problem stringlengths 10 5.15k | answer dict |
|---|---|
An infinite geometric series has a first term of $15$ and a second term of $5$. A second infinite geometric series has the same first term of $15$, a second term of $5+n$, and a sum of three times that of the first series. Find the value of $n$. | {
"answer": "6.67",
"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 vectors $\overrightarrow{m} = (2\sin B, -\sqrt{3})$ and $\overrightarrow{n} = (\cos 2B, 2\cos^2 B - 1)$ and $\overrightarrow{m} \parallel \overrightarrow{n}$:
(1) Find the measure of acute angle $B$;
(2) If $b = 2$, find the maximum value of the area $S_{\triangle ABC}$ of triangle $ABC$. | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The vertex of the parabola $y^2 = 4x$ is $O$, and the coordinates of point $A$ are $(5, 0)$. A line $l$ with an inclination angle of $\frac{\pi}{4}$ intersects the line segment $OA$ (but does not pass through points $O$ and $A$) and intersects the parabola at points $M$ and $N$. The maximum area of $\triangle AMN$ is __________. | {
"answer": "8\\sqrt{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 foci $F_{1}$ and $F_{2}$, point $A$ lies on $C$, point $B$ lies on the $y$-axis, and satisfies $\overrightarrow{A{F}_{1}}⊥\overrightarrow{B{F}_{1}}$, $\overrightarrow{A{F}_{2}}=\frac{2}{3}\overrightarrow{{F}_{2}B}$. What is the eccentricity of $C$? | {
"answer": "\\frac{\\sqrt{5}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given real numbers $x$ and $y$ satisfying $x^{2}+y^{2}-4x-2y-4=0$, calculate the maximum value of $x-y$. | {
"answer": "1+3\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $ABC$, with $BC=15$, $AC=10$, and $\angle A=60^\circ$, find $\cos B$. | {
"answer": "\\frac{\\sqrt{6}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If a positive four-digit number's thousand digit \\(a\\), hundred digit \\(b\\), ten digit \\(c\\), and unit digit \\(d\\) satisfy the relation \\((a-b)(c-d) < 0\\), then it is called a "Rainbow Four-Digit Number", for example, \\(2012\\) is a "Rainbow Four-Digit Number". How many "Rainbow Four-Digit Numbers" are there among the positive four-digit numbers? (Answer with a number directly) | {
"answer": "3645",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the value of the expression
\[\log_3 (64 + \log_3 (64 + \log_3 (64 + \cdots))),\]
assuming it is positive. | {
"answer": "3.8",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=\sin (2x+ \frac {\pi}{6})+\cos 2x$.
(I) Find the interval of monotonic increase for the function $f(x)$;
(II) In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are respectively $a$, $b$, and $c$. Given that $f(A)= \frac { \sqrt {3}}{2}$, $a=2$, and $B= \frac {\pi}{3}$, find the area of $\triangle ABC$. | {
"answer": "\\frac {3+ \\sqrt {3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the two foci of a hyperbola are respectively $F_1(-3,0)$ and $F_2(3,0)$, and one asymptotic line is $y=\sqrt{2}x$, calculate the length of the chord that passes through the foci and is perpendicular to the $x$-axis. | {
"answer": "4\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=2\sin \omega x\cos \omega x-2\sqrt{3}\cos^{2}\omega x+\sqrt{3}$ ($\omega > 0$), and the distance between two adjacent axes of symmetry of the graph of $y=f(x)$ is $\frac{\pi}{2}$.
(Ⅰ) Find the interval of monotonic increase for the function $f(x)$;
(Ⅱ) Given that 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 the function $f(x)=ax^{2}+bx+c(a\\neq 0)$, its graph intersects with line $l$ at two points $A(t,t^{3}-t)$, $B(2t^{2}+3t,t^{3}+t^{2})$, where $t\\neq 0$ and $t\\neq -1$. Find the value of $f{{'}}(t^{2}+2t)$. | {
"answer": "\\dfrac {1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Triangle $DEF$ has side lengths $DE = 15$, $EF = 39$, and $FD = 36$. Rectangle $WXYZ$ has vertex $W$ on $\overline{DE}$, vertex $X$ on $\overline{DF}$, and vertices $Y$ and $Z$ on $\overline{EF}$. In terms of the side length $WX = \theta$, the area of $WXYZ$ can be expressed as the quadratic polynomial \[Area(WXYZ) = \gamma \theta - \delta \theta^2.\]
Then the coefficient $\delta = \frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$. | {
"answer": "229",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Mrs. Everett recorded the performance of her students in a chemistry test. However, due to a data entry error, 5 students who scored 60% were mistakenly recorded as scoring 70%. Below is the corrected table after readjusting these students. Using the data, calculate the average percent score for these $150$ students.
\begin{tabular}{|c|c|}
\multicolumn{2}{c}{}\\\hline
\textbf{$\%$ Score}&\textbf{Number of Students}\\\hline
100&10\\\hline
95&20\\\hline
85&40\\\hline
70&40\\\hline
60&20\\\hline
55&10\\\hline
45&10\\\hline
\end{tabular} | {
"answer": "75.33",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given Lucy starts with an initial term of 8 in her sequence, where each subsequent term is generated by either doubling the previous term and subtracting 2 if a coin lands on heads, or halving the previous term and subtracting 2 if a coin lands on tails, determine the probability that the fourth term in Lucy's sequence is an integer. | {
"answer": "\\frac{3}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the sequence $\{a\_n\}$ that satisfies $a\_n-(-1)^{n}a\_{n-1}=n$ $(n\geqslant 2)$, and $S\_n$ is the sum of the first $n$ terms of the sequence, find the value of $S\_{40}$. | {
"answer": "440",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A mineralogist is hosting a competition to guess the age of an ancient mineral sample. The age is provided by the digits 2, 2, 3, 3, 5, and 9, with the condition that the age must start with an odd number. | {
"answer": "120",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There exist constants $b_1,$ $b_2,$ $b_3,$ $b_4$ such that
\[\sin^4 \theta = b_1 \sin \theta + b_2 \sin 2 \theta + b_3 \sin 3 \theta + b_4 \sin 4 \theta\]for all angles $\theta.$ Find $b_1^2 + b_2^2 + b_3^2 + b_4^2.$ | {
"answer": "\\frac{17}{64}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let a line passing through the origin \\(O\\) intersect a circle \\((x-4)^{2}+y^{2}=16\\) at point \\(P\\), and let \\(M\\) be the midpoint of segment \\(OP\\). Establish a polar coordinate system with the origin \\(O\\) as the pole and the positive half-axis of \\(x\\) as the polar axis.
\\((\\)Ⅰ\\()\\) Find the polar equation of the trajectory \\(C\\) of point \\(M\\);
\\((\\)Ⅱ\\()\\) Let the polar coordinates of point \\(A\\) be \\((3, \dfrac {π}{3})\\), and point \\(B\\) lies on curve \\(C\\). Find the maximum area of \\(\\triangle OAB\\). | {
"answer": "3+ \\dfrac {3}{2} \\sqrt {3}",
"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 foci $F_{1}$ and $F_{2}$, point $A$ lies on $C$, point $B$ lies on the $y$-axis, and satisfies $\overrightarrow{AF_{1}}⊥\overrightarrow{BF_{1}}$, $\overrightarrow{AF_{2}}=\frac{2}{3}\overrightarrow{F_{2}B}$. Determine the eccentricity of $C$. | {
"answer": "\\frac{\\sqrt{5}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system $(xOy)$, the parametric equation of line $l$ is given by $\begin{cases}x=1+t\cos a \\ y= \sqrt{3}+t\sin a\end{cases} (t\text{ is a parameter})$, where $0\leqslant α < π$. In the polar coordinate system with $O$ as the pole and the positive half of the $x$-axis as the polar axis, the curve $C_{1}$ is defined by $ρ=4\cos θ$. The line $l$ is tangent to the curve $C_{1}$.
(1) Convert the polar coordinate equation of curve $C_{1}$ to Cartesian coordinates and determine the value of $α$;
(2) Given point $Q(2,0)$, line $l$ intersects with curve $C_{2}$: $x^{2}+\frac{{{y}^{2}}}{3}=1$ at points $A$ and $B$. Calculate the area of triangle $ABQ$. | {
"answer": "\\frac{6 \\sqrt{2}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two circles have radius $2$ and $3$ , and the distance between their centers is $10$ . Let $E$ be the intersection of their two common external tangents, and $I$ be the intersection of their two common internal tangents. Compute $EI$ .
(A *common external tangent* is a tangent line to two circles such that the circles are on the same side of the line, while a *common internal tangent* is a tangent line to two circles such that the circles are on opposite sides of the line).
*Proposed by Connor Gordon)* | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given vectors $\overrightarrow{m}=(\sqrt{3}\cos x,-\cos x)$ and $\overrightarrow{n}=(\cos (x-\frac{π}{2}),\cos x)$, satisfying the function $f\left(x\right)=\overrightarrow{m}\cdot \overrightarrow{n}+\frac{1}{2}$.
$(1)$ Find the interval on which $f\left(x\right)$ is monotonically increasing on $[0,\frac{π}{2}]$.
$(2)$ If $f\left(\alpha \right)=\frac{5}{13}$, where $\alpha \in [0,\frac{π}{4}]$, find the value of $\cos 2\alpha$. | {
"answer": "\\frac{12\\sqrt{3}-5}{26}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $f(x)$ be a differentiable function, and $\lim_{\Delta x \to 0} \frac{{f(1)-f(1-2\Delta x)}}{{\Delta x}}=-1$, determine the slope of the tangent line to the curve $y=f(x)$ at the point $(1,f(1))$. | {
"answer": "-\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the function $f(x)$ satisfies: $4f(x)f(y)=f(x+y)+f(x-y)$ $(x,y∈R)$ and $f(1)= \frac{1}{4}$, find $f(2014)$. | {
"answer": "- \\frac{1}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( M = 35 \cdot 36 \cdot 65 \cdot 280 \). Calculate the ratio of the sum of the odd divisors of \( M \) to the sum of the even divisors of \( M \). | {
"answer": "1:62",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let the function $f(x)=(x+a)\ln x$, $g(x)= \frac {x^{2}}{e^{x}}$, it is known that the tangent line of the curve $y=f(x)$ at the point $(1,f(1))$ is parallel to the line $2x-y=0$.
(Ⅰ) If the equation $f(x)=g(x)$ has a unique root within $(k,k+1)$ $(k\in\mathbb{N})$, find the value of $k$.
(Ⅱ) Let the function $m(x)=\min\{f(x),g(x)\}$ (where $\min\{p,q\}$ represents the smaller value between $p$ and $q$), find the maximum value of $m(x)$. | {
"answer": "\\frac {4}{e^{2}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $a>0$, the minimum value of the function $f(x) = e^{x-a} - \ln(x+a) - 1$ $(x>0)$ is 0. Determine the range of values for the real number $a$. | {
"answer": "\\{\\frac{1}{2}\\}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $\triangle ABC$, it is known that the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, $c$, with $a=\sqrt{6}$, $\sin ^{2}B+\sin ^{2}C=\sin ^{2}A+\frac{2\sqrt{3}}{3}\sin A\sin B\sin C$. Choose one of the following conditions to determine whether triangle $\triangle ABC$ exists. If it exists, find the area of $\triangle ABC$; if it does not exist, please explain the reason.
① The length of the median $AD$ of side $BC$ is $\frac{\sqrt{10}}{2}$
② $b+c=2\sqrt{3}$
③ $\cos B=-\frac{3}{5}$. | {
"answer": "\\frac{\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A certain item has a cost price of $4$ yuan and is sold at a price of $5$ yuan. The merchant is planning to offer a discount on the selling price, but the profit margin must not be less than $10\%$. Find the maximum discount rate that can be offered. | {
"answer": "12\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The minimum value of \\(f(x)=\sin x+\cos x-\sin x\cos x\\) is | {
"answer": "- \\frac{1}{2}- \\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Among 7 students, 6 are to be arranged to participate in social practice activities in two communities on Saturday, with 3 people in each community. How many different arrangements are there? (Answer with a number) | {
"answer": "140",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the actual lighthouse's cylindrical base is 60 meters high, and the spherical top's volume is approximately 150,000 liters, and the miniature model's top holds around 0.15 liters, determine the height of Lara’s model lighthouse, in centimeters. | {
"answer": "60",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many numbers are in the list $-58, -51, -44, \ldots, 71, 78$? | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Regarding the value of \\(\pi\\), the history of mathematics has seen many creative methods for its estimation, such as the famous Buffon's Needle experiment and the Charles' experiment. Inspired by these, we can also estimate the value of \\(\pi\\) through designing the following experiment: ask \\(200\\) students, each to randomly write down a pair of positive real numbers \\((x,y)\\) both less than \\(1\\); then count the number of pairs \\((x,y)\\) that can form an obtuse triangle with \\(1\\) as the third side, denoted as \\(m\\); finally, estimate the value of \\(\pi\\) based on the count \\(m\\). If the result is \\(m=56\\), then \\(\pi\\) can be estimated as \_\_\_\_\_\_ (expressed as a fraction). | {
"answer": "\\dfrac {78}{25}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the smallest integer $k$ such that $k>1$ and $k$ has a remainder of $3$ when divided by any of $11,$ $4,$ and $3.$ | {
"answer": "135",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A copper cube with an edge length of $l = 5 \text{ cm}$ is heated to a temperature of $t_{1} = 100^{\circ} \text{C}$. Then, it is placed on ice, which has a temperature of $t_{2} = 0^{\circ} \text{C}$. Determine the maximum depth the cube can sink into the ice. The specific heat capacity of copper is $c_{\text{s}} = 400 \text{ J/(kg}\cdot { }^{\circ} \text{C})$, the latent heat of fusion of ice is $\lambda = 3.3 \times 10^{5} \text{ J/kg}$, the density of copper is $\rho_{m} = 8900 \text{ kg/m}^3$, and the density of ice is $\rho_{n} = 900 \text{ kg/m}^3$. (10 points) | {
"answer": "0.06",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two cylindrical poles, with diameters of $10$ inches and $30$ inches respectively, are placed side by side and bound together with a wire. Calculate the length of the shortest wire that will go around both poles.
**A)** $20\sqrt{3} + 24\pi$
**B)** $20\sqrt{3} + \frac{70\pi}{3}$
**C)** $30\sqrt{3} + 22\pi$
**D)** $16\sqrt{3} + 25\pi$
**E)** $18\sqrt{3} + \frac{60\pi}{3}$ | {
"answer": "20\\sqrt{3} + \\frac{70\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two cards are dealt at random from a standard deck of 52 cards. What is the probability that the first card is a Queen and the second card is a $\diamondsuit$? | {
"answer": "\\frac{52}{221}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $\triangle ABC$, $\angle BAC = \frac{π}{3}$, $D$ is the midpoint of $AB$, $P$ is a point on segment $CD$, and satisfies $\overrightarrow{AP} = t\overrightarrow{AC} + \frac{1}{3}\overrightarrow{AB}$. If $|\overrightarrow{BC}| = \sqrt{6}$, then the maximum value of $|\overrightarrow{AP}|$ is ______. | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $F_{1}$ and $F_{2}$ be the two foci of the hyperbola $\dfrac {x^{2}}{4}- \dfrac {y^{2}}{b^{2}}=1$. Point $P$ is on the hyperbola and satisfies $\angle F_{1}PF_{2}=90^{\circ}$. If the area of $\triangle F_{1}PF_{2}$ is $2$, find the value of $b$. | {
"answer": "\\sqrt {2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given real numbers $a$, $b$, $c$, and $d$ satisfy $(b + 2a^2 - 6\ln a)^2 + |2c - d + 6| = 0$, find the minimum value of $(a - c)^2 + (b - d)^2$. | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the list where each integer $n$ appears $n$ times for $1 \leq n \leq 300$, find the median of the numbers. | {
"answer": "212",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$x_{n+1}= \left ( 1+\frac2n \right )x_n+\frac4n$ , for every positive integer $n$ . If $x_1=-1$ , what is $x_{2000}$ ? | {
"answer": "2000998",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\Delta ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively. Given that $b=2$, $c=2\sqrt{2}$, and $C=\frac{\pi}{4}$, find the area of $\Delta ABC$. | {
"answer": "\\sqrt{3} +1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given 50 feet of fencing, where 5 feet is used for a gate that does not contribute to the enclosure area, what is the greatest possible number of square feet in the area of a rectangular pen enclosed by the remaining fencing? | {
"answer": "126.5625",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A conference hall is setting up seating for a workshop. Each row must contain $13$ chairs, and initially, there are $169$ chairs in total. The organizers expect $95$ participants to attend the workshop. To ensure all rows are completely filled with minimal empty seats, how many chairs should be removed? | {
"answer": "65",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For the inequality system about $y$ $\left\{\begin{array}{l}{2y-6≤3(y-1)}\\{\frac{1}{2}a-3y＞0}\end{array}\right.$, if it has exactly $4$ integer solutions, then the product of all integer values of $a$ that satisfy the conditions is ______. | {
"answer": "720",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
(1) Given $\sin \left( \frac{\pi }{3}-\alpha \right)=\frac{1}{2}$, find the value of $\cos \left( \frac{\pi }{6}+\alpha \right)$;
(2) Given $\cos \left( \frac{5\pi }{12}+\alpha \right)=\frac{1}{3}$ and $-\pi < \alpha < -\frac{\pi }{2}$, find the value of $\cos \left( \frac{7\pi }{12}-\alpha \right)+\sin \left( \alpha -\frac{7\pi }{12} \right)$: | {
"answer": "- \\frac{1}{3}+ \\frac{2 \\sqrt{2}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that point $(a, b)$ moves on the line $x + 2y + 3 = 0$, find the maximum or minimum value of $2^a + 4^b$. | {
"answer": "\\frac{\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system $xOy$, the parametric equation of curve $C$ is $\begin{cases} & x=3\cos \alpha \\ & y=\sin \alpha \end{cases}$ ($\alpha$ is the parameter), and in the polar coordinate system with the origin as the pole and the positive $x$-axis as the polar axis, the polar equation of line $l$ is $\rho \sin (\theta -\dfrac{\pi }{4})=\sqrt{2}$.
$(1)$ Find the general equation of curve $C$ and the inclination angle of line $l$;
$(2)$ Let point $P(0,2)$, line $l$ intersects curve $C$ at points $A$ and $B$, find $|PA|+|PB|$． | {
"answer": "\\dfrac {18 \\sqrt {2}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given plane vectors $\overrightarrow{a}$, $\overrightarrow{b}$, $\overrightarrow{c}$ satisfy $\langle \overrightarrow{a}, \overrightarrow{b} \rangle = 60^\circ$, and $\{|\overrightarrow{a}|, |\overrightarrow{b}|, |\overrightarrow{c}|\} = \{1, 2, 3\}$, calculate the maximum value of $|\overrightarrow{a}+ \overrightarrow{b}+ \overrightarrow{c}|$. | {
"answer": "\\sqrt{7}+3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The new PUMaC tournament hosts $2020$ students, numbered by the following set of labels $1, 2, . . . , 2020$ . The students are initially divided up into $20$ groups of $101$ , with each division into groups equally likely. In each of the groups, the contestant with the lowest label wins, and the winners advance to the second round. Out of these $20$ students, we chose the champion uniformly at random. If the expected value of champion’s number can be written as $\frac{a}{b}$ , where $a, b$ are relatively prime integers, determine $a + b$ . | {
"answer": "2123",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, $2\sin 2A\cos A-\sin 3A+\sqrt{3}\cos A=\sqrt{3}$.
(1) Find the measure of angle $A$;
(2) Given that $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively, if $a=1$ and $\sin A+\sin (B-C)=2\sin 2C$, find the area of $\triangle ABC$. | {
"answer": "\\frac{\\sqrt{3}}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $α∈(0,π)$, and $\sin α + \cos α = \frac{\sqrt{2}}{2}$, find the value of $\sin α - \cos α$. | {
"answer": "\\frac{\\sqrt{6}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There is a solid iron cone with a base radius of $3cm$ and a slant height of $5cm$. After melting it at high temperature and casting it into a solid iron sphere (without considering any loss), the radius of this iron sphere is _______ $cm$. | {
"answer": "\\sqrt[3]{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If a worker receives a 30% cut in wages, calculate the percentage raise he needs to regain his original pay. | {
"answer": "42.857\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=\frac{1}{3}x^{3}+ax^{2}+bx$ $(a,b\in \mathbb{R})$ attains a maximum value of $9$ at $x=-3$.
$(I)$ Find the values of $a$ and $b$;
$(II)$ Find the maximum and minimum values of the function $f(x)$ in the interval $[-3,3]$. | {
"answer": "- \\frac{5}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A triangle is divided into 1000 triangles. What is the maximum number of distinct points that can be vertices of these triangles? | {
"answer": "1002",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Define the determinant $D_1$ = $|1|$ , the determinant $D_2$ = $|1 1|$ $|1 3|$
, and the determinant $D_3=$
|1 1 1|
|1 3 3|
|1 3 5|
.
In general, for positive integer n, let the determinant $D_n$ have 1s in every position of its first row and first
column, 3s in the remaining positions of the second row and second column, 5s in the remaining positions of the third row and third column, and so forth. Find the least n so that $D_n$ $\geq$ 2015. | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate: $|1-\sqrt{2}|+(\frac{1}{2})^{-2}-\left(\pi -2023\right)^{0}$. | {
"answer": "\\sqrt{2} + 2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system $xOy$, a moving line $l$: $y=x+m$ intersects the parabola $C$: $x^2=2py$ ($p>0$) at points $A$ and $B$, and $\overrightarrow {OA}\cdot \overrightarrow {OB}=m^{2}-2m$.
1. Find the equation of the parabola $C$.
2. Let $P$ be the point where the line $y=x$ intersects $C$ (and $P$ is different from the origin), and let $D$ be the intersection of the tangent line to $C$ at $P$ and the line $l$. Define $t= \frac {|PD|^{2}}{|DA|\cdot |DB|}$. Is $t$ a constant value? If so, compute its value; otherwise, explain why it's not constant. | {
"answer": "\\frac{5}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The diagram shows a large triangle divided into squares and triangles. Let \( S \) be the number of squares of any size in the diagram and \( T \) be the number of triangles of any size in the diagram. What is the value of \( S \times T \)?
A) 30
B) 35
C) 48
D) 70
E) 100 | {
"answer": "70",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the rectangular coordinate system $(xOy)$, there is a circle $C_{1}$: $(x+ \sqrt {3})^{2}+y^{2}=4$, and a curve $C_{2}$ with parametric equations $\begin{cases}x=2+2\cos θ \\\\ y=2\sin θ\end{cases} (θ \text{ is the parameter})$. Establish a polar coordinate system with $O$ as the pole and the positive $x$-axis as the polar axis.
1. Write the polar equation of $C_{1}$ and convert $C_{2}$ to a Cartesian equation.
2. If the polar equation of the line $C_{3}$ is $θ= \dfrac {π}{3}(ρ∈R)$, and $C_{2}$ intersects with $C_{3}$ at points $A$ and $B$, find the area of $△ABC_{1}$ ($C_{1}$ is the center of circle $C_{1}$). | {
"answer": "\\dfrac {3}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the 6-digit repetend in the decimal representation of $\frac{7}{29}$. | {
"answer": "241379",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are $2017$ distinct points in the plane. For each pair of these points, construct the midpoint of the segment joining the pair of points. What is the minimum number of distinct midpoints among all possible ways of placing the points? | {
"answer": "2016",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle \( \triangle ABC \), point \( E \) is on side \( AB \) with \( AE = 1 \) and \( EB = 2 \). Suppose points \( D \) and \( F \) are on sides \( AC \) and \( BC \) respectively, and \( DE \parallel BC \) and \( EF \parallel AC \). What is the ratio of the area of quadrilateral \( CDEF \) to the area of triangle \( \triangle ABC \)? | {
"answer": "4: 9",
"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 a vertex coordinate at $B(0,1)$, and the eccentricity of the ellipse being $\frac{\sqrt{3}}{2}$,
$(I)$ find the equation of the ellipse;
$(II)$ let point $Q$ be a point on ellipse $C$ below the x-axis, and $F_1$, $F_2$ are the left and right foci of the ellipse, respectively. If the slope of line $QF_1$ is $\frac{\pi}{6}$, find the area of $\triangle QF_1F_2$. | {
"answer": "\\frac{\\sqrt{3}}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Luke wants to fence a rectangular piece of land with an area of at least 450 square feet. The length of the land is 1.5 times the width. What should the width of the rectangle be if he wants to use the least amount of fencing? | {
"answer": "10\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $x, y, a \in \mathbb{R}^*$, and when $x + 2y = 1$, the minimum value of $\frac{3}{x} + \frac{a}{y}$ is $6\sqrt{3}$. Determine the minimum value of $3x + ay$ when $\frac{1}{x} + \frac{2}{y} = 1$. | {
"answer": "6\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that \(7^{-1} \equiv 55 \pmod{101}\), find \(49^{-1} \pmod{101}\), as a residue modulo 101. (Answer should be between 0 and 100, inclusive.) | {
"answer": "96",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $$f(x)=\cos 2x + 2\sqrt{3}\sin x\cos x$$
(1) Find the range of the function $f(x)$ and write down the intervals where $f(x)$ is monotonically increasing;
(2) If $$0 < \theta < \frac{\pi}{6}$$ and $$f(\theta) = \frac{4}{3}$$, calculate the value of $\cos 2\theta$. | {
"answer": "\\frac{\\sqrt{15} + 2}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are five positive integers that are divisors of each number in the list $$60, 120, -30, 180, 240$$. Find the sum of these five positive integers. | {
"answer": "17",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
At a robot racing competition, a certain number of mechanisms were presented. The robots were paired to race the same distance. The protocol recorded the differences in the finishing times between the winner and the loser in each of the races. All the differences were distinct: 1 sec, 2 sec, 3 sec, 4 sec, 5 sec, 6 sec, 7 sec, 8 sec, 9 sec, 13 sec. It is known that during the races, each robot competed against each other robot exactly once, and each robot always ran at the same speed. Determine the time of the slowest mechanism if the best time to complete the distance was 50 seconds. | {
"answer": "63",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Rodney is trying to guess a secret number based on the following clues:
1. It is a two-digit integer.
2. The tens digit is odd.
3. The units digit is one of the following: 2, 4, 6, 8.
4. The number is greater than 50.
What is the probability that Rodney will guess the correct number? Express your answer as a common fraction. | {
"answer": "\\frac{1}{12}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the sequence $\{a_{n}\}$ is an arithmetic sequence, $a_{9}+a_{12} \lt 0$, $a_{10}\cdot a_{11} \lt 0$, and the sum of the first $n$ terms of the sequence $\{a_{n}\}$, denoted as $S_{n}$, has a maximum value, determine the maximum value of $n$ when $S_{n} \gt 0$. | {
"answer": "19",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For a geometric sequence $\{b_n\}$ with a common ratio of $4$, if $T_n$ is the product of the first $n$ terms of the sequence $\{b_n\}$, then $\frac{T_{20}}{T_{10}}$, $\frac{T_{30}}{T_{20}}$, $\frac{T_{40}}{T_{30}}$ also form a geometric sequence with a common ratio of $4^{100}$. Similarly, in an arithmetic sequence $\{a_n\}$ with a common difference of $3$, if $S_n$ is the sum of the first $n$ terms of $\{a_n\}$, then \_\_\_\_\_\_ also form an arithmetic sequence with a common difference of \_\_\_\_\_\_. | {
"answer": "300",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two distinct numbers a and b are chosen randomly from the set $\{3, 3^2, 3^3, ..., 3^{15}\}$. What is the probability that $\mathrm{log}_a b$ is an integer?
A) $\frac{1}{10}$
B) $\frac{2}{7}$
C) $\frac{1}{7}$
D) $\frac{1}{5}$ | {
"answer": "\\frac{2}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A cuboid has an integer volume. Three of the faces have different areas, namely $7, 27$ , and $L$ . What is the smallest possible integer value for $L$ ? | {
"answer": "21",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $\cos \alpha= \frac {1}{7}$, $\cos (\alpha+\beta)=- \frac {11}{14}$, and $\alpha\in(0, \frac {\pi}{2})$, $\alpha+\beta\in( \frac {\pi}{2},\pi)$, find the value of $\cos \beta$. | {
"answer": "\\frac {1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given two parallel lines \\(l_{1}\\) and \\(l_{2}\\) passing through points \\(P_{1}(1,0)\\) and \\(P_{2}(0,5)\\) respectively, and the distance between \\(l_{1}\\) and \\(l_{2}\\) is \\(5\\), then the slope of line \\(l_{1}\\) is \_\_\_\_\_\_. | {
"answer": "\\dfrac {5}{12}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a directed graph with $2013$ vertices, there is exactly one edge between any two vertices and for every vertex there exists an edge outwards this vertex. We know that whatever the arrangement of the edges, from every vertex we can reach $k$ vertices using at most two edges. Find the maximum value of $k$ . | {
"answer": "2012",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given vectors $\overrightarrow {a}$ and $\overrightarrow {b}$ that satisfy $| \overrightarrow {a} |=5$, $| \overrightarrow {b} |=3$, and $( \overrightarrow {a} - \overrightarrow {b} )(2 \overrightarrow {a} + 3 \overrightarrow {b} )=13$.
1. Find the cosine value of the angle between $\overrightarrow {a}$ and $\overrightarrow {b}$.
2. Find the magnitude of $| \overrightarrow {a} + 2 \overrightarrow {b} |$. | {
"answer": "\\sqrt{21}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A right pyramid has a square base where each side measures 15 cm. The height of the pyramid, measured from the center of the base to the peak, is 15 cm. Calculate the total length of all edges of the pyramid. | {
"answer": "60 + 4\\sqrt{337.5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
When the square root of \( x \) is cubed, the result is 100. What is the value of \( x \)? | {
"answer": "10^{\\frac{4}{3}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A river boat travels at a constant speed from point A to point B. Along the riverbank, there is a road. The boat captain observes that every 30 minutes, a bus overtakes the boat from behind, and every 10 minutes, a bus approaches from the opposite direction. Assuming that the buses depart from points A and B uniformly and travel at a constant speed, what is the interval time (in minutes) between each bus departure? | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A high school's 11th-grade class 1 has 45 male students and 15 female students. The teacher uses stratified sampling to form a 4-person extracurricular interest group. Calculate the probability of a student being selected for this group and the number of male and female students in the extracurricular interest group.
After a month of study and discussion, this interest group decides to select two students to conduct an experiment. The method is to first select one student from the group to conduct the experiment, and after that student has completed it, select another student from the remaining group members to conduct the experiment. Calculate the probability that among the two selected students, exactly one is a female student.
After the experiment, the first student obtained experimental data of 68, 70, 71, 72, 74. The second student obtained experimental data of 69, 70, 70, 72, 74. Which student's experiment is more stable? And explain why. | {
"answer": "0.5",
"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. Given that $a=3$, $b=\sqrt{3}$, and $A=\dfrac{\pi}{3}$, find the measure of angle $B$. | {
"answer": "\\frac{\\pi}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given positive numbers \(a,b\) satisfying \(2ab=\dfrac{2a-b}{2a+3b},\) then the maximum value of \(b\) is \_\_\_\_\_ | {
"answer": "\\dfrac{1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A $44$-gon $Q_1$ is constructed in the Cartesian plane, and the sum of the squares of the $x$-coordinates of the vertices equals $176$. The midpoints of the sides of $Q_1$ form another $44$-gon, $Q_2$. Finally, the midpoints of the sides of $Q_2$ form a third $44$-gon, $Q_3$. Find the sum of the squares of the $x$-coordinates of the vertices of $Q_3$. | {
"answer": "44",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the smallest solution to the equation \[\lfloor x^2 \rfloor - \lfloor x \rfloor^2 = 19.\] | {
"answer": "\\sqrt{119}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Four steel balls, each with a radius of 1, are completely packed into a container in the shape of a regular tetrahedron. The minimum height of this regular tetrahedron is: | {
"answer": "2 + \\frac{2 \\sqrt{6}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Integers $1, 2, 3, ... ,n$ , where $n > 2$ , are written on a board. Two numbers $m, k$ such that $1 < m < n, 1 < k < n$ are removed and the average of the remaining numbers is found to be $17$ . What is the maximum sum of the two removed numbers? | {
"answer": "51",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the result when $\frac{2}{3}$ is subtracted from $2\frac{1}{4}$? | {
"answer": "1\\frac{7}{12}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A number from the set $\{30, 31, 32, \ldots, 500\}$ is randomly selected. What is the probability that the number is greater than 100 but less than or equal to 200? Express your answer as a common fraction. | {
"answer": "\\frac{100}{471}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $ABC$, $AB=2$, $AC=5$, and $\cos A= \frac {4}{5}$. A point $P$ is chosen randomly inside triangle $ABC$. The probability that the area of triangle $PAB$ is greater than $1$ and less than or equal to $2$ is __________. | {
"answer": "\\frac {1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If a worker receives a $20$% cut in wages, calculate the exact percentage or dollar amount of a raise needed for the worker to regain their original pay. | {
"answer": "25\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \\(\alpha\\) be an acute angle. If \\(\cos (\alpha+ \dfrac {\pi}{6})= \dfrac {3}{5}\\), then \\(\sin (\alpha- \dfrac {\pi}{6})=\\) \_\_\_\_\_\_. | {
"answer": "\\dfrac {4-3 \\sqrt {3}}{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
To protect the ecological environment, a mountainous area in our city has started to implement the policy of returning farmland to forest since 2005. It is known that at the end of 2004, the forest coverage area of this mountainous area was $a$ acres.
(1) Assuming the annual natural growth rate of forest coverage area after returning farmland to forest is 2%, write the function relationship between the forest coverage area $y$ (in acres) and the number of years $x$ (in years) since the implementation of returning farmland to forest, and calculate the forest coverage area of this mountainous area at the end of 2009.
(2) If by the end of 2014, the forest coverage area of this mountainous area needs to be at least twice that at the end of 2004, additional artificial greening projects must be implemented. What is the minimum annual average growth rate of the forest coverage area required to meet this goal by the end of 2014?
(Reference data: $1.02^{4}=1.082$, $1.02^{5}=1.104$, $1.02^{6}=1.126$, $\lg2=0.301$, $\lg1.072=0.0301$) | {
"answer": "7.2\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the set of numbers 1, 2, 3, 4, 5, select an even number a and an odd number b to form a vector $\overrightarrow{a} = (a, b)$ with the origin as the starting point. From all the vectors obtained with the origin as the starting point, select any two vectors as adjacent sides to form a parallelogram. Let the total number of parallelograms formed be n, and among them, let the number of parallelograms with an area not exceeding 4 be m. Calculate the value of $\frac{m}{n}$. | {
"answer": "\\frac{1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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