problem stringlengths 10 5.15k | answer dict |
|---|---|
Chelsea goes to La Verde's at MIT and buys 100 coconuts, each weighing 4 pounds, and 100 honeydews, each weighing 5 pounds. She wants to distribute them among \( n \) bags, so that each bag contains at most 13 pounds of fruit. What is the minimum \( n \) for which this is possible? | {
"answer": "75",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that in triangle ABC, the lengths of the sides opposite to angles A, B, and C are a, b, and c respectively, and b = 3, c = 1, and A = 2B.
(1) Find the value of a;
(2) Find the value of sin(A + $\frac{\pi}{4}$). | {
"answer": "\\frac{4 - \\sqrt{2}}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Triangle $DEF$ has side lengths $DE = 15$, $EF = 36$, and $FD = 39$. 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 $WX = \epsilon$, the area of $WXYZ$ can be expressed as the quadratic polynomial \[Area(WXYZ) = \gamma \epsilon - \delta \epsilon^2.\]
Determine the coefficient $\delta = \frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$. | {
"answer": "17",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A square has a side length of $40\sqrt{3}$ cm. Calculate the length of the diagonal of the square and the area of a circle inscribed within it. | {
"answer": "1200\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a complex number $z=3+bi\left(b=R\right)$, and $\left(1+3i\right)\cdot z$ is an imaginary number.<br/>$(1)$ Find the complex number $z$;<br/>$(2)$ If $ω=\frac{z}{{2+i}}$, find the complex number $\omega$ and its modulus $|\omega|$. | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $B = (20, 14)$ and $C = (18, 0)$ be two points in the plane. For every line $\ell$ passing through $B$ , we color red the foot of the perpendicular from $C$ to $\ell$ . The set of red points enclose a bounded region of area $\mathcal{A}$ . Find $\lfloor \mathcal{A} \rfloor$ (that is, find the greatest integer not exceeding $\mathcal A$ ).
*Proposed by Yang Liu* | {
"answer": "157",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the sum of the last three digits of each term in the following part of the Fibonacci Factorial Series: $1!+2!+3!+5!+8!+13!+21!$? | {
"answer": "249",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given vectors $\overrightarrow{a}=(\sqrt{2}\cos \omega x,1)$ and $\overrightarrow{b}=(2\sin (\omega x+ \frac{\pi}{4}),-1)$ where $\frac{1}{4}\leqslant \omega\leqslant \frac{3}{2}$, and the function $f(x)= \overrightarrow{a}\cdot \overrightarrow{b}$, and the graph of $f(x)$ has an axis of symmetry at $x= \frac{5\pi}{8}$.
$(1)$ Find the value of $f( \frac{3}{4}\pi)$;
$(2)$ If $f( \frac{\alpha}{2}- \frac{\pi}{8})= \frac{\sqrt{2}}{3}$ and $f( \frac{\beta}{2}- \frac{\pi}{8})= \frac{2\sqrt{2}}{3}$, and $\alpha,\beta\in(-\frac{\pi}{2}, \frac{\pi}{2})$, find the value of $\cos (\alpha-\beta)$. | {
"answer": "\\frac{2\\sqrt{10}+2}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that there are 25 cities in the County of Maplewood, and the average population per city lies between $6,200$ and $6,800$, estimate the total population of all the cities in the County of Maplewood. | {
"answer": "162,500",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For real numbers $a$ and $b$ , define $$ f(a,b) = \sqrt{a^2+b^2+26a+86b+2018}. $$ Find the smallest possible value of the expression $$ f(a, b) + f (a,-b) + f(-a, b) + f (-a, -b). $$ | {
"answer": "4 \\sqrt{2018}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle ABC, where AB = 24 and BC = 18, find the largest possible value of $\tan A$. | {
"answer": "\\frac{3\\sqrt{7}}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that a certain product requires $6$ processing steps, where $2$ of these steps must be consecutive and another $2$ steps cannot be consecutive, calculate the number of possible processing sequences. | {
"answer": "144",
"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, with $c=2$ and $A \neq B$.
1. Find the value of $\frac{a \sin A - b \sin B}{\sin (A-B)}$.
2. If the area of $\triangle ABC$ is $1$ and $\tan C = 2$, find the value of $a+b$. | {
"answer": "\\sqrt{5} + 1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Take a standard set of dominoes and remove all duplicates and blanks. Then consider the remaining 15 dominoes as fractions. The dominoes are arranged in such a way that the sum of all fractions in each row equals 10. However, unlike proper fractions, you are allowed to use as many improper fractions (such as \(\frac{4}{3}, \frac{5}{2}, \frac{6}{1}\)) as you wish, as long as the sum in each row equals 10. | {
"answer": "10",
"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 $b=3$, $c=2\sqrt{3}$, and $A=30^{\circ}$, find the values of angles $B$, $C$, and side $a$. | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Reading material: After studying square roots, Kang Kang found that some expressions containing square roots can be written as the square of another expression, such as $3+2\sqrt{2}=({1+\sqrt{2}})^2$. With his good thinking skills, Kang Kang made the following exploration: Let $a+b\sqrt{2}=({m+n\sqrt{2}})^2$ (where $a$, $b$, $m$, $n$ are all positive integers), then $a+b\sqrt{2}=m^2+2n^2+2mn\sqrt{2}$ (rational and irrational numbers correspondingly equal), therefore $a=m^{2}+2n^{2}$, $b=2mn$. In this way, Kang Kang found a method to transform the expression $a+b\sqrt{2}$ into a square form. Please follow Kang Kang's method to explore and solve the following problems:
$(1)$ When $a$, $b$, $m$, $n$ are all positive integers, if $a+b\sqrt{3}=({c+d\sqrt{3}})^2$, express $a$ and $b$ in terms of $c$ and $d$: $a=$______, $b=$______;
$(2)$ If $7-4\sqrt{3}=({e-f\sqrt{3}})^2$, and $e$, $f$ are both positive integers, simplify $7-4\sqrt{3}$;
$(3)$ Simplify: $\sqrt{7+\sqrt{21-\sqrt{80}}}$. | {
"answer": "1+\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the hexagon $ABCDEF$ with vertices $A, B, C, D, E, F$, all internal triangles dividing the hexagon are similar to isosceles triangle $ABC$, where $AB = AC$. Among these triangles, there are $10$ smallest triangles each with area $2$, and the area of triangle $ABC$ is $80$. Determine the area of the quadrilateral $DBCE$.
A) 60
B) 65
C) 70
D) 75
E) 80 | {
"answer": "70",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A sphere is inscribed in a right circular cylinder. The height of the cylinder is 12 inches, and the diameter of its base is 10 inches. Find the volume of the inscribed sphere. Express your answer in terms of $\pi$. | {
"answer": "\\frac{500}{3} \\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Any six points are taken inside or on a rectangle with dimensions $2 \times 1$. Let $b$ be the smallest possible number with the property that it is always possible to select one pair of points from these six such that the distance between them is equal to or less than $b$. Determine the value of $b$. | {
"answer": "\\frac{\\sqrt{5}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that \\(\alpha\\) and \\(\beta\\) are acute angles, and \\(\cos \alpha= \frac{\sqrt{5}}{5}\\), \\(\sin (\alpha+\beta)= \frac{3}{5}\\), find the value of \\(\cos \beta\\. | {
"answer": "\\frac{2\\sqrt{5}}{25}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many squares are shown in the drawing? | {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider a $5 \times 5$ grid of squares, where each square is either colored blue or left blank. The design on the grid is considered symmetric if it remains unchanged under a 90° rotation around the center. How many symmetric designs can be created if there must be at least one blue square but not all squares can be blue? | {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
(1) Given $0 < x < \frac{1}{2}$, find the maximum value of $y= \frac{1}{2}x(1-2x)$;
(2) Given $x > 0$, find the maximum value of $y=2-x- \frac{4}{x}$;
(3) Given $x$, $y\in\mathbb{R}_{+}$, and $x+y=4$, find the minimum value of $\frac{1}{x}+ \frac{3}{y}$. | {
"answer": "1+ \\frac{ \\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the distance from the foci of the hyperbola $C$ to its asymptotes is equal to the length of $C$'s real semi-axis, then the eccentricity of $C$ is \_\_\_\_\_\_. | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a right prism with all vertices on the same sphere, with a height of $4$ and a volume of $32$, the surface area of this sphere is ______. | {
"answer": "32\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The eccentricity of the ellipse given that the slope of line $l$ is $2$, and it intersects the ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ $(a > b > 0)$ at two different points, where the projections of these two intersection points on the $x$-axis are exactly the two foci of the ellipse. | {
"answer": "\\sqrt{2}-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the sum of the polynomials $A$ and $B$ is $12x^{2}y+2xy+5$, where $B=3x^{2}y-5xy+x+7$. Find:<br/>$(1)$ The polynomial $A$;<br/>$(2)$ When $x$ takes any value, the value of the expression $2A-\left(A+3B\right)$ is a constant. Find the value of $y$. | {
"answer": "\\frac{2}{11}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the area of a quadrilateral with vertices at $(0,0)$, $(4,3)$, $(7,0)$, and $(4,4)$? | {
"answer": "3.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two circles, both with the same radius $r$ , are placed in the plane without intersecting each other. A line in the plane intersects the first circle at the points $A,B$ and the other at points $C,D$ , so that $|AB|=|BC|=|CD|=14\text{cm}$ . Another line intersects the circles at $E,F$ , respectively $G,H$ so that $|EF|=|FG|=|GH|=6\text{cm}$ . Find the radius $r$ . | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A pentagon is obtained by joining, in order, the points \((0,0)\), \((1,2)\), \((3,3)\), \((4,1)\), \((2,0)\), and back to \((0,0)\). The perimeter of the pentagon can be written in the form \(a + b\sqrt{c} + d\sqrt{e}\), where \(a\), \(b\), \(c\), \(d\), and \(e\) are whole numbers. Find \(a+b+c+d+e\). | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $a = \log 8$ and $b = \log 25,$ compute
\[5^{a/b} + 2^{b/a}.\] | {
"answer": "2 \\sqrt{2} + 5^{2/3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an ellipse C centered at the origin with its left focus F($-\sqrt{3}$, 0) and right vertex A(2, 0).
(1) Find the standard equation of ellipse C;
(2) A line l with a slope of $\frac{1}{2}$ intersects ellipse C at points A and B. Find the maximum value of the chord length |AB| and the equation of line l at this time. | {
"answer": "\\sqrt{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $P$, $Q$, and $R$ be points on a circle of radius $24$. If $\angle PRQ = 40^\circ$, what is the circumference of the minor arc $PQ$? Express your answer in terms of $\pi$. | {
"answer": "\\frac{32\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( S = \left\{\left(s_{1}, s_{2}, \cdots, s_{6}\right) \mid s_{i} \in \{0, 1\}\right\} \). For any \( x, y \in S \) where \( x = \left(x_{1}, x_{2}, \cdots, x_{6}\right) \) and \( y = \left(y_{1}, y_{2}, \cdots, y_{6}\right) \), define:
(1) \( x = y \) if and only if \( \sum_{i=1}^{6}\left(x_{i} - y_{i}\right)^{2} = 0 \);
(2) \( x y = x_{1} y_{1} + x_{2} y_{2} + \cdots + x_{6} y_{6} \).
If a non-empty set \( T \subseteq S \) satisfies \( u v \neq 0 \) for any \( u, v \in T \) where \( u \neq v \), then the maximum number of elements in set \( T \) is: | {
"answer": "32",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A line passes through point $Q(\frac{1}{3}, \frac{4}{3})$ and intersects the hyperbola $x^{2}- \frac{y^{2}}{4}=1$ at points $A$ and $B$. Point $Q$ is the midpoint of chord $AB$.
1. Find the equation of the line containing $AB$.
2. Find the length of $|AB|$. | {
"answer": "\\frac{8\\sqrt{2}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $$x \in \left(- \frac{\pi}{2}, \frac{\pi}{2}\right)$$ and $$\sin x + \cos x = \frac{1}{5}$$, calculate the value of $\tan 2x$. | {
"answer": "-\\frac{24}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\theta$ is an angle in the second quadrant, and $\tan 2\theta = -2\sqrt{2}$.
(1) Find the value of $\tan \theta$.
(2) Calculate the value of $\frac {2\cos^{2} \frac {\theta}{2}-\sin\theta-\tan \frac {5\pi}{4}}{\sqrt {2}\sin(\theta + \frac {\pi}{4})}$. | {
"answer": "3 + 2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $ABCDEF$ be a regular hexagon with each side length $s$. Points $G$, $H$, $I$, $J$, $K$, and $L$ are the midpoints of sides $AB$, $BC$, $CD$, $DE$, $EF$, and $FA$, respectively. The segments $\overline{AH}$, $\overline{BI}$, $\overline{CJ}$, $\overline{DK}$, $\overline{EL}$, and $\overline{FG}$ form another hexagon inside $ABCDEF$. Find the ratio of the area of this inner hexagon to the area of hexagon $ABCDEF$, expressed as a fraction in its simplest form. | {
"answer": "\\frac{3}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
(In the preliminaries of optimal method and experimental design) When using the 0.618 method to find the optimal amount to add in an experiment, if the current range of excellence is $[628, 774]$ and the good point is 718, then the value of the addition point for the current experiment is ________. | {
"answer": "684",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let line $l_1: x + my + 6 = 0$ and line $l_2: (m - 2)x + 3y + 2m = 0$. When $m = \_\_\_\_\_\_$, $l_1 \parallel l_2$. | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If parallelogram ABCD has an area of 100 square meters, and E and G are the midpoints of sides AD and CD, respectively, while F is the midpoint of side BC, find the area of quadrilateral DEFG. | {
"answer": "25",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Convert the following radians to degrees and degrees to radians:
(1) $$\frac {\pi}{12}$$ = \_\_\_\_\_\_ ; (2) $$\frac {13\pi}{6}$$ = \_\_\_\_\_\_ ; (3) -$$\frac {5}{12}$$π = \_\_\_\_\_\_ .
(4) 36° = \_\_\_\_\_\_ rad; (5) -105° = \_\_\_\_\_\_ rad. | {
"answer": "- \\frac {7}{12}\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=x^{3}-x^{2}+1$.
$(1)$ Find the equation of the tangent line to the function $f(x)$ at the point $(1,f(1))$;
$(2)$ Find the extreme values of the function $f(x)$. | {
"answer": "\\dfrac {23}{27}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let the complex number $z=-3\cos \theta + i\sin \theta$ (where $i$ is the imaginary unit).
(1) When $\theta= \frac {4}{3}\pi$, find the value of $|z|$;
(2) When $\theta\in\left[ \frac {\pi}{2},\pi\right]$, the complex number $z_{1}=\cos \theta - i\sin \theta$, and $z_{1}z$ is a pure imaginary number, find the value of $\theta$. | {
"answer": "\\frac {2\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Define $f\left(n\right)=\textrm{LCM}\left(1,2,\ldots,n\right)$ . Determine the smallest positive integer $a$ such that $f\left(a\right)=f\left(a+2\right)$ .
*2017 CCA Math Bonanza Lightning Round #2.4* | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ in $\triangle ABC$, respectively, and $\sqrt{3}c\sin A = a\cos C$.
$(I)$ Find the value of $C$;
$(II)$ If $c=2a$ and $b=2\sqrt{3}$, find the area of $\triangle ABC$. | {
"answer": "\\frac{\\sqrt{15} - \\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider a modified octahedron with an additional ring of vertices. There are 4 vertices on the top ring, 8 on the middle ring, and 4 on the bottom ring. An ant starts at the highest top vertex and walks down to one of four vertices on the next level down (the middle ring). From there, without returning to the previous vertex, the ant selects one of the 3 adjacent vertices (excluding the one it came from) and continues to the next level. What is the probability that the ant reaches the bottom central vertex? | {
"answer": "\\frac{1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The real numbers $a$, $b$, and $c$ satisfy the equation $({a}^{2}+\frac{{b}^{2}}{4}+\frac{{c}^{2}}{9}=1)$. Find the maximum value of $a+b+c$. | {
"answer": "\\sqrt{14}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
John borrows $2000$ from Mary, who charges an interest rate of $6\%$ per month (which compounds monthly). What is the least integer number of months after which John will owe more than triple what he borrowed? | {
"answer": "19",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle DEF$ with sides $5$, $12$, and $13$, a circle with center $Q$ and radius $2$ rolls around inside the triangle, always keeping tangency to at least one side of the triangle. When $Q$ first returns to its original position, through what distance has $Q$ traveled? | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sides opposite to the internal angles $A$, $B$, $C$ of $\triangle ABC$ are $a$, $b$, $c$ respectively. Given $\frac{a-b+c}{c}= \frac{b}{a+b-c}$ and $a=2$, the maximum area of $\triangle ABC$ is __________. | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Cagney can frost a cupcake every 15 seconds, Lacey can frost a cupcake every 25 seconds, and Hardy can frost a cupcake every 50 seconds. Calculate the number of cupcakes that Cagney, Lacey, and Hardy can frost together in 6 minutes. | {
"answer": "45",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given two lines $l_{1}: ax-y+a=0$ and $l_{2}: (2a-3)x+ay-a=0$ are parallel, determine the value of $a$. | {
"answer": "-3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\omega \in \mathbb{C}$ , and $\left | \omega \right | = 1$ . Find the maximum length of $z = \left( \omega + 2 \right) ^3 \left( \omega - 3 \right)^2$ . | {
"answer": "108",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a positive integer \( n \geqslant 2 \), positive real numbers \( a_1, a_2, \ldots, a_n \), and non-negative real numbers \( b_1, b_2, \ldots, b_n \), which satisfy the following conditions:
(a) \( a_1 + a_2 + \cdots + a_n + b_1 + b_2 + \cdots + b_n = n \);
(b) \( a_1 a_2 \cdots a_n + b_1 b_2 \cdots b_n = \frac{1}{2} \).
Find the maximum value of \( a_1 a_2 \cdots a_n \left( \frac{b_1}{a_1} + \frac{b_2}{a_2} + \cdots + \frac{b_n}{a_n} \right) \). | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Define a $\it{good\ word}$ as a sequence of letters that consists only of the letters $A$, $B$, $C$, and $D$ --- some of these letters may not appear in the sequence --- and in which $A$ is never immediately followed by $B$, $B$ is never immediately followed by $C$, $C$ is never immediately followed by $D$, and $D$ is never immediately followed by $A$. How many eight-letter good words are there? | {
"answer": "8748",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the sum of the distances from one vertex of a rectangle with length $3$ and width $4$ to the centers of the opposite sides. | {
"answer": "\\sqrt{13} + 2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $PQR$, $\cos(2P-Q) + \sin(P+Q) = 2$ and $PQ = 5$. What is $QR$? | {
"answer": "5\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the equation of a circle $(x-1)^{2}+(y-1)^{2}=9$, point $P(2,2)$ lies inside the circle. The longest and shortest chords passing through point $P$ are $AC$ and $BD$ respectively. Determine the product $AC \cdot BD$. | {
"answer": "12\\sqrt{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the square root of $\dfrac{10!}{210}$. | {
"answer": "72\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $\triangle ABC$, the length of the side opposite angle $A$ is equal to 2, the vector $\overrightarrow {m} = (2, 2\cos^2 \frac {B+C}{2}-1)$, and the vector $\overrightarrow {n} = (\sin \frac {A}{2}, -1)$.
(1) Find the size of angle $A$ when the dot product $\overrightarrow {m} \cdot \overrightarrow {n}$ reaches its maximum value;
(2) Under the condition of (1), find the maximum area of $\triangle ABC$. | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Oil, as an important strategic reserve commodity, has always been of concern to countries. According to reports from relevant departments, it is estimated that the demand for oil in China in 2022 will be 735,000,000 tons. Express 735,000,000 in scientific notation as ____. | {
"answer": "7.35 \\times 10^{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Moe has a new, larger rectangular lawn measuring 120 feet by 180 feet. He uses a mower with a swath width of 30 inches. However, he overlaps each cut by 6 inches to ensure no grass is missed. Moe walks at a rate of 6000 feet per hour while pushing the mower. What is the closest estimate of the number of hours it will take Moe to mow the lawn? | {
"answer": "1.8",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle \( \triangle ABC \), the sides opposite to angles \( A \), \( B \), and \( C \) are \( a \), \( b \), and \( c \) respectively. Given that \( a^2 - (b - c)^2 = (2 - \sqrt{3})bc \) and \( \sin A \sin B = \cos^2 \frac{C}{2} \), and the length of the median \( AM \) from \( A \) to side \( BC \) is \( \sqrt{7} \):
1. Find the measures of angles \( A \) and \( B \);
2. Find the area of \( \triangle ABC \). | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given: $$\frac { A_{ n }^{ 3 }}{6}=n$$ (where $n\in\mathbb{N}^{*}$), and $(2-x)^{n}=a_{0}+a_{1}x+a_{2}x^{2}+\ldots+a_{n}x^{n}$
Find the value of $a_{0}-a_{1}+a_{2}-\ldots+(-1)^{n}a_{n}$. | {
"answer": "81",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The opposite of $-23$ is ______; the reciprocal is ______; the absolute value is ______. | {
"answer": "23",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Mrs. Carter's algebra class consists of 48 students. Due to a schedule conflict, 40 students took the Chapter 5 test, averaging 75%, while the remaining 8 students took it the following day, achieving an average score of 82%. What is the new overall mean score of the class on the Chapter 5 test? Express the answer as a percent. | {
"answer": "76.17\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A class has 54 students, and there are 4 tickets for the Shanghai World Expo to be distributed among the students using a systematic sampling method. If it is known that students with numbers 3, 29, and 42 have already been selected, then the student number of the fourth selected student is ▲. | {
"answer": "16",
"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 $y=f(x)$, calculate the value of $f\left(-\frac{{5π}}{{12}}\right)$. | {
"answer": "\\frac{\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Sixteen 6-inch wide square posts are evenly spaced with 4 feet between them to enclose a square field. What is the outer perimeter, in feet, of the fence? | {
"answer": "56",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\cos 78^{\circ}$ is approximately $\frac{1}{5}$, $\sin 66^{\circ}$ is approximately: | {
"answer": "0.92",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \(\mathbf{v}\) be a vector such that
\[
\left\| \mathbf{v} + \begin{pmatrix} 4 \\ 2 \end{pmatrix} \right\| = 10.
\]
Find the smallest possible value of \(\|\mathbf{v}\|\). | {
"answer": "10 - 2\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Four consecutive even integers have a product of 6720. What is the largest of these four integers? | {
"answer": "14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The probability that a randomly chosen divisor of $25!$ is odd. | {
"answer": "\\frac{1}{23}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In acute triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. Given that $a \neq b$, $c = \sqrt{3}$, and $\sqrt{3} \cos^2 A - \sqrt{3} \cos^2 B = \sin A \cos A - \sin B \cos B$.
(I) Find the measure of angle $C$;
(II) If $\sin A = \frac{4}{5}$, find the area of $\triangle ABC$. | {
"answer": "\\frac{24\\sqrt{3} + 18}{25}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given two similar triangles $\triangle ABC\sim\triangle FGH$, where $BC = 24 \text{ cm}$ and $FG = 15 \text{ cm}$. If the length of $AC$ is $18 \text{ cm}$, find the length of $GH$. Express your answer as a decimal to the nearest tenth. | {
"answer": "11.3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $| \overrightarrow{a}|=| \overrightarrow{b}|=| \overrightarrow{c}|=1$, and $ \overrightarrow{a}+ \overrightarrow{b}+ \sqrt {3} \overrightarrow{c}=0$, find the value of $ \overrightarrow{a} \overrightarrow{b}+ \overrightarrow{b} \overrightarrow{c}+ \overrightarrow{c} \overrightarrow{a}$. | {
"answer": "\\dfrac {1}{2}- \\sqrt {3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Michael read on average 30 pages each day for the first two days, then increased his average to 50 pages each day for the next four days, and finally read 70 pages on the last day. Calculate the total number of pages in the book. | {
"answer": "330",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Sector $OAB$ is a quarter of a circle with a radius of 6 cm. A circle is inscribed within this sector, tangent to the two radii and the arc at three points. Determine the radius of the inscribed circle, expressed in simplest radical form. | {
"answer": "6\\sqrt{2} - 6",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If Fang Fang cuts a piece of paper into 9 pieces, then selects one of the resulting pieces to cut into 9 pieces again, and so on, determine the number of cuts made to achieve a total of 2009 paper pieces. | {
"answer": "251",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $\tan (\alpha-\beta)= \frac {1}{2}$, $\tan \beta=- \frac {1}{7}$, and $\alpha$, $\beta\in(0,\pi)$, find the value of $2\alpha-\beta$. | {
"answer": "- \\frac {3\\pi}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a complex number $Z = x + yi$ ($x, y \in \mathbb{R}$) such that $|Z - 4i| = |Z + 2|$, find the minimum value of $2^x + 4^y$. | {
"answer": "4\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Sixty cards are placed into a box, each bearing a number 1 through 15, with each number represented on four cards. Four cards are drawn from the box at random without replacement. Let \(p\) be the probability that all four cards bear the same number. Let \(q\) be the probability that three of the cards bear a number \(a\) and the other bears a number \(b\) that is not equal to \(a\). What is the value of \(q/p\)? | {
"answer": "224",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, the sides corresponding to the internal angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. Given that $b = a \sin C + c \cos A$,
(1) Find the value of $A + B$;
(2) If $c = \sqrt{2}$, find the maximum area of $\triangle ABC$. | {
"answer": "\\frac{1 + \\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $n$ be a $5$-digit number, and let $q$ and $r$ be the quotient and the remainder, respectively, when $n$ is divided by $50$. Determine how many values of $n$ make $q+r$ divisible by $13$.
A) 7000
B) 7200
C) 7400
D) 7600 | {
"answer": "7200",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
We color some cells in $10000 \times 10000$ square, such that every $10 \times 10$ square and every $1 \times 100$ line have at least one coloring cell. What minimum number of cells we should color ? | {
"answer": "10000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system $xOy$, the parametric equation of line $l$ is $\begin{cases} x=t \\ y= \sqrt {2}+2t \end{cases}$ (where $t$ is the parameter), with point $O$ as the pole and the positive $x$-axis as the polar axis, the polar coordinate equation of curve $C$ is $\rho=4\cos\theta$.
(1) Find the Cartesian coordinate equation of curve $C$ and the general equation of line $l$;
(2) If the $x$-coordinates of all points on curve $C$ are shortened to $\frac {1}{2}$ of their original length, and then the resulting curve is translated 1 unit to the left, obtaining curve $C_1$, find the maximum distance from the points on curve $C_1$ to line $l$. | {
"answer": "\\frac {3 \\sqrt {10}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Grace has $\$4.80$ in U.S. coins. She has the same number of dimes and pennies. What is the greatest number of dimes she could have? | {
"answer": "43",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A trapezoid $ABCD$ has bases $AD$ and $BC$. If $BC = 60$ units, and altitudes from $B$ and $C$ to line $AD$ divide it into segments of lengths $AP = 20$ units and $DQ = 19$ units, with the length of the altitude itself being $30$ units, what is the perimeter of trapezoid $ABCD$?
**A)** $\sqrt{1300} + 159$
**B)** $\sqrt{1261} + 159$
**C)** $\sqrt{1300} + \sqrt{1261} + 159$
**D)** $259$
**E)** $\sqrt{1300} + 60 + \sqrt{1261}$ | {
"answer": "\\sqrt{1300} + \\sqrt{1261} + 159",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A list of $3042$ positive integers has a unique mode, which occurs exactly $15$ times. Calculate the least number of distinct values that can occur in the list. | {
"answer": "218",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\cos \alpha + \sin \alpha = \frac{2}{3}$, find the value of $\frac{\sqrt{2}\sin(2\alpha - \frac{\pi}{4}) + 1}{1 + \tan \alpha}$. | {
"answer": "-\\frac{5}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two cards are dealt at random from two standard decks of 104 cards mixed together. What is the probability that the first card drawn is an ace and the second card drawn is also an ace? | {
"answer": "\\dfrac{7}{1339}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A factory must filter its emissions before discharging them. The relationship between the concentration of pollutants $p$ (in milligrams per liter) and the filtration time $t$ (in hours) during the filtration process is given by the equation $p(t) = p_0e^{-kt}$. Here, $e$ is the base of the natural logarithm, and $p_0$ is the initial pollutant concentration. After filtering for one hour, it is observed that the pollutant concentration has decreased by $\frac{1}{5}$.
(Ⅰ) Determine the function $p(t)$.
(Ⅱ) To ensure that the pollutant concentration does not exceed $\frac{1}{1000}$ of the initial value, for how many additional hours must the filtration process be continued? (Given that $\lg 2 \approx 0.3$) | {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two books of different subjects are taken from three shelves, each having 10 Chinese books, 9 math books, and 8 English books. Calculate the total number of different ways to do this. | {
"answer": "242",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a regular decagon $ABCDEFGHIJ$ with area $n$, calculate the area $m$ of pentagon $ACEGI$, which is defined using every second vertex of the decagon, and then determine the value of $\frac{m}{n}$. | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=- \sqrt {3}\sin ^{2}x+\sin x\cos x$.
(1) Find the value of $f( \dfrac {25π}{6})$;
(2) Let $α∈(0,π)$, $f( \dfrac {α}{2})= \dfrac {1}{4}- \dfrac { \sqrt {3}}{2}$, find the value of $\sin α$. | {
"answer": "\\dfrac {1+3 \\sqrt {5}}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let the circles $S_1$ and $S_2$ meet at the points $A$ and $B$. A line through $B$ meets $S_1$ at a point $D$ other than $B$ and meets $S_2$ at a point $C$ other than $B$. The tangent to $S_1$ through $D$ and the tangent to $S_2$ through $C$ meet at $E$. If $|AD|=15$, $|AC|=16$, $|AB|=10$, what is $|AE|$? | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Define the determinant operation $\begin{vmatrix} a_{1} & a_{2} \\ b_{1} & b_{2}\end{vmatrix} =a_{1}b_{2}-a_{2}b_{1}$, and consider the function $f(x)= \begin{vmatrix} \sqrt {3} & \sin x \\ 1 & \cos x\end{vmatrix}$. If the graph of this function is translated to the left by $t(t > 0)$ units, and the resulting graph corresponds to an even function, then find the minimum value of $t$. | {
"answer": "\\dfrac{5\\pi}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the complex number $z=a^{2}-1+(a+1)i (a \in \mathbb{R})$ is a purely imaginary number, find the imaginary part of $\dfrac{1}{z+a}$. | {
"answer": "-\\dfrac{2}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In Tranquility Town, the streets are all $30$ feet wide and the blocks they enclose are rectangles with lengths of side $500$ feet and $300$ feet. Alice walks around the rectangle on the $500$-foot side of the street, while Bob walks on the opposite side of the street. How many more feet than Alice does Bob walk for every lap around the rectangle? | {
"answer": "240",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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