problem stringlengths 10 5.15k | answer dict |
|---|---|
Calculate the surface integrals of the first kind:
a) \(\iint_{\sigma}|x| dS\), where \(\sigma\) is defined by \(x^2 + y^2 + z^2 = 1\), \(z \geqslant 0\).
b) \(\iint_{\sigma} (x^2 + y^2) dS\), where \(\sigma\) is defined by \(x^2 + y^2 = 2z\), \(z = 1\).
c) \(\iint_{\sigma} (x^2 + y^2 + z^2) dS\), where \(\sigma\) is the part of the cone defined by \(z^2 - x^2 - y^2 = 0\), \(z \geqslant 0\), truncated by the cylinder \(x^2 + y^2 - 2x = 0\). | {
"answer": "3\\sqrt{2} \\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A regular octagon's perimeter is given as $P=16\sqrt{2}$. If $R_i$ denotes the midpoint of side $V_iV_{i+1}$ (with $V_8V_1$ as the last side), calculate the area of the quadrilateral $R_1R_3R_5R_7$.
A) $8$
B) $4\sqrt{2}$
C) $10$
D) $8 + 4\sqrt{2}$
E) $12$ | {
"answer": "8 + 4\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the line $l_1: x + ay + 6 = 0$ is parallel to the line $l_2: (a-2)x + 3y + 2a = 0$, calculate the distance between lines $l_1$ and $l_2$. | {
"answer": "\\frac{8\\sqrt{2}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Solve the equations:
(1) $(x-3)^2+2x(x-3)=0$
(2) $x^2-4x+1=0$. | {
"answer": "2-\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system $xOy$, with the origin as the pole and the positive half-axis of the $x$-axis as the polar axis, the polar coordinate equation of the curve $C_{1}$ is $\rho \cos \theta = 4$.
$(1)$ Let $M$ be a moving point on the curve $C_{1}$, point $P$ lies on the line segment $OM$, and satisfies $|OP| \cdot |OM| = 16$. Find the rectangular coordinate equation of the locus $C_{2}$ of point $P$.
$(2)$ Suppose the polar coordinates of point $A$ are $({2, \frac{π}{3}})$, point $B$ lies on the curve $C_{2}$. Find the maximum value of the area of $\triangle OAB$. | {
"answer": "2 + \\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In square $ABCD$, points $P$ and $Q$ lie on $\overline{AD}$ and $\overline{AB}$ respectively. Segments $\overline{BP}$ and $\overline{CQ}$ intersect at point $R$, with $BR = 8$ and $PR = 9$. If $\triangle BRP$ is a right triangle with $\angle BRP = 90^\circ$, what is the area of the square $ABCD$?
A) 144
B) 169
C) 225
D) 256
E) 289 | {
"answer": "225",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose there are 15 dogs including Rex and Daisy. We need to divide them into three groups of sizes 6, 5, and 4. How many ways can we form the groups such that Rex is in the 6-dog group and Daisy is in the 4-dog group? | {
"answer": "72072",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \(A\) and \(B\) be two moving points on the ellipse \(x^2 + 3y^2 = 1\), and \(OA \perp OB\) (where \(O\) is the origin). Find the product of the maximum and minimum values of \( |AB| \). | {
"answer": "\\frac{2 \\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, $c$ respectively. The radius of the circumcircle of $\triangle ABC$ is $1$, and $b = acosC - \frac{{\sqrt{3}}}{6}ac$.
$(Ⅰ)$ Find the value of $a$;
$(Ⅱ)$ If $b = 1$, find the area of $\triangle ABC$. | {
"answer": "\\frac{\\sqrt{3}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the function $f(x)$ defined on $\mathbb{R}$ satisfies $f(4)=2-\sqrt{3}$, and for any $x$, $f(x+2)=\frac{1}{-f(x)}$, find $f(2018)$. | {
"answer": "-2-\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an ellipse $E$: $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)$ with left focus $F_{1}$ and right focus $F_{2}$, and the focal distance $F_{1}F_{2}$ is $2$. A line passing through $F_{1}$ intersects the ellipse $E$ at points $A$ and $B$, and the perimeter of $\triangle ABF_{2}$ is $4\sqrt{3}$.
$(1)$ Find the equation of the ellipse $E$;
$(2)$ If the slope of line $AB$ is $2$, find the area of $\triangle ABF_{2}$. | {
"answer": "\\frac{4\\sqrt{15}}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the lines $l_1: ax+2y-1=0$ and $l_2: 8x+ay+2-a=0$, if $l_1 \parallel l_2$, find the value of the real number $a$. | {
"answer": "-4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A traffic light follows a cycle of green for 45 seconds, yellow for 5 seconds, and red for 40 seconds. Sam observes the light for a random five-second interval. What is the probability that the light changes from one color to another during his observation? | {
"answer": "\\frac{1}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The number 2015 can be represented as a sum of consecutive integers in several ways, for example, $2015 = 1007 + 1008$ or $2015 = 401 + 402 + 403 + 404 + 405$. How many ways can this be done? | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For a positive integer $n>1$ , let $g(n)$ denote the largest positive proper divisor of $n$ and $f(n)=n-g(n)$ . For example, $g(10)=5, f(10)=5$ and $g(13)=1,f(13)=12$ . Let $N$ be the smallest positive integer such that $f(f(f(N)))=97$ . Find the largest integer not exceeding $\sqrt{N}$ | {
"answer": "19",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The shape shown is made up of three similar right-angled triangles. The smallest triangle has two sides of side-length 2, as shown. What is the area of the shape? | {
"answer": "14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A projectile is launched with an initial velocity of $u$ at an angle of $\phi$ from the horizontal. The trajectory of the projectile is given by the parametric equations:
\[
x = ut \cos \phi,
\]
\[
y = ut \sin \phi - \frac{1}{2} gt^2,
\]
where $t$ is time and $g$ is the acceleration due to gravity. Suppose $u$ is constant but $\phi$ varies from $0^\circ$ to $180^\circ$. As $\phi$ changes, the highest points of the trajectories trace a closed curve. The area enclosed by this curve can be expressed as $d \cdot \frac{u^4}{g^2}$. Find the value of $d$. | {
"answer": "\\frac{\\pi}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A person rolled a fair six-sided die $100$ times and obtained a $6$ $19$ times. What is the approximate probability of rolling a $6$? | {
"answer": "0.19",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the fifth-largest divisor of 2,500,000,000. | {
"answer": "156,250,000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $\sin\alpha + \cos\alpha = \frac{\sqrt{2}}{3}$, where $\alpha \in (0, \pi)$, calculate the value of $\sin\left(\alpha + \frac{\pi}{12}\right)$. | {
"answer": "\\frac{2\\sqrt{2} + \\sqrt{3}}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the parametric equation of line $l$ as $$\begin{cases} x= \sqrt {3}+t \\ y=7+ \sqrt {3}t\end{cases}$$ ($t$ is the parameter), a coordinate system is established with the origin as the pole and the positive half of the $x$-axis as the polar axis. The polar equation of curve $C$ is $\rho \sqrt {a^{2}\sin^{2}\theta+4\cos^{2}\theta}=2a$ ($a>0$).
1. Find the Cartesian equation of curve $C$.
2. Given point $P(0,4)$, line $l$ intersects curve $C$ at points $M$ and $N$. If $|PM|\cdot|PN|=14$, find the value of $a$. | {
"answer": "\\frac{2\\sqrt{21}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute \[ \left\lfloor \dfrac {1007^3}{1005 \cdot 1006} - \dfrac {1005^3}{1006 \cdot 1007} + 5 \right\rfloor,\] where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x.$ | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For all positive reals $ a$ , $ b$ , and $ c$ , what is the value of positive constant $ k$ satisfies the following inequality?
$ \frac{a}{c\plus{}kb}\plus{}\frac{b}{a\plus{}kc}\plus{}\frac{c}{b\plus{}ka}\geq\frac{1}{2007}$ . | {
"answer": "6020",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The perpendicular bisectors of the sides of triangle $PQR$ meet its circumcircle at points $P',$ $Q',$ and $R',$ respectively. If the perimeter of triangle $PQR$ is 30 and the radius of the circumcircle is 7, then find the area of hexagon $PQ'RP'QR'.$ | {
"answer": "105",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the expression $10 \square 10 \square 10 \square 10 \square 10$, fill in the four spaces with each of the operators "+", "-", "×", and "÷" exactly once. The maximum possible value of the resulting expression is: | {
"answer": "109",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that Josie jogs parallel to a canal along which a boat is moving at a constant speed in the same direction and counts 130 steps to reach the front of the boat from behind it, and 70 steps from the front to the back, find the length of the boat in terms of Josie's steps. | {
"answer": "91",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Point \(P\) is inside an equilateral \(\triangle ABC\) such that the measures of \(\angle APB, \angle BPC, \angle CPA\) are in the ratio 5:6:7. Determine the ratio of the measures of the angles of the triangle formed by \(PA, PB, PC\) (in increasing order). | {
"answer": "2: 3: 4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the parameter equation of line $l$ and the equation of circle $C$ in the polar coordinate system, find the rectangular coordinate equation of circle $C$ and the minimum value of $\frac{1}{|PA|} + \frac{1}{|PB|}$, where $P(1, 2)$ and $A$, $B$ are the intersection points of line $l$ and circle $C$.
The parameter equation of line $l$ in the rectangular coordinate system is $\begin{cases} x = 1 + t\cos\alpha\\ y = 2 + t\sin\alpha \end{cases}$ ($t$ is the parameter), and the equation of circle $C$ in the polar coordinate system (with the same unit length and origin as the rectangular coordinate system, and the positive $x$-axis as the polar axis) is $\rho = 6\sin\theta$. | {
"answer": "\\frac{2\\sqrt{7}}{7}",
"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 $b\tan A = (2c-b)\tan B$.
$(1)$ Find angle $A$;
$(2)$ If $\overrightarrow{m}=(0,-1)$ and $\overrightarrow{n}=(\cos B, 2\cos^2\frac{C}{2})$, find the minimum value of $|\overrightarrow{m}+\overrightarrow{n}|$. | {
"answer": "\\frac{\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $A$ is an interior angle of $\triangle ABC$, when $x= \frac {5\pi}{12}$, the function $f(x)=2\cos x\sin (x-A)+\sin A$ attains its maximum value. The sides opposite to the angles $A$, $B$, $C$ of $\triangle ABC$ are $a$, $b$, $c$ respectively.
$(1)$ Find the angle $A$;
$(2)$ If $a=7$ and $\sin B + \sin C = \frac {13 \sqrt {3}}{14}$, find the area of $\triangle ABC$. | {
"answer": "10\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Sarah is leading a class of $35$ students. Initially, all students are standing. Each time Sarah waves her hands, a prime number of standing students sit down. If no one is left standing after Sarah waves her hands $3$ times, what is the greatest possible number of students that could have been standing before her third wave? | {
"answer": "31",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Vasya has 9 different books by Arkady and Boris Strugatsky, each containing a single work by the authors. Vasya wants to arrange these books on a shelf in such a way that:
(a) The novels "Beetle in the Anthill" and "Waves Extinguish the Wind" are next to each other (in any order).
(b) The stories "Restlessness" and "A Story About Friendship and Non-friendship" are next to each other (in any order).
In how many ways can Vasya do this?
Choose the correct answer:
a) \(4 \cdot 7!\);
b) \(9!\);
c) \(\frac{9!}{4!}\);
d) \(4! \cdot 7!\);
e) another answer. | {
"answer": "4 \\cdot 7!",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $F$ is the focus of the parabola $x^{2}=8y$, $P$ is a moving point on the parabola, and the coordinates of $A$ are $(0,-2)$, find the minimum value of $\frac{|PF|}{|PA|}$. | {
"answer": "\\frac{\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Inside the square \(ABCD\), points \(K\) and \(M\) are marked (point \(M\) is inside triangle \(ABD\), point \(K\) is inside \(BMC\)) such that triangles \(BAM\) and \(DKM\) are congruent \((AM = KM, BM = MD, AB = KD)\). Find \(\angle KCM\) if \(\angle AMB = 100^\circ\). | {
"answer": "35",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a class of 120 students, the teacher recorded the following scores for an exam. Calculate the average score for the class.
\begin{tabular}{|c|c|}
\multicolumn{2}{c}{}\\\hline
\textbf{Score (\%)}&\textbf{Number of Students}\\\hline
95&12\\\hline
85&24\\\hline
75&30\\\hline
65&20\\\hline
55&18\\\hline
45&10\\\hline
35&6\\\hline
\end{tabular} | {
"answer": "69.83",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the equations $z^2 = 1 + 3\sqrt{10}i$ and $z^2 = 2 - 2\sqrt{2}i$, where $i = \sqrt{-1}$, find the vertices formed by the solutions of these equations on the complex plane and compute the area of the quadrilateral they form.
A) $17\sqrt{6} - 2\sqrt{2}$
B) $18\sqrt{6} - 3\sqrt{2}$
C) $19\sqrt{6} - 2\sqrt{2}$
D) $20\sqrt{6} - 2\sqrt{2}$ | {
"answer": "19\\sqrt{6} - 2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For real numbers \( x \) and \( y \), simplify the equation \(\cfrac{\frac{1}{x} + \frac{1}{y}}{\frac{1}{x} + 2\frac{1}{y}} = 4\) and express it as \(\frac{x+y}{x+2y}\). | {
"answer": "\\frac{4}{11}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Out of 100 externally identical marbles, one is radioactive, but I don't know which one it is. A friend of mine would buy only non-radioactive marbles from me, at a price of 1 forint each. Another friend of mine has an instrument that can determine whether or not there is a radioactive marble among any number of marbles. He charges 1 forint per measurement, but if there is a radioactive marble among those being measured, all of the marbles in the measurement will become radioactive.
What is the maximum profit I can absolutely achieve? | {
"answer": "92",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the equation of line $l$ is $ax+by+c=0$, where $a$, $b$, and $c$ form an arithmetic sequence, the maximum distance from the origin $O$ to the line $l$ is ______. | {
"answer": "\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the diagram, \( Z \) lies on \( XY \) and the three circles have diameters \( XZ \), \( ZY \), and \( XY \). If \( XZ = 12 \) and \( ZY = 8 \), calculate the ratio of the area of the shaded region to the area of the unshaded region. | {
"answer": "\\frac{12}{13}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the probability of selecting the letter "$s$" in the word "statistics". | {
"answer": "\\frac{3}{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find an integer $n$ such that the decimal representation of the number $5^{n}$ contains at least 1968 consecutive zeros. | {
"answer": "1968",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$F$ is the right focus of the hyperbola $C: \dfrac{x^{2}}{a^{2}} - \dfrac{y^{2}}{b^{2}} = 1 \left(a > 0, b > 0\right)$. A perpendicular line is drawn from point $F$ to asymptote $C$, with the foot of the perpendicular denoted as $A$, intersecting another asymptote at point $B$. If $2\overrightarrow {AF} = \overrightarrow {FB}$, then find the eccentricity of $C$. | {
"answer": "\\dfrac{2\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Joe has exactly enough paint to paint the surface (excluding the bases) of a cylinder with radius 3 and height 4. It turns out this is also exactly enough paint to paint the entire surface of a cube. The volume of this cube is \( \frac{48}{\sqrt{K}} \). What is \( K \)? | {
"answer": "\\frac{36}{\\pi^3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given real numbers $x$ and $y$ satisfying $x^{2}+2y^{2}-2xy=4$, find the maximum value of $xy$. | {
"answer": "2\\sqrt{2} + 2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the sum of the first $n$ terms of the sequence $\{a_n\}$ is $S_n = -a_n - \left(\frac{1}{2}\right)^{n-1} + 2$, and $(1) b_n = 2^n a_n$, find the general term formula for $\{b_n\}$. Also, $(2)$ find the maximum term of $\{a_n\}$. | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $x \gt 0$, $y \gt 0$, and $x+y=1$, find the minimum value of $\frac{2{x}^{2}-x+1}{xy}$. | {
"answer": "2\\sqrt{2}+1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In Dr. Strange's laboratory, there are some bacteria. Each day, 11 bacteria are eliminated, and each night, 5 bacteria are added. If there are 50 bacteria on the morning of the first day, on which day will all the bacteria be eliminated? | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Point $C(0,p)$ lies on the $y$-axis between $Q(0,15)$ and $O(0,0)$. Point $B$ has coordinates $(15,0)$. Determine an expression for the area of $\triangle COB$ in terms of $p$, and compute the length of segment $QB$. Your answer should be simplified as much as possible. | {
"answer": "15\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the plane rectangular coordinate system $xOy$, the parametric equations of the line $l$ are $\left\{{\begin{array}{l}{x=t}\\{y=-1+\sqrt{3}t}\end{array}}\right.$ (where $t$ is a parameter). Establish a polar coordinate system with the coordinate origin $O$ as the pole and the positive half-axis of the $x$-axis as the polar axis. The polar coordinate equation of the curve $C$ is $\rho =2\sin \theta +2\cos \theta$.
$(1)$ Find the general equation of the line $l$ and the rectangular coordinate equation of the curve $C$;
$(2)$ Let point $P(0,-1)$. If the line $l$ intersects the curve $C$ at points $A$ and $B$, find the value of $\frac{1}{{|{PA}|}}+\frac{1}{{|{PB}|}}$. | {
"answer": "\\frac{2\\sqrt{3}+1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two sides of a right triangle have the lengths 8 and 15. What is the product of the possible lengths of the third side? Express the product as a decimal rounded to the nearest tenth. | {
"answer": "215.7",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=e^{x}(ax+b)-x^{2}-4x$, the equation of the tangent line to the curve $y=f(x)$ at the point $(0, f(0))$ is $y=4x+4$.
(1) Find the values of $a$ and $b$;
(2) Determine the intervals of monotonicity for $f(x)$ and find the maximum value of $f(x)$. | {
"answer": "4(1-e^{-2})",
"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 $\angle A=45^{\circ}$, $a=6$.
(1) If $\angle C=105^{\circ}$, find $b$;
(2) Find the maximum area of $\triangle ABC$. | {
"answer": "9(1+\\sqrt{2})",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\theta=\frac{2\pi}{2015}$ , and suppose the product \[\prod_{k=0}^{1439}\left(\cos(2^k\theta)-\frac{1}{2}\right)\] can be expressed in the form $\frac{b}{2^a}$ , where $a$ is a non-negative integer and $b$ is an odd integer (not necessarily positive). Find $a+b$ .
*2017 CCA Math Bonanza Tiebreaker Round #3* | {
"answer": "1441",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A circle has a radius of 3 units. A line segment of length 3 units is tangent to the circle at its midpoint. Determine the area of the region consisting of all such line segments.
A) $1.5\pi$
B) $2.25\pi$
C) $3\pi$
D) $4.5\pi$ | {
"answer": "2.25\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that 3 females and 2 males participate in a performance sequence, and the 2 males cannot appear consecutively, and female A cannot be the first to appear, determine the total number of different performance sequences. | {
"answer": "60",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the plane rectangular coordinate system $xOy$, the parametric equations of the line $l$ are $\left\{{\begin{array}{l}{x=t}\\{y=-1+\sqrt{3}t}\end{array}}\right.$ (where $t$ is a parameter). Establish a polar coordinate system with the coordinate origin $O$ as the pole and the positive half-axis of the $x$-axis as the polar axis. The polar coordinate equation of the curve $C$ is $\rho =2\sin \theta +2\cos \theta$.
$(1)$ Find the general equation of the line $l$ and the rectangular coordinate equation of the curve $C$;
$(2)$ Let point $P(0,-1)$. If the line $l$ intersects the curve $C$ at points $A$ and $B$, find the value of $\frac{1}{{|{PA}|}}+\frac{1}{{|{PB}|}}$. | {
"answer": "\\frac{2\\sqrt{3} + 1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given triangle ABC, the sides opposite to angles A, B, and C are denoted as a, b, and c, respectively, and a = 6. Find the maximum value of the area of triangle ABC given that $\sqrt{7}bcosA = 3asinB$. | {
"answer": "9\\sqrt{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given \( a_{n} = 4^{2n - 1} + 3^{n - 2} \) (for \( n = 1, 2, 3, \cdots \)), where \( p \) is the smallest prime number dividing infinitely many terms of the sequence \( a_{1}, a_{2}, a_{3}, \cdots \), and \( q \) is the smallest prime number dividing every term of the sequence, find the value of \( p \cdot q \). | {
"answer": "5 \\times 13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The inclination angle of the line $x-y+1=0$ can be calculated. | {
"answer": "\\frac{\\pi}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $b$ is an odd multiple of 9, find the greatest common divisor of $8b^2 + 81b + 289$ and $4b + 17$. | {
"answer": "17",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given two non-zero planar vectors $\overrightarrow{a}, \overrightarrow{b}$ that satisfy the condition: for any $λ∈R$, $| \overrightarrow{a}-λ \overrightarrow{b}|≥slant | \overrightarrow{a}- \frac {1}{2} \overrightarrow{b}|$, then:
$(①)$ If $| \overrightarrow{b}|=4$, then $\overrightarrow{a}· \overrightarrow{b}=$ _______ ;
$(②)$ If the angle between $\overrightarrow{a}, \overrightarrow{b}$ is $\frac {π}{3}$, then the minimum value of $\frac {|2 \overrightarrow{a}-t· \overrightarrow{b}|}{| \overrightarrow{b}|}$ is _______ . | {
"answer": "\\sqrt {3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $a$, $b$, $c$ be positive numbers, and $a+b+9c^2=1$. Find the maximum value of $\sqrt{a}+ \sqrt{b}+ \sqrt{3}c$. | {
"answer": "\\frac{\\sqrt{21}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two circles of radius 3 are centered at $(3,0)$ and at $(0,3)$. What is the area of the intersection of the interiors of these two circles? Express your answer in fully expanded form in terms of $\pi$. | {
"answer": "\\frac{9\\pi}{2} - 9",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=\sin (\omega x+\varphi)$ with $\omega > 0$ and $|\varphi| < \frac {\pi}{2}$, the function has a minimum period of $4\pi$ and, after being shifted to the right by $\frac {2\pi}{3}$ units, becomes symmetric about the $y$-axis. Determine the value of $\varphi$. | {
"answer": "-\\frac{\\pi}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a regular hexagon $A B C D E F$, the diagonals $A C$ and $C E$ are divided by points $M$ and $N$ respectively in the following ratios: $\frac{A M}{A C} = \frac{C N}{C E} = r$. If points $B$, $M$, and $N$ are collinear, determine the ratio $r$. | {
"answer": "\\frac{\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In an isosceles triangle $ABC$ with $AB = AC = 6$ units and $BC = 5$ units, a point $P$ is randomly selected inside the triangle $ABC$. What is the probability that $P$ is closer to vertex $C$ than to either vertex $A$ or vertex $B$? | {
"answer": "\\frac{1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $f(x)$ and $g(x)$ be two monic cubic polynomials, and let $s$ be a real number. Assume two of the roots of $f(x)$ are $s + 2$ and $s + 5,$ and two of the roots of $g(x)$ are $s + 4$ and $s + 8.$ Given that:
\[ f(x) - g(x) = 2s \] for all real numbers $x.$ Find $s.$ | {
"answer": "3.6",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The letters \( A, J, H, S, M, E \) and the numbers \( 1, 9, 8, 9 \) are "rotated" as follows:
\begin{tabular}{rrr}
AJHSME & 1989 & \\
1. JHSMEA & 9891 & (1st rotation) \\
2. HSMEAJ & 8919 & (2nd rotation) \\
3. SMEAJH & 9198 & (3rd rotation) \\
..... & &
\end{tabular}
To make AJHSME1989 reappear, the minimum number of rotations needed is: | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \(C\) be the circle with the equation \(x^2 - 4y - 18 = -y^2 + 6x + 26\). Find the center \((a, b)\) and radius \(r\) of the circle, and compute \(a + b + r\). | {
"answer": "5 + \\sqrt{57}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given two circles $A:(x+4)^2+y^2=25$ and $B:(x-4)^2+y^2=1$, a moving circle $M$ is externally tangent to both fixed circles. Let the locus of the center of moving circle $M$ be curve $C$.
(I) Find the equation of curve $C$;
(II) If line $l$ intersects curve $C$ at points $P$ and $Q$, and $OP \perp OQ$. Is $\frac{1}{|OP|^2}+\frac{1}{|OQ|^2}$ a constant value? If it is, find the value; if not, explain the reason. | {
"answer": "\\frac{1}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
(1) Given the hyperbola $C$: $\dfrac{x^{2}}{a^{2}} - \dfrac{y^{2}}{b^{2}} = 1$ $(a > 0, b > 0)$, its right vertex is $A$, and a circle $A$ with center $A$ and radius $b$ intersects one of the asymptotes of the hyperbola $C$ at points $M$ and $N$. If $\angle MAN = 60^{\circ}$, then the eccentricity of $C$ is ______.
(2) The equation of one of the asymptotes of the hyperbola $\dfrac{x^{2}}{a^{2}} - \dfrac{y^{2}}{9} = 1$ $(a > 0)$ is $y = \dfrac{3}{5}x$, then $a=$ ______.
(3) A tangent line to the circle $x^{2} + y^{2} = \dfrac{1}{4}a^{2}$ passing through the left focus $F$ of the hyperbola $\dfrac{x^{2}}{a^{2}} - \dfrac{y^{2}}{b^{2}} = 1$ $(a > 0, b > 0)$ intersects the right branch of the hyperbola at point $P$. If $\overrightarrow{OE} = \dfrac{1}{2}(\overrightarrow{OF} + \overrightarrow{OP})$, then the eccentricity of the hyperbola is ______.
(4) A line passing through the focus $F$ of the parabola $y^{2} = 2px$ $(p > 0)$ with an inclination angle of $\dfrac{\pi}{4}$ intersects the parabola at points $A$ and $B$. If the perpendicular bisector of chord $AB$ passes through point $(0,2)$, then $p=$ ______. | {
"answer": "\\dfrac{4}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given non-zero vectors \\(a\\) and \\(b\\) satisfying \\(|b|=2|a|\\) and \\(a \perp (\sqrt{3}a+b)\\), find the angle between \\(a\\) and \\(b\\). | {
"answer": "\\dfrac{5\\pi}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given quadrilateral $ABCD$ where $AC \perp BD$ and $AC=2$, $BD=3$, find the minimum value of $\overrightarrow{AB} \cdot \overrightarrow{CD}$. | {
"answer": "- \\dfrac{13}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the lateral area of a cylinder with a square cross-section is $4\pi$, calculate the volume of the cylinder. | {
"answer": "2\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=2\sin x\cos x+2\sqrt{3}\cos^{2}x-\sqrt{3}$, $x\in R$.
(1) Find the smallest positive period and the monotonically increasing interval of the function $f(x)$;
(2) In acute triangle $ABC$, if $f(A)=1$, $\overrightarrow{AB}\cdot\overrightarrow{AC}=\sqrt{2}$, find the area of $\triangle ABC$. | {
"answer": "\\frac{\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Gavrila is in an elevator cabin which is moving downward with a deceleration of 5 m/s². Find the force with which Gavrila presses on the floor. Gavrila's mass is 70 kg, and the acceleration due to gravity is 10 m/s². Give the answer in newtons, rounding to the nearest whole number if necessary. | {
"answer": "350",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the plane rectangular coordinate system $(xOy)$, with $O$ as the pole and the positive semi-axis of $x$ as the polar axis, establish a polar coordinate system. The polar coordinate equation of the curve $C_{1}$ is $\rho = 4\cos \theta$, and the line $l$: $\begin{cases} x = 1 - \frac{2\sqrt{5}}{5}t \\ y = 1 + \frac{\sqrt{5}}{5}t \end{cases}$ ($t$ is a parameter)
(I) Find the rectangular coordinate equation of the curve $C_{1}$ and the general equation of the line $l$;
(II) If the parametric equation of the curve $C_{2}$ is $\begin{cases} x = 2\cos \alpha \\ y = \sin \alpha \end{cases}$ ($\alpha$ is a parameter), $P$ is a point on the curve $C_{1}$ with a polar angle of $\frac{\pi}{4}$, and $Q$ is a moving point on the curve $C_{2}$, find the maximum value of the distance from the midpoint $M$ of $PQ$ to the line $l$. | {
"answer": "\\frac{\\sqrt{10}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $F_{1}$ and $F_{2}$ are the foci of a pair of related curves, and $P$ is the intersection point of the ellipse and the hyperbola in the first quadrant, when $\angle F_{1}PF_{2}=60^{\circ}$, determine the eccentricity of the ellipse in this pair of related curves. | {
"answer": "\\dfrac{\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the value of $\frac{2468_{10}}{111_{3}} - 3471_{9} + 1234_{7}$. Express your answer in base 10. | {
"answer": "-1919",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the two distinct square roots of a positive number $a$ are $2x-2$ and $6-3x$, find the cube root of $a$. | {
"answer": "\\sqrt[3]{36}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A triangle with side lengths 8, 13, and 17 has an incircle. The side length of 8 is divided by the point of tangency into segments \( r \) and \( s \), with \( r < s \). Find the ratio \( r : s \). | {
"answer": "1: 3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $O$ be the origin, the parabola $C_{1}$: $y^{2}=2px\left(p \gt 0\right)$ and the hyperbola $C_{2}$: $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\left(a \gt 0,b \gt 0\right)$ have a common focus $F$. The line passing through $F$ and perpendicular to the $x$-axis intersects $C_{1}$ at points $A$ and $B$, and intersects $C_{2}$ in the first quadrant at point $M$. If $\overrightarrow{OM}=m\overrightarrow{OA}+n\overrightarrow{OB}\left(m,n\in R\right)$ and $mn=\frac{1}{8}$, find the eccentricity of the hyperbola $C_{2}$. | {
"answer": "\\frac{\\sqrt{6} + \\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the rectangular coordinate system $xOy$, a polar coordinate system is established with the coordinate origin as the pole and the positive semi-axis of the $x$-axis as the polar axis. The polar coordinate equation of circle $C$ is $\rho^2 - 2m\rho\cos\theta + 4\rho\sin\theta = 1 - 2m$.
(1) Find the rectangular coordinate equation of $C$ and its radius.
(2) When the radius of $C$ is the smallest, the curve $y = \sqrt{3}|x - 1| - 2$ intersects $C$ at points $A$ and $B$, and point $M(1, -4)$. Find the area of $\triangle MAB$. | {
"answer": "2 + \\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A two-digit number, when three times the sum of its units and tens digits is subtracted by -2, still results in the original number. Express this two-digit number algebraically. | {
"answer": "28",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the equation of line $l_{1}$ is $y=x$, and the equation of line $l_{2}$ is $y=kx-k+1$, find the value of $k$ for which the area of triangle $OAB$ is $2$. | {
"answer": "\\frac{1}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $y < 1$ and \[(\log_{10} y)^2 - \log_{10}(y^3) = 75,\] compute the value of \[(\log_{10}y)^3 - \log_{10}(y^4).\] | {
"answer": "\\frac{2808 - 336\\sqrt{309}}{8} - 6 + 2\\sqrt{309}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that point \( P \) lies on the hyperbola \(\frac{x^{2}}{16} - \frac{y^{2}}{9} = 1\), and the distance from \( P \) to the right directrix of this hyperbola is the arithmetic mean of the distances from \( P \) to the two foci of this hyperbola, find the x-coordinate of \( P \). | {
"answer": "-\\frac{64}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\alpha$ is an angle in the third quadrant, the function $f(\alpha)$ is defined as:
$$f(\alpha) = \frac {\sin(\alpha - \frac {\pi}{2}) \cdot \cos( \frac {3\pi}{2} + \alpha) \cdot \tan(\pi - \alpha)}{\tan(-\alpha - \pi) \cdot \sin(-\alpha - \pi)}.$$
1. Simplify $f(\alpha)$.
2. If $\cos(\alpha - \frac {3\pi}{2}) = \frac {1}{5}$, find $f(\alpha + \frac {\pi}{6})$. | {
"answer": "\\frac{6\\sqrt{2} - 1}{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider a polynomial with integer coefficients given by:
\[8x^5 + b_4 x^4 + b_3 x^3 + b_2 x^2 + b_1 x + 24 = 0.\]
Find the number of different possible rational roots of this polynomial. | {
"answer": "28",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( N = 34 \times 34 \times 63 \times 270 \). The ratio of the sum of all odd factors of \( N \) to the sum of all even factors of \( N \) is ( ). | {
"answer": "1: 14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $PQR$, $PQ = 8$, $PR = 17$, and the length of median $PM$ is 12. Additionally, the angle $\angle QPR = 60^\circ$. Find the area of triangle $PQR$. | {
"answer": "34\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the distance between the vertices of the hyperbola given by the equation $4x^2 + 16x - 9y^2 + 18y - 23 = 0.$ | {
"answer": "\\sqrt{30}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A circle is divided into six equal sections. Each section is to be coloured with a single colour so that three sections are red, one is blue, one is green, and one is yellow. Two circles have the same colouring if one can be rotated to match the other. How many different colourings are there for the circle? | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\alpha$ is an angle in the third quadrant, $f(\alpha) = \frac {\sin(\pi-\alpha)\cdot \cos(2\pi-\alpha)\cdot \tan(-\alpha-\pi)}{\tan(-\alpha )\cdot \sin(-\pi -\alpha)}$.
1. Simplify $f(\alpha)$;
2. If $\cos\left(\alpha- \frac {3}{2}\pi\right) = \frac {1}{5}$, find the value of $f(\alpha)$;
3. If $\alpha=-1860^\circ$, find the value of $f(\alpha)$. | {
"answer": "\\frac {1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A population consists of $20$ individuals numbered $01$, $02$, $\ldots$, $19$, $20$. Using the following random number table, select $5$ individuals. The selection method is to start from the numbers in the first row and first two columns of the random number table, and select two numbers from left to right each time. If the two selected numbers are not within the population, remove them and continue selecting two numbers to the right. Then, the number of the $4$th individual selected is ______.<br/><table><tbody><tr><td width="84" align="center">$7816$</td><td width="84" align="center">$6572$</td><td width="84" align="center">$0802$</td><td width="84" align="center">$6314$</td><td width="84" align="center">$0702$</td><td width="84" align="center">$4369$</td><td width="84" align="center">$9728$</td><td width="84" align="center">$0198$</td></tr><tr><td align="center">$3204$</td><td align="center">$9234$</td><td align="center">$4935$</td><td align="center">$8200$</td><td align="center">$3623$</td><td align="center">$4869$</td><td align="center">$6938$</td><td align="center">$7481$</td></tr></tbody></table> | {
"answer": "14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The eccentricity of the ellipse given by the parametric equations $\begin{cases} x=3\cos\theta \\ y=4\sin\theta\end{cases}$ is $\frac{\sqrt{7}}{\sqrt{3^2+4^2}}$, calculate this value. | {
"answer": "\\frac { \\sqrt {7}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Selected Exercise $4-4$: Coordinate Systems and Parametric Equations
In the rectangular coordinate system $xOy$, the parametric equations of the curve $C$ are given by $\begin{cases} & x=\cos \theta \\ & y=\sin \theta \end{cases}$, where $\theta$ is the parameter. In the polar coordinate system with the same unit length as the rectangular coordinate system $xOy$, taking the origin $O$ as the pole and the non-negative half of the $x$-axis as the polar axis, the equation of the line $l$ is given by $\sqrt{2}p \sin (\theta - \frac{\pi}{4}) = 3$.
(I) Find the Cartesian equation of the curve $C$ and the equation of the line $l$ in rectangular coordinates.
(II) Let $P$ be any point on the curve $C$. Find the maximum distance from the point $P$ to the line $l$. | {
"answer": "\\frac{3\\sqrt{2}}{2} + 1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose that \( f(x) \) and \( g(x) \) are functions which satisfy \( f(g(x)) = x^3 \) and \( g(f(x)) = x^4 \) for all \( x \ge 1 \). If \( g(81) = 81 \), compute \( [g(3)]^4 \). | {
"answer": "81",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the diagram below, $ABCD$ is a trapezoid such that $\overline{AB}\parallel \overline{CD}$ and $\overline{AC}\perp\overline{CD}$. If $CD = 15$, $\tan D = 2$, and $\tan B = 3$, then what is $BC$? | {
"answer": "10\\sqrt{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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