problem stringlengths 10 5.15k | answer dict |
|---|---|
Given that $(a + \frac{1}{a})^3 = 3$, find the value of $a^4 + \frac{1}{a^4}$.
A) $9^{1/3} - 4 \cdot 3^{1/3} + 2$
B) $9^{1/3} - 2 \cdot 3^{1/3} + 2$
C) $9^{1/3} + 4 \cdot 3^{1/3} + 2$
D) $4 \cdot 3^{1/3} - 9^{1/3}$ | {
"answer": "9^{1/3} - 4 \\cdot 3^{1/3} + 2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
(1) Simplify: $\dfrac {\tan (3\pi-\alpha)\cos (2\pi-\alpha)\sin (-\alpha+ \dfrac {3\pi}{2})}{\cos (-\alpha-\pi)\sin (-\pi+\alpha)\cos (\alpha+ \dfrac {5\pi}{2})}$;
(2) Given $\tan \alpha= \dfrac {1}{4}$, find the value of $\dfrac {1}{2\cos ^{2}\alpha -3\sin \alpha \cos \alpha }$. | {
"answer": "\\dfrac {17}{20}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The volume of a given sphere is \(72\pi\) cubic inches. Find the surface area of the sphere. Express your answer in terms of \(\pi\). | {
"answer": "36\\pi 2^{2/3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Add $36_7 + 274_7.$ Express your answer in base 7. | {
"answer": "343_7",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two concentric circles have radii $1$ and $4$ . Six congruent circles form a ring where each of the six circles is tangent to the two circles adjacent to it as shown. The three lightly shaded circles are internally tangent to the circle with radius $4$ while the three darkly shaded circles are externally tangent to the circle with radius $1$ . The radius of the six congruent circles can be written $\textstyle\frac{k+\sqrt m}n$ , where $k,m,$ and $n$ are integers with $k$ and $n$ relatively prime. Find $k+m+n$ .
[asy]
size(150);
defaultpen(linewidth(0.8));
real r = (sqrt(133)-9)/2;
draw(circle(origin,1)^^circle(origin,4));
for(int i=0;i<=2;i=i+1)
{
filldraw(circle(dir(90 + i*120)*(4-r),r),gray);
}
for(int j=0;j<=2;j=j+1)
{
filldraw(circle(dir(30+j*120)*(1+r),r),darkgray);
}
[/asy] | {
"answer": "126",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many positive integer multiples of $210$ can be expressed in the form $6^{j} - 6^{i}$, where $i$ and $j$ are integers and $0 \leq i < j \leq 49$? | {
"answer": "600",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( A \) be a point on the parabola \( y = x^2 - 4x \), and let \( B \) be a point on the line \( y = 2x - 3 \). Find the shortest possible distance \( AB \). | {
"answer": "\\frac{6\\sqrt{5}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given triangle $\triangle ABC$ with angles $A$, $B$, $C$ and their respective opposite sides $a$, $b$, $c$, such that $\frac{\sqrt{3}c}{\cos C} = \frac{a}{\cos(\frac{3\pi}{2} + A)}$.
(I) Find the value of $C$;
(II) If $\frac{c}{a} = 2$, $b = 4\sqrt{3}$, find the area of $\triangle ABC$. | {
"answer": "2\\sqrt{15} - 2\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The equations of the asymptotes of the hyperbola \\( \frac {x^{2}}{3}- \frac {y^{2}}{6}=1 \\) are \_\_\_\_\_\_, and its eccentricity is \_\_\_\_\_\_. | {
"answer": "\\sqrt {3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $a = \sin (2015\pi - \frac {\pi}{6})$ and the function $f(x) = \begin{cases} a^{x}, & x > 0 \\ f(-x), & x < 0 \end{cases}$, calculate the value of $f(\log_{2} \frac {1}{6})$. | {
"answer": "\\frac {1}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the xy-plane, what is the length of the shortest path from $(0,0)$ to $(15,20)$ that does not go inside the circle $(x-7)^2 + (y-9)^2 = 36$?
A) $2\sqrt{94} + 3\pi$
B) $2\sqrt{130} + 3\pi$
C) $2\sqrt{94} + 6\pi$
D) $2\sqrt{130} + 6\pi$
E) $94 + 3\pi$ | {
"answer": "2\\sqrt{94} + 3\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A school has eight identical copies of a particular book. At any given time, some of these copies are in the school library and some are with students. How many different ways are there for some of the books to be in the library and the rest to be with students if at least one book is in the library and at least one is with students? | {
"answer": "254",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the smallest \( n > 1 \) for which the average of the first \( n \) (non-zero) squares is a square? | {
"answer": "337",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the sequence $\{a_n\}$ is an arithmetic sequence, and if $\frac{a_{12}}{a_{11}} < -1$, find the maximum value of $n$ for which the sum of its first $n$ terms, $s_n$, is greater than $0$. | {
"answer": "21",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that point $P$ moves on the ellipse $\frac{x^{2}}{4}+y^{2}=1$, find the minimum distance from point $P$ to line $l$: $x+y-2\sqrt{5}=0$. | {
"answer": "\\frac{\\sqrt{10}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Ket $f(x) = x^{2} +ax + b$ . If for all nonzero real $x$ $$ f\left(x + \dfrac{1}{x}\right) = f\left(x\right) + f\left(\dfrac{1}{x}\right) $$ and the roots of $f(x) = 0$ are integers, what is the value of $a^{2}+b^{2}$ ? | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On a line \( l \) in space, points \( A \), \( B \), and \( C \) are sequentially located such that \( AB = 18 \) and \( BC = 14 \). Find the distance between lines \( l \) and \( m \) if the distances from points \( A \), \( B \), and \( C \) to line \( m \) are 12, 15, and 20, respectively. | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\triangle ABC$ have sides $a$, $b$, and $c$ opposite to angles $A$, $B$, and $C$ respectively. Given that $\frac{{a^2 + c^2 - b^2}}{{\cos B}} = 4$. Find:<br/>
$(1)$ $ac$;<br/>
$(2)$ If $\frac{{2b\cos C - 2c\cos B}}{{b\cos C + c\cos B}} - \frac{c}{a} = 2$, find the area of $\triangle ABC$. | {
"answer": "\\frac{\\sqrt{15}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given two arithmetic sequences $\{a_n\}$ and $\{b_n\}$, whose sums of the first $n$ terms are $A_n$ and $B_n$ respectively, and $\frac {A_{n}}{B_{n}} = \frac {7n+1}{4n+27}$, determine $\frac {a_{6}}{b_{6}}$. | {
"answer": "\\frac{78}{71}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A set of numbers $\{-3, 1, 5, 8, 10, 14\}$ needs to be rearranged with new rules:
1. The largest isn't in the last position, but it is in one of the last four places.
2. The smallest isn’t in the first position, but it is in one of the first four places.
3. The median isn't in the middle positions.
What is the product of the second and fifth numbers after rearrangement?
A) $-21$
B) $24$
C) $-24$
D) $30$
E) None of the above | {
"answer": "-24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a geometric sequence $\{a_n\}$ with the sum of its first n terms denoted as $S_n$, if $S_5$, $S_4$, and $S_6$ form an arithmetic sequence, determine the common ratio $q$ of the sequence $\{a_n\}$. | {
"answer": "-2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $F\_1$ and $F\_2$ are the left and right foci of the hyperbola $C: \frac{x^{2}}{a^{2}}- \frac{y^{2}}{b^{2}}=1 (a > 0, b > 0)$, if a point $P$ on the right branch of the hyperbola $C$ satisfies $|PF\_1|=3|PF\_2|$ and $\overrightarrow{PF\_1} \cdot \overrightarrow{PF\_2}=a^{2}$, calculate the eccentricity of the hyperbola $C$. | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The product underwent a price reduction from 25 yuan to 16 yuan. Calculate the average percentage reduction for each price reduction. | {
"answer": "20\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the parametric equation of line C1 as $$\begin{cases} x=2+t \\ y=t \end{cases}$$ (where t is the parameter), and the polar coordinate equation of the ellipse C2 as ρ²cos²θ + 9ρ²sin²θ = 9. Establish a rectangular coordinate system with the origin O as the pole and the positive semi-axis of the x-axis as the polar axis.
1. Find the general equation of line C1 and the standard equation of ellipse C2 in rectangular coordinates.
2. If line C1 intersects with ellipse C2 at points A and B, and intersects with the x-axis at point E, find the value of |EA + EB|. | {
"answer": "\\frac{6\\sqrt{3}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two concentric circles have the same center, labeled $C$. The larger circle has a radius of $12$ units while the smaller circle has a radius of $7$ units. Determine the area of the ring formed between these two circles and also calculate the circumference of the larger circle. | {
"answer": "24\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In \\(\triangle ABC\\), \\(AB=BC\\), \\(\cos B=-\dfrac{7}{18}\\). If an ellipse with foci at points \\(A\\) and \\(B\\) passes through point \\(C\\), find the eccentricity of the ellipse. | {
"answer": "\\dfrac{3}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Among the subsets of the set $\{1,2, \cdots, 100\}$, calculate the maximum number of elements a subset can have if it does not contain any pair of numbers where one number is exactly three times the other. | {
"answer": "67",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Seven distinct integers are picked at random from $\{1,2,3,\ldots,12\}$. What is the probability that, among those selected, the third smallest is $4$? | {
"answer": "\\frac{7}{33}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The parametric equation of line $l$ is
$$
\begin{cases}
x= \frac { \sqrt {2}}{2}t \\
y= \frac { \sqrt {2}}{2}t+4 \sqrt {2}
\end{cases}
$$
(where $t$ is the parameter), and the polar equation of circle $c$ is $\rho=2\cos(\theta+ \frac{\pi}{4})$. Tangent lines are drawn from the points on the line to the circle; find the minimum length of these tangent lines. | {
"answer": "2\\sqrt{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On March 12, 2016, the fourth Beijing Agriculture Carnival opened in Changping. The event was divided into seven sections: "Three Pavilions, Two Gardens, One Belt, and One Valley." The "Three Pavilions" refer to the Boutique Agriculture Pavilion, the Creative Agriculture Pavilion, and the Smart Agriculture Pavilion; the "Two Gardens" refer to the Theme Carnival Park and the Agricultural Experience Park; the "One Belt" refers to the Strawberry Leisure Experience Belt; and the "One Valley" refers to the Yanshou Ecological Sightseeing Valley. Due to limited time, a group of students plans to visit "One Pavilion, One Garden, One Belt, and One Valley." How many different routes can they take for their visit? (Answer with a number) | {
"answer": "144",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A circle is inscribed in a regular hexagon. A smaller hexagon has two non-adjacent sides coinciding with the sides of the larger hexagon and the remaining vertices touching the circle. What percentage of the area of the larger hexagon is the area of the smaller hexagon? Assume the side of the larger hexagon is twice the radius of the circle. | {
"answer": "25\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. If $\sin (A-B)+ \sin C= \sqrt {2}\sin A$.
(I) Find the value of angle $B$;
(II) If $b=2$, find the maximum value of $a^{2}+c^{2}$, and find the values of angles $A$ and $C$ when the maximum value is obtained. | {
"answer": "8+4 \\sqrt {2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider the graph on $1000$ vertices $v_1, v_2, ...v_{1000}$ such that for all $1 \le i < j \le 1000$ , $v_i$ is connected to $v_j$ if and only if $i$ divides $j$ . Determine the minimum number of colors that must be used to color the vertices of this graph such that no two vertices sharing an edge are the same color. | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the area of $\triangle ABC$ is $2 \sqrt {3}$, $BC=2$, $C=120^{\circ}$, find the length of side $AB$. | {
"answer": "2 \\sqrt {7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $\overrightarrow{a}=(1+\cos \omega x,-1)$, $\overrightarrow{b}=( \sqrt {3},\sin \omega x)$ ($\omega > 0$), and the function $f(x)= \overrightarrow{a}\cdot \overrightarrow{b}$, with the smallest positive period of $f(x)$ being $2\pi$.
(1) Find the expression of the function $f(x)$.
(2) Let $\theta\in(0, \frac {\pi}{2})$, and $f(\theta)= \sqrt {3}+ \frac {6}{5}$, find the value of $\cos \theta$. | {
"answer": "\\frac {3 \\sqrt {3}+4}{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Quadrilateral $ABCD$ has right angles at $B$ and $D$. The length of the diagonal $AC$ is $5$. If two sides of $ABCD$ have integer lengths and one of these lengths is an odd integer, determine the area of $ABCD$. | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
James used a calculator to find the product $0.005 \times 3.24$. He forgot to enter the decimal points, and the calculator showed $1620$. If James had entered the decimal points correctly, what would the answer have been?
A) $0.00162$
B) $0.0162$
C) $0.162$
D) $0.01620$
E) $0.1620$ | {
"answer": "0.0162",
"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. It is known that $c^{2}=a^{2}+b^{2}-4bc\cos C$, and $A-C= \frac {\pi}{2}$.
(Ⅰ) Find the value of $\cos C$;
(Ⅱ) Find the value of $\cos \left(B+ \frac {\pi}{3}\right)$. | {
"answer": "\\frac {4-3 \\sqrt {3}}{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Evaluate the expression \[(5^{1001} + 6^{1002})^2 - (5^{1001} - 6^{1002})^2\] and express it in the form \(k \cdot 30^{1001}\) for some integer \(k\). | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A 12-hour digital clock has a glitch such that whenever it is supposed to display a 5, it mistakenly displays a 7. For example, when it is 5:15 PM the clock incorrectly shows 7:77 PM. What fraction of the day will the clock show the correct time? | {
"answer": "\\frac{33}{40}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $\tan \alpha = 3$, evaluate the expression $2\sin ^{2}\alpha + 4\sin \alpha \cos \alpha - 9\cos ^{2}\alpha$. | {
"answer": "\\dfrac{21}{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the smallest positive value of $m$ so that the equation $10x^2 - mx + 1980 = 0$ has integral solutions? | {
"answer": "290",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, it is known that $\cos C+(\cos A- \sqrt {3}\sin A)\cos B=0$.
(1) Find the measure of angle $B$;
(2) If $\sin (A- \frac {π}{3})= \frac {3}{5}$, find $\sin 2C$. | {
"answer": "\\frac {24+7 \\sqrt {3}}{50}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the singing scores 9.4, 8.4, 9.4, 9.9, 9.6, 9.4, 9.7, calculate the average and variance of the remaining data after removing the highest and lowest scores. | {
"answer": "0.016",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the maximum value of $\frac{(3^t-2t)t}{9^t}$ for real values of $t$?
A) $\frac{1}{10}$
B) $\frac{1}{12}$
C) $\frac{1}{8}$
D) $\frac{1}{6}$
E) $\frac{1}{4}$ | {
"answer": "\\frac{1}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $f(x)= \sqrt {3}\sin \dfrac {x}{4}\cos \dfrac {x}{4}+ \cos ^{2} \dfrac {x}{4}+ \dfrac {1}{2}$.
(1) Find the period of $f(x)$;
(2) In $\triangle ABC$, sides $a$, $b$, and $c$ correspond to angles $A$, $B$, and $C$ respectively, and satisfy $(2a-c)\cos B=b\cos C$, find the value of $f(B)$. | {
"answer": "\\dfrac{\\sqrt{3}}{2} + 1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, $\frac{1}{2}\cos 2A = \cos^2 A - \cos A$.
(I) Find the measure of angle $A$;
(II) If $a=3$, $\sin B = 2\sin C$, find the area of $\triangle ABC$. | {
"answer": "\\frac{3\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A novice economist-cryptographer received a cryptogram from a ruler which contained a secret decree about implementing an itemized tax on a certain market. The cryptogram specified the amount of tax revenue that needed to be collected, emphasizing that a greater amount could not be collected in that market. Unfortunately, the economist-cryptographer made an error in decrypting the cryptogram—the digits of the tax revenue amount were identified in the wrong order. Based on erroneous data, a decision was made to introduce an itemized tax on producers of 90 monetary units per unit of goods. It is known that the market demand is represented by \( Q_d = 688 - 4P \), and the market supply is linear. When there are no taxes, the price elasticity of market supply at the equilibrium point is 1.5 times higher than the modulus of the price elasticity of the market demand function. After the tax was introduced, the producer price fell to 64 monetary units.
1) Restore the market supply function.
2) Determine the amount of tax revenue collected at the chosen rate.
3) Determine the itemized tax rate that would meet the ruler's decree.
4) What is the amount of tax revenue specified by the ruler to be collected? | {
"answer": "6480",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $α∈(0, \dfrac {π}{2})$ and $β∈(0, \dfrac {π}{2})$, with $cosα= \dfrac {1}{7}$ and $cos(α+β)=- \dfrac {11}{14}$, find the value of $sinβ$. | {
"answer": "\\dfrac{\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the coefficient of the $x^3$ term in the expansion of $\left(x+a\right)\left(x-2\right)^5$ is $-60$, find the value of $a$. | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that point P(x, y) satisfies the equation (x-4 cos θ)^{2} + (y-4 sin θ)^{2} = 4, where θ ∈ R, find the area of the region that point P occupies. | {
"answer": "32 \\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If five squares of a $3 \times 3$ board initially colored white are chosen at random and blackened, what is the expected number of edges between two squares of the same color?
| {
"answer": "\\frac{16}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On the first day, 1 bee brings back 5 companions. On the second day, 6 bees (1 from the original + 5 brought back on the first day) fly out, each bringing back 5 companions. Determine the total number of bees in the hive after the 6th day. | {
"answer": "46656",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider an isosceles right triangle with leg lengths of 1 each. Inscribed in this triangle is a square in such a way that one vertex of the square coincides with the right-angle vertex of the triangle. Another square with side length $y$ is inscribed in an identical isosceles right triangle where one side of the square lies on the hypotenuse of the triangle. What is $\dfrac{x}{y}$?
A) $\frac{1}{\sqrt{2}}$
B) $1$
C) $\sqrt{2}$
D) $\frac{\sqrt{2}}{2}$ | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Derek is deciding between two different-sized pizzas at his favorite restaurant. The menu lists a 14-inch pizza and an 18-inch pizza. Calculate the percent increase in area if Derek chooses the 18-inch pizza over the 14-inch pizza. | {
"answer": "65.31\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the center of the hyperbola is at the origin and one focus is F<sub>1</sub>(-$$\sqrt{5}$$, 0), if point P is on the hyperbola and the midpoint of segment PF<sub>1</sub> has coordinates (0, 2), then the equation of this hyperbola is _________ and its eccentricity is _________. | {
"answer": "\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $\left(a-c\right)\left(a+c\right)\sin C=c\left(b-c\right)\sin B$.
$(1)$ Find angle $A$;
$(2)$ If the area of $\triangle ABC$ is $\sqrt{3}$, $\sin B\sin C=\frac{1}{4}$, find the value of $a$. | {
"answer": "2\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a parabola $y=x^{2}-7$, find the length of the line segment $|AB|$ where $A$ and $B$ are two distinct points on it that are symmetric about the line $x+y=0$. | {
"answer": "5 \\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
To popularize knowledge of fire safety, a certain school organized a competition on related knowledge. The competition is divided into two rounds, and each participant must participate in both rounds. If a participant wins in both rounds, they are considered to have won the competition. It is known that in the first round of the competition, the probabilities of participants A and B winning are $\frac{4}{5}$ and $\frac{3}{5}$, respectively; in the second round, the probabilities of A and B winning are $\frac{2}{3}$ and $\frac{3}{4}$, respectively. A and B's outcomes in each round are independent of each other.<br/>$(1)$ The probability of A winning exactly one round in the competition;<br/>$(2)$ If both A and B participate in the competition, find the probability that at least one of them wins the competition. | {
"answer": "\\frac{223}{300}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The owner of an individual clothing store purchased 30 dresses for $32 each. The selling price of the 30 dresses varies for different customers. Using $47 as the standard price, any excess amount is recorded as positive and any shortfall is recorded as negative. The results are shown in the table below:
| Number Sold | 7 | 6 | 3 | 5 | 4 | 5 |
|-------------|---|---|---|---|---|---|
| Price/$ | +3 | +2 | +1 | 0 | -1 | -2 |
After selling these 30 dresses, how much money did the clothing store earn? | {
"answer": "472",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)= \sqrt{2}\sin \left( 2x- \frac{\pi}{4} \right)$, where $x\in\mathbb{R}$, if the maximum and minimum values of $f(x)$ in the interval $\left[ \frac{\pi}{8}, \frac{3\pi}{4} \right]$ are $a$ and $b$ respectively, then the value of $a+b$ is ______. | {
"answer": "\\sqrt{2}-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Five glass bottles can be recycled to make a new bottle. Additionally, for every 20 new bottles created, a bonus bottle can be made from residual materials. Starting with 625 glass bottles, how many total new bottles can eventually be made from recycling and bonuses? (Keep counting recycled and bonus bottles until no further bottles can be manufactured. Do not include the original 625 bottles in your count.) | {
"answer": "163",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Read the following material and then answer the following questions: When simplifying algebraic expressions, sometimes we encounter expressions like $\frac{5}{\sqrt{3}}$, $\frac{2}{\sqrt{3}+1}$, in fact, we can further simplify them:<br/>$($1) $\frac{5}{\sqrt{3}}=\frac{5×\sqrt{3}}{\sqrt{3}×\sqrt{3}}=\frac{5}{3}\sqrt{3}$;<br/>$($2) $\frac{2}{\sqrt{3}+1}=\frac{2×(\sqrt{3}-1)}{(\sqrt{3}+1)(\sqrt{3}-1)}=\frac{2(\sqrt{3}-1)}{(\sqrt{3})^{2}-1}=\sqrt{3}-1$;<br/>$($3) $\frac{2}{\sqrt{3}+1}=\frac{3-1}{\sqrt{3}+1}=\frac{(\sqrt{3})^{2}-{1}^{2}}{\sqrt{3}+1}=\frac{(\sqrt{3}+1)(\sqrt{3}-1)}{\sqrt{3}+1}=\sqrt{3}-1$. The method of simplification mentioned above is called rationalizing the denominator.<br/>$(1)$ Simplify $\frac{2}{\sqrt{5}+\sqrt{3}}$ using different methods:<br/>① Refer to formula (2) to simplify $\frac{2}{\sqrt{5}+\sqrt{3}}=\_\_\_\_\_\_.$<br/>② Refer to formula (3) to simplify $\frac{2}{\sqrt{5}+\sqrt{3}}=\_\_\_\_\_\_.$<br/>$(2)$ Simplify: $\frac{1}{\sqrt{3}+1}+\frac{1}{\sqrt{5}+\sqrt{3}}+\frac{1}{\sqrt{7}+\sqrt{5}}+\ldots +\frac{1}{\sqrt{99}+\sqrt{97}}$. | {
"answer": "\\frac{3\\sqrt{11}-1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Grandma Wang has 6 stools that need to be painted by a painter. Each stool needs to be painted twice. The first coat takes 2 minutes, but there must be a 10-minute wait before applying the second coat. How many minutes will it take to paint all 6 stools? | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $y = \cos\left(x+ \frac {\pi}{5}\right)$, where $x\in\mathbb{R}$, determine the horizontal shift required to obtain this function's graph from the graph of $y=\cos x$. | {
"answer": "\\frac {\\pi}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given four points on a sphere, $A$, $B$, $C$, $D$, with the center of the sphere being point $O$, and $O$ is on $CD$. If the maximum volume of the tetrahedron $A-BCD$ is $\frac{8}{3}$, then the surface area of sphere $O$ is ______. | {
"answer": "16\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Alex, Bonnie, and Chris each have $3$ blocks, colored red, blue, and green; and there are $3$ empty boxes. Each person independently places one of their blocks into each box. Each block placement by Bonnie and Chris is picked such that there is a 50% chance that the color matches the color previously placed by Alex or Bonnie respectively. Calculate the probability that at least one box receives $3$ blocks all of the same color.
A) $\frac{27}{64}$
B) $\frac{29}{64}$
C) $\frac{37}{64}$
D) $\frac{55}{64}$
E) $\frac{63}{64}$ | {
"answer": "\\frac{37}{64}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Country $X$ has $40\%$ of the world's population and $60\%$ of the world's wealth. Country $Y$ has $20\%$ of the world's population but $30\%$ of its wealth. Country $X$'s top $50\%$ of the population owns $80\%$ of the wealth, and the wealth in Country $Y$ is equally shared among its citizens. Determine the ratio of the wealth of an average citizen in the top $50\%$ of Country $X$ to the wealth of an average citizen in Country $Y$. | {
"answer": "1.6",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In acute triangle ABC, the sides opposite to angles A, B, C are a, b, c respectively, and a = 2b*sin(A).
(1) Find the measure of angle B.
(2) If a = $3\sqrt{3}$, c = 5, find b. | {
"answer": "\\sqrt{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider a city grid with intersections labeled A, B, C, and D. Assume a student walks from intersection A to intersection B every morning, always walking along the designated paths and only heading east or south. The student passes through intersections C and D along the way. The intersections are placed such that A to C involves 3 eastward moves and 2 southward moves, and C to D involves 2 eastward moves and 1 southward move, and finally from D to B requires 1 eastward move and 2 southward moves. Each morning, at each intersection where he has a choice, he randomly chooses whether to go east or south with probability $\frac{1}{2}$. Determine the probability that the student walks through C, and then D on any given morning.
A) $\frac{15}{77}$
B) $\frac{10}{462}$
C) $\frac{120}{462}$
D) $\frac{3}{10}$
E) $\frac{64}{462}$ | {
"answer": "\\frac{15}{77}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A regular tetrahedron with four equilateral triangular faces has a sphere inscribed within it and another sphere circumscribed about it. Each of the four faces of the tetrahedron is tangent to a unique external sphere which is also tangent to the circumscribed sphere, but now these external spheres have radii larger than those in the original setup. Assume new radii are 50% larger than the radius of the inscribed sphere. A point $P$ is selected at random inside the circumscribed sphere. Compute the probability that $P$ lies inside one of these external spheres. | {
"answer": "0.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, let the sides opposite to angles $A$, $B$, and $C$ be $a$, $b$, and $c$ respectively. Given that $a\cos B=3$ and $b\sin A=4$.
(I) Find $\tan B$ and the value of side $a$;
(II) If the area of $\triangle ABC$ is $S=10$, find the perimeter $l$ of $\triangle ABC$. | {
"answer": "10 + 2\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Point $P$ is a moving point on the curve $C_{1}$: $(x-2)^{2}+y^{2}=4$. Taking the origin $O$ as the pole and the positive half-axis of $x$ as the polar axis, a polar coordinate system is established. Point $P$ is rotated counterclockwise by $90^{\circ}$ around the pole $O$ to obtain point $Q$. Suppose the trajectory equation of point $Q$ is the curve $C_{2}$.
$(1)$ Find the polar equations of curves $C_{1}$ and $C_{2}$;
$(2)$ The ray $\theta= \dfrac {\pi}{3}(\rho > 0)$ intersects curves $C_{1}$ and $C_{2}$ at points $A$ and $B$ respectively. Given the fixed point $M(2,0)$, find the area of $\triangle MAB$. | {
"answer": "3- \\sqrt {3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Celine has 240 feet of fencing. She needs to enclose a rectangular area such that the area is eight times the perimeter of the rectangle. If she uses up all her fencing material, how many feet is the largest side of the enclosure? | {
"answer": "101",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a circle with center $O$, $\overline{AB}$ and $\overline{CD}$ are diameters perpendicular to each other. Chord $\overline{DF}$ intersects $\overline{AB}$ at $E$. Given that $DE = 8$ and $EF = 4$, determine the area of the circle.
A) $24\pi$
B) $30\pi$
C) $32\pi$
D) $36\pi$
E) $40\pi$ | {
"answer": "32\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A large square has each side divided into four equal parts. A square is inscribed such that its vertices touch these division points, as illustrated below. Determine the ratio of the area of the inscribed square to the large square.
[asy]
draw((0,0)--(4,0)--(4,4)--(0,4)--cycle);
draw((1,0)--(1,0.1)); draw((2,0)--(2,0.1)); draw((3,0)--(3,0.1));
draw((4,1)--(3.9,1)); draw((4,2)--(3.9,2)); draw((4,3)--(3.9,3));
draw((1,4)--(1,3.9)); draw((2,4)--(2,3.9)); draw((3,4)--(3,3.9));
draw((0,1)--(0.1,1)); draw((0,2)--(0.1,2)); draw((0,3)--(0.1,3));
draw((1,0)--(4,3)--(3,4)--(0,1)--cycle);
[/asy]
A) $\frac{1}{2}$
B) $\frac{1}{3}$
C) $\frac{5}{8}$
D) $\frac{3}{4}$ | {
"answer": "\\frac{5}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An ancient Greek was born on January 7, 40 B.C., and died on January 7, 40 A.D. Calculate the number of years he lived. | {
"answer": "79",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the curve E with the polar coordinate equation 4(ρ^2^-4)sin^2^θ=(16-ρ^2)cos^2^θ, establish a rectangular coordinate system with the non-negative semi-axis of the polar axis as the x-axis and the pole O as the coordinate origin.
(1) Write the rectangular coordinate equation of the curve E;
(2) If point P is a moving point on curve E, point M is the midpoint of segment OP, and the parameter equation of line l is $$\begin{cases} x=- \sqrt {2}+ \frac {2 \sqrt {5}}{5}t \\ y= \sqrt {2}+ \frac { \sqrt {5}}{5}t\end{cases}$$ (t is the parameter), find the maximum value of the distance from point M to line l. | {
"answer": "\\sqrt{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $a \in \{0,1,2\}$ and $b \in \{-1,1,3,5\}$, the probability that the function $f(x) = ax^2 - 2bx$ is an increasing function in the interval $(1, +\infty)$ is $(\quad\quad)$. | {
"answer": "\\frac{1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a square, points \(P\) and \(Q\) are the midpoints of the top and right sides, respectively. What fraction of the interior of the square is shaded when the region outside the triangle \(OPQ\) (assuming \(O\) is the bottom-left corner of the square) is shaded? Express your answer as a common fraction.
[asy]
filldraw((0,0)--(2,0)--(2,2)--(0,2)--cycle,gray,linewidth(1));
filldraw((0,2)--(2,1)--(2,2)--cycle,white,linewidth(1));
label("P",(0,2),N);
label("Q",(2,1),E);
[/asy] | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Twenty pairs of integers are formed using each of the integers \( 1, 2, 3, \ldots, 40 \) once. The positive difference between the integers in each pair is 1 or 3. If the resulting differences are added together, what is the greatest possible sum? | {
"answer": "58",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the rectangular coordinate system, the line $l$ passes through the origin with an inclination angle of $\frac{\pi}{4}$; the parametric equations of the curve $C\_1$ are $\begin{cases} x = \frac{\sqrt{3}}{3} \cos \alpha \\ y = \sin \alpha \end{cases}$ (where $\alpha$ is the parameter); the parametric equations of the curve $C\_2$ are $\begin{cases} x = 3 + \sqrt{13} \cos \alpha \\ y = 2 + \sqrt{13} \sin \alpha \end{cases}$ (where $\alpha$ is the parameter).
(1) Find the polar coordinate equation of line $l$, and the Cartesian equations of curves $C\_1$ and $C\_2$;
(2) If the intersection points of line $l$ with curves $C\_1$ and $C\_2$ in the first quadrant are $M$ and $N$ respectively, find the distance between $M$ and $N$. | {
"answer": "\\frac{9\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In an 11x11 grid making up a square, there are 121 uniformly spaced grid points including those on the edges. The point P is located in the very center of the square. A point Q is randomly chosen from the other 120 points. What is the probability that the line PQ is a line of symmetry for the square?
A) $\frac{1}{6}$
B) $\frac{1}{4}$
C) $\frac{1}{3}$
D) $\frac{1}{2}$
E) $\frac{2}{3}$ | {
"answer": "\\frac{1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are three spheres and a cube. The first sphere is tangent to each face of the cube, the second sphere is tangent to each edge of the cube, and the third sphere passes through each vertex of the cube. What is the ratio of the surface areas of these three spheres? | {
"answer": "1:2:3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given tetrahedron $P-ABC$, if one line is randomly selected from the lines connecting the midpoints of each edge, calculate the probability that this line intersects plane $ABC$. | {
"answer": "\\frac{3}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $A_n^2 = 132$, calculate the value of $n$. | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the function $f(x)=\ln (e^{x}+a+1)$ ($a$ is a constant) is an odd function on the set of real numbers $R$.
(1) Find the value of the real number $a$;
(2) If the equation $\frac{\ln x}{f(x)}=x^{2}-2ex+m$ has exactly one real root, find the value of $m$. | {
"answer": "e^{2}+\\frac{1}{e}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Three congruent cones, each with a radius of 8 cm and a height of 8 cm, are enclosed within a cylinder. The base of each cone is consecutively stacked and forms a part of the cylinder’s interior base, while the height of the cylinder is 24 cm. Calculate the volume of the cylinder that is not occupied by the cones, and express your answer in terms of $\pi$. | {
"answer": "1024\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $(1-2)^{9}=a_{9}x^{9}+a_{8}x^{8}+\ldots+a_{1}x+a_{0}$, then the sum of $a_1+a_2+\ldots+a$ is \_\_\_\_\_\_. | {
"answer": "-2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a speech competition, judges will score participants based on the content, delivery, and effectiveness of the speech, with weights of $4:4:2$ respectively. If a student receives scores of $91$, $94$, and $90$ in these three aspects, then the student's total score is ______ points. | {
"answer": "92",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given positive numbers $x$ and $y$ satisfying $x^2+y^2=1$, find the maximum value of $\frac {1}{x}+ \frac {1}{y}$. | {
"answer": "2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the following system of equations: $$ \begin{cases} R I +G +SP = 50 R I +T + M = 63 G +T +SP = 25 SP + M = 13 M +R I = 48 N = 1 \end{cases} $$
Find the value of L that makes $LMT +SPR I NG = 2023$ true.
| {
"answer": "\\frac{341}{40}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a right triangular prism $ABC-A_{1}B_{1}C_{1}$, where $\angle BAC=90^{\circ}$, the area of the side face $BCC_{1}B_{1}$ is $16$. Find the minimum value of the radius of the circumscribed sphere of the right triangular prism $ABC-A_{1}B_{1}C_{1}$. | {
"answer": "2 \\sqrt {2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Point \( A \) lies on the line \( y = \frac{5}{12} x - 7 \), and point \( B \) lies on the parabola \( y = x^2 \). What is the minimum length of segment \( AB \)? | {
"answer": "\\frac{4007}{624}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sequence $\{a\_n\}$ is a geometric sequence with the first term $a\_1=4$, and $S\_3$, $S\_2$, $S\_4$ form an arithmetic sequence.
(1) Find the general term formula of the sequence $\{a\_n\}$;
(2) If $b\_n=\log \_{2}|a\_n|$, let $T\_n$ be the sum of the first $n$ terms of the sequence $\{\frac{1}{b\_n b\_{n+1}}\}$. If $T\_n \leqslant \lambda b\_{n+1}$ holds for all $n \in \mathbb{N}^*$, find the minimum value of the real number $\lambda$. | {
"answer": "\\frac{1}{16}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=ax^{2}-2x+1$.
$(1)$ When $a\neq 0$, discuss the monotonicity of the function $f(x)$;
$(2)$ If $\frac {1}{3}\leqslant a\leqslant 1$, and the maximum value of $f(x)$ on $[1,3]$ is $M(a)$, the minimum value is $N(a)$, let $g(a)=M(a)-N(a)$, find the expression of $g(a)$;
$(3)$ Under the condition of $(2)$, find the minimum value of $g(a)$. | {
"answer": "\\frac {1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the least integer value of $x$ for which $3|x| - 2 > 13$. | {
"answer": "-6",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given four numbers $101010_{(2)}$, $111_{(5)}$, $32_{(8)}$, and $54_{(6)}$, the smallest among them is \_\_\_\_\_\_. | {
"answer": "32_{(8)}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the sum of the first $n$ terms of the sequence ${a_n}$ is $S_n$, and the sum of the first $n$ terms of the sequence ${b_n}$ is $T_n$. It is known that $a_1=2$, $3S_n=(n+m)a_n$, ($m\in R$), and $a_nb_n=\frac{1}{2}$. If for any $n\in N^*$, $\lambda>T_n$ always holds true, then the minimum value of the real number $\lambda$ is $\_\_\_\_\_\_$. | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Assume that the scores $X$ of 400,000 students in a math mock exam in Yunnan Province approximately follow a normal distribution $N(98,100)$. It is known that a certain student's score ranks among the top 9100 in the province. Then, the student's math score will not be lower than ______ points. (Reference data: $P(\mu -\sigma\ \ < X < \mu +\sigma )=0.6827, P(\mu -2\sigma\ \ < X < \mu +2\sigma )=0.9545$) | {
"answer": "118",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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