problem stringlengths 14 4.09k | solution stringlengths 1 802k | task_type stringclasses 3
values | problem_tokens int64 5 895 |
|---|---|---|---|
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;
$(2)$ Given that line $l$ intersects the ellipse and the midpoint of chord $BC$... | 1 + \frac{\sqrt{3}}{2} | math | 130 |
Simplify the expression $2 - \frac{2}{2 + \sqrt{5}} + \frac{2}{2 - \sqrt{5}}$.
A) $2 - 4\sqrt{5}$
B) $2 + 4\sqrt{5}$
C) $2$
D) $-4\sqrt{5}$ | 2 - 4\sqrt{5} | math | 75 |
Given $a \gt 0$, $b\in R$, if the inequality $\left(ax-2\right)(-x^{2}-bx+4)\leqslant 0$ holds for all $x \gt 0$, then the minimum value of $b+\frac{3}{a}$ is ______. | 2\sqrt{2} | math | 68 |
A graph has \( 12k \) vertices. Each vertex is connected by \( 3k+6 \) edges. For any two vertices, the number of vertices connected to both of them is the same. Determine the value of \( k \). | 3 | math | 53 |
How many positive integers less than $100$ yield a remainder of $2$ when divided by $7$? What is the sum of these numbers? | 665 | math | 33 |
Given a geometric sequence $\{a_n\}$ with a sum of the first $n$ terms as $S_n$, and $S_{10}:S_5 = 1:2$, find the value of $\frac{S_5 + S_{10} + S_{15}}{S_{10} - S_5}$. | -\frac{9}{2} | math | 75 |
Given the function $f(x)=a\ln x- \frac {1}{2}x^{2}+x$, $g(x)= \frac {1}{2}x^{2}-2x+1$.
(I) When $a=2$, find the maximum and minimum values of $f(x)$ for $x\in[1,e^{2}]$ (Reference data: $e^{2}\approx7.4$);
(II) If for all $x\in(0,+\infty)$, $f(x)+g(x)\leqslant 0$ always holds, find the value of the real number $a$. | 1 | math | 139 |
Given that the circumcenter O of triangle $ABC$ satisfies $\overrightarrow{AO} = \frac{1}{3}(\overrightarrow{AB} + \overrightarrow{AC})$, find the value of $\cos A$. | \frac{1}{2} | math | 48 |
Five students (including A, B, C) are arranged in a row. A must be adjacent to B, and A must not be adjacent to C. The number of different ways to arrange them is _____. (Provide your answer in numerical form) | 36 | math | 51 |
Team A and Team B play a series of games where the first team to win three games wins the series. Each team is equally likely to win each game, there are no ties, and the outcomes of the individual games are independent. Additionally, it's given that Team A never wins two consecutive games until they win the final game... | 1 | math | 107 |
Given the price of an item increased by 30% in February, decreased by 10% in March, increased by 15% in April, and then decreased by $y\%$ in May, determine the percentage $y$ such that the price of the item at the end of May is the same as it had been at the beginning of February. | 26 | math | 76 |
What is the greatest common divisor of 1729 and 1337? | 7 | math | 19 |
Given an arithmetic sequence $\{a_n\}$ with the sum of the first $n$ terms denoted as $S_n$ and a common difference of $1$, if $S_6=3S_3$, then calculate the value of $a_9$. | a_9=a_1+8d=2+8=10 | math | 56 |
Given a circle $C: x^{2}+y^{2}=16$ and a line $l: (a-b)x + (3b-2a)y - a = 0$ where $a$ and $b$ are not both zero. As $a$ and $b$ vary, the minimum value of the chord length intercepted by the circle $C$ and the line $l$ is ____. | 2\sqrt{6} | math | 88 |
Calculate the slope angle of the line $x+2=0$. | \frac {π}{2} | math | 14 |
Ms. Chen has two options to commute to her office via routes X and Y. Route X is 8 miles long and she travels at an average speed of 40 miles per hour. Route Y is a total of 7 miles, consisting of 5.5 miles traveled at 50 miles per hour, a 1-mile stretch at 10 miles per hour due to construction, and a 0.5-mile stretch ... | 2.1 | math | 113 |
Given the function $f(x)=|x|+2^{|x|}$, and it satisfies $f(a-1) < f(2)$, the range of values for the real number $a$ is _____. | (-1,3) | math | 47 |
How many lattice points lie on the graph of the equation $x^2 + y^2 = 72$? | 12 | math | 25 |
Given \( f(u) = u^{2} + au + (b-2) \), where \( u = x + \frac{1}{x} \) (with \( x \in \mathbb{R} \) and \( x \neq 0 \)). If \( a \) and \( b \) are real numbers such that the equation \( f(u) = 0 \) has at least one real root, find the minimum value of \( a^{2} + b^{2} \). | 4/5 | math | 109 |
Solve the following equation:
$$
2x^{4} + 2y^{4} - 4x^{3}y + 6x^{2}y^{2} - 4xy^{3} + 7y^{2} + 7z^{2} - 14yz - 70y + 70z + 175 = 0.
$$ | (x, y, z) = (0, 0, -5) | math | 85 |
Given an ellipse C: $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ with $a > b > 0$, draw a perpendicular line from the right focus F<sub>2</sub> to the x-axis that intersects ellipse C at point P. If $\sin\angle PF_1F_2 = \frac{1}{3}$, determine the relationship between $a$ and $b$. | \sqrt{2}b | math | 100 |
For every real number \( x \), evaluate the expression \( (x+1)^{2} - x^{2} \). | 2x + 1 | math | 27 |
Given the parabola $y^{2}=8x$ whose directrix intersects with the hyperbola $\dfrac {x^{2}}{a^{2}}- \dfrac {y^{2}}{b^{2}}=1$ ($a > 0,b > 0$) at points $A$ and $B$, and one of the asymptotes of the hyperbola is $y= \dfrac {4 \sqrt {3}}{3}x$, find the standard equation of the hyperbola. | \dfrac {x^{2}}{3}- \dfrac {y^{2}}{16}=1 | math | 112 |
Let \( A \) be the set of any 20 points on the circumference of a circle. Joining any two points in \( A \) produces one chord of this circle. Suppose every three such chords are not concurrent. Find the number of regions within the circle which are divided by all these chords. | 5036 | math | 63 |
How many increasing arithmetic progressions are there consisting of 22 different natural numbers, where all numbers are not greater than 1000? | 23312 | math | 30 |
For which integer $c$ does $x^2 + x + c$ divide $x^{13} - x + 106$? | 2 | math | 32 |
The point that is $4$ units away from the origin on the number line represents the number $x$ such that $|x|=4$. | 4 \text{ or } -4 | math | 30 |
Consider an equilateral triangle $ABC$ with side length $3$. Right triangle $CBD$ is constructed outwardly on side $BC$ of triangle $ABC$ such that $CB = BD$ and $BCD$ is a right angle at $B$. Find $\sin^2\left(\angle CAD\right)$.
A) $\frac{3}{4}$
B) $\frac{1}{4}$
C) $\frac{1}{2}$
D) $\frac{\sqrt{2}}{2}$
E) $\f... | \frac{1}{2} | math | 122 |
The product of the digits of 1423 is 24. Find how many distinct four-digit positive integers have a product of their digits equal to 18. | 36 | math | 36 |
Find all positive integers \( m \), \( n \) such that \( \frac{1}{m} + \frac{1}{n} - \frac{1}{mn} = \frac{2}{5} \). | (3, 10), (10, 3), (4, 5), (5, 4) | math | 48 |
Given the function $f\left( x \right)=a{{x}^{2}}+x-\ln x$, determine the range of values for the real number $a$ such that the function is monotonically increasing on $\left( 2,+\infty \right)$. | \left[ 0, +\infty \right) | math | 60 |
In the quadratic equation $x^{2}-2ax+b=0$, if $a^{2}-b \gt 0$, then $a$ is called the midpoint value of the equation.<br/>$(1)$ The midpoint value of the equation $x^{2}-8x+3=0$ is ______.<br/>$(2)$ Given that the midpoint value of the equation $x^{2}-mx+n=0$ is $3$, and one of the roots is exactly equal to $n$, find t... | n = 5 | math | 112 |
Given the ellipse $C: \frac{x^{2}}{2}+y^{2}=1$ and the line $l: y=x+\sqrt{3}$, calculate the maximum distance from a point on the ellipse C to the line l. | \sqrt{6} | math | 52 |
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 $2b\cdot\cos A=c\cdot\cos A+a\cdot\cos C$.
$(1)$ Find the magnitude of angle $A$;
$(2)$ If $a=\sqrt{7}$ and $b+c=4$, find the value of $bc$. | 3 | math | 101 |
Elective 4-4: Coordinate System and Parametric Equations
In the Cartesian coordinate system $xOy$, the curve $C$ is given by the parametric equations:
\[
\begin{cases}
x=1+ \sqrt{3}\cos \varphi \\
y= \sqrt{3}\sin \varphi
\end{cases}
\]
where $\varphi$ is the parameter, $0 \leqslant \varphi \leqslant \pi$. A polar coo... | 5 | math | 249 |
Given that the points $(1, -2)$, $(3, 4)$, and $(6, m/3)$ are on the same straight line, find the value(s) of $m$.
**A)** $27$
**B)** $39$
**C)** $52$
**D)** $15$ | 39 | math | 71 |
Given that $P$ is a point on the parabola $y^2=4x$, let the distance from $P$ to the directrix be $d_1$, and the distance from $P$ to point $A(1, 4)$ be $d_2$. Find the minimum value of $d_1+d_2$. | 4 | math | 74 |
Mr. $X$ owns a home worth $$20,000$. He sells it to Mr. $Y$ at a $15\%$ profit. Mr. $Y$ then sells the house back to Mr. $X$ at a $15\%$ loss. How much did Mr. $X$ gain or lose in the end?
A) Mr. X gains $3500$
B) Mr. X gains $3450$
C) Mr. X loses $3450$
D) Mr. X gains $3200$ | 3450 | math | 125 |
Given point $A(1,2)$ and circle $C: x^{2}+y^{2}+2mx+2y+2=0$.
$(1)$ If there are two tangents passing through point $A$, find the range of $m$.
$(2)$ When $m=-2$, a point $P$ on the line $2x-y+3=0$ is chosen to form two tangents $PM$ and $PN$ to the circle. Find the minimum area of quadrilateral $PMCN$. | \frac{7\sqrt{15}}{5} | math | 113 |
Given that $y$ is a quadratic function of $x$ in the form of $y=ax^{2}-bx+2 (a\ne 0).$
(1) When $a=-2$, $b=-4$, find the axis of symmetry and the vertex coordinates of the function graph.
(2) Under the conditions in (1), $Q(m,t)$ is a point on the graph, and its symmetric point $P$ about the origin also falls on the ... | y_{1} > y_{2} | math | 186 |
From a group of 4 male students and 6 female students, determine the number of different ways to select 3 people to participate in a club activity, such that both male and female students are included. | 96 | math | 42 |
Fill in the blanks with appropriate numbers.
4 liters 25 milliliters = ___ milliliters
6.09 cubic decimeters = ___ cubic centimeters
4.9 cubic decimeters = ___ liters ___ milliliters
2.03 cubic meters = ___ cubic meters ___ cubic decimeters. | 30 | math | 65 |
Sally has 12 blue marbles and 8 red marbles in a bag. She removes a marble at random, records the color, puts it back, and then repeats this process until she has withdrawn 8 marbles. What is the probability that exactly five of the marbles that she removes are blue? Express your answer as a decimal rounded to the near... | 0.279 | math | 78 |
What is the value of the following expression: $2 - 5 + 8 - 11 + 14 - \cdots - 47 + 50 - 53 + 56$ ? | 29 | math | 47 |
For a natural number $A$, define the product of its digits in base 10 as $p(A)$. Find all values of $A$ that satisfy the equation $A = 1.5 \cdot p(A)$. | 48 | math | 48 |
Express 1.59 million in scientific notation. | 1.59 \times 10^{6} | math | 11 |
Sides $\overline{AJ}$ and $\overline{EF}$ of regular decagon $ABCDEFGHIJ$ are extended to meet at point $Q$. What is the degree measure of angle $Q$? | 72^\circ | math | 43 |
The diagonal of a particular square is 6 inches. The diameter of a particular circle is 8 inches. By how many square inches is the area of the circle greater than the area of the square? Express your answer as a decimal to the nearest tenth. | 32.3 | math | 52 |
The square of an integer is 140 greater than three times the integer itself. What is the sum of all integers for which this is true? | 4 | math | 31 |
For the given ellipse, calculate the distance between the foci. The ellipse is centered at point (2,2) with a semi-major axis of 5 units and a semi-minor axis of 3 units. | 8 | math | 44 |
Given $|a| = 4$, $|b| = 5$, and $|a + b| = -(a + b)$, find the value of $2a - b$. | 2a - b = 13 \text{ or } -3 | math | 41 |
There are 100 silver coins, ordered by weight, and 101 gold coins, also ordered by weight. It is known that all coins have different weights. We have a two-pan balance scale that allows us to determine which of any two coins is heavier. How can we find the coin that ranks 101st in weight among all the coins with the mi... | 8 \text{ weighings} | math | 84 |
The constant term in the expansion of ${(1+x+\frac{1}{{x}^{2}})}^{5}$ can be found by calculating the coefficient of the term with no $x$ in it. | 31 | math | 43 |
Let $a$ and $b$ be positive integers such that $a-b=8$ and $\text{gcd}\left(\frac{a^3+b^3}{a+b}, ab\right) = 16$. Find the smallest possible value of $b$. | 4 | math | 57 |
Calculate the indefinite integral:
$$
\int \frac{x \cdot \cos x + \sin x}{(x \cdot \sin x)^{2}} \, dx
$$ | -\frac{1}{x \sin x} + C | math | 38 |
Given that each player requires one pair of shoes and one uniform for local matches, and the same pair for international games, where shoes cost $10 per pair and each uniform costs $15 more than a pair of shoes, and the total expenditure for the gear is $5600, calculate the number of players in the Club. | 80 | math | 69 |
For a finite sequence \(P = \left(p_1, p_2, \cdots, p_n\right)\), the Cesaro sum (named after the mathematician Cesaro) is defined as \(\frac{1}{n}(S_1 + S_2 + \cdots + S_n)\), where \(S_k = p_1 + p_2 + \cdots + p_k\) for \(1 \leq k \leq n\). If a sequence \(\left(p_1, p_2, \cdots, p_{99}\right)\) of 99 terms has a Ces... | 991 | math | 182 |
For a positive integer \( n \), let \( x_n \) be the real root of the equation \( n x^{3} + 2 x - n = 0 \). Define \( a_n = \left[ (n+1) x_n \right] \) (where \( [x] \) denotes the greatest integer less than or equal to \( x \)) for \( n = 2, 3, \ldots \). Then find \( \frac{1}{1005} \left( a_2 + a_3 + a_4 + \cdots + a... | 2013 | math | 138 |
Given that segment AB passes through the center of the ellipse $\frac{x^2}{8} + \frac{y^2}{4} = 1$ with points A and B on the ellipse, and $F_1$, $F_2$ being the two foci of the ellipse, find the maximum area of the quadrilateral $F_1AF_2B$. | 8 | math | 79 |
Given positive numbers $a$ and $b$ that satisfy the equation $3ab - 3 = a + 3b$, find the minimum value of $a + 3b$. | 6 | math | 39 |
A tourist travels from point \(A\) to point \(B\) in 1 hour and 56 minutes. The route from \(A\) to \(B\) first goes uphill, then on flat terrain, and finally downhill. What is the length of the flat terrain if the tourist's speed downhill is 6 km/h, uphill is 4 km/h, and on flat terrain is 5 km/h, and the total distan... | 3 | math | 115 |
The length of the chord cut by the line $y=2x-2$ on the circle $(x-2)^{2}+(y-2)^{2}=25$ can be calculated. | 10 | math | 43 |
Given a set of sample data with $9$ numbers, the average is $8$, and the variance is $12$. After adding one more data point to this set, the new average of the sample data becomes $9$. Calculate the variance of the new sample data. | 19.8 | math | 56 |
From the numbers 1 to 200, one or more numbers were selected to form a group with the following property: if the group contains at least two numbers, then the sum of any two numbers in this group is divisible by 5. What is the maximum number of numbers that can be in the group with this property? | 40 | math | 68 |
How many ordered pairs of real numbers $(x, y)$ satisfy the following system of equations?
\[
\left\{
\begin{aligned}
x + 2y &= 2 \\
\left| |x| - 2|y| \right| &= 2
\end{aligned}
\right.
\] | 2 | math | 73 |
Given a function $f(x)$ defined on $\mathbb{R}$ that satisfies the equation $f(x)+f(1-x)=1$ for all $x$, and for $x \geqslant 0$ the relation $f\left(\dfrac{x}{3}\right)=\dfrac{1}{2}f(x)$ holds, and for $0 \leqslant x_1 < x_2 \leqslant 1$, $f(x_1)\leqslant f(x_2)$, find the value of $f\left(\dfrac{1}{2018}\right)$. | \dfrac{1}{128} | math | 137 |
A cylinder has a radius of 5 cm and a height of 12 cm. What is the longest segment that would fit inside this cylinder? | 2\sqrt{61} \text{ cm} | math | 30 |
The height of the pillar at D is 17 meters, but reformatted question is:
What is the height of the pillar at D? | 17 | math | 29 |
What is the sum of $\left(\dfrac{1}{4}\right) + \left(\dfrac{1}{4}\right)^2 + \left(\dfrac{1}{4}\right)^3 + \left(\dfrac{1}{4}\right)^4 + \left(\dfrac{1}{4}\right)^5 + \left(\dfrac{1}{4}\right)^6$? | \frac{4095}{12288} | math | 89 |
A factory produces a certain product, and the defective rate $p$ is related to the daily output $x$ (in ten thousand units) as follows: $P= \begin{cases} \dfrac {1}{6-x}, & 0 < x\leqslant c \\ \dfrac {2}{3}, & x > c\end{cases}$ ($c$ is a constant, and $0 < c < 6$). It is known that for every qualified product produced,... | 3 | math | 215 |
Suppose $X$ is a random variable that takes on only nonnegative integer values, with $E\left[ X \right] = 1$, $E\left[ X^2 \right] = 2$, and $E \left[ X^3 \right] = 5$. Determine the smallest possible value of the probability of the event $X=0$. | \frac{1}{3} | math | 79 |
In \\(\Delta ABC\\), with \\(A=60^\circ\\), \\(b=1\\), and the area being \\(\sqrt{3}\\), find the value of \\(\dfrac{a+b+c}{\sin A+\sin B+\sin C}\\). | \dfrac{2\sqrt{39}}{3} | math | 60 |
The graph of the parabola defined by the equation \( y = (x-2)^2 + 3 \) is rotated 180 degrees about its vertex, then shifted 4 units to the left, and then shifted 3 units down. Find the sum of the zeros of the resulting parabola. | -4 | math | 66 |
Consider the line integral
$$
\int_{L} \frac{-y \, dx}{x^{2}+y^{2}}+\frac{x \, dy}{x^{2}+y^{2}}
$$ | 2\pi | math | 46 |
Angle $EAB$ is a right angle, and $BE = 12$ units. Additionally, a point $H$ is placed such that triangle $EAH$ is also a right triangle with $AH = 5$ units. Determine the number of square units in the sum of the areas of the three squares $ABCD$, $AEFG$, and $AHIJ$. | 169 | math | 80 |
A frequency distribution of the scores for Mr. Sampson's algebra class is shown. What percent of the class received a score in the $60\%$-$69\%$ range? \begin{tabular}{|c|c|}
Test Scores & Frequencies\\
\hline
$90\% - 100\%$& IIII\\
$80\% - 89\%$& IIII IIII\\
$70\% - 79\%$& IIII II\\
$60\% - 69\%$ & IIII I\\
Below $60\... | 20\% | math | 152 |
Let $h_1$ and $h_2$ be the altitudes of a triangle drawn to the sides with length 5 and $2\sqrt 6$, respectively. If $5 + h_1 \leq 2\sqrt 6 + h_2$, determine the length of the third side of the triangle. | 7 | math | 69 |
Given that $3^a = 4^b = 5^c = 6$, find the value of $\frac {1}{a}+ \frac {1}{b}+ \frac {1}{c}$. | \log_{6} 60 | math | 48 |
Factor the following expression: $74a^2 + 222a + 148a^3$. | 74a(2a^2 + a + 3) | math | 26 |
Given the function $f(x)=x^{3}- \frac {3}{2}x^{2}+ \frac {3}{4}x+ \frac {1}{8}$, find the value of $\sum\limits_{k=1}^{2016}f( \frac {k}{2017})$. | 504 | math | 72 |
Given the expansion of \\((x^3+ \frac{1}{ \sqrt{x}})^n \\) contains a constant term and the minimum value of \\(n\\) is \\(a\\), evaluate \\(\int\_{-a}^{a} \sqrt{{a}^{2}-{x}^{2}}dx \\). | \frac{49π}{2} | math | 71 |
Given the functions $f(x)=2(x+1)$ and $g(x)=x+ \ln x$, points $A$ and $B$ are located on the graphs of $f(x)$ and $g(x)$ respectively, and their y-coordinates are always equal. Calculate the minimum distance between points $A$ and $B$. | \frac{3}{2} | math | 70 |
Given an isosceles triangle \(XYZ\) with \(XY = YZ\) and an angle at the vertex equal to \(96^{\circ}\). Point \(O\) is located inside triangle \(XYZ\) such that \(\angle OZX = 30^{\circ}\) and \(\angle OXZ = 18^{\circ}\). Find the measure of angle \(\angle YOX\). | 78 | math | 88 |
Find all integers \( n \) such that \( n^5 + 3 \) is divisible by \( n^2 + 1 \). | -3, -1, 0, 1, 2 | math | 30 |
In the diagram, \( PQR \) is a line segment, \( \angle PQS = 125^\circ \), \( \angle QSR = x^\circ \), and \( SQ = SR \). What is the value of \( x \)? | 70 | math | 55 |
Expand the binomial ${(\sqrt{x}-\frac{2}{x})^{n}}$:
(1) If $n=6$, find the second to last term.
(2) If the coefficient ratio of the 5th term to the 3rd term is $56:3$, find the sum of all binomial coefficients. | 1024 | math | 72 |
The quadratic function \( f(x) = ax^2 + bx + c \) (where \( a, b, c \in \mathbb{R} \) and \( a \neq 0 \)) satisfies the following conditions:
1. \( f(-1) = 0 \);
2. For all \( x \in \mathbb{R} \), \( f(x) \geq x \);
3. For all \( x \in (0, 2) \), \( f(x) \leq \frac{(x + 1)^2}{4} \).
Given \( x_1, x_2, x_3 \in (0, 2) \... | 1 | math | 207 |
The entry fee for a concert is \$30 per adult and \$15 per child. On a particular day, the concert collected \$2250 in entry fees, with at least one adult and one child attending. Determine the ratio of adults to children on that day, such that the ratio is closest to 1. | 1 | math | 67 |
Given that the three vertices of $\triangle ABC$ are $A(4,0)$, $B(6,7)$, and $C(0,3)$.
$(1)$ Find the equation of the line containing the altitude of side $BC$.
$(2)$ Find the equation of the line that bisects the area of $\triangle ABC$ and passes through point $B$. | 11x - 8y - 10 = 0 | math | 82 |
A certain school's second-year junior high students from classes A and B participated in the Hua Cup competition, totaling $a$ students. The average score of class A was 71 points per student, and the average score of class B was 69 points per student. The total score of the two classes was 3480 points. How many studen... | 50 | math | 85 |
Suppose $g(x)$ is a function defined for all real $x$, and suppose $g$ is invertible. Consider the graphs of $y=g(x^3)$ and $y=g(x^6)$. How many points do they intersect? | 2 | math | 52 |
Given that $f(x)$ is an odd function defined on $\mathbb{R}$, for $x \in [0, +\infty)$, $f(x) = 2^x + x - m$ (where $m$ is a constant).
(1) Determine the value of the constant $m$.
(2) Determine the explicit formula for $f(x)$.
(3) If it holds true that for any $x \in [-3, -2]$, $f(k \cdot 4^x) + f(1 - 2^{x+1}) > 0$, f... | k \in (-8, +\infty) | math | 139 |
Given that the sum of the first n terms of a geometric sequence {a_{n}} is S_{n}, if S_{3} = 7 and S_{6} = 63, find the value of a_{1}. | 1 | math | 49 |
If one focus of the hyperbola $\left({{x}^{2}}-\dfrac{{{y}^{2}}}{m}=1\right)$ is at $(-3,0)$, determine the value of $m$. | 8 | math | 49 |
The equation of the line through $(4,3)$ and $(12,-3)$ is $\frac{x}{a}+\frac{y}{b}=1$. Find $a$. | 8 | math | 38 |
A starship enters an extraordinary meteor shower. Some of the meteors travel along a straight line at the same speed, equally spaced. Another group of meteors travels similarly along another straight line, parallel to the first, with the same speed but in the opposite direction, also equally spaced. The ship travels p... | 9.1 | math | 171 |
What is the positive difference between the $2000^{\mathrm{th}}$ term and the $2005^{\mathrm{th}}$ term of the arithmetic sequence $-8,$ $-2,$ $4,$ $10,$ $\ldots$? | 30 | math | 59 |
\( A_1, A_2, A_3, A_4 \) are consecutive vertices of a regular \( n \)-gon. Given the equation \( \frac{1}{A_1A_2} = \frac{1}{A_1A_3} + \frac{1}{A_1A_4} \), what are the possible values of \( n \)? | n = 7 | math | 84 |
Choose three people from five, including A and B, to form a line. What is the probability that person A is not in the first position and person B is not in the last position? | \frac {13}{20} | math | 39 |
Given the sequence {a<sub>n</sub>} with the sum of its first n terms S<sub>n</sub> = n<sup>2</sup> - 2n + 2, find a<sub>1</sub> and a<sub>n</sub>. | 1 | math | 60 |
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