problem stringlengths 10 5.15k | answer dict |
|---|---|
Veronica put on five rings: one on her little finger, one on her middle finger, and three on her ring finger. In how many different orders can she take them all off one by one?
A) 16
B) 20
C) 24
D) 30
E) 45 | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that α is an acute angle, cos(α+π/6) = 2/3, find the value of sinα. | {
"answer": "\\dfrac{\\sqrt{15} - 2}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the x-coordinate of point Q, given that point P has coordinates $(\frac{3}{5}, \frac{4}{5})$, point Q is in the third quadrant with $|OQ| = 1$ and $\angle POQ = \frac{3\pi}{4}$. | {
"answer": "-\\frac{7\\sqrt{2}}{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the area of a triangle, given that the radius of the inscribed circle is 1, and the lengths of all three altitudes are integers. | {
"answer": "3\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the quadratic function $f(x)=x^{2}+mx+n$.
(1) If $f(x)$ is an even function with a minimum value of $1$, find the analytic expression of $f(x)$;
(2) Under the condition of (1), for the function $g(x)= \frac {6x}{f(x)}$, solve the inequality $g(2^{x}) > 2^{x}$ with respect to $x$;
(3) For the function $h(x)=|f(x)|$, if the maximum value of $h(x)$ is $M$ when $x\in[-1,1]$, and $M\geqslant k$ holds for any real numbers $m$ and $n$, find the maximum value of $k$. | {
"answer": "\\frac {1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Math City has ten streets, none of which are parallel, and some of which can intersect more than once due to their curved nature. There are two curved streets which each make an additional intersection with three other streets. Calculate the maximum number of police officers needed at intersections. | {
"answer": "51",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an ellipse in the Cartesian coordinate system $xoy$, its center is at the origin, the left focus is $F(-\sqrt{3},0)$, and the right vertex is $D(2,0)$. Let point $A(1,\frac{1}{2})$.
(1) Find the standard equation of the ellipse;
(2) If $P$ is a moving point on the ellipse, find the trajectory equation of the midpoint $M$ of the line segment $PA$;
(3) A line passing through the origin $O$ intersects the ellipse at points $B$ and $C$. Find the maximum area of $\triangle ABC$. | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the ellipse $\frac{x^2}{3} + y^2 = 1$ and the line $l: y = kx + m$ intersecting the ellipse at two distinct points $A$ and $B$.
(1) If $m = 1$ and $\overrightarrow{OA} \cdot \overrightarrow{OB} = 0$ ($O$ is the origin), find the value of $k$.
(2) If the distance from the origin $O$ to the line $l$ is $\frac{\sqrt{3}}{2}$, find the maximum area of $\triangle AOB$. | {
"answer": "\\frac{\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A polynomial with integer coefficients is of the form
\[8x^4 + b_3 x^3 + b_2 x^2 + b_1 x + 24 = 0.\]Find the number of different possible rational roots for this polynomial. | {
"answer": "28",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, $|AB|=5$, $|AC|=6$, if $B=2C$, then calculate the length of edge $BC$. | {
"answer": "\\frac {11}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Twelve people arrive at dinner, but the circular table only seats eight. If two seatings, such that one is a rotation of the other, are considered the same, then in how many different ways can we choose eight people, divide them into two groups of four each, and seat each group at two separate circular tables? | {
"answer": "1247400",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the length of the interval of solutions of the inequality $a \le 3x + 6 \le b$ where the length of the interval is $15$. | {
"answer": "45",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A certain department store sells suits and ties, with each suit priced at $1000$ yuan and each tie priced at $200 yuan. During the "National Day" period, the store decided to launch a promotion offering two discount options to customers.<br/>Option 1: Buy one suit and get one tie for free;<br/>Option 2: Pay 90% of the original price for both the suit and the tie.<br/>Now, a customer wants to buy 20 suits and $x$ ties $\left(x > 20\right)$.<br/>$(1)$ If the customer chooses Option 1, the payment will be ______ yuan (expressed as an algebraic expression in terms of $x$). If the customer chooses Option 2, the payment will be ______ yuan (expressed as an algebraic expression in terms of $x$).<br/>$(2)$ If $x=30$, calculate and determine which option is more cost-effective at this point.<br/>$(3)$ When $x=30$, can you come up with a more cost-effective purchasing plan? Please describe your purchasing method. | {
"answer": "21800",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
When Tanya excluded all numbers from 1 to 333 that are divisible by 3 but not by 7, and all numbers that are divisible by 7 but not by 3, she ended up with 215 numbers. Did she solve the problem correctly? | {
"answer": "205",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a recipe that prepares $8$ servings of fruit punch requires $3$ oranges, $2$ liters of juice, and $1$ liter of soda, and Kim has $10$ oranges, $12$ liters of juice, and $5$ liters of soda, determine the greatest number of servings of fruit punch that she can prepare by maintaining the same ratio of ingredients. | {
"answer": "26",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the functions $f(x)=(x-2)e^{x}$ and $g(x)=kx^{3}-x-2$,
(1) Find the range of $k$ such that the function $g(x)$ is not monotonic in the interval $(1,2)$;
(2) Find the maximum value of $k$ such that the inequality $f(x)\geqslant g(x)$ always holds when $x\in[0,+\infty)$. | {
"answer": "\\frac{1}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given points $A$, $B$, $C$ with coordinates $(4,0)$, $(0,4)$, $(3\cos \alpha,3\sin \alpha)$ respectively, and $\alpha\in\left( \frac {\pi}{2}, \frac {3\pi}{4}\right)$. If $\overrightarrow{AC} \perp \overrightarrow{BC}$, find the value of $\frac {2\sin ^{2}\alpha-\sin 2\alpha}{1+\tan \alpha}$. | {
"answer": "- \\frac {7 \\sqrt {23}}{48}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the vertex of angle \\(θ\\) is at the origin of coordinates, its initial side coincides with the positive half-axis of \\(x\\), and its terminal side is on the line \\(4x+3y=0\\), then \\( \dfrac{\cos \left( \left. \dfrac{π}{2}+θ \right. \right)-\sin (-π-θ)}{\cos \left( \left. \dfrac{11π}{2}-θ \right. \right)+\sin \left( \left. \dfrac{9π}{2}+θ \right. \right)}=\)\_\_\_\_\_\_\_\_． | {
"answer": "\\dfrac{8}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the constant term in the expansion of $\left( \sqrt {x}+ \dfrac {1}{2 \sqrt {x}}\right)^{8}$. | {
"answer": "\\dfrac {35}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $z$ be a complex number. If the equation \[x^3 + (4-i)x^2 + (2+5i)x = z\] has two roots that form a conjugate pair, find the absolute value of the real part of $z$ .
*Proposed by Michael Tang* | {
"answer": "423",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the parabola passing through points $A(2-3b, m)$ and $B(4b+c-1, m)$ is $y=-\frac{1}{2}x^{2}+bx-b^{2}+2c$, if the parabola intersects the $x$-axis, calculate the length of segment $AB$. | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A triangle ABC has vertices at points $A = (0,2)$, $B = (0,0)$, and $C = (10,0)$. A vertical line $x = a$ divides the triangle into two regions. Find the value of $a$ such that the area to the left of the line is one-third of the total area of triangle ABC.
A) $\frac{10}{3}$
B) $5$
C) $\frac{15}{4}$
D) $2$ | {
"answer": "\\frac{10}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The left and right foci of the ellipse $\dfrac{x^{2}}{16} + \dfrac{y^{2}}{9} = 1$ are $F_{1}$ and $F_{2}$, respectively. There is a point $P$ on the ellipse such that $\angle F_{1}PF_{2} = 30^{\circ}$. Find the area of triangle $F_{1}PF_{2}$. | {
"answer": "18 - 9\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given vectors $\overrightarrow{a}=(x,3)$ and $\overrightarrow{b}=(-1,y-1)$, and $\overrightarrow{a}+2\overrightarrow{b}=(0,1)$, find the value of $|\overrightarrow{a}+\overrightarrow{b}|$. | {
"answer": "\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system, given that point $P(3,4)$ is a point on the terminal side of angle $\alpha$, if $\cos(\alpha+\beta)=\frac{1}{3}$, where $\beta \in (0,\pi)$, then $\cos \beta =\_\_\_\_\_\_.$ | {
"answer": "\\frac{3 + 8\\sqrt{2}}{15}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $a$, $b$, and $c$ are positive real numbers satisfying $a + 2b + 3c = 6$, find the maximum value of $\sqrt{a + 1} + \sqrt{2b + 1} + \sqrt{3c + 1}$. | {
"answer": "3\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Each of the integers \(1, 2, 3, \ldots, 9\) is assigned to a vertex of a regular 9-sided polygon (every vertex receives exactly one integer from \(\{1, 2, \ldots, 9\}\), and no two vertices receive the same integer) so that the sum of the integers assigned to any three consecutive vertices does not exceed some positive integer \(n\). What is the least possible value of \(n\) for which this assignment can be done? | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Three cards are dealt at random from a standard deck of 52 cards. What is the probability that the first card is an Ace, the second card is a diamond, and the third card is a Jack? | {
"answer": "\\frac{1}{650}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $\tan\alpha= \dfrac {1}{3}$, find the values of the following expressions:
1. $\dfrac {\sin \alpha+\cos\alpha}{5\cos\alpha-\sin\alpha}$
2. $\dfrac {1}{2\sin\alpha\cdot \cos\alpha+\cos ^{2}\alpha}$. | {
"answer": "\\dfrac {2}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $k$ be a positive integer, and the coefficient of the fourth term in the expansion of $(1+ \frac{x}{k})^{k}$ is $\frac{1}{16}$. Consider the functions $y= \sqrt{8x-x^{2}}$ and $y= \frac{1}{4}kx$, and let $S$ be the shaded region enclosed by their graphs. Calculate the probability that the point $(x,y)$ lies within the shaded region $S$ for any $x \in [0,4]$ and $y \in [0,4]$. | {
"answer": "\\frac{\\pi}{4} - \\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the circle $(x+1)^2+(y-2)^2=1$ and the origin O, find the minimum value of the distance |PM| if the tangent line from point P to the circle has a point of tangency M such that |PM|=|PO|. | {
"answer": "\\frac {2 \\sqrt {5}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$1.$ A bottle in the shape of a cone lies on its base. Water is poured into the bottle until its level reaches a distance of 8 centimeters from the vertex of the cone (measured vertically). We now turn the bottle upside down without changing the amount of water it contains; This leaves an empty space in the upper part of the cone that is 2 centimeters high.
Find the height of the bottle. | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $O$ is the circumcenter of $\triangle ABC$, and $D$ is the midpoint of $BC$, if $\overrightarrow{AO} \cdot \overrightarrow{AD} = 4$ and $BC = 2\sqrt{6}$, find the length of $AD$. | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute the value of $p$ such that the equation
\[\frac{2x + 3}{px - 2} = x\]
has exactly one solution. | {
"answer": "-\\frac{4}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the function $f(x)=\sin x+a\cos x$ has a symmetry axis on $x=\frac{5π}{3}$, determine the maximum value of the function $g(x)=a\sin x+\cos x$. | {
"answer": "\\frac {2\\sqrt {3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a lottery game, the host randomly selects one of the four identical empty boxes numbered $1$, $2$, $3$, $4$, puts a prize inside, and then closes all four boxes. The host knows which box contains the prize. When a participant chooses a box, before opening the chosen box, the host randomly opens another box without the prize and asks the participant if they would like to change their selection to increase the chances of winning. Let $A_{i}$ represent the event that box $i$ contains the prize $(i=1,2,3,4)$, and let $B_{i}$ represent the event that the host opens box $i$ $(i=2,3,4)$. Now, if it is known that the participant chose box $1$, then $P(B_{3}|A_{2})=$______; $P(B_{3})=______.$ | {
"answer": "\\frac{1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Simplify the expression:
\[
\frac{4 + 2i}{4 - 2i} + \frac{4 - 2i}{4 + 2i} + \frac{4i}{4 - 2i} - \frac{4i}{4 + 2i}.
\] | {
"answer": "\\frac{2}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Several energy-saving devices with a total weight of 120 kg were delivered to the factory. It is known that the total weight of the three lightest devices is 31 kg, and the total weight of the three heaviest devices is 41 kg. How many energy-saving devices were delivered to the factory if the weights of any two devices are different? | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\tan \alpha +\tan \beta -\tan \alpha \tan \beta +1=0$, and $\alpha ,\beta \in \left(\frac{\pi }{2},\pi \right)$, calculate $\alpha +\beta$. | {
"answer": "\\frac{7\\pi}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the hyperbola $\dfrac {x^{2}}{9}- \dfrac {y^{2}}{27}=1$ with its left and right foci denoted as $F_{1}$ and $F_{2}$ respectively, and $F_{2}$ being the focus of the parabola $y^{2}=2px$, find the area of $\triangle PF_{1}F_{2}$. | {
"answer": "36 \\sqrt {6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that a four-digit integer $MMMM$, with all identical digits, is multiplied by the one-digit integer $M$, the result is the five-digit integer $NPMPP$. Assuming $M$ is the largest possible single-digit integer that maintains the units digit property of $M^2$, find the greatest possible value of $NPMPP$. | {
"answer": "89991",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A sphere with volume $V$ is inside a closed right triangular prism $ABC-A_{1}B_{1}C_{1}$, where $AB \perp BC$, $AB=6$, $BC=8$, and $AA_{1}=3$. Find the maximum value of $V$. | {
"answer": "\\frac{9\\pi}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are five students, A, B, C, D, and E, arranged to participate in the volunteer services for the Shanghai World Expo. Each student is assigned one of four jobs: translator, guide, etiquette, or driver. Each job must be filled by at least one person. Students A and B cannot drive but can do the other three jobs, while students C, D, and E are capable of doing all four jobs. The number of different arrangements for these tasks is _________. | {
"answer": "108",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let the number $9999\cdots 99$ be denoted by $N$ with $94$ nines. Then find the sum of the digits in the product $N\times 4444\cdots 44$. | {
"answer": "846",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Aws plays a solitaire game on a fifty-two card deck: whenever two cards of the same color are adjacent, he can remove them. Aws wins the game if he removes all the cards. If Aws starts with the cards in a random order, what is the probability for him to win? | {
"answer": "\\frac{\\left( \\binom{26}{13} \\right)^2}{\\binom{52}{26}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=2\cos ^{2} \frac{x}{2}- \sqrt {3}\sin x$.
(I) Find the smallest positive period and the range of the function;
(II) If $a$ is an angle in the second quadrant and $f(a- \frac {π}{3})= \frac {1}{3}$, find the value of $\frac {\cos 2a}{1+\cos 2a-\sin 2a}$. | {
"answer": "\\frac{1-2\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Five positive integers (not necessarily all different) are written on five cards. Boris calculates the sum of the numbers on every pair of cards. He obtains only three different totals: 57, 70, and 83. What is the largest integer on any card? | {
"answer": "48",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle \(ABC\), the sides are known: \(AB = 6\), \(BC = 4\), and \(AC = 8\). The angle bisector of \(\angle C\) intersects side \(AB\) at point \(D\). A circle is drawn through points \(A\), \(D\), and \(C\), intersecting side \(BC\) at point \(E\). Find the area of triangle \(ADE\). | {
"answer": "\\frac{3 \\sqrt{15}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system $xOy$, the equation of circle $C$ is $(x- \sqrt {3})^{2}+(y+1)^{2}=9$. Establish a polar coordinate system with $O$ as the pole and the non-negative half-axis of $x$ as the polar axis.
$(1)$ Find the polar equation of circle $C$;
$(2)$ The line $OP$: $\theta= \frac {\pi}{6}$ ($p\in R$) intersects circle $C$ at points $M$ and $N$. Find the length of segment $MN$. | {
"answer": "2 \\sqrt {6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that rectangle ABCD has dimensions AB = 7 and AD = 8, and right triangle DCE shares the same height as rectangle side DC = 7 and extends horizontally from D towards E, and the area of the right triangle DCE is 28, find the length of DE. | {
"answer": "\\sqrt{113}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $F_1$ and $F_2$ are the common foci of the ellipse $C_1: \frac{x^2}{4} + y^2 = 1$ and the hyperbola $C_2$, and $A, B$ are the common points of $C_1$ and $C_2$ in the second and fourth quadrants, respectively. If the quadrilateral $AF_1BF_2$ is a rectangle, determine the eccentricity of $C_2$. | {
"answer": "\\frac{\\sqrt{6}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the polar coordinate system, the polar equation of curve \\(C\\) is given by \\(\rho = 6\sin \theta\\). The polar coordinates of point \\(P\\) are \\((\sqrt{2}, \frac{\pi}{4})\\). Taking the pole as the origin and the positive half-axis of the \\(x\\)-axis as the polar axis, a Cartesian coordinate system is established.
\\((1)\\) Find the Cartesian equation of curve \\(C\\) and the Cartesian coordinates of point \\(P\\);
\\((2)\\) A line \\(l\\) passing through point \\(P\\) intersects curve \\(C\\) at points \\(A\\) and \\(B\\). If \\(|PA| = 2|PB|\\), find the value of \\(|AB|\\). | {
"answer": "3\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given in the Cartesian coordinate system $xOy$, the parametric equation of curve $C$ is $\begin{cases} x=3+5\cos \alpha \\ y=4+5\sin \alpha \end{cases}$, ($\alpha$ is the parameter), points $A$ and $B$ are on curve $C$. With the origin $O$ as the pole and the positive half-axis of $x$ as the polar axis, the polar coordinates of points $A$ and $B$ are respectively $A(\rho_{1}, \frac{\pi}{6})$, $B(\rho_{2}, \frac{\pi}{2})$
(Ⅰ) Find the polar equation of curve $C$;
(Ⅱ) Suppose the center of curve $C$ is $M$, find the area of $\triangle MAB$. | {
"answer": "\\frac{25\\sqrt{3}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the line $l_{1}$: $x+my-2=0$ intersects the line $l_{2}$: $mx-y+2=0$ at point $P$, and a tangent line passing through point $P$ is drawn to the circle $C: (x+2)^{2} + (y+2)^{2} = 1$, with the point of tangency being $M$, then the maximum value of $|PM|$ is ____. | {
"answer": "\\sqrt{31}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Evaluate the expression $\log_{10} 60 + \log_{10} 80 - \log_{10} 15$. | {
"answer": "2.505",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $i$ is the imaginary unit, $a\in\mathbb{R}$, if $\frac{1-i}{a+i}$ is a pure imaginary number, calculate the modulus of the complex number $z=(2a+1)+ \sqrt{2}i$. | {
"answer": "\\sqrt{11}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the terminal side of angle \\(\alpha\\) passes through the point \\(P(m,2\sqrt{2})\\), \\(\sin \alpha= \frac{2\sqrt{2}}{3}\\) and \\(\alpha\\) is in the second quadrant.
\\((1)\\) Find the value of \\(m\\);
\\((2)\\) If \\(\tan \beta= \sqrt{2}\\), find the value of \\( \frac{\sin \alpha\cos \beta+3\sin \left( \frac{\pi}{2}+\alpha\right)\sin \beta}{\cos (\pi+\alpha)\cos (-\beta)-3\sin \alpha\sin \beta}\\). | {
"answer": "\\frac{\\sqrt{2}}{11}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\sin x \cdot \cos x = -\frac{1}{4}$ and $\frac{3\pi}{4} < x < \pi$, find the value of $\sin x + \cos x$. | {
"answer": "-\\frac{\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $α$ is an acute angle and $\sin α= \frac {3}{5}$, find the value of $\cos α$ and $\cos (α+ \frac {π}{6})$. | {
"answer": "\\frac {4\\sqrt {3}-3}{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the sum of distinct residues of the number $2012^n+m^2$ on $\mod 11$ where $m$ and $n$ are positive integers. | {
"answer": "39",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A shooter's probability of hitting the 10, 9, and 8 rings in a single shot are respectively 0.2, 0.3, and 0.1. Express the probability that the shooter scores no more than 8 in a single shot as a decimal. | {
"answer": "0.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a circle that is tangent to the side \(DC\) of a regular pentagon \(ABCDE\) at point \(D\) and tangent to the side \(AB\) at point \(A\), what is the degree measure of the minor arc \(AD\)? | {
"answer": "144",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 10 steps, and one can take 1, 2, or 3 steps at a time to complete them in 7 moves. Calculate the total number of different ways to do this. | {
"answer": "77",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the triangle \( \triangle ABC \), if \(\sin^2 A + \sin^2 B + \sin^2 C = 2\), calculate the maximum value of \(\cos A + \cos B + 2 \cos C\). | {
"answer": "\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( n \in \mathbf{Z}_{+} \), and
$$
\begin{array}{l}
a, b, c \in \{x \mid x \in \mathbf{Z} \text{ and } x \in [1,9]\}, \\
A_{n} = \underbrace{\overline{a a \cdots a}}_{n \text{ digits}}, B_{n} = \underbrace{b b \cdots b}_{2n \text{ digits}}, C_{n} = \underbrace{c c \cdots c}_{2n \text{ digits}}.
\end{array}
$$
The maximum value of \(a + b + c\) is ( ), given that there exist at least two \(n\) satisfying \(C_{n} - B_{n} = A_{n}^{2}\). | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The integers that can be expressed as a sum of three distinct numbers chosen from the set $\{4,7,10,13, \ldots,46\}$. | {
"answer": "37",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a certain country, there are 47 cities. Each city has a bus station from which buses travel to other cities in the country and possibly abroad. A traveler studied the schedule and determined the number of internal bus routes originating from each city. It turned out that if we do not consider the city of Ozerny, then for each of the remaining 46 cities, the number of internal routes originating from it differs from the number of routes originating from other cities. Find out how many cities in the country have direct bus connections with the city of Ozerny.
The number of internal bus routes for a given city is the number of cities in the country that can be reached from that city by a direct bus without transfers. Routes are symmetric: if you can travel by bus from city $A$ to city $B$, you can also travel by bus from city $B$ to city $A$. | {
"answer": "23",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $F_n$ be the nth Fibonacci number, defined by $F_1 = F_2 = 1$ and $F_n = F_{n-1} + F_{n-2}$ for $n > 2$ . Let $A_0, A_1, A_2,\cdots$ be a sequence of points on a circle of radius $1$ such that the minor arc from $A_{k-1}$ to $A_k$ runs clockwise and such that
\[\mu(A_{k-1}A_k)=\frac{4F_{2k+1}}{F_{2k+1}^2+1}\]
for $k \geq 1$ , where $\mu(XY )$ denotes the radian measure of the arc $XY$ in the clockwise direction. What is the limit of the radian measure of arc $A_0A_n$ as $n$ approaches infinity? | {
"answer": "2\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the minimum possible value of the sum
\[\frac{a}{3b} + \frac{b}{5c} + \frac{c}{7a},\]
where $a,$ $b,$ and $c$ are positive real numbers. | {
"answer": "\\frac{3}{\\sqrt[3]{105}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x) = 2\sin x\cos x - 2\sin^2 x + 1$, determine the smallest positive value of $\varphi$ such that the graph of $f(x)$ shifted to the right by $\varphi$ units is symmetric about the y-axis. | {
"answer": "\\frac{3\\pi}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the base of a triangle is $24$ inches, two lines are drawn parallel to the base, with one of the lines dividing the triangle exactly in half by area, and the other line dividing one of the resulting triangles further into equal areas. If the total number of areas the triangle is divided into is four, find the length of the parallel line closer to the base. | {
"answer": "12\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Warehouse A and Warehouse B originally stored whole bags of grain. If 90 bags are transferred from Warehouse A to Warehouse B, then the grain in Warehouse B will be twice that in Warehouse A. If a certain number of bags are transferred from Warehouse B to Warehouse A, then the grain in Warehouse A will be six times that in Warehouse B. What is the minimum number of bags originally stored in Warehouse A? | {
"answer": "153",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\{1, a, \frac{b}{a}\} = \{0, a^2, a+b\}$, find the value of $a^{2015} + b^{2014}$. | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the line $mx - y + m + 2 = 0$ intersects with circle $C\_1$: $(x + 1)^2 + (y - 2)^2 = 1$ at points $A$ and $B$, and point $P$ is a moving point on circle $C\_2$: $(x - 3)^2 + y^2 = 5$. Determine the maximum area of $\triangle PAB$. | {
"answer": "3\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, where $a=2$, $c=3$, and it satisfies $(2a-c)\cdot\cos B=b\cdot\cos C$. Find the value of $\overrightarrow{AB}\cdot\overrightarrow{BC}$. | {
"answer": "-3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In obtuse triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. Given $a=7$, $b=3$, and $\cos C= \frac{ 11}{14}$.
1. Find the values of $c$ and angle $A$.
2. Find the value of $\sin (2C- \frac{ \pi }{6})$. | {
"answer": "\\frac{ 71}{98}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the set \( M = \{1, 3, 5, 7, 9\} \), find the non-empty set \( A \) such that:
1. Adding 4 to each element in \( A \) results in a subset of \( M \).
2. Subtracting 4 from each element in \( A \) also results in a subset of \( M \).
Determine the set \( A \). | {
"answer": "{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $f(x)=2\sqrt{3}\sin(\pi-x)\sin x-(\sin x-\cos x)^{2}$.
(I) Determine the intervals on which $f(x)$ is increasing;
(II) On the graph of $y=f(x)$, if every horizontal coordinate of the points is stretched to twice its original value (vertical coordinate remains unchanged) and then the resulting graph is translated to the left by $\frac{\pi}{3}$ units, we get the graph of the function $y=g(x)$. Find the value of $g\left(\frac{\pi}{6}\right)$. | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $ABC$, it is known that $BC = 1$, $\angle B = \frac{\pi}{3}$, and the area of $\triangle ABC$ is $\sqrt{3}$. The length of $AC$ is __________. | {
"answer": "\\sqrt{13}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $ABC$ be a triangle with sides $51, 52, 53$ . Let $\Omega$ denote the incircle of $\bigtriangleup ABC$ . Draw tangents to $\Omega$ which are parallel to the sides of $ABC$ . Let $r_1, r_2, r_3$ be the inradii of the three corener triangles so formed, Find the largest integer that does not exceed $r_1 + r_2 + r_3$ . | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An integer is called "octal" if it is divisible by $8$ or if at least one of its digits is $8$ .
How many integers between $1$ and $100$ are octal? | {
"answer": "27",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\frac{\cos \alpha}{1 + \sin \alpha} = \sqrt{3}$, find the value of $\frac{\cos \alpha}{\sin \alpha - 1}$. | {
"answer": "-\\frac{\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, and it is given that $(2a-c)\cos B=b\cos C$.
(Ⅰ) Find the magnitude of angle $B$;
(Ⅱ) If $a=2$ and $c=3$, find the value of $\sin C$. | {
"answer": "\\frac {3 \\sqrt {21}}{14}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the sample 7, 8, 9, x, y has an average of 8, and xy=60, then the standard deviation of this sample is \_\_\_\_\_\_. | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two boxes of candies have a total of 176 pieces. If 16 pieces are taken out from the second box and put into the first box, the number of pieces in the first box is 31 more than m times the number of pieces in the second box (m is an integer greater than 1). Then, determine the number of pieces originally in the first box. | {
"answer": "131",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
(12 points) A workshop has a total of 12 workers and needs to equip two types of machines. Each type A machine requires 2 people to operate, consumes 30 kilowatt-hours of electricity per day, and can produce products worth 40,000 yuan; each type B machine requires 3 people to operate, consumes 20 kilowatt-hours of electricity per day, and can produce products worth 30,000 yuan. Now, the daily electricity supply to the workshop is no more than 130 kilowatt-hours. How should the workshop equip these two types of machines to maximize the daily output value? What is the maximum output value in ten thousand yuan? | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Danial went to a fruit stall that sells apples, mangoes, and papayas. Each apple costs $3$ RM ,each mango costs $4$ RM , and each papaya costs $5$ RM . He bought at least one of each fruit, and paid exactly $50$ RM. What is the maximum number of fruits that he could have bought? | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $M = 36 \cdot 36 \cdot 77 \cdot 330$. Find the ratio of the sum of the odd divisors of $M$ to the sum of the even divisors of $M$. | {
"answer": "1 : 62",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=|2x-a|+|x+ \frac {2}{a}|$
$(1)$ When $a=2$, solve the inequality $f(x)\geqslant 1$;
$(2)$ Find the minimum value of the function $g(x)=f(x)+f(-x)$. | {
"answer": "4 \\sqrt {2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider a square where each side measures 1 unit. At each vertex of the square, a quarter circle is drawn outward such that each side of the square serves as the radius for two adjoining quarter circles. Calculate the total perimeter formed by these quarter circles. | {
"answer": "2\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A fair six-sided die has faces numbered $1, 2, 3, 4, 5, 6$. The die is rolled four times, and the results are $a, b, c, d$. What is the probability that one of the numbers in the set $\{a, a+b, a+b+c, a+b+c+d\}$ is equal to 4? | {
"answer": "$\\frac{343}{1296}$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a polar coordinate system, the polar equation of curve C is $\rho=2\cos\theta+2\sin\theta$. Establish a Cartesian coordinate system with the pole as the origin and the positive x-axis as the polar axis. The parametric equation of line l is $\begin{cases} x=1+t \\ y= \sqrt{3}t \end{cases}$ (t is the parameter). Find the length of the chord that curve C cuts off on line l. | {
"answer": "\\sqrt{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
At the end of $1997$, Jason was one-third as old as his grandmother. The sum of the years in which they were born was $3852$. How old will Jason be at the end of $2004$?
A) $41.5$
B) $42.5$
C) $43.5$
D) $44.5$ | {
"answer": "42.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\mathbf{v}$ be a vector such that
\[\left\| \mathbf{v} + \begin{pmatrix} 4 \\ -2 \end{pmatrix} \right\| = 10.\]
Find the smallest possible value of $\|\mathbf{v}\|$. | {
"answer": "10 - 2\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Wei decides to modify the design of his logo by using a larger square and three tangent circles instead. Each circle remains tangent to two sides of the square and to one adjacent circle. If each side of the square is now 24 inches, calculate the number of square inches that will be shaded. | {
"answer": "576 - 108\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If for any $x\in R$, $2x+2\leqslant ax^{2}+bx+c\leqslant 2x^{2}-2x+4$ always holds, then the maximum value of $ab$ is ______. | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $cos\left( \frac{\pi}{6}-\alpha\right) = \frac{\sqrt{3}}{2}$, find the value of $cos\left( \frac{5\pi}{6}+\alpha\right) - sin^2\left(-\alpha+\frac{7\pi}{6}\right)$.
If $cos\alpha = \frac{2}{3}$ and $\alpha$ is an angle in the fourth quadrant, find the value of $\frac{sin(\alpha-2\pi)+sin(-\alpha-3\pi)cos(-\alpha-3\pi)}{cos(\pi -\alpha)-cos(-\pi-\alpha)cos(\alpha-4\pi)}$. | {
"answer": "\\frac{\\sqrt{5}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In convex quadrilateral \(EFGH\), \(\angle E = \angle G\), \(EF = GH = 150\), and \(EH \neq FG\). The perimeter of \(EFGH\) is 580. Find \(\cos E\). | {
"answer": "\\frac{14}{15}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the measure of an orthogonal trihedral angle? What is the sum of the measures of polyhedral angles that share a common vertex, have no common interior points, and together cover the entire space? | {
"answer": "4\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The volume of a regular octagonal prism is $8 \, \mathrm{m}^{3}$, and its height is $2.2 \, \mathrm{m}$. Find the lateral surface area of the prism. | {
"answer": "16 \\sqrt{2.2 \\cdot (\\sqrt{2}-1)}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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