f(x) + f(x+4) = 23$. When $x \in (0,4]$, $f(x) = x^2 - 2^x$. Find the number of zeros of the function $f(x)$ on the interval $(-4,2023]$. | {
"answer": "506",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $f(\alpha)= \dfrac {\sin (\pi-\alpha)\cos (2\pi-\alpha)\cos (-\alpha+ \dfrac {3}{2}\pi)}{\cos ( \dfrac {\pi}{2}-\alpha)\sin (-\pi-\alpha)}$
$(1)$ Simplify $f(\alpha)$;
$(2)$ If $\alpha$ is an angle in the third quadrant, and $\cos (\alpha- \dfrac {3}{2}\pi)= \dfrac {1}{5}$, find the value of $f(\alpha)$;
$(3)$ If $\alpha=- \dfrac {31}{3}\pi$, find the value of $f(\alpha)$. | {
"answer": "-\\dfrac {1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an ellipse $C:\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1(a>b>0)$ with eccentricity $\frac{1}{2}$, a circle $\odot E$ with center at the origin and radius equal to the minor axis of the ellipse is tangent to the line $x-y+\sqrt{6}=0$. <br/>$(1)$ Find the equation of the ellipse $C$; <br/>$(2)$ A line passing through the fixed point $Q(1,0)$ with slope $k$ intersects the ellipse $C$ at points $M$ and $N$. If $\overrightarrow{OM}•\overrightarrow{ON}=-2$, find the value of the real number $k$ and the area of $\triangle MON$. | {
"answer": "\\frac{6\\sqrt{6}}{11}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=\sin ^{2}x+a\sin x\cos x-\cos ^{2}x$, and $f(\frac{\pi }{4})=1$.
(1) Find the value of the constant $a$;
(2) Find the smallest positive period and minimum value of $f(x)$. | {
"answer": "-\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A system of inequalities defines a region on a coordinate plane as follows:
$$
\begin{cases}
x+y \leq 5 \\
3x+2y \geq 3 \\
x \geq 1 \\
y \geq 1
\end{cases}
$$
Determine the number of units in the length of the longest side of the quadrilateral formed by the region satisfying all these conditions. Express your answer in simplest radical form. | {
"answer": "3\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the line $2x+my=2m-4$ is parallel to the line $mx+2y=m-2$, find the value of $m$. | {
"answer": "-2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given two lines $$l_{1}: \sqrt {3}x+y-1=0$$ and $$l_{2}: ax+y=1$$, and $l_{1}$ is perpendicular to $l_{2}$, then the slope angle of $l_{1}$ is \_\_\_\_\_\_, and the distance from the origin to $l_{2}$ is \_\_\_\_\_\_. | {
"answer": "\\frac { \\sqrt {3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The graph of the function $f(x)=\sin({ωx-\frac{π}{6}})$, where $0<ω<6$, is shifted to the right by $\frac{π}{6}$ units to obtain the graph of the function $g(x)$. If $\left(0,\frac{π}{ω}\right)$ is a monotone interval of $g(x)$, and $F(x)=f(x)+g(x)$, determine the maximum value of $F(x)$. | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The line $y=2b$ intersects the left and right branches of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \ (a > 0, b > 0)$ at points $B$ and $C$ respectively, with $A$ being the right vertex and $O$ the origin. If $\angle AOC = \angle BOC$, then calculate the eccentricity of the hyperbola. | {
"answer": "\\frac{\\sqrt{19}}{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 axes of symmetry 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"
} |
A sphere with center $O$ has radius $10$. A right triangle with sides $8, 15,$ and $17$ is situated in 3D space such that each side is tangent to the sphere. What is the distance between $O$ and the plane determined by the triangle?
- **A)** $\sqrt{84}$
- **B)** $\sqrt{85}$
- **C)** $\sqrt{89}$
- **D)** $\sqrt{91}$
- **E)** $\sqrt{95}$ | {
"answer": "\\sqrt{91}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A certain store sells a batch of helmets for $80 each. It can sell 200 helmets per month. During the "Creating a Civilized City" period, the store plans to reduce the price of the helmets for sale. After investigation, it was found that for every $1 decrease in price, an additional 20 helmets are sold per month. It is known that the cost price of the helmets is $50 each.
$(1)$ If the price of each helmet is reduced by $10, the store can sell ______ helmets per month, and the monthly profit from sales is ______ dollars.
$(2)$ If the store plans to reduce the price of these helmets to reduce inventory while ensuring a monthly profit of $7500, find the selling price of the helmets. | {
"answer": "65",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
As shown in the diagram, rectangle \(ABCD\) is inscribed in a semicircle, with \(EF\) as the diameter of the semicircle. Given that \(DA = 16\), and \(FD = AE = 9\), find the area of rectangle \(ABCD\). | {
"answer": "240",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A flower shop buys a number of roses from a farm at a price of 5 yuan per rose each day and sells them at a price of 10 yuan per rose. If the roses are not sold by the end of the day, they are discarded.
(1) If the shop buys 16 roses in one day, find the profit function \( y \) (in yuan) with respect to the demand \( n \) for that day (in roses, \( n \in \mathbf{N} \)).
(2) The shop recorded the daily demand for roses (in roses) for 100 days and summarized the data in Table 1.
Using the frequencies of the demands recorded over the 100 days as probabilities for each demand:
(i) If the shop buys 16 roses in one day, let \( X \) represent the profit (in yuan) for that day. Find the distribution, expected value, and variance of \( X \).
(ii) If the shop plans to buy either 16 or 17 roses in one day, which would you recommend they buy? Please explain your reasoning. | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\dfrac{\pi}{2} < \alpha < \beta < \dfrac{3\pi}{4}, \cos(\alpha - \beta) = \dfrac{12}{13}, \sin(\alpha + \beta) = -\dfrac{3}{5}$, find the value of $\sin 2\alpha$. | {
"answer": "-\\dfrac{56}{65}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
As shown in the diagram, circles \( \odot O_{1} \) and \( \odot O_{2} \) are externally tangent. The line segment \( O_{1}O_{2} \) intersects \( \odot O_{1} \) at points \( A \) and \( B \), and intersects \( \odot O_{2} \) at points \( C \) and \( D \). Circle \( \odot O_{3} \) is internally tangent to \( \odot O_{1} \) at point \( B \), and circle \( \odot O_{4} \) is internally tangent to \( \odot O_{2} \) at point \( C \). The common external tangent of \( \odot O_{2} \) and \( \odot O_{3} \) passes through point \( A \), tangent to \( \odot O_{3} \) at point \( E \) and tangent to \( \odot O_{2} \) at point \( F \). The common external tangent of \( \odot O_{1} \) and \( \odot O_{4} \) passes through point \( D \). If the radius of circle \( \odot O_{3} \) is 1.2, what is the radius of circle \( \odot O_{4} \)? | {
"answer": "1.2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Elective 4-4: Coordinate System and Parametric Equations
In the Cartesian coordinate system $xOy$, with the origin $O$ as the pole and the positive half-axis of $x$ as the polar axis, a polar coordinate system is established. If the polar equation of curve $C$ is $\rho\cos^2\theta-4\sin\theta=0$, and the polar coordinates of point $P$ are $(3, \frac{\pi}{2})$, in the Cartesian coordinate system, line $l$ passes through point $P$ with a slope of $\sqrt{3}$.
(Ⅰ) Write the Cartesian coordinate equation of curve $C$ and the parametric equation of line $l$;
(Ⅱ) Suppose line $l$ intersects curve $C$ at points $A$ and $B$, find the value of $\frac{1}{|PA|}+ \frac{1}{|PB|}$. | {
"answer": "\\frac{\\sqrt{6}}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the value of $x$ in the following expressions:
(1) $8x^3 = 27$; (2) $(x-2)^2 = 3$. | {
"answer": "-\\sqrt{3} + 2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A factory produces a type of instrument. Due to limitations in production capacity and technical level, some defective products are produced. According to experience, the defect rate $p$ of the factory producing this instrument is generally related to the daily output $x$ (pieces) as follows:
$$
P= \begin{cases}
\frac {1}{96-x} & (1\leq x\leq 94, x\in \mathbb{N}) \\
\frac {2}{3} & (x>94, x\in \mathbb{N})
\end{cases}
$$
It is known that for every qualified instrument produced, a profit of $A$ yuan can be made, but for every defective product produced, a loss of $\frac {A}{2}$ yuan will be incurred. The factory wishes to determine an appropriate daily output.
(1) Determine whether producing this instrument can be profitable when the daily output (pieces) exceeds 94 pieces, and explain the reason;
(2) When the daily output $x$ pieces does not exceed 94 pieces, try to express the daily profit $T$ (yuan) of producing this instrument as a function of the daily output $x$ (pieces);
(3) To obtain the maximum profit, how many pieces should the daily output $x$ be? | {
"answer": "84",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two sectors of a circle of radius $15$ overlap in the same manner as the original problem, with $P$ and $R$ as the centers of the respective circles. The angle at the centers for both sectors is now $45^\circ$. Determine the area of the shaded region. | {
"answer": "\\frac{225\\pi - 450\\sqrt{2}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the inequality system about $x$ is $\left\{\begin{array}{l}{\frac{x+3}{2}≥x-1}\\{3x+6＞a+4}\end{array}\right.$ has exactly $3$ odd solutions, and the solution to the equation about $y$ is $3y+6a=22-y$ is a non-negative integer, then the product of all integers $a$ that satisfy the conditions is ____. | {
"answer": "-3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Sunshine High School is planning to order a batch of basketballs and jump ropes from an online store. After checking on Tmall, they found that each basketball is priced at $120, and each jump rope is priced at $25. There are two online stores, Store A and Store B, both offering free shipping and their own discount schemes:<br/>Store A: Buy one basketball and get one jump rope for free;<br/>Store B: Pay 90% of the original price for both the basketball and jump rope.<br/>It is known that they want to buy 40 basketballs and $x$ jump ropes $\left(x \gt 40\right)$.<br/>$(1)$ If they purchase from Store A, the payment will be ______ yuan; if they purchase from Store B, the payment will be ______ yuan; (express in algebraic expressions with $x$)<br/>$(2)$ If $x=80$, through calculation, determine which store is more cost-effective to purchase from at this point.<br/>$(3)$ If $x=80$, can you provide a more cost-effective purchasing plan? Write down your purchasing method and calculate the amount to be paid. | {
"answer": "5700",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Quadrilateral $EFGH$ has right angles at $F$ and $H$, and $EG = 5$. If $EFGH$ has two sides with distinct integer lengths, and each side length is greater than 1, what is the area of $EFGH$? Express your answer in simplest radical form. | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find $B^2$, where $B$ is the sum of the absolute values of all roots of the equation:
\[x = \sqrt{26} + \frac{119}{{\sqrt{26}+\frac{119}{{\sqrt{26}+\frac{119}{{\sqrt{26}+\frac{119}{{\sqrt{26}+\frac{119}{x}}}}}}}}}.\] | {
"answer": "502",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given two numbers, a and b, are randomly selected within the interval (-π, π), determine the probability that the function f(x) = x^2 + 2ax - b^2 + π has a root. | {
"answer": "\\dfrac{3}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given vectors $\overrightarrow {m}$=(cosx, sinx) and $\overrightarrow {n}$=(cosx, $\sqrt {3}$cosx), where x∈R, define the function f(x) = $\overrightarrow {m}$$\cdot \overrightarrow {n}$+ $\frac {1}{2}$.
(1) Find the analytical expression and the interval where the function is strictly increasing;
(2) Let a, b, and c be the sides opposite to angles A, B, and C of △ABC, respectively. If f(A)=2, b+c=$2 \sqrt {2}$, and the area of △ABC is $\frac {1}{2}$, find the value of a. | {
"answer": "\\sqrt {3}-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Observe the following equations:
\\(① \dfrac {1}{ \sqrt {2}+1}= \dfrac { \sqrt {2}-1}{( \sqrt {2}+1)( \sqrt {2}-1)}= \sqrt {2}-1\\);
\\(② \dfrac {1}{ \sqrt {3}+ \sqrt {2}}= \dfrac { \sqrt {3}- \sqrt {2}}{( \sqrt {3}+ \sqrt {2})( \sqrt {3}- \sqrt {2})}= \sqrt {3}- \sqrt {2}\\);
\\(③ \dfrac {1}{ \sqrt {4}+ \sqrt {3}}= \dfrac { \sqrt {4}- \sqrt {3}}{( \sqrt {4}+ \sqrt {3})( \sqrt {4}- \sqrt {3})}= \sqrt {4}- \sqrt {3}\\);\\(…\\)
Answer the following questions:
\\((1)\\) Following the pattern of the equations above, write the \\(n\\)th equation: \_\_\_\_\_\_ ;
\\((2)\\) Using the pattern you observed, simplify: \\( \dfrac {1}{ \sqrt {8}+ \sqrt {7}}\\);
\\((3)\\) Calculate: \\( \dfrac {1}{1+ \sqrt {2}}+ \dfrac {1}{ \sqrt {2}+ \sqrt {3}}+ \dfrac {1}{ \sqrt {3}+2}+…+ \dfrac {1}{3+ \sqrt {10}}\\). | {
"answer": "\\sqrt {10}-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In \\(\Delta ABC\\), given that \\(a= \sqrt{3}, b= \sqrt{2}, B=45^{\circ}\\), find \\(A, C\\) and \\(c\\). | {
"answer": "\\frac{\\sqrt{6}- \\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $F\_1$ and $F\_2$ are the left and right foci of the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$, perpendicular lines to the $x$-axis are drawn through points $F\_1$ and $F\_2$ intersecting the ellipse at four points to form a square, determine the eccentricity $e$ of the ellipse. | {
"answer": "\\frac{\\sqrt{5} - 1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the function $f(x)$ is an odd function defined on $\mathbb{R}$, and when $x > 0$, $f(x) = -x^{2} + 4x$.
- (Ⅰ) Find the analytical expression of the function $f(x)$.
- (Ⅱ) Find the minimum value of the function $f(x)$ on the interval $\left[-2,a\right]$ where $\left(a > -2\right)$. | {
"answer": "-4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\cos(75^\circ + \alpha) = \frac{1}{3}$, where $\alpha$ is an angle in the third quadrant, find the value of $\cos(105^\circ - \alpha) + \sin(\alpha - 105^\circ)$. | {
"answer": "\\frac{2\\sqrt{2} - 1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that α and β are acute angles, and $\tan \alpha = \frac{2}{t}$, $\tan \beta = \frac{t}{15}$. When $10\tan \alpha + 3\tan \beta$ reaches its minimum value, the value of $\alpha + \beta$ is \_\_\_\_\_\_. | {
"answer": "\\frac{\\pi}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A standard die is rolled eight times. What is the probability that the product of all eight rolls is odd and consists only of prime numbers? Express your answer as a common fraction. | {
"answer": "\\frac{1}{6561}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $a_n = -n^2 + 10n + 11$, then find the value of $n$ for which the sum of the sequence $\{a_n\}$ from the first term to the nth term is maximized. | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a moving point $E$ such that the product of the slopes of the lines from $E$ to points $A(2,0)$ and $B(-2,0)$ is $- \frac {1}{4}$, and the trajectory of point $E$ is curve $C$.
$(1)$ Find the equation of curve $C$;
$(2)$ Draw a line $l$ through point $D(1,0)$ that intersects curve $C$ at points $P$ and $Q$. Find the maximum value of $\overrightarrow{OP} \cdot \overrightarrow{OQ}$. | {
"answer": "\\frac {1}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A deck consists of six red cards and six green cards, each with labels $A$, $B$, $C$, $D$, $E$ corresponding to each color. Two cards are dealt from this deck. A winning pair consists of cards that either share the same color or the same label. Calculate the probability of drawing a winning pair.
A) $\frac{1}{2}$
B) $\frac{10}{33}$
C) $\frac{30}{66}$
D) $\frac{35}{66}$
E) $\frac{40}{66}$ | {
"answer": "\\frac{35}{66}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the set of integers $\{1, 2, 3, \dots, 9\}$, from which three distinct numbers are arbitrarily selected as the coefficients of the quadratic function $f_{(x)} = ax^2 + bx + c$, determine the total number of functions $f_{(x)}$ that satisfy $\frac{f(1)}{2} \in \mathbb{Z}$. | {
"answer": "264",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $f\left(\alpha \right)=\frac{\mathrm{sin}\left(\pi -\alpha \right)\mathrm{cos}\left(2\pi -\alpha \right)\mathrm{cos}\left(-\alpha +\frac{3\pi }{2}\right)}{\mathrm{cos}\left(\frac{\pi }{2}-\alpha \right)\mathrm{sin}\left(-\pi -\alpha \right)}$.
(1) Simplify $f(\alpha )$.
(2) If $\alpha$ is an angle in the third quadrant and $\mathrm{cos}(\alpha -\frac{3\pi }{2})=\frac{1}{5}$, find the value of $f(\alpha )$. | {
"answer": "\\frac{2\\sqrt{6}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $F_{1}$ and $F_{2}$ are the left and right foci of the hyperbola $C: \frac {x^{2}}{a^{2}} - \frac {y^{2}}{b^{2}} = 1$ $(a > 0, b > 0)$, point $P$ is on the hyperbola $C$, $PF_{2}$ is perpendicular to the x-axis, and $\sin \angle PF_{1}F_{2} = \frac {1}{3}$, determine the eccentricity of the hyperbola $C$. | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the smallest positive real number $x$ such that
\[\lfloor x^2 \rfloor - x \lfloor x \rfloor = 10.\] | {
"answer": "\\frac{131}{11}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
From the five numbers \\(1, 2, 3, 4, 5\\), select any \\(3\\) to form a three-digit number without repeating digits. When the three digits include both \\(2\\) and \\(3\\), \\(2\\) must be placed before \\(3\\) (not necessarily adjacent). How many such three-digit numbers are there? | {
"answer": "51",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
When the numbers \(\sqrt{5}, 2.1, \frac{7}{3}, 2.0 \overline{5}, 2 \frac{1}{5}\) are arranged in order from smallest to largest, the middle number is: | {
"answer": "2 \\frac{1}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Sabrina has a fair tetrahedral die whose faces are numbered 1, 2, 3, and 4, respectively. She creates a sequence by rolling the die and recording the number on its bottom face. However, she discards (without recording) any roll such that appending its number to the sequence would result in two consecutive terms that sum to 5. Sabrina stops the moment that all four numbers appear in the sequence. Find the expected (average) number of terms in Sabrina's sequence. | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system $xOy$, the parametric equation of line $l$ is $\begin{cases} x=3+t\cos \alpha \\ y=1+t\sin \alpha\end{cases}$ (where $t$ 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 curve $C$ is $\rho=4\cos \theta$.
$(1)$ Find the equation of curve $C$ in the Cartesian coordinate system;
$(2)$ If point $P(3,1)$, suppose circle $C$ intersects line $l$ at points $A$ and $B$, find the minimum value of $|PA|+|PB|$. | {
"answer": "2 \\sqrt {2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a function defined on $\mathbb{R}$, $f(x)=A\sin (\omega x+\varphi)$ where $A > 0$, $\omega > 0$, and $|\varphi| \leqslant \frac {\pi}{2}$, the minimum value of the function is $-2$, and the distance between two adjacent axes of symmetry is $\frac {\pi}{2}$. After the graph of the function is shifted to the left by $\frac {\pi}{12}$ units, the resulting graph corresponds to an even function.
$(1)$ Find the expression for the function $f(x)$.
$(2)$ If $f\left( \frac {x_{0}}{2}\right)=- \frac {3}{8}$, and $x_{0}\in\left[ \frac {\pi}{2},\pi\right]$, find the value of $\cos \left(x_{0}+ \frac {\pi}{6}\right)$. | {
"answer": "- \\frac { \\sqrt {741}}{32}- \\frac {3}{32}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The shortest distance from a point on the curve $y=\ln x$ to the line $y=x+2$ is what value? | {
"answer": "\\frac{3\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose that \(g(x)\) is a function such that
\[g(xy) + 2x = xg(y) + g(x)\] for all real numbers \(x\) and \(y.\) If \(g(-1) = 3\) and \(g(1) = 1\), then compute \(g(-101).\) | {
"answer": "103",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Describe how to place the vertices of a triangle in the faces of a cube in such a way that the shortest side of the triangle is the biggest possible. | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=\sin({ωx+φ})$ $({ω＞0,|φ|≤\frac{π}{2}})$, $f(0)=\frac{{\sqrt{2}}}{2}$, and the function $f\left(x\right)$ is monotonically decreasing on the interval $({\frac{π}{{16}},\frac{π}{8}})$, then the maximum value of $\omega$ is ______. | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose $x$ and $y$ satisfy the system of inequalities $\begin{cases} & x-y \geqslant 0 \\ & x+y-2 \geqslant 0 \\ & x \leqslant 2 \end{cases}$, calculate the minimum value of $x^2+y^2-2x$. | {
"answer": "-\\dfrac{1}{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 $y=f(x)$, determine the value of $f(-\frac{{5π}}{{12}})$. | {
"answer": "\\frac{\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
From a large sheet of aluminum, triangular sheets (with each cell side equal to 1) are cut with vertices at marked points. What is the minimum area of the triangle that can be obtained? | {
"answer": "$\\frac{1}{2}$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine how many ordered pairs of positive integers $(x, y)$, where $x < y$, have a harmonic mean of $5^{20}$. | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The function $g$ is defined on the set of integers and satisfies \[g(n)= \begin{cases} n-5 & \mbox{if }n\ge 1200 \\ g(g(n+7)) & \mbox{if }n<1200. \end{cases}\] Find $g(70)$. | {
"answer": "1195",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x) = \sin x \cos x - \sqrt{3} \cos (x+\pi) \cos x, x \in \mathbb{R}$.
(Ⅰ) Find the minimal positive period of $f(x)$;
(Ⅱ) If the graph of the function $y = f(x)$ is translated by $\overrightarrow{b}=\left( \frac{\pi}{4}, \frac{\sqrt{3}}{2} \right)$ to obtain the graph of the function $y = g(x)$, find the maximum value of $y=g(x)$ on the interval $\left[0, \frac{\pi}{4}\right]$. | {
"answer": "\\frac{3\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The number halfway between $\dfrac{1}{8}$ and $\dfrac{1}{3}$ is
A) $\dfrac{11}{48}$
B) $\dfrac{11}{24}$
C) $\dfrac{5}{24}$
D) $\dfrac{1}{4}$
E) $\dfrac{1}{5}$ | {
"answer": "\\dfrac{11}{48}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A new dump truck delivered sand to a construction site, forming a conical pile with a diameter of $12$ feet. The height of the cone was $50\%$ of its diameter. However, the pile was too large, causing some sand to spill, forming a cylindrical layer directly around the base of the cone. The height of this cylindrical layer was $2$ feet and the thickness was $1$ foot. Calculate the total volume of sand delivered, expressing your answer in terms of $\pi$. | {
"answer": "98\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system $(xOy)$, a pole is established at the origin $O$ with the non-negative semi-axis of the $x$-axis as the polar axis, forming a polar coordinate system. Given that the equation of line $l$ is $4ρ\cos θ-ρ\sin θ-25=0$, and the curve $W$ is defined by the parametric equations $x=2t, y=t^{2}-1$.
1. Find the Cartesian equation of line $l$ and the general equation of curve $W$.
2. If point $P$ is on line $l$, and point $Q$ is on curve $W$, find the minimum value of $|PQ|$. | {
"answer": "\\frac{8\\sqrt{17}}{17}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=\sin (x+ \frac{7\pi}{4})+\cos (x- \frac{3\pi}{4})$, where $x\in R$.
(1) Find the smallest positive period and the minimum value of $f(x)$;
(2) Given that $f(\alpha)= \frac{6}{5}$, where $0 < \alpha < \frac{3\pi}{4}$, find the value of $f(2\alpha)$. | {
"answer": "\\frac{31\\sqrt{2}}{25}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that triangle $PQR$ is a right triangle, each side being the diameter of a semicircle, the area of the semicircle on $\overline{PQ}$ is $18\pi$, and the arc of the semicircle on $\overline{PR}$ has length $10\pi$, calculate the radius of the semicircle on $\overline{QR}$. | {
"answer": "\\sqrt{136}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Acute-angled $\triangle ABC$ is inscribed in a circle with center at $O$; $\stackrel \frown {AB} = 100^\circ$ and $\stackrel \frown {BC} = 80^\circ$.
A point $E$ is taken in minor arc $AC$ such that $OE$ is perpendicular to $AC$. The task is to determine the ratio of the magnitudes of $\angle OBE$ and $\angle BAC$. | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $x_{0}$ is a zero of the function $f(x)=2a\sqrt{x}+b-{e}^{\frac{x}{2}}$, and $x_{0}\in [\frac{1}{4}$,$e]$, find the minimum value of $a^{2}+b^{2}$. | {
"answer": "\\frac{{e}^{\\frac{3}{4}}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the smallest positive integer $n$ such that $\frac{n}{n+150}$ is equal to a terminating decimal? | {
"answer": "50",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Simplify the expression $\dfrac {\cos 40 ^{\circ} }{\cos 25 ^{\circ} \sqrt {1-\sin 40 ^{\circ} }}$. | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The minimum positive period of the function $f(x)=\sin x$ is $\pi$. | {
"answer": "2\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $$\frac {\pi}{2} < \alpha < \pi$$, $$0 < \beta < \frac {\pi}{2}$$, $$\tan\alpha = -\frac {3}{4}$$, and $$\cos(\beta-\alpha) = \frac {5}{13}$$, find the value of $\sin\beta$. | {
"answer": "\\frac {63}{65}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that Connie adds $3$ to a number and gets $45$ as her answer, but she should have subtracted $3$ from the number to get the correct answer, determine the correct number. | {
"answer": "39",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Regular hexagon $PQRSTU$ has vertices $P$ and $R$ at $(0,0)$ and $(8,2)$, respectively. What is its area? | {
"answer": "102\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
From milk with a fat content of $5\%$, cottage cheese with a fat content of $15.5\%$ is produced, while there remains whey with a fat content of $0.5\%$. How much cottage cheese is obtained from 1 ton of milk? | {
"answer": "0.3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the positive value of $k$ such that the equation $4x^3 + 9x^2 + kx + 4 = 0$ has exactly one real solution in $x$. | {
"answer": "6.75",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Rationalize the denominator of $\frac{2+\sqrt{5}}{3-\sqrt{5}}$. When you write your answer in the form $A+B\sqrt{C}$, where $A$, $B$, and $C$ are integers, what is $ABC$? | {
"answer": "275",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( ABC \) be a triangle with \( AB = 5 \), \( AC = 4 \), \( BC = 6 \). The angle bisector of \( \angle C \) intersects side \( AB \) at \( X \). Points \( M \) and \( N \) are drawn on sides \( BC \) and \( AC \), respectively, such that \( \overline{XM} \parallel \overline{AC} \) and \( \overline{XN} \parallel \overline{BC} \). Compute the length \( MN \). | {
"answer": "\\frac{3 \\sqrt{14}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $α$ and $β ∈ ( \frac{π}{2},π)$, and $sinα + cosα = a$, $cos(β - α) = \frac{3}{5}$.
(1) If $a = \frac{1}{3}$, find the value of $sinαcosα + tanα - \frac{1}{3cosα}$;
(2) If $a = \frac{7}{13}$, find the value of $sinβ$. | {
"answer": "\\frac{16}{65}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given points $M(4,0)$ and $N(1,0)$, any point $P$ on curve $C$ satisfies: $\overset{→}{MN} \cdot \overset{→}{MP} = 6|\overset{→}{PN}|$.
(I) Find the trajectory equation of point $P$;
(II) A line passing through point $N(1,0)$ intersects curve $C$ at points $A$ and $B$, and intersects the $y$-axis at point $H$. If $\overset{→}{HA} = λ_1\overset{→}{AN}$ and $\overset{→}{HB} = λ_2\overset{→}{BN}$, determine whether $λ_1 + λ_2$ is a constant value. If it is, find this value; if not, explain the reason. | {
"answer": "-\\frac{8}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given two points $A(-2, 0)$ and $B(0, 2)$, point $C$ is any point on the circle $x^2 + y^2 - 2x = 0$, the minimum value of the area of $\triangle ABC$ is \_\_\_\_\_\_. | {
"answer": "3 - \\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the minimum value of
\[\sqrt{x^2 + (2 - x)^2} + \sqrt{(2 - x)^2 + (2 + x)^2}\]over all real numbers $x.$ | {
"answer": "2\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $ABCDE$ be a regular pentagon. The star $ACEBD$ has area 1. $AC$ and $BE$ meet at $P$ , while $BD$ and $CE$ meet at $Q$ . Find the area of $APQD$ . | {
"answer": "1/2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given vectors $\overrightarrow{m}=(\sin x, -1)$ and $\overrightarrow{n}=(\sqrt{3}\cos x, -\frac{1}{2})$, let $f(x)=(\overrightarrow{m}+\overrightarrow{n})\cdot \overrightarrow{m}$.
(1) Find the analytic expression for $f(x)$ and its intervals of monotonic increase;
(2) Given that $a$, $b$, and $c$ are the sides opposite angles $A$, $B$, and $C$ in triangle $\triangle ABC$, respectively, and $A$ is an acute angle with $a=2\sqrt{3}$ and $c=4$. If $f(A)$ is the maximum value of $f(x)$ on the interval $[0, \frac{\pi}{2}]$, find $A$, $b$, and the area $S$ of $\triangle ABC$. | {
"answer": "2\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the number of positive integers less than or equal to $1200$ that are neither $5$-nice nor $6$-nice. | {
"answer": "800",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $|x+2|+|1-x|=9-|y-5|-|1+y|$, find the maximum and minimum values of $x+y$. | {
"answer": "-3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the smallest positive integer that has eight positive odd integer divisors and sixteen positive even integer divisors? | {
"answer": "3000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Petya and Vasya are playing the following game. Petya thinks of a natural number \( x \) with a digit sum of 2012. On each turn, Vasya chooses any natural number \( a \) and finds out the digit sum of the number \( |x-a| \) from Petya. What is the minimum number of turns Vasya needs to determine \( x \) with certainty? | {
"answer": "2012",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the area of the parallelogram formed by the vectors $\begin{pmatrix} 4 \\ 2 \\ -3 \end{pmatrix}$ and $\begin{pmatrix} 2 \\ -4 \\ 5 \end{pmatrix}$. | {
"answer": "6\\sqrt{30}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider the function $g(x) = \frac{x^2}{2} + 2x - 1$. Determine the sum of all distinct numbers $x$ such that $g(g(g(x))) = 1$. | {
"answer": "-4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=2\sin (\pi-x)\cos x$.
- (I) Find the smallest positive period of $f(x)$;
- (II) Find the maximum and minimum values of $f(x)$ in the interval $\left[- \frac {\pi}{6}, \frac {\pi}{2}\right]$. | {
"answer": "- \\frac{ \\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=(2-a)(x-1)-2\ln x$ $(a\in \mathbb{R})$.
(Ⅰ) If the tangent line at the point $(1,g(1))$ on the curve $g(x)=f(x)+x$ passes through the point $(0,2)$, find the interval where the function $g(x)$ is decreasing;
(Ⅱ) If the function $y=f(x)$ has no zeros in the interval $\left(0, \frac{1}{2}\right)$, find the minimum value of $a$. | {
"answer": "2-4\\ln 2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $m$ is a positive integer, and given that $\mathop{\text{lcm}}[40,m]=120$ and $\mathop{\text{lcm}}[m,45]=180$, what is $m$? | {
"answer": "60",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \(a\), \(b\), \(c\), and \(d\) be distinct positive integers such that \(a+b\), \(a+c\), and \(a+d\) are all odd and are all squares. Let \(L\) be the least possible value of \(a + b + c + d\). What is the value of \(10L\)? | {
"answer": "670",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
28 apples weigh 3 kilograms. If they are evenly divided into 7 portions, each portion accounts for $\boxed{\frac{1}{7}}$ of all the apples, and each portion weighs $\boxed{\frac{3}{7}}$ kilograms. | {
"answer": "\\frac{3}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A rectangle has dimensions $4$ and $2\sqrt{3}$. Two equilateral triangles are contained within this rectangle, each with one side coinciding with the longer side of the rectangle. The triangles intersect, forming another polygon. What is the area of this polygon?
A) $2\sqrt{3}$
B) $4\sqrt{3}$
C) $6$
D) $8\sqrt{3}$ | {
"answer": "4\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a large circle with a radius of 11 and small circles with a radius of 1, determine the maximum number of small circles that can be placed inside the large circle, such that each small circle is internally tangent to the large circle and the small circles do not overlap. | {
"answer": "31",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Among the following functions, identify which pairs represent the same function.
1. $f(x) = |x|, g(x) = \sqrt{x^2}$;
2. $f(x) = \sqrt{x^2}, g(x) = (\sqrt{x})^2$;
3. $f(x) = \frac{x^2 - 1}{x - 1}, g(x) = x + 1$;
4. $f(x) = \sqrt{x + 1} \cdot \sqrt{x - 1}, g(x) = \sqrt{x^2 - 1}$. | {
"answer": "(1)",
"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"
} |
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