problem stringlengths 10 5.15k | answer dict |
|---|---|
What is the largest six-digit number that can be obtained by removing nine digits from the number 778157260669103, without changing the order of its digits?
(a) 778152
(b) 781569
(c) 879103
(d) 986103
(e) 987776 | {
"answer": "879103",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the focal length of the hyperbola that shares the same asymptotes with the hyperbola $\frac{x^{2}}{9} - \frac{y^{2}}{16} = 1$ and passes through the point $A(-3, 3\sqrt{2})$. | {
"answer": "\\frac{5\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the minimum value of
\[\sqrt{x^2 + (x-2)^2} + \sqrt{(x-2)^2 + (x+2)^2}\] over all real numbers $x$. | {
"answer": "2\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sum of the first 2015 digits of the decimal part of the repeating decimal \(0.0142857\) is $\qquad$ | {
"answer": "9065",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Simplify $\sqrt{\frac{1}{{49}}}=$____; $|{2-\sqrt{5}}|=$____. | {
"answer": "\\sqrt{5}-2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a three-dimensional Cartesian coordinate system, the vertices of triangle ∆ABC are A(3,4,1), B(0,4,5), and C(5,2,0). Find the value of tan A/2. | {
"answer": "\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system $xOy$, the equation of curve $C_{1}$ is $(x-1)^{2}+y^{2}=1$, and the parametric equation of curve $C_{2}$ is:
$$
\begin{cases}
x= \sqrt {2}\cos \theta \\
y=\sin \theta
\end{cases}
$$
($\theta$ is the parameter), with $O$ as the pole and the positive half-axis of $x$ as the polar axis in the polar coordinate system.
(1) Find the polar equations of $C_{1}$ and $C_{2}$.
(2) The ray $y= \frac { \sqrt {3}}{3}x(x\geqslant 0)$ intersects with $C_{1}$ at a point $A$ different from the origin, and intersects with $C_{2}$ at point $B$. Find $|AB|$. | {
"answer": "\\sqrt {3}- \\frac {2 \\sqrt {10}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a trapezoid \(ABCD\) with bases \(AB\) and \(CD\), and angles \(\angle C = 30^\circ\) and \(\angle D = 80^\circ\). Find \(\angle ACB\), given that \(DB\) is the bisector of \(\angle D\). | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Arrange 3 male students and 4 female students in a row. Under the following different requirements, calculate the number of different arrangement methods:
(1) Person A and Person B must stand at the two ends;
(2) All male students must be grouped together;
(3) Male students must not stand next to each other;
(4) Exactly one person stands between Person A and Person B. | {
"answer": "1200",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the value of the polynomial f(x) = 7x^7 + 6x^6 + 5x^5 + 4x^4 + 3x^3 + 2x^2 + x using the Qin Jiushao algorithm when x = 3. Find the value of V₄. | {
"answer": "789",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the ellipse $\frac{x^{2}}{4} + \frac{y^{2}}{2} = 1$ with two foci $F_{1}$ and $F_{2}$. A point $P$ lies on the ellipse such that $| PF_{1} | - | PF_{2} | = 2$. Determine the area of $\triangle PF_{1}F_{2}$. | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Translate the graph of $y= \sqrt {2}\sin (2x+ \frac {\pi}{3})$ to the right by $\phi(0 < \phi < \pi)$ units to obtain the graph of the function $y=2\sin x(\sin x-\cos x)-1$. Then, $\phi=$ ______. | {
"answer": "\\frac {13\\pi}{24}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A person rolls two dice simultaneously and gets the scores $a$ and $b$. The eccentricity $e$ of the ellipse $\frac{y^2}{a^2} + \frac{x^2}{b^2} = 1$ satisfies $e \geq \frac{\sqrt{3}}{2}$. Calculate the probability that this event occurs. | {
"answer": "\\frac{1}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $ABCD$ be a convex quadrilateral with $\angle ABD = \angle BCD$ , $AD = 1000$ , $BD = 2000$ , $BC = 2001$ , and $DC = 1999$ . Point $E$ is chosen on segment $DB$ such that $\angle ABD = \angle ECD$ . Find $AE$ . | {
"answer": "1000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A geometric sequence of positive integers has its first term as 5 and its fourth term as 480. What is the second term of the sequence? | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The pentagon \(PQRST\) is divided into four triangles with equal perimeters. The triangle \(PQR\) is equilateral. \(PTU\), \(SUT\), and \(RSU\) are congruent isosceles triangles. What is the ratio of the perimeter of the pentagon \(PQRST\) to the perimeter of the triangle \(PQR\)? | {
"answer": "5:3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $ABC$, let $AB = 4$, $AC = 7$, $BC = 9$, and $D$ lies on $\overline{BC}$ such that $\overline{AD}$ bisects $\angle BAC$. Find $\cos \angle BAD$. | {
"answer": "\\sqrt{\\frac{5}{14}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $α\in\mathbb{R}$ and $\sin α + 2\cos α = \frac{\sqrt{10}}{2}$, find the value of $\tan α$. | {
"answer": "-\\frac{1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the largest $K$ satisfying the following:
Given any closed intervals $A_1,\ldots, A_N$ of length $1$ where $N$ is an arbitrary positive integer. If their union is $[0,2021]$ , then we can always find $K$ intervals from $A_1,\ldots, A_N$ such that the intersection of any two of them is empty. | {
"answer": "1011",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the red light lasts for $45$ seconds, determine the probability that a pedestrian will have to wait at least $20$ seconds before the light turns green. | {
"answer": "\\dfrac{5}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ respectively. Given that $b=1$, $c= \sqrt {3}$, and $\angle C= \frac {2}{3}\pi$, find the area $S_{\triangle ABC}$. | {
"answer": "\\frac { \\sqrt {3}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A sequence $ (S_n), n \geq 1$ of sets of natural numbers with $ S_1 = \{1\}, S_2 = \{2\}$ and
\[{ S_{n + 1} = \{k \in }\mathbb{N}|k - 1 \in S_n \text{ XOR } k \in S_{n - 1}\}.
\]
Determine $ S_{1024}.$ | {
"answer": "1024",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $ABCD$ be a square of side length $1$ , and let $P$ be a variable point on $\overline{CD}$ . Denote by $Q$ the intersection point of the angle bisector of $\angle APB$ with $\overline{AB}$ . The set of possible locations for $Q$ as $P$ varies along $\overline{CD}$ is a line segment; what is the length of this segment? | {
"answer": "3 - 2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a right square prism $ABCD-A_{1}B_{1}C_{1}D_{1}$ with a base edge length of $1$, and $AB_{1}$ forms a $60^{\circ}$ angle with the base $ABCD$, find the distance from $A_{1}C_{1}$ to the base $ABCD$. | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is GCF(LCM(16, 21), LCM(14, 18))? | {
"answer": "14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the sum of all integers $n$ not less than $3$ such that the measure, in degrees, of an interior angle of a regular $n$ -gon is an integer.
*2016 CCA Math Bonanza Team #3* | {
"answer": "1167",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\mathcal{T}$ be the set of real numbers that can be represented as repeating decimals of the form $0.\overline{ab}$ where $a$ and $b$ are distinct digits. Find the sum of the elements of $\mathcal{T}$. | {
"answer": "\\frac{90}{11}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The ecology club at a school has 30 members: 12 boys and 18 girls. A 4-person committee is to be chosen at random. What is the probability that the committee has at least 1 boy and at least 1 girl? | {
"answer": "\\dfrac{530}{609}",
"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 $CD = 20$, $\tan D = 2$, and $\tan B = 2.5$, then what is $BC$?
[asy]
pair A,B,C,D;
C = (0,0);
D = (20,0);
A = (20,40);
B= (30,40);
draw(A--B--C--D--A);
label("$A$",A,N);
label("$B$",B,N);
label("$C$",C,S);
label("$D$",D,S);
[/asy] | {
"answer": "4\\sqrt{116}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The surface of a clock is circular, and on its circumference, there are 12 equally spaced points representing the hours. Calculate the total number of rectangles that can have these points as vertices. | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that there are 6 teachers with IDs $A$, $B$, $C$, $D$, $E$, $F$ and 4 different schools, with the constraints that each school must have at least 1 teacher and $B$ and $D$ must be arranged in the same school, calculate the total number of different arrangements. | {
"answer": "240",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The graph of the function $y=\sin (2x+\varphi) (0 < \varphi < \pi)$ is shifted to the right by $\frac{\pi}{8}$ and then is symmetric about the $y$-axis. Determine the possible value(s) of $\varphi$. | {
"answer": "\\frac{3\\pi}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Bonnie constructs a rectangular prism frame using 12 pieces of wire, each 8 inches long. Meanwhile, Roark uses 2-inch-long pieces of wire to construct a series of unit rectangular prism frames that are not connected to each other. The total volume of Roark's prisms is equal to the volume of Bonnie's prism. Find the ratio of the total length of Bonnie's wire to Roark's wire. Express your answer as a common fraction. | {
"answer": "\\frac{1}{16}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=\sin \frac {x}{2}\cos \frac {x}{2}+\cos ^{2} \frac {x}{2}-1$.
$(1)$ Find the smallest positive period of the function $f(x)$ and the interval where it is monotonically decreasing;
$(2)$ Find the minimum value of the function $f(x)$ on the interval $\left[ \frac {\pi}{4}, \frac {3\pi}{2}\right]$. | {
"answer": "- \\frac { \\sqrt {2}+1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a certain number quiz, the test score of a student with seat number $n$ ($n=1,2,3,4$) is denoted as $f(n)$. If $f(n) \in \{70,85,88,90,98,100\}$ and it satisfies $f(1)<f(2) \leq f(3)<f(4)$, then the total number of possible combinations of test scores for these 4 students is \_\_\_\_\_\_\_\_. | {
"answer": "35",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an arithmetic sequence $\{a_{n}\}$ and $\{b_{n}\}$, where the sums of the first $n$ terms are $S_{n}$ and $T_{n}$, respectively, and $\left(2n+3\right)S_{n}=nT_{n}$, calculate the value of $\frac{{{a_5}}}{{{b_6}}}$. | {
"answer": "\\frac{9}{25}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $a$ and $b$ are real numbers, and $\frac{a}{1-i} + \frac{b}{2-i} = \frac{1}{3-i}$, find the sum of the first 100 terms of the arithmetic sequence ${an + b}$. | {
"answer": "-910",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A book has a total of 100 pages, numbered sequentially from 1, 2, 3, 4…100. The digit “2” appears in the page numbers a total of \_\_\_\_\_\_ times. | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the ellipse $C:\dfrac{x^2}{b^2}+\dfrac{y^2}{a^2}=1 \left( a > b > 0 \right)$ has an eccentricity of $\dfrac{\sqrt{2}}{2}$, and point $A(1,\sqrt{2})$ is on the ellipse.
$(1)$ Find the equation of ellipse $C$;
$(2)$ If a line $l$ with a slope of $\sqrt{2}$ intersects the ellipse $C$ at two distinct points $B$ and $C$, find the maximum area of $\Delta ABC$. | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A hotpot restaurant in Chongqing operates through three methods: dining in, takeout, and setting up a stall outside (referred to as stall). In June, the ratio of revenue from dining in, takeout, and stall for this hotpot restaurant was $3:5:2$. With the introduction of policies to promote consumption, the owner of the hotpot restaurant expects the total revenue in July to increase. It is projected that the increase in revenue from the stall will account for $\frac{2}{5}$ of the total revenue increase. The revenue from the stall in July will then reach $\frac{7}{20}$ of the total revenue in July. In order for the ratio of revenue from dining in to takeout in July to be $8:5$, the additional revenue from takeout in July compared to the total revenue in July will be ______. | {
"answer": "\\frac{1}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 6 locked suitcases and 6 keys for them. However, it is unknown which key opens which suitcase. What is the minimum number of attempts needed to ensure that all suitcases are opened? How many attempts are needed if there are 10 suitcases and 10 keys? | {
"answer": "45",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
When $\frac{1}{2222}$ is expressed as a decimal, what is the sum of the first 60 digits after the decimal point? | {
"answer": "114",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a new sequence, the first term is \(a_1 = 5000\) and the second term is \(a_2 = 5001\). Furthermore, the values of the remaining terms are designed so that \(a_n + a_{n+1} + a_{n+2} = 2n\) for all \( n \geq 1 \). Determine \(a_{1000}\). | {
"answer": "5666",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Xiaoli decides which subject among history, geography, or politics to review during tonight's self-study session based on the outcome of a mathematical game. The rules of the game are as follows: in the Cartesian coordinate system, starting from the origin $O$, and then ending at points $P_{1}(-1,0)$, $P_{2}(-1,1)$, $P_{3}(0,1)$, $P_{4}(1,1)$, $P_{5}(1,0)$, to form $5$ vectors. By randomly selecting any two vectors and calculating the dot product $y$ of these two vectors, if $y > 0$, she will review history; if $y=0$, she will review geography; if $y < 0$, she will review politics.
$(1)$ List all possible values of $y$;
$(2)$ Calculate the probability of Xiaoli reviewing history and the probability of reviewing geography. | {
"answer": "\\dfrac{3}{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The school committee has organized a "Chinese Dream, My Dream" knowledge speech competition. There are 4 finalists, and each contestant can choose any one topic from the 4 backup topics to perform their speech. The number of scenarios where exactly one of the topics is not selected by any of the 4 contestants is ______. | {
"answer": "324",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Inside a pentagon, 1000 points were marked and the pentagon was divided into triangles such that each of the marked points became a vertex of at least one of them. What is the minimum number of triangles that could be formed? | {
"answer": "1003",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Four spheres of radius 1 are placed so that each touches the other three. What is the radius of the smallest sphere that contains all four spheres? | {
"answer": "\\sqrt{\\frac{3}{2}} + 1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given points P(-2, -2), Q(0, -1), and a point R(2, m) is chosen such that PR + PQ is minimized. What is the value of the real number $m$? | {
"answer": "-2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The line $ax+2by=1$ intersects the circle $x^{2}+y^{2}=1$ at points $A$ and $B$ (where $a$ and $b$ are real numbers), and $\triangle AOB$ is a right-angled triangle ($O$ is the origin). The maximum distance between point $P(a,b)$ and point $Q(0,0)$ is ______. | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Six students are to be arranged into two classes, with two students in each class, and there are six classes in total. Calculate the number of different arrangement plans. | {
"answer": "90",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A certain store sells a batch of thermal shirts, with an average daily sales of 20 pieces and a profit of $40 per piece. In order to increase sales and profits, the store has taken appropriate price reduction measures. After investigation, it was found that within a certain range, for every $1 decrease in the unit price of the thermal shirts, the store can sell an additional 2 pieces per day on average. If the store aims to make a daily profit of $1200 by selling this batch of thermal shirts and minimizing inventory, the unit price of the thermal shirts should be reduced by ______ dollars. | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
John is cycling east at a speed of 8 miles per hour, while Bob is also cycling east at a speed of 12 miles per hour. If Bob starts 3 miles west of John, determine the time it will take for Bob to catch up to John. | {
"answer": "45",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $XYZ$, the sides are in the ratio $3:4:5$. If segment $XM$ bisects the largest angle at $X$ and divides side $YZ$ into two segments, find the length of the shorter segment given that the length of side $YZ$ is $12$ inches. | {
"answer": "\\frac{9}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many positive integers less than $800$ are either a perfect cube or a perfect square? | {
"answer": "35",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the positive value of $x$ which satisfies
\[\log_5 (x + 2) + \log_{\sqrt{5}} (x^2 + 2) + \log_{\frac{1}{5}} (x + 2) = 3.\] | {
"answer": "\\sqrt{\\sqrt{125} - 2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The lengths of the three sides of $\triangle ABC$ are 5, 7, and 8, respectively. The radius of its circumcircle is ______, and the radius of its incircle is ______. | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If \( \frac{10+11+12}{3} = \frac{2010+2011+2012+N}{4} \), then find the value of \(N\). | {
"answer": "-5989",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $f(x)=\sin \left( 2x+ \frac{π}{6} \right)+ \frac{3}{2}$, $x\in R$.
(1) Find the minimum positive period of the function $f(x)$;
(2) Find the interval(s) where the function $f(x)$ is monotonically decreasing;
(3) Find the maximum value of the function and the corresponding $x$ value(s). | {
"answer": "\\frac{5}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An ellipse has a focus at coordinates $\left(0,-\sqrt {2}\right)$ and is represented by the equation $2x^{2}-my^{2}=1$. Find the value of the real number $m$. | {
"answer": "-\\dfrac{2}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the greatest common factor of 180, 240, and 300? | {
"answer": "60",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the terminal side of angle θ passes through point P(-x, -6) and $$cosθ=- \frac {5}{13}$$, find the value of $$tan(θ+ \frac {π}{4})$$. | {
"answer": "-\\frac {17}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The square $BCDE$ is inscribed in circle $\omega$ with center $O$ . Point $A$ is the reflection of $O$ over $B$ . A "hook" is drawn consisting of segment $AB$ and the major arc $\widehat{BE}$ of $\omega$ (passing through $C$ and $D$ ). Assume $BCDE$ has area $200$ . To the nearest integer, what is the length of the hook?
*Proposed by Evan Chen* | {
"answer": "67",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $\sin\left( \frac{\pi}{3} + a \right) = \frac{5}{13}$, and $a \in \left( \frac{\pi}{6}, \frac{2\pi}{3} \right)$, find the value of $\sin\left( \frac{\pi}{12} + a \right)$. | {
"answer": "\\frac{17\\sqrt{2}}{26}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a 4 by 4 grid, each of the 16 small squares measures 3 cm by 3 cm and is shaded. Four unshaded circles are then placed on top of the grid, one in each quadrant. The area of the visible shaded region can be written in the form $A-B\pi$ square cm. What is the value of $A+B$? | {
"answer": "180",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A larger square contains two non-overlapping shapes: a circle with diameter $2$ and a rectangle with side lengths $2$ and $4$. Find the smallest possible side length of the larger square such that these shapes can fit without overlapping, and then, find the area of the square $S$ that can be inscribed precisely in the remaining free space inside the larger square.
A) $\sqrt{16-2\pi}$
B) $\pi - \sqrt{8}$
C) $\sqrt{8 - \pi}$
D) $\sqrt{10 - \pi}$ | {
"answer": "\\sqrt{8 - \\pi}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given triangle \( \triangle ABC \) with internal angles \( A, B, C \) opposite to sides \( a, b, c \) respectively, vectors \( \vec{m} = (1, 1 - \sqrt{3} \sin A) \) and \( \vec{n} = (\cos A, 1) \), and \( \vec{m} \perp \vec{n} \):
(1) Find angle \( A \);
(2) Given \( b + c = \sqrt{3} a \), find the value of \( \sin \left(B + \frac{\pi}{6} \right) \). | {
"answer": "\\frac{\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A rectangular yard contains two flower beds in the shape of congruent isosceles right triangles. The yard's remaining area forms a trapezoidal shape, as shown. The lengths of the parallel sides of the trapezoid are $20$ and $30$ meters, respectively. What fraction of the yard is occupied by the flower beds?
A) $\frac{1}{8}$
B) $\frac{1}{6}$
C) $\frac{1}{5}$
D) $\frac{1}{4}$
E) $\frac{1}{3}$ | {
"answer": "\\frac{1}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given vectors $\overrightarrow{m}=(\sin x,-1)$ and $\overrightarrow{n}=(\sqrt{3}\cos x,-\frac{1}{2})$, and the function $f(x)=(\overrightarrow{m}+\overrightarrow{n})\cdot\overrightarrow{m}$.
- (I) Find the interval where $f(x)$ is monotonically decreasing;
- (II) Given $a$, $b$, and $c$ are respectively the sides opposite to angles $A$, $B$, and $C$ in $\triangle ABC$, with $A$ being an acute angle, $a=2\sqrt{3}$, $c=4$, and $f(A)$ is exactly the maximum value of $f(x)$ on the interval $\left[0, \frac{\pi}{2}\right]$, find $A$, $b$, and the area $S$ of $\triangle ABC$. | {
"answer": "2\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $g(x) = ax^7 + bx^3 + dx^2 + cx - 8$. If $g(-7) = 3$, then find $g(7)$. | {
"answer": "-19",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A'Niu is riding a horse to cross a river. There are four horses named A, B, C, and D. It takes 2 minutes for horse A to cross the river, 3 minutes for horse B, 7 minutes for horse C, and 6 minutes for horse D. Only two horses can be driven across the river at a time. The question is: what is the minimum number of minutes required to get all four horses across the river? | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given two lines $l_1: x+3y-3m^2=0$ and $l_2: 2x+y-m^2-5m=0$ intersect at point $P$ ($m \in \mathbb{R}$).
(1) Express the coordinates of the intersection point $P$ of lines $l_1$ and $l_2$ in terms of $m$.
(2) For what value of $m$ is the distance from point $P$ to the line $x+y+3=0$ the shortest? And what is the shortest distance? | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider the sequence $\sqrt{2}, \sqrt{5}, 2\sqrt{2}, \sqrt{11}, \ldots$. Determine the position of $\sqrt{41}$ in this sequence. | {
"answer": "14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given positive numbers $a$, $b$, $c$ satisfying: $a^2+ab+ac+bc=6+2\sqrt{5}$, find the minimum value of $3a+b+2c$. | {
"answer": "2\\sqrt{10}+2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the curve
\[
(x - \arcsin \alpha)(x - \arccos \alpha) + (y - \arcsin \alpha)(y + \arccos \alpha) = 0
\]
is intersected by the line \( x = \frac{\pi}{4} \), determine the minimum value of the length of the chord intercepted as \( \alpha \) varies. | {
"answer": "\\frac{\\pi}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a high school's "Campus Microfilm Festival" event, the school will evaluate the films from two aspects: "viewing numbers" and "expert ratings". If a film A is higher than film B in at least one of these aspects, then film A is considered not inferior to film B. It is known that there are 10 microfilms participating. If a film is not inferior to the other 9 films, it is considered an excellent film. Therefore, the maximum possible number of excellent films among these 10 microfilms is __________. | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $T$ be the set of all rational numbers $r$, $0<r<1$, that have a repeating decimal expansion in the form $0.efghefgh\ldots=0.\overline{efgh}$, where the digits $e$, $f$, $g$, and $h$ are not necessarily distinct. To write the elements of $T$ as fractions in lowest terms, how many different numerators are required? | {
"answer": "6000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the number of multiplication and addition operations needed to compute the value of the polynomial $f(x) = 3x^6 + 4x^5 + 5x^4 + 6x^3 + 7x^2 + 8x + 1$ at $x = 0.7$ using the Horner's method. | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If P and Q are points on the line y = 1 - x and the curve y = -e^x, respectively, find the minimum value of |PQ|. | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Three congruent isosceles triangles are constructed inside an equilateral triangle with a side length of $\sqrt{2}$. Each base of the isosceles triangle is placed on one side of the equilateral triangle. If the total area of the isosceles triangles equals $\frac{1}{2}$ the area of the equilateral triangle, find the length of one of the two congruent sides of one of the isosceles triangles.
A) $\frac{1}{4}$
B) $\frac{1}{3}$
C) $\frac{2\sqrt{2}}{5}$
D) $\frac{1}{2}$
E) $\frac{\sqrt{3}}{4}$ | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $\triangle ABC$, if $\cos B = \frac{{\sqrt{2}}}{2}$, then the minimum value of $(\tan ^{2}A-3)\sin 2C$ is ______. | {
"answer": "4\\sqrt{2} - 6",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Each segment with endpoints at the vertices of a regular 100-gon is painted red if there is an even number of vertices between its endpoints, and blue otherwise (in particular, all sides of the 100-gon are red). Numbers are placed at the vertices such that the sum of their squares equals 1, and the products of the numbers at the endpoints are placed on the segments. Then the sum of the numbers on the red segments is subtracted from the sum of the numbers on the blue segments. What is the largest possible result? | {
"answer": "1/2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the value of the following expressions without using a calculator:
\\((1)\\lg 5^{2}+ \frac {2}{3}\lg 8+\lg 5\lg 20+(\lg 2)^{2}\\)
\\((2)\\) Let \(2^{a}=5^{b}=m\), and \(\frac {1}{a}+ \frac {1}{b}=2\), find \(m\). | {
"answer": "\\sqrt {10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system $xOy$, with $O$ as the pole and the non-negative half-axis of the $x$-axis as the polar axis, a polar coordinate system is established. The polar coordinates of point $P$ are $(3, \frac{\pi}{4})$. The parametric equation of curve $C$ is $\rho=2\cos (\theta- \frac{\pi}{4})$ (with $\theta$ as the parameter).
(Ⅰ) Write the Cartesian coordinates of point $P$ and the Cartesian coordinate equation of curve $C$;
(Ⅱ) If $Q$ is a moving point on curve $C$, find the minimum distance from the midpoint $M$ of $PQ$ to the line $l$: $2\rho\cos \theta+4\rho\sin \theta= \sqrt{2}$. | {
"answer": "\\frac{\\sqrt{10}-1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the random variable $X \sim N(1, \sigma^{2})$, if $P(0 < x < 3)=0.5$, $P(0 < X < 1)=0.2$, then $P(X < 3)=$\_\_\_\_\_\_\_\_\_\_\_ | {
"answer": "0.8",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\alpha$ is an angle in the second quadrant, and $P(x, 4)$ is a point on its terminal side, with $\cos\alpha= \frac {1}{5}x$, then $x= \_\_\_\_\_\_$, and $\tan\alpha= \_\_\_\_\_\_$. | {
"answer": "-\\frac {4}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the number of different patterns that can be created by shading exactly three of the nine small triangles, no two of which can share a side, considering patterns that can be matched by rotations or by reflections as the same. | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $$\overrightarrow {a} = (x-1, y)$$, $$\overrightarrow {b} = (x+1, y)$$, and $$|\overrightarrow {a}| + |\overrightarrow {b}| = 4$$
(1) Find the equation of the trajectory C of point M(x, y).
(2) Let P be a moving point on curve C, and F<sub>1</sub>(-1, 0), F<sub>2</sub>(1, 0), find the maximum and minimum values of $$\overrightarrow {PF_{1}} \cdot \overrightarrow {PF_{2}}$$.
(3) If a line l intersects curve C at points A and B, and a circle with AB as its diameter passes through the origin O, investigate whether the distance from point O to line l is constant. If yes, find the constant value; if no, explain why. | {
"answer": "\\frac {2 \\sqrt {21}}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A rectangular prism has edges $a=b=8$ units and $c=27$ units. Divide the prism into four parts from which a cube can be assembled. | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the sequence ${a_n}$ is an arithmetic sequence with first term $1$ and common difference $2$,
(1) Find the general term formula for ${a_n}$;
(2) Let $b_n = \frac{1}{a_n \cdot a_{n-1}}$. Denote the sum of the first $n$ terms of the sequence ${b_n}$ as $T_n$. Find the minimum value of $T_n$. | {
"answer": "\\frac{1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $α \in \left(0, \frac{\pi}{2}\right)$, $\cos \left(α+ \frac{\pi}{3}\right) = -\frac{2}{3}$, then $\cos α =$ \_\_\_\_\_\_. | {
"answer": "\\frac{\\sqrt{15}-2}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If \(\frac{\left(\frac{a}{c}+\frac{a}{b}+1\right)}{\left(\frac{b}{a}+\frac{b}{c}+1\right)}=11\), and \(a, b\), and \(c\) are positive integers, then the number of ordered triples \((a, b, c)\), such that \(a+2b+c \leq 40\), is: | {
"answer": "42",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a box, there are 6 cards labeled with numbers 1, 2, ..., 6. Now, one card is randomly drawn from the box, and its number is denoted as $a$. After adjusting the cards in the box to keep only those with numbers greater than $a$, a second card is drawn from the box. The probability of drawing an odd-numbered card in the first draw and an even-numbered card in the second draw is __________. | {
"answer": "\\frac{17}{45}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In acute triangle $\triangle ABC$, the lengths of the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given vectors $\overrightarrow{m} = (2, c)$ and $\overrightarrow{n} = (\frac{b}{2}\cos C - \sin A, \cos B)$, with $b = \sqrt{3}$ and $\overrightarrow{m} \perp \overrightarrow{n}$.
(1) Find angle $B$;
(2) Find the maximum area of $\triangle ABC$ and the lengths of the other two sides, $a$ and $c$, when the area is maximum. | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\alpha \in (0, \frac{\pi}{2})$ and $\beta \in (0, \frac{\pi}{2})$, and $\sin(2\alpha + \beta) = \frac{3}{2} \sin(\beta)$, find the minimum value of $\cos(\beta)$. | {
"answer": "\\frac{\\sqrt{5}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $(a_n)_{n \equal{} 1}^\infty$ is defined on real numbers with $a_n \not \equal{} 0$, $a_na_{n \plus{} 3} = a_{n \plus{} 2}a_{n \plus{} 5}$, and $a_1a_2 + a_3a_4 + a_5a_6 = 6$. Find the value of $a_1a_2 + a_3a_4 + \cdots + a_{41}a_{42}$. | {
"answer": "42",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $x \gt -1$, $y \gt 0$, and $x+2y=1$, find the minimum value of $\frac{1}{x+1}+\frac{1}{y}$. | {
"answer": "\\frac{3+2\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let the sequence $a_{1}, a_{2}, \cdots$ be defined recursively as follows: $a_{n}=11a_{n-1}-n$ . If all terms of the sequence are positive, the smallest possible value of $a_{1}$ can be written as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. What is $m+n$ ? | {
"answer": "121",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let
\[f(x)=\int_0^1 |t-x|t \, dt\]
for all real $x$ . Sketch the graph of $f(x)$ . What is the minimum value of $f(x)$ ? | {
"answer": "\\frac{2 - \\sqrt{2}}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The distance from Stockholm to Malmö on a map is 120 cm. The scale on the map is 1 cm: 20 km. If there is a stop in between at Lund, which is 30 cm away from Malmö on the same map, how far is it from Stockholm to Malmö passing through Lund, in kilometers? | {
"answer": "2400",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Define: For any three-digit natural number $m$, if $m$ satisfies that the tens digit is $1$ greater than the hundreds digit, and the units digit is $1$ greater than the tens digit, then this three-digit number is called an "upward number"; for any three-digit natural number $n$, if $n$ satisfies that the tens digit is $1$ less than the hundreds digit, and the units digit is $1$ less than the tens digit, then this three-digit number is called a "downward number." The multiple of $7$ of an "upward number" $m$ is denoted as $F(m)$, and the multiple of $8$ of a "downward number" $n$ is denoted as $G(n)$. If $\frac{F(m)+G(n)}{18}$ is an integer, then each pair of $m$ and $n$ is called a "seven up eight down number pair." In all "seven up eight down number pairs," the maximum value of $|m-n|$ is ______. | {
"answer": "531",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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