problem stringlengths 10 5.15k | answer dict |
|---|---|
The Tianfu Greenway is a popular check-in spot for the people of Chengdu. According to statistics, there is a linear relationship between the number of tourists on the Tianfu Greenway, denoted as $x$ (in units of 10,000 people), and the economic income of the surrounding businesses, denoted as $y$ (in units of 10,000 yuan). It is known that the regression line equation is $\hat{y}=12.6x+0.6$. The statistics of the number of tourists on the Tianfu Greenway and the economic income of the surrounding businesses for the past five months are shown in the table below:
| $x$ | 2 | 3 | 3.5 | 4.5 | 7 |
|-----|-----|-----|-----|-----|-----|
| $y$ | 26 | 38 | 43 | 60 | $a$ |
The value of $a$ in the table is ______. | {
"answer": "88",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Pompous Vova has an iPhone XXX, and on that iPhone, he has a calculator with voice commands: "Multiply my number by two and subtract two from the result," "Multiply my number by three and then add four," and lastly, "Add seven to my number!" The iPhone knows that initially, Vova's number was 1. How many four-digit numbers could the iPhone XXX theoretically achieve by obediently following Vova's commands? | {
"answer": "9000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Alan, Jason, and Shervin are playing a game with MafsCounts questions. They each start with $2$ tokens. In each round, they are given the same MafsCounts question. The first person to solve the MafsCounts question wins the round and steals one token from each of the other players in the game. They all have the same probability of winning any given round. If a player runs out of tokens, they are removed from the game. The last player remaining wins the game.
If Alan wins the first round but does not win the second round, what is the probability that he wins the game?
*2020 CCA Math Bonanza Individual Round #4* | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$F, G, H, I,$ and $J$ are collinear in that order such that $FG = 2, GH = 1, HI = 3,$ and $IJ = 7$. If $P$ can be any point in space, what is the smallest possible value of $FP^2 + GP^2 + HP^2 + IP^2 + JP^2$? | {
"answer": "102.8",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $x$ is the median of the data set $1$, $2$, $3$, $x$, $5$, $6$, $7$, and the average of the data set $1$, $2$, $x^{2}$, $-y$ is $1$, find the minimum value of $y- \frac {1}{x}$. | {
"answer": "\\frac {23}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The line $l_{1}: x+a^{2}y+6=0$ and the line $l_{2}: (a-2)x+3ay+2a=0$ are parallel, find the value of $a$. | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On a map, a rhombus-shaped park is represented where the scale is given as 1 inch equals 100 miles. The long diagonal of the park on the map measures 10 inches, and the angle between the diagonals of the rhombus is 60 degrees. Calculate the actual area of the park in square miles.
A) $100000\sqrt{3}$ square miles
B) $200000\sqrt{3}$ square miles
C) $300000\sqrt{3}$ square miles
D) $400000\sqrt{3}$ square miles | {
"answer": "200000\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a mathematics class, the probability of earning an A is 0.6 times the probability of earning a B, and the probability of earning a C is 1.6 times the probability of earning a B. The probability of earning a D is 0.3 times the probability of earning a B. Assuming that all grades are A, B, C, or D, how many B's will there be in a mathematics class of 50 students? | {
"answer": "14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Real numbers $u$ and $v$ are each chosen independently and uniformly at random from the interval $(0, 2)$. What is the probability that $\lfloor \log_3 u \rfloor = \lfloor \log_3 v \rfloor$?
A) $\frac{1}{9}$
B) $\frac{1}{3}$
C) $\frac{4}{9}$
D) $\frac{5}{9}$
E) $\frac{1}{2}$ | {
"answer": "\\frac{5}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider the equation $x^2 + 14x = 32$. Find the values of $a$ and $b$ such that the positive solution of the equation has the form $\sqrt{a}-b$, where $a$ and $b$ are positive natural numbers. Calculate $a+b$. | {
"answer": "88",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute $\cos 105^\circ$. | {
"answer": "\\frac{\\sqrt{2} - \\sqrt{6}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that sin(α + $\frac {π}{4}$) = $\frac { \sqrt {3}}{3}$, and α ∈ (0, π), find the value of cosα. | {
"answer": "\\frac {-2 \\sqrt {3}+ \\sqrt {6}}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The line passing through the points (3, 9) and (-1, 1) intersects the x-axis at a point whose x-coordinate is $\frac{9-1}{3-(-1)}$ | {
"answer": "- \\frac{3}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the price savings of buying the computer at store A is $15 more than buying it at store B, and store A offers a 15% discount followed by a $90 rebate, while store B offers a 25% discount and no rebate, calculate the sticker price of the computer. | {
"answer": "750",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A triangle has a base of 20 inches. Two lines are drawn parallel to the base, intersecting the other two sides and dividing the triangle into four regions of equal area. Determine the length of the parallel line closer to the base. | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a right triangle JKL, where $\angle J$ is $90^\circ$, side JL is known to be 12 units, and the hypotenuse KL is 13 units. Calculate $\tan K$ and $\cos L$. | {
"answer": "\\frac{5}{13}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Among all pairs of real numbers $(x, y)$ such that $\cos \sin x = \cos \sin y$ with $-\frac{15\pi}{2} \le x, y \le \frac{15\pi}{2}$, Ana randomly selects a pair $(X, Y)$. Compute the probability that $X = Y$. | {
"answer": "\\frac{1}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that events $A$ and $B$ are independent, and $P(A)=\frac{1}{2}$, $P(B)=\frac{2}{3}$, find $P(\overline{AB})$. | {
"answer": "\\frac{1}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The diagram shows a quadrilateral \(PQRS\) made from two similar right-angled triangles, \(PQR\) and \(PRS\). The length of \(PQ\) is 3, the length of \(QR\) is 4, and \(\angle PRQ = \angle PSR\).
What is the perimeter of \(PQRS\)? | {
"answer": "22",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $DEF$, $DE = 8$, $EF = 6$, and $FD = 10$.
[asy]
defaultpen(1);
pair D=(0,0), E=(0,6), F=(8,0);
draw(D--E--F--cycle);
label("\(D\)",D,SW);
label("\(E\)",E,N);
label("\(F\)",F,SE);
[/asy]
Point $Q$ is arbitrarily placed inside triangle $DEF$. What is the probability that $Q$ lies closer to $D$ than to either $E$ or $F$? | {
"answer": "\\frac{1}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If point $P$ is the golden section point of segment $AB$, and $AP < BP$, $BP=10$, then $AP=\_\_\_\_\_\_$. | {
"answer": "5\\sqrt{5} - 5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle \( \triangle ABC \), \( BD \) is a median, \( CF \) intersects \( BD \) at \( E \), and \( BE = ED \). Point \( F \) is on \( AB \), and if \( BF = 5 \), then the length of \( BA \) is: | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $y=\left(m-2\right)x+(m^{2}-4)$ is a direct proportion function, find the possible values of $m$. | {
"answer": "-2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two people, A and B, play a guessing game. First, A thinks of a number denoted as $a$, then B guesses the number A thought of, denoting B's guess as $b$. Both $a$ and $b$ belong to the set $\{0,1,2,…,9\}$. If $|a-b|=1$, then A and B are said to have a "telepathic connection". If two people are randomly chosen to play this game, the probability that they have a "telepathic connection" is ______. | {
"answer": "\\dfrac {9}{50}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $a,b$ are positive real numbers, and $({(a-b)}^{2}=4{{(ab)}^{3}})$, find the minimum value of $\dfrac{1}{a}+\dfrac{1}{b}$ . | {
"answer": "2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that there are 6 male doctors and 3 female nurses who need to be divided into three medical teams, where each team consists of two male doctors and 1 female nurse, find the number of different arrangements. | {
"answer": "540",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle \(ABC\), the sides \(a\), \(b\), and \(c\) are opposite to angles \(A\), \(B\), and \(C\) respectively, with \(a-2b=0\).
1. If \(B= \dfrac{\pi}{6}\), find \(C\).
2. If \(C= \dfrac{2}{3}\pi\) and \(c=14\), find the area of \(\triangle ABC\). | {
"answer": "14 \\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Max sold glasses of lemonade for 25 cents each. He sold 41 glasses on Saturday and 53 glasses on Sunday. What were his total sales for these two days? | {
"answer": "$23.50",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a regular decagon, calculate the number of distinct points in the interior of the decagon where two or more diagonals intersect. | {
"answer": "210",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the number of non-positive integers for which the values of the quadratic polynomial \(2x^2 + 2021x + 2019\) are non-positive. | {
"answer": "1010",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a 2x3 rectangle with six unit squares, the lower left corner at the origin, find the value of $c$ such that a slanted line extending from $(c,0)$ to $(4,4)$ divides the entire region into two regions of equal area. | {
"answer": "\\frac{5}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
When two fair 12-sided dice are tossed, the numbers $a$ and $b$ are obtained. What is the probability that both the two-digit number $ab$ (where $a$ and $b$ are digits) and each of $a$ and $b$ individually are divisible by 4? | {
"answer": "\\frac{1}{16}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The first few rows of a new sequence are given as follows:
- Row 1: $3$
- Row 2: $6, 6, 6, 6$
- Row 3: $9, 9, 9, 9, 9, 9$
- Row 4: $12, 12, 12, 12, 12, 12, 12, 12$
What is the value of the $40^{\mathrm{th}}$ number if this arrangement were continued? | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given vectors $\overrightarrow{a}=(\sin x,\cos x),\overrightarrow{b}=(2\sqrt{3}\cos x-\sin x,\cos x)$, and $f(x)=\overrightarrow{a}\cdot\overrightarrow{b}$.
$(1)$ Find the interval where the function $f(x)$ is monotonically decreasing.
$(2)$ If $f(x_0)=\frac{2\sqrt{3}}{3}$ and $x_0\in\left[\frac{\pi}{6},\frac{\pi}{2}\right]$, find the value of $\cos 2x_0$. | {
"answer": "\\frac{\\sqrt{3}-3\\sqrt{2}}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For how many integers \( n \) between 1 and 20 (inclusive) is \( \frac{n}{18} \) a repeating decimal? | {
"answer": "14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a circle centered at $O$, points $A$ and $C$ lie on the circle, each of $\overline{BA}$ and $\overline{BC}$ are tangent to the circle. Triangle $ABC$ is isosceles with $AB = BC$ and $\angle ABC = 100^\circ$. The circle intersects $\overline{BO}$ at $D$. Determine $\frac{BD}{BO}$.
A) $\frac{1}{3}$
B) $\frac{1}{2}$
C) $\frac{2}{3}$
D) $\frac{3}{4}$ | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the smallest constant $n$, such that for any positive real numbers $x$, $y$, and $z$,
\[\sqrt{\frac{x}{y + 2z}} + \sqrt{\frac{y}{2x + z}} + \sqrt{\frac{z}{x + 2y}} > n.\] | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In cube \( ABCD A_{1} B_{1} C_{1} D_{1} \), with an edge length of 6, points \( M \) and \( N \) are the midpoints of edges \( AB \) and \( B_{1} C_{1} \) respectively. Point \( K \) is located on edge \( DC \) such that \( D K = 2 K C \). Find:
a) The distance from point \( N \) to line \( AK \);
b) The distance between lines \( MN \) and \( AK \);
c) The distance from point \( A_{1} \) to the plane of triangle \( MNK \). | {
"answer": "\\frac{66}{\\sqrt{173}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \(x\) and \(y\) be positive real numbers such that
\[
\frac{1}{x + 1} + \frac{1}{y + 1} = \frac{1}{2}.
\]
Find the minimum value of \(x + 3y.\) | {
"answer": "4 + 4 \\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are $168$ primes below $1000$ . Then sum of all primes below $1000$ is, | {
"answer": "76127",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $f(\alpha)= \frac{\sin(\pi - \alpha)\cos(-\alpha)\cos(-\alpha + \frac{3\pi}{2})}{\cos(\frac{\pi}{2} - \alpha)\sin(-\pi - \alpha)}$.
(1) Find the value of $f(-\frac{41\pi}{6})$;
(2) If $\alpha$ is an angle in the third quadrant and $\cos(\alpha - \frac{3\pi}{2}) = \frac{1}{3}$, find the value of $f(\alpha)$. | {
"answer": "\\frac{2\\sqrt{2}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system $xOy$, line $l_{1}$: $kx-y+2=0$ intersects with line $l_{2}$: $x+ky-2=0$ at point $P$. When the real number $k$ varies, the maximum distance from point $P$ to the line $x-y-4=0$ is \_\_\_\_\_\_. | {
"answer": "3\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the plane rectangular coordinate system $xOy$, the parametric equations of the line $l_{1}$ are $\left\{\begin{array}{l}{x=t}\\{y=kt}\end{array}\right.$ (where $t$ is the parameter), and the parametric equations of the line $l_{2}$ are $\left\{\begin{array}{l}{x=-km+2}\\{y=m}\end{array}\right.$ (where $m$ is the parameter). Let $P$ be the intersection point of the lines $l_{1}$ and $l_{2}$. As $k$ varies, the locus of point $P$ is curve $C_{1}$. <br/>$(Ⅰ)$ Find the equation of the locus of curve $C_{1}$; <br/>$(Ⅱ)$ Using the origin as the pole and the positive $x$-axis as the polar axis, the polar coordinate equation of line $C_{2}$ is $\rho \sin (\theta +\frac{π}{4})=3\sqrt{2}$. Point $Q$ is a moving point on curve $C_{1}$. Find the maximum distance from point $Q$ to line $C_{2}$. | {
"answer": "1+\\frac{5\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Simplify \[\frac{1}{\dfrac{3}{\sqrt{5}+2} + \dfrac{4}{\sqrt{7}-2}}.\] | {
"answer": "\\frac{3}{9\\sqrt{5} + 4\\sqrt{7} - 10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sequence \\(\{a_n\}\) consists of numbers \\(1\\) or \\(2\\), with the first term being \\(1\\). Between the \\(k\\)-th \\(1\\) and the \\(k+1\\)-th \\(1\\), there are \\(2k-1\\) \\(2\\)s, i.e., the sequence \\(\{a_n\}\) is \\(1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, \ldots\\). Let the sum of the first \\(n\\) terms of the sequence \\(\{a_n\}\) be \\(S_n\\), then \\(S_{20} =\\) , \\(S_{2017} =\\) . | {
"answer": "3989",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A and B plan to meet between 8:00 and 9:00 in the morning, and they agreed that the person who arrives first will wait for the other for 10 minutes before leaving on their own. Calculate the probability that they successfully meet. | {
"answer": "\\dfrac{11}{36}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a quadratic equation \( x^{2} + bx + c = 0 \) with roots 98 and 99, within the quadratic function \( y = x^{2} + bx + c \), if \( x \) takes on values 0, 1, 2, 3, ..., 100, how many of the values of \( y \) are divisible by 6? | {
"answer": "67",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Fold a 10m long rope in half 5 times, then cut it in the middle with scissors. How many segments is the rope cut into? | {
"answer": "33",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the sequence $\{a\_n\}$ satisfies $a\_1=1$, and for any $n∈N^∗$, $a_{n+1}=a\_n+n+1$, find the value of $$\frac {1}{a_{1}}+ \frac {1}{a_{2}}+…+ \frac {1}{a_{2017}}+ \frac {1}{a_{2016}}+ \frac {1}{a_{2019}}$$. | {
"answer": "\\frac{2019}{1010}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the parametric equations of line $l$ as $\begin{cases} x=t\cos α \\ y=1+t\sin α \end{cases}\left(t \text{ is a parameter, } \frac{π}{2}\leqslant α < π\right)$, a polar coordinate system is established with the coordinate origin as the pole and the positive semi-axis of $x$ as the polar axis. The polar coordinate equation of circle $C$ is $ρ =2\cos θ$.
(I) Discuss the number of common points between line $l$ and circle $C$;
(II) Draw a perpendicular line to line $l$ passing through the pole, with the foot of the perpendicular denoted as $P$, find the length of the chord formed by the intersection of the trajectory of point $P$ and circle $C$. | {
"answer": "\\frac{2\\sqrt{5}}{5}",
"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 $\overrightarrow{m}=(\sin C,\sin B\cos A)$ and $\overrightarrow{n}=(b,2c)$ with $\overrightarrow{m}\cdot \overrightarrow{n}=0$.
(1) Find angle $A$;
(2) If $a=2 \sqrt {3}$ and $c=2$, find the area of $\triangle ABC$. | {
"answer": "\\sqrt {3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let the function \( g(x) \) take positive real numbers to real numbers such that
\[ xg(y) - yg(x) = g \left( \frac{x}{y} \right) + x - y \]
for all positive real numbers \( x \) and \( y \). Find all possible values of \( g(50) \). | {
"answer": "-24.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A parallelogram-shaped paper WXYZ with an area of 7.17 square centimeters is placed on another parallelogram-shaped paper EFGH, as shown in the diagram. The intersection points A, C, B, and D are formed, and AB // EF and CD // WX. What is the area of the paper EFGH in square centimeters? Explain the reasoning. | {
"answer": "7.17",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a large square of area 100 square units, points $P$, $Q$, $R$, and $S$ are the midpoints of the sides of the square. A line is drawn from each corner of the square to the midpoint of the opposite side, creating a new, smaller, central polygon. What is the area of this central polygon? | {
"answer": "25",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Either increasing the radius of a cylinder by 4 inches or the height by 10 inches results in the same volume. The original height of the cylinder is 5 inches. What is the original radius in inches? | {
"answer": "2 + 2\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a circle, parallel chords of lengths 5, 12, and 13 determine central angles of $\theta$, $\phi$, and $\theta + \phi$ radians, respectively, where $\theta + \phi < \pi$. If $\sin \theta$, which is a positive rational number, is expressed as a fraction in lowest terms, what is the sum of its numerator and denominator? | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $ABC$, $BC = 40$ and $\angle C = 45^\circ$. Let the perpendicular bisector of $BC$ intersect $BC$ at $D$ and extend to meet an extension of $AB$ at $E$. Find the length of $DE$. | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given two points $A(-2,0)$ and $B(0,2)$, and point $C$ is any point on the circle $x^{2}+y^{2}-2x=0$, find the minimum area of $\triangle ABC$. | {
"answer": "3 - \\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many natural numbers from 1 to 700, inclusive, contain the digit 7 at least once? | {
"answer": "133",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $\sqrt{20} \approx 4.472, \sqrt{2} \approx 1.414$, find $-\sqrt{0.2} \approx$____. | {
"answer": "-0.4472",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If \(x\) and \(y\) are positive real numbers such that \(6x^2 + 12xy + 6y^2 = x^3 + 3x^2 y + 3xy^2\), find the value of \(x\). | {
"answer": "\\frac{24}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the ellipse $\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1\ (a > b > 0)$, with $F\_{1}$ as the left focus, $A$ as the right vertex, and $B\_{1}$, $B\_{2}$ as the upper and lower vertices respectively. If the four points $F\_{1}$, $A$, $B\_{1}$, and $B\_{2}$ lie on the same circle, find the eccentricity of this ellipse. | {
"answer": "\\dfrac{\\sqrt{5}-1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A line $l$ in the coordinate plane has the equation $2x - 3y + 30 = 0$. This line is rotated $90^\circ$ counterclockwise about the point $(15,10)$ to obtain line $k'$. Find the $x$-coordinate of the $x$-intercept of $k'$. | {
"answer": "\\frac{65}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $\triangle ABC$, $A=60^{\circ}$, $a=\sqrt{6}$, $b=2$.
$(1)$ Find $\angle B$;
$(2)$ Find the area of $\triangle ABC$. | {
"answer": "\\frac{3 + \\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given two circles $C\_1$: $x^{2}+y^{2}=1$ and $C\_2$: $(x-2)^{2}+(y-4)^{2}=1$, a moving point $P(a,b)$ passes through and forms tangent lines $PM$ and $PN$ to circles $C\_1$ and $C\_2$ respectively with $M$ and $N$ being the points of tangency. If $PM=PN$, find the minimum value of $\sqrt{a^{2}+b^{2}}+\sqrt{(a-5)^{2}+(b+1)^{2}}$. | {
"answer": "\\sqrt{34}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $ABCD$ be a parallelogram with vertices $A(0, 0)$, $B(0, 5)$, $C(x, 5)$, and $D(x, 0)$ where $x > 0$. The parallelogram is titled such that $AD$ and $BC$ make an angle of $30^\circ$ with the horizontal axis. If the area of the parallelogram is 35 square units, find the value of $x$. | {
"answer": "14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A bag contains four balls of identical shape and size, numbered $1$, $2$, $3$, $4$.
(I) A ball is randomly drawn from the bag, its number recorded as $a$, then another ball is randomly drawn from the remaining three, its number recorded as $b$. Find the probability that the quadratic equation $x^{2}+2ax+b^{2}=0$ has real roots.
(II) A ball is randomly drawn from the bag, its number recorded as $m$, then the ball is returned to the bag, and another ball is randomly drawn, its number recorded as $n$. If $(m,n)$ is used as the coordinates of point $P$, find the probability that point $P$ falls within the region $\begin{cases} x-y\geqslant 0 \\ x+y-5 < 0\end{cases}$. | {
"answer": "\\frac {1}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the curves $y=x^2-1$ and $y=1+x^3$ have perpendicular tangents at $x=x_0$, find the value of $x_0$. | {
"answer": "-\\frac{1}{\\sqrt[3]{6}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given vectors $\overrightarrow{m}=(1,3\cos \alpha)$ and $\overrightarrow{n}=(1,4\tan \alpha)$, where $\alpha \in (-\frac{\pi}{2}, \;\;\frac{\pi}{2})$, and their dot product $\overrightarrow{m} \cdot \overrightarrow{n} = 5$.
(I) Find the magnitude of $|\overrightarrow{m}+ \overrightarrow{n}|$;
(II) Let the angle between vectors $\overrightarrow{m}$ and $\overrightarrow{n}$ be $\beta$, find the value of $\tan(\alpha + \beta)$. | {
"answer": "\\frac{\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two lines are perpendicular and intersect at point $O$. Points $A$ and $B$ move along these two lines at a constant speed. When $A$ is at point $O$, $B$ is 500 yards away from point $O$. After 2 minutes, both points $A$ and $B$ are equidistant from $O$. After another 8 minutes, they are still equidistant from $O$. Find the ratio of the speed of $A$ to the speed of $B$. | {
"answer": "\\frac{2}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine, with proof, the smallest positive integer \( n \) with the following property: For every choice of \( n \) integers, there exist at least two whose sum or difference is divisible by 2009. | {
"answer": "1006",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $α∈(0, \dfrac{π}{2})$, $\cos ( \dfrac{π}{4}-α)=2 \sqrt{2}\cos 2α$, then $\sin 2α=$____. | {
"answer": "\\dfrac{15}{16}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Marla has a large white cube that has an edge of 8 feet. She also has enough green paint to cover 200 square feet. Marla uses all the paint to create a white circle centered on each face, surrounded by a green border. What is the area of one of the white circles, in square feet?
A) $15.34$ sq ft
B) $30.67$ sq ft
C) $45.23$ sq ft
D) $60.89$ sq ft | {
"answer": "30.67",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider all polynomials of the form
\[x^7 + b_6 x^6 + b_5 x^5 + \dots + b_2 x^2 + b_1 x + b_0,\]
where \( b_i \in \{0,1\} \) for all \( 0 \le i \le 6 \). Find the number of such polynomials that have exactly two different integer roots, -1 and 0. | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $x$ be Walter's age in 1996. If Walter was one-third as old as his grandmother in 1996, then his grandmother's age in 1996 is $3x$. The sum of their birth years is $3864$, and the sum of their ages in 1996 plus $x$ and $3x$ is $1996+1997$. Express Walter's age at the end of 2001 in terms of $x$. | {
"answer": "37",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A person bought a bond for 1000 yuan with a maturity of one year. After the bond matured, he spent 440 yuan and then used the remaining money to buy the same type of bond again for another year. After the bond matured the second time, he received 624 yuan. Calculate the annual interest rate of this bond. | {
"answer": "4\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
50 businessmen - Japanese, Koreans, and Chinese - are sitting at a round table. It is known that between any two nearest Japanese, there are exactly as many Chinese as there are Koreans at the table. How many Chinese can be at the table? | {
"answer": "32",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The distance from the origin to the line $3x+2y-13=0$ is $\sqrt {13}$. | {
"answer": "\\sqrt{13}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, with $D$ on $AC$ and $F$ on $BC$, given $AB \perp AC$, $AF \perp BC$, and $BD = DF = FC = 1$. If also $D$ is the midpoint of $AC$, find the length of $AC$.
A) 1
B) $\sqrt{2}$
C) $\sqrt{3}$
D) 2
E) $\sqrt[3]{4}$ | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=2ax- \frac {3}{2}x^{2}-3\ln x$, where $a\in\mathbb{R}$ is a constant,
$(1)$ If $f(x)$ is a decreasing function on $x\in[1,+\infty)$, find the range of the real number $a$;
$(2)$ If $x=3$ is an extremum point of $f(x)$, find the maximum value of $f(x)$ on $x\in[1,a]$. | {
"answer": "\\frac {33}{2}-3\\ln 3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sum of the series $\frac{3}{4} + \frac{5}{8} + \frac{9}{16} + \frac{17}{32} + \frac{33}{64} + \frac{65}{128} - 3.5$. | {
"answer": "\\frac{-1}{128}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The graph of the function $y=g(x)$ is shown below. For all $x > 5$, it is true that $g(x) > 0.5$. If $g(x) = \frac{x^2}{Dx^2 + Ex + F}$, where $D, E,$ and $F$ are integers, then find $D+E+F$. Assume the function has vertical asymptotes at $x = -3$ and $x = 4$ and a horizontal asymptote below 1 but above 0.5. | {
"answer": "-24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the hotel has 80 suites, the daily rent is 160 yuan, and for every 20 yuan increase in rent, 3 guests are lost, determine the optimal daily rent to set in order to maximize profits, considering daily service and maintenance costs of 40 yuan for each occupied room. | {
"answer": "360",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose you have an unlimited number of pennies, nickels, dimes, and quarters. Determine the number of ways to make 30 cents using these coins. | {
"answer": "17",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function f(x) = 2sin(x - π/6)sin(x + π/3), x ∈ R.
(1) Find the smallest positive period of the function f(x) and the center of symmetry of its graph.
(2) In △ABC, if A = π/4, and acute angle C satisfies f(C/2 + π/6) = 1/2, find the value of BC/AB. | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $M \Theta N$ represent the remainder when the larger of $M$ and $N$ is divided by the smaller one. For example, $3 \Theta 10 = 1$. For non-zero natural numbers $A$ less than 40, given that $20 \Theta(A \bigodot 20) = 7$, find $A$. | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are two equilateral triangles with a vertex at $(0, 1)$ , with another vertex on the line $y = x + 1$ and with the final vertex on the parabola $y = x^2 + 1$ . Find the area of the larger of the two triangles. | {
"answer": "26\\sqrt{3} + 45",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are $8$ balls of the same size, including $4$ different black balls, $2$ different red balls, and $2$ different yellow balls.<br/>$(1)$ Arrange these $8$ balls in a line, with the black balls together, the 2 red balls adjacent, and the 2 yellow balls not adjacent. Find the number of ways to arrange them;<br/>$(2)$ Take out $4$ balls from these $8$ balls, ensuring that balls of each color are taken. Find the number of ways to do so;<br/>$(3)$ Divide these $8$ balls into three groups, each group having at least $2$ balls. Find the number of ways to divide them. | {
"answer": "490",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many positive 3-digit numbers are multiples of 25, but not of 60? | {
"answer": "33",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $m$ be the smallest positive integer such that $m^2+(m+1)^2+\cdots+(m+10)^2$ is the square of a positive integer $n$ . Find $m+n$ | {
"answer": "95",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The probability that the random variable $X$ follows a normal distribution $N\left( 3,{{\sigma }^{2}} \right)$ and $P\left( X\leqslant 4 \right)=0.84$ can be expressed in terms of the standard normal distribution $Z$ as $P(Z\leqslant z)=0.84$, where $z$ is the z-score corresponding to the upper tail probability $1-0.84=0.16$. Calculate the value of $P(2<X<4)$. | {
"answer": "0.68",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two distinct positive integers $a$ and $b$ are factors of 48. If $a\cdot b$ is not a factor of 48, what is the smallest possible value of $a\cdot b$? | {
"answer": "32",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $ab= \frac{1}{4}$, $a$, $b \in (0,1)$, find the minimum value of $\frac{1}{1-a}+ \frac{2}{1-b}$. | {
"answer": "4+ \\frac{4 \\sqrt{2}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are integers $x$ that satisfy the inequality $|x-2000|+|x| \leq 9999$. Find the number of such integers $x$. | {
"answer": "9999",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given real numbers \( a, b, c \) satisfy
\[
a^{2}+b^{2}-4a \leqslant 1, \quad b^{2}+c^{2}-8b \leqslant -3, \quad c^{2}+a^{2}-12c \leqslant -26,
\]
what is the value of \( (a+b)^{c} \)? | {
"answer": "27",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many distinct four-digit positive integers are there such that the product of their digits equals 18? | {
"answer": "48",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The perimeter of a square with side length $y$ inches is equal to the circumference of a circle with radius 5 centimeters. If 1 inch equals 2.54 centimeters, what is the value of $y$ in inches? Express your answer as a decimal to the nearest hundredth. | {
"answer": "3.09",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a geometric sequence $\{a_n\}$ with the first term $\frac{3}{2}$ and common ratio $-\frac{1}{2}$, and the sum of the first $n$ terms is $S_n$, then when $n\in N^*$, the sum of the maximum and minimum values of $S_n - \frac{1}{S_n}$ is ______. | {
"answer": "\\frac{1}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
(1) Compute: $\sin 6^\circ \sin 42^\circ \sin 66^\circ \sin 78^\circ$
(2) Given that $\alpha$ is an angle in the second quadrant, and $\sin \alpha = \frac{\sqrt{15}}{4}$, find the value of $\frac{\sin(\alpha + \frac{\pi}{4})}{\sin 2\alpha + \cos 2\alpha + 1}$. | {
"answer": "-\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sequence $b_1, b_2, b_3, \dots$ satisfies $b_1 = 25$, $b_9 = 125$, and for $n \ge 3$, $b_n$ is the geometric mean of the first $n - 1$ terms. Find $b_2$. | {
"answer": "625",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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