task_type stringclasses 4
values | problem stringlengths 14 5.23k | solution stringlengths 1 8.29k | problem_tokens int64 9 1.02k | solution_tokens int64 1 1.98k |
|---|---|---|---|---|
math | The graph of the quadratic function $y=x^{2}-2x+1$ intersects the $x$-axis how many times? | 1 | 28 | 1 |
math | Rational Man and Irrational Man both buy new cars, and they decide to drive around two racetracks from time \( t = 0 \) to \( t = \infty \). Rational Man drives along the path parameterized by
\[
x = \cos t, \quad y = \sin t,
\]
and Irrational Man drives along the path parameterized by
\[
x = \cos \left(\frac{t}{2}\rig... | 0 | 159 | 1 |
math | Ninety percent of adults drink coffee and eighty percent drink tea, while seventy percent drink soda. What is the smallest possible percentage of adults who drink both coffee and tea, but not soda? | 0\% | 39 | 3 |
math | A master services 5 machines. He spends $20\%$ of his working time at the first machine, 10\% at the second machine, 15\% at the third machine, 25\% at the fourth machine, and 30\% at the fifth machine. Find the probability that at a randomly chosen moment, he is located: 1) at the first or third machine; 2) at the sec... | 0.35, 0.40, 0.45, 0.45, 0.55 | 132 | 28 |
math | What is the shortest distance on the surface of a given right parallelepiped (with edges $a > b > c$) between one vertex and its opposite vertex? Consider vertex $N$ to be the opposite of vertex $M$, where three faces meeting at $N$ are parallel to the three faces meeting at $M$. | \sqrt{a^2 + (b+c)^2} | 67 | 13 |
math | A traffic light cycles as follows: green for 40 seconds, then yellow for 5 seconds, followed by red for 40 seconds, and then blue for 5 seconds. Mary picks a random five-second time interval to observe the light. What is the probability that the color changes while she is watching? | \frac{2}{9} | 64 | 7 |
math | Given that one asymptote of the hyperbola $\frac{x^{2}}{a^{2}}-y^{2}=1$ is perpendicular to the line $l$: $3x+y+1=0$, determine the focal length of this hyperbola. | 2 \sqrt {10} | 55 | 7 |
math | Given a hyperbola $$C: \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 (a > 0, b > 0)$$ with its left focus $F_1$, a line passing through $F_1$ intersects the left branch of the hyperbola $C$ at two points $M$ and $N$ (with $M$ above $N$), and $MN$ is parallel to the $y$-axis. Point $B$ is located at $(0, b)$. If $\angle... | (1, \sqrt{2}) | 149 | 8 |
math | Given that the function $y = f(x)$ is an even function, and the function $y = f(x - 2)$ is monotonically decreasing on the interval $[0,2]$, determine the relationship between $f(-1)$, $f(0)$, and $f(2)$. | f(0) < f(-1) < f(2) | 65 | 14 |
math | If $AB \lt 0$ and $BC \lt 0$, determine the quadrant in which the line $Ax+By+C=0$ does not pass. | 4 | 35 | 1 |
math | In the sequence \(1^{2}, 2^{2}, 3^{2}, \cdots, 2005^{2}\), add a "+" or "-" sign before each number to make the algebraic sum the smallest non-negative number. Write the resulting expression. | 1 | 57 | 1 |
math | Given the sequence $\{a_n\}$ that satisfies $\dfrac{1}{{a_{n+1}}} - \dfrac{1}{{a_n}} = d$ ($n \in \mathbb{N}^*$), where $d$ is a constant, and the sequence $\left\{ \dfrac{1}{{x_n}}\right\}$ is a harmonic sequence with $x_1 + x_2 + \ldots + x_{20} = 200$, find the value of $x_5 + x_{16}$. | 20 | 124 | 2 |
math | Given that the lengths of each side and diagonal of quadrilateral $ABCD$ in space are all equal to $1$, and points $E$ and $F$ are the midpoints of $BC$ and $AD$ respectively, then the value of $\overrightarrow{AE} \cdot \overrightarrow{CF}$ is ____. | -\frac{1}{2} | 69 | 7 |
math | Given $z_1 = 2 + i$ and $z_2 = 1 + 2i$, determine the quadrant in which the point corresponding to the complex number $z = z_2 - z_1$ is located. | 2 | 50 | 1 |
math | Fifty slips are placed into a hat, each bearing a number from 1 to 10, with each number entered on five slips. Five slips are drawn from the hat at random and without replacement. Let $p$ be the probability that all five slips bear the same number. Let $q$ be the probability that three slips bear a number $a$ and the o... | 450 | 97 | 3 |
math | Calculate the value of $x$ when the arithmetic mean of the following five expressions is 30: $$x + 10 \hspace{.5cm} 3x - 5 \hspace{.5cm} 2x \hspace{.5cm} 18 \hspace{.5cm} 2x + 6$$ | 15.125 | 78 | 6 |
math | Given the sets \( A = \{(x, y) \mid ax + y = 1, x, y \in \mathbb{Z}\} \), \( B = \{(x, y) \mid x + ay = 1, x, y \in \mathbb{Z}\} \), and \( C = \{(x, y) \mid x^2 + y^2 = 1\} \), find the value of \( a \) when \( (A \cup B) \cap C \) is a set with four elements. | -1 | 121 | 2 |
math | Given that $2\sin\beta\sin\left(\alpha-\frac{\pi}{4}\right)=\sin\left(\alpha-\beta+\frac{\pi}{4}\right)$, calculate $\tan\left(\alpha+\beta\right)$. | -1 | 54 | 2 |
math | Given that 11 positive numbers are to be inserted between 1 and 2 to form an increasing geometric sequence, determine the value of the fourth number to be inserted. | 2^{\frac{1}{3}} | 35 | 9 |
math | In quadrilateral $EFGH$, the internal angles form an arithmetic sequence. Furthermore, triangles $EFG$ and $HGF$ are similar with $\angle EFG = \angle HGF$ and $\angle EGF = \angle HFG$. Each of these triangles' angles also forms an arithmetic sequence. In degrees, what is the largest possible sum of the largest and sm... | 180 | 86 | 3 |
math | Let $S$ be the set of positive integer divisors of $18^7$. Three numbers are chosen independently and at random with replacement from the set $S$ and labeled $a_1, a_2,$ and $a_3$ in the order they are chosen. The probability that $a_1$ divides $a_2$ and $a_2$ divides $a_3$ is $\tfrac{m}{n},$ where $m$ and $n$ are rela... | 17 | 113 | 2 |
math | Given two circles with radii \( R \) and \( r \), one outside the other, two common external tangents are drawn to both circles. Find the length between the points of tangency, given that the extensions of these tangents form a right angle. ( \( R > r \) ). | 2\sqrt{Rr} | 62 | 7 |
math | Given that {a<sub>n</sub>} is an arithmetic sequence, if a<sub>1</sub>+a<sub>5</sub>+a<sub>9</sub>=8π, find the sum of the first 9 terms S<sub>9</sub> and the value of cos(a<sub>3</sub>+a<sub>7</sub>). | - \frac {1}{2} | 81 | 8 |
math | Given line $l_{1}: 4x-3y+6=0$ and line $l_{2}: x=0$, the minimum sum of distances from a moving point $P$ on the parabola $y^{2}=4x$ to lines $l_{1}$ and $l_{2}$ is __________. | 1 | 71 | 1 |
math | Triangle ABC has $\angle BAC = 45^\circ$, $\angle CBA \leq 120^\circ$, $BC=2$, and $AC = 2 \cdot AB$. Let H, I, and O be the orthocenter, incenter, and circumcenter of $\triangle ABC$, respectively. Find the measure of ∠CBA if the area of pentagon BCOIH is the maximum possible. | 60^\circ | 91 | 4 |
math | A triangle with angles 15° and 60° is inscribed in a circle of radius \( R \). Find the area of the triangle. | \frac{\sqrt{3}}{4} R^2 | 32 | 13 |
math | Given points $A(-3,0)$, $B(-1,-2)$, if there are exactly two points $M$, $N$ on the circle $(x-2)^{2}+y^{2}=r^{2}(r > 0)$ such that the areas of $\triangle MAB$ and $\triangle NAB$ are both $4$, then the range of $r$ is \_\_\_\_\_\_. | ( \frac { \sqrt {2}}{2}, \frac {9 \sqrt {2}}{2}) | 91 | 24 |
math | A frustum of a cone has a volume of $78$. One base area is 9 times the other. Find the volume of the cone that cuts this frustum. | 81 | 36 | 2 |
math | Compute the value of
\[
\frac{\binom{1/3}{2013} \cdot 4^{2013}}{\binom{4027}{2013}} \, .
\] | \frac{-2^{4026}}{3^{2013} \cdot 4027} | 52 | 26 |
math | Each of the corner fields of the outer square should be filled with one of the numbers \(2, 4, 6\), and \( 8 \), with different numbers in different fields. In the four fields of the inner square, there should be the products of the numbers from the adjacent fields of the outer square. In the circle, there should be th... | 84, 96, 100 | 125 | 11 |
math | Find all numbers \(a, b, c, d\) such that:
(i) The function \(f(x)=4x^{3} - dx\) satisfies the inequality \( |f(x)| \leq 1 \) for \(x \in [-1,1]\);
(ii) The function \(g(x)=4x^{3} + ax^{2} + bx + c\) satisfies the inequality \( |g(x)| \leq 1 \) for \(x \in [-1,1]\). | d=3, b=-3, a=0, c=0 | 106 | 15 |
math | Find the smallest positive integer \( n \) such that \(\underbrace{2^{2 \cdot \cdot}}_{n} > 3^{3^{3^{3}}}\). (The notation \(\underbrace{2^{2 \cdot \cdot}}_{n}\) is used to denote a power tower with \( n \) 2's. For example, \(\underbrace{2^{2^{2}}}_{n}\) with \( n=4 \) would equal \( 2^{2^{2^{2}}} \).) | 6 | 115 | 1 |
math | Given that \( n \) is a positive integer and \( x \) a real number, compute the sum:
\[ \sum_{k=0}^{n} k^{2} \cdot C_{n}^{k} \cdot x^{k} \cdot(1-x)^{n-k}. \]
(Please write the answer in its simplest form.) | n \cdot (n-1) \cdot x^2 + n \cdot x | 76 | 18 |
math | In triangle $△ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. It is known that $\frac {b}{c}= \frac {2 \sqrt {3}}{3}$ and $A + 3C = \pi$.
(1) Find the value of $\cos C + \cos B$;
(2) If $b = 3\sqrt {3}$, find the area of triangle $△ABC$. | \frac{9\sqrt{2}}{4} | 108 | 12 |
math | A circle with a radius of 10 cm has rays drawn from point $A$ that are tangent to the circle at points $B$ and $C$ such that triangle $ABC$ is equilateral. Find the area of the triangle. | 75 \sqrt{3} | 50 | 7 |
math | The Miami Heat and the San Antonio Spurs are playing a best-of-five series basketball championship, in which the team that first wins three games wins the whole series. Assume that the probability that the Heat wins a given game is $x$ (there are no ties). The expected value for the total number of games played can ... | 21 | 94 | 2 |
math | Let \( P\left(x+a, y_{1}\right) \), \( Q\left(x, y_{2}\right) \), and \( R\left(2+a, y_{3}\right) \) be three distinct points on the graph of the inverse function of \( f(x)=2^x+a \). If \( y_{1} \), \( y_{2} \), and \( y_{3} \) form an arithmetic sequence with exactly one real value of \( x \), determine the range of ... | S = \frac{1}{4} | 146 | 9 |
math | A certain organization has 840 staff members. Now, 42 individuals are chosen using systematic sampling for a questionnaire survey. If all 840 individuals are randomly assigned numbers from 1 to 840, determine the number of people among the 42 sampled whose numbers fall within the interval $[61, 120]$. | 3 | 75 | 1 |
math | Find an array of \( n \) different positive integers \(\left(a_{1}, a_{2}, \cdots, a_{n}\right)\) such that \( a_{1} < a_{2} < \cdots < a_{n} \), and for any non-negative integer \( d \), the following condition holds:
$$
\frac{\left(a_{1}+d\right)\left(a_{2}+d\right) \cdots \left(a_{n}+d\right)}{a_{1} a_{2} \cdots a_{... | (1, 2, \ldots, n) | 135 | 12 |
math | In a certain city, a bus route has six stops, named A<sub>0</sub>, A<sub>1</sub>, A<sub>2</sub>, A<sub>3</sub>, A<sub>4</sub>, and A<sub>5</sub>. Two passengers, A and B, board the bus at stop A<sub>0</sub>. Each of them has an equal probability of getting off at any of the stops A<sub>i</sub> (i=1, 2, 3, 4, 5).
(Ⅰ) ... | \frac{4}{5} | 166 | 7 |
math | Given that $\tan \alpha +\tan \beta -\tan \alpha \tan \beta +1=0$, and $\alpha ,\beta \in \left(\frac{\pi }{2},\pi \right)$, calculate $\alpha +\beta$. | \frac{7\pi}{4} | 56 | 9 |
math | Let $g$ be a function taking the positive integers to the positive integers, such that:
(i) $g$ is increasing (i.e., $g(n + 1) > g(n)$ for all positive integers $n$)
(ii) $g(mn) = g(m) g(n)$ for all positive integers $m$ and $n$
(iii) if $m \neq n$ and $m^n = n^m$, then $g(m) = n$ or $g(n) = m$.
Find the sum of all p... | 324 | 126 | 3 |
math | Compute: $$\left\lfloor\frac{2005^{3}}{2003 \cdot 2004}-\frac{2003^{3}}{2004 \cdot 2005}\right\rfloor$$ | 8 | 59 | 1 |
math | Convert the quadratic equation $\left(1+3x\right)\left(x-3\right)=2x^{2}+1$ into general form: ____, with the constant term being ____. | -4 | 42 | 2 |
math | The admission fee for a concert is $ \$30$ per adult and $ \$15$ per child. On a certain day, the concert collected $ \$2550$ in admission fees with at least one adult and one child attending. Of all the possible ratios of adults to children on that day, which one is closest to $1$? | \frac{57}{56} | 73 | 9 |
math | Given the function $f(x)=\left\{{\begin{array}{l}{{{({x-a})}^2},x≤0}\\{x+\frac{1}{x}+a,x＞0}\end{array}}\right.$, if $a=0$, the interval where $f(x)$ is monotonically increasing is ______; if $f(0)$ is the minimum value of the function $f(x)$, the range of real number $a$ is ______. | [0, 2] | 104 | 6 |
math | Given that $|\overrightarrow{a}|=6$, $|\overrightarrow{b}|=3$, and $\overrightarrow{a}\cdot\overrightarrow{b}=-12$, find the projection of vector $\overrightarrow{a}$ in the direction of vector $\overrightarrow{b}$. | -4 | 63 | 2 |
math | Given the function $f(x)=\cos (2x-\varphi )-\sqrt{3}\sin (2x-\varphi )(|\varphi | < \frac{\pi }{2})$, the graph of which is translated to the right by $\frac{\pi }{12}$ units and then becomes symmetric about the $y$-axis, determine the minimum value of $f(x)$ in the interval $\left[ -\frac{\pi }{2},0 \right]$. | -\sqrt{3} | 105 | 5 |
math | Given an arithmetic-geometric sequence $\{a\_n\}$ with the sum of its first $n$ terms denoted as $S\_n$, it satisfies the equation $2S\_n = 2^{n+1} + λ (λ ∈ R)$.
(I) Find the general term formula for the sequence $\{a\_n\}$.
(II) If sequence $\{b\_n\}$ satisfies $b\_n = \frac{1}{(2n+1)\log\_4(a\_na\_n+1)}$, find the s... | \frac{2n}{2n+1} | 144 | 11 |
math | A triangle is composed of circular cells arranged in $5784$ rows: the first row has one cell, the second has two cells, and so on (see the picture). The cells are divided into pairs of adjacent cells (circles touching each other), so that each cell belongs to exactly one pair. A pair of adjacent cells is called **dia... | 2892 | 116 | 4 |
math | It is known that in the Cartesian coordinate system \(xOy\), line \(l\) passing through the point \((1,0)\) is perpendicular to the line \(x-y+1=0\), and \(l\) intersects the circle \(C: x^2+y^2=-2y+3\) at points \(A\) and \(B\). The area of triangle \(\triangle OAB\) is \_\_\_\_\_\_. | 2 | 94 | 1 |
math | A circle is inscribed in a rectangle with length $2m$ and width $m$, then a rectangle is inscribed in that circle, then a circle is inscribed in the latter rectangle, and so on. If $S_n$ is the sum of the areas of the first $n$ inscribed circles as $n$ grows, find the limit of $S_n$ as $n$ increases indefinitely.
A) $\... | \frac{\pi m^2}{2} | 151 | 10 |
math | Calculate the greatest integer less than or equal to \[\frac{3^{110}+2^{110}}{3^{106}+2^{106}}.\] | 80 | 42 | 2 |
math | Find the derivative of the following functions:
(1) $y=x\left(1+ \frac{2}{x}+ \frac{2}{x^{2}}\right)$
(2) $y=x^{4}-3x^{2}-5x+6$ | 4x^{3}-6x-5 | 60 | 9 |
math | Given that in $\triangle ABC$, $\sin A + \cos A = \frac{1}{5}$.
(1) Find the value of $\sin A \cdot \cos A$;
(2) Determine whether $\triangle ABC$ is an acute triangle or an obtuse triangle;
(3) Find the value of $\tan A$. | -\frac{4}{3} | 70 | 7 |
math | A straight line is tangent to the graphs of the functions $y=\ln x$ and $y=e^{x}$ at points $P(x_{1}, y_{1})$ and $Q(x_{2}, y_{2})$ respectively. The value of $(1-e^{y_1})(1+x_2)$ is ______. | 2 | 70 | 1 |
math | Given the line defined by the parametric equations $\begin{cases} x=2-t\sin 30^{\circ} \\ y=-1+t\sin 30^{\circ} \end{cases}$ and the circle ${x}^{2}+{y}^{2}=8$, the line intersects the circle at points $B$ and $C$. Point $O$ is the origin. Calculate the area of triangle $BOC$. | \frac{\sqrt{15}}{2} | 95 | 11 |
math | Find in explicit form all ordered pairs of positive integers $(m, n)$ such that $mn-1$ divides $m^2 + n^2$ . | (a_1, b_1) = (1, 2), (1, 3), (2, 1), (3, 1) \text{ and the explicit formula is } 5a_n - b_n = a_{n+1}, b_n = a_{n-1} | 38 | 65 |
math | Let the complex number \( z = x + yi \) satisfy that the ratio of the real part to the imaginary part of \(\frac{z+1}{z+2}\) is \(\sqrt{3}\), where \( \mathrm{i} \) is the imaginary unit, \( x, y \in \mathbb{R} \). Then determine the range of \(\frac{y}{x}\). | \left[ \frac{-3\sqrt{3}-4\sqrt{2}}{5}, \frac{-3\sqrt{3}+4\sqrt{2}}{5} \right] | 87 | 43 |
math | In this scenario, two regular polygons are given: a pentagon and a triangle. Determine the sum of the angles \(PQR\) and \(PQS\) where \(PQ\) is a common side of both polygons. | 168^\circ | 45 | 5 |
math | 2sin15°cos15°= | \sin30^{\circ}=\frac{1}{2} | 10 | 15 |
math | A frame for three square photos has an equal width all around (see the diagram). The perimeter of one photo opening is 60 cm, and the total perimeter of the entire frame is 180 cm. What is the width of the frame? | 5 \text{ cm} | 52 | 6 |
math | What is the greatest number of consecutive integers whose sum is $45$? | 90 | 16 | 2 |
math | Given the algebraic expression
$$
\frac{bx(a^{2}x^{2} + 2a^{2}y^{2} + b^{2}y^{2})}{bx + ay} + \frac{ay(a^{2}x^{2} + 2b^{2}x^{2} + b^{2}y^{2})}{bx + ay}
$$
simplify the numerator. | (ax + by)^{2} | 90 | 7 |
math | Determine the volume of the body obtained by cutting the ball of radius \( R \) by the trihedron with vertex in the center of that ball if its dihedral angles are \( \alpha, \beta, \gamma \). | V = \frac{1}{3} R^3 (\alpha + \beta + \gamma - \pi) | 49 | 24 |
math | Given that a full circle is 800 clerts on Venus and is 360 degrees, calculate the number of clerts in an angle of 60 degrees. | 133.\overline{3} | 37 | 9 |
math | Given that the cube root of \( m \) is a number in the form \( n + r \), where \( n \) is a positive integer and \( r \) is a positive real number less than \(\frac{1}{1000}\). When \( m \) is the smallest positive integer satisfying the above condition, find the value of \( n \). | 19 | 78 | 2 |
math | Let \( x \) and \( y \) be non-zero real numbers such that
\[ \frac{x \sin \frac{\pi}{5} + y \cos \frac{\pi}{5}}{x \cos \frac{\pi}{5} - y \sin \frac{\pi}{5}} = \tan \frac{9 \pi}{20}. \]
(1) Find the value of \(\frac{y}{x}\).
(2) In triangle \( \triangle ABC \), if \( \tan C = \frac{y}{x} \), find the maximum value o... | \frac{3}{2} | 140 | 7 |
math | In right triangle $PQR$, $PQ=15$, $QR=8$, and angle $R$ is a right angle. A semicircle is inscribed in the triangle such that it touches $PQ$ and $QR$ at their midpoints and the hypotenuse $PR$. What is the radius of the semicircle?
A) $\frac{24}{5}$
B) $\frac{12}{5}$
C) $\frac{17}{4}$
D) $\frac{15}{3}$ | \frac{24}{5} | 116 | 8 |
math | Given a non-zero sequence \(\{a_n\}\) that satisfies \(a_1 = 1\), \(a_2 = 3\), and \(a_n (a_{n+2} - 1) = a_{n+1} (a_{n+1} - 1)\) for \(n \in \mathbf{N}^*\), find the value of \(\mathrm{C}_{2023}^{0} a_1 + \mathrm{C}_{2023}^{1} a_2 + \mathrm{C}_{2023}^{2} a_3 + \cdots + \mathrm{C}_{2023}^{2023} a_{2024}\) expressed as a... | 2 \cdot 3^{2023} - 2^{2023} | 172 | 20 |
math | On every kilometer marker along the highway between the villages of Yolkino and Palkino, there is a post with a sign. On one side of the sign, the distance to Yolkino is indicated, and on the other side, the distance to Palkino is indicated. Borya noticed that the sum of all the digits on each sign equals 13. What is t... | 49 | 91 | 2 |
math | Find the values of $x$ in the following equations.
$(1) (2x-1)^{2}-25=0$;
$(2) \frac{1}{3}(x+3)^{3}-9=0$. | x = 0 | 53 | 4 |
math | Two identical rulers are placed together. Each ruler is exactly 10 cm long and is marked in centimeters from 0 to 10. The 3 cm mark on each ruler is aligned with the 4 cm mark on the other. The overall length is \( L \) cm. What is the value of \( L \)?
A) 13
B) 14
C) 15
D) 16
E) 17 | 13 | 98 | 2 |
math | The mean, median, and unique mode of the positive integers 3, 4, 5, 6, 6, 7, and $x$ are all equal. What is the value of $x$? | 11 | 47 | 2 |
math | Find all integer values that the fraction $\frac{8n + 157}{4n + 7}$ can take for natural numbers \(n\). In the answer, write down the sum of the values found. | 18 | 46 | 2 |
math | Let \( ABC \) be an isosceles triangle with \( AB = AC \). Let \( D \) and \( E \) be the midpoints of segments \( AB \) and \( AC \), respectively. Suppose that there exists a point \( F \) on ray \( \overrightarrow{DE} \) outside of \( ABC \) such that triangle \( BFA \) is similar to triangle \( ABC \). Compute \( \... | \sqrt{2} | 100 | 5 |
math | A group of n friends wrote a math contest consisting of eight short-answer problem $S_1, S_2, S_3, S_4, S_5, S_6, S_7, S_8$ , and four full-solution problems $F_1, F_2, F_3, F_4$ . Each person in the group correctly solved exactly 11 of the 12 problems. We create an 8 x 4 table. Inside the square located in the $i... | 32 | 174 | 2 |
math | Given a rectangular grid of dots with 5 rows and 6 columns, determine how many different squares can be formed using these dots as vertices. | 40 | 29 | 2 |
math | In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given vectors $\overrightarrow{m}=(\sin B+\sin C,\sin A+\sin B)$, $\overrightarrow{n}=(\sin B-\sin C,\sin A)$, and $\overrightarrow{m}\perp \overrightarrow{n}$.
(1) Find the measure of angle $C$;
(2) If $\triangle A... | 2+\sqrt{3} | 135 | 6 |
math | Given the function $f(x)=\log_{k}x$ ($k$ is a constant, $k>0$ and $k\neq 1$).<br/>$(1)$ Choose one of the following conditions _____ to make the sequence $\{a_{n}\}$ a geometric sequence, and explain the reason;<br/>① The sequence $\{f(a_{n})\}$ is a geometric sequence with the first term being $2$ and the common ratio... | \frac{n}{2n+1} | 264 | 9 |
math | In the plane Cartesian coordinate system $xOy$, a line $l$ passes through the origin, and $\vec{n}=(3,1)$ is a normal vector to $l$. Given that the sequence $\left\{a_{n}\right\}$ satisfies: for any positive integer $n$, the point $\left(a_{n+1}, a_{n}\right)$ lies on $l$. If $a_{2}=6$, then the value of $a_{1} a_{2} a... | -32 | 122 | 3 |
math | With all angles measured in degrees, the product $\prod_{k=1}^{22} \sec^2(4k)^\circ=p^q$, where $p$ and $q$ are integers greater than 1. Find the value of $p+q$. | 46 | 58 | 2 |
math | A tangent line MN is drawn from a moving point M to the circle: $(x-2)^{2}+(y-2)^{2}=1$, where N is the point of tangency. If $|MN|=|MO|$ (O is the origin of coordinates), then the minimum value of $|MN|$ is \_\_\_\_\_\_. | \dfrac {7 \sqrt {2}}{8} | 75 | 12 |
math | Given right $\triangle DEF$ with legs $DE=5$, and $EF=12$. Find the length of the shorter angle trisector from $F$ to the hypotenuse.
A) $\frac{35\sqrt{3} - 60}{50}$
B) $\frac{1440\sqrt{3} - 600}{407}$
C) $\frac{720\sqrt{5} - 360}{203}$
D) $\frac{840\sqrt{11} - 480}{407}$ | \frac{1440\sqrt{3} - 600}{407} | 134 | 22 |
math | Imagine a dodecahedron (a polyhedron with 12 pentagonal faces) and an ant starts at one of the top vertices. The ant will randomly walk to one of three adjacent vertices, denoted as vertex A. From vertex A, the ant then walks to one of another three randomly selected adjacent vertices, signified as vertex B. Calculate ... | \frac{1}{3} | 102 | 7 |
math | (1) Use the "Euclidean algorithm" to find the greatest common divisor of 459 and 357.
(2) Use Horner's method to calculate the value of the polynomial $f(x) = 12 + 35x - 8x^2 + 79x^3 + 6x^4 + 5x^5 + 3x^6$ when $x = -4$. | 3392 | 95 | 4 |
math | Determine the value of $\frac{1+\tan 15^{\circ}}{1-\tan 15^{\circ}}$. | \sqrt{3} | 30 | 5 |
math | Given a real number $a$ chosen arbitrarily within the interval $[-5,5]$, determine the probability that the line $x+y+a=0$ intersects the circle $(x-1)^{2}+(y+2)^{2}=2$. | \frac{2}{5} | 53 | 7 |
math | The store owner bought 2000 pens at $0.15 each and plans to sell them at $0.30 each, calculate the number of pens he needs to sell to make a profit of exactly $150. | 1000 | 50 | 4 |
math | Consider an isosceles triangle $XYZ$ where $XY = YZ$ and $\angle YXZ = \angle YZX = 50^\circ$. Line segments $\overline{JK}$, $\overline{LM}$, and $\overline{NO}$ are drawn parallel to base $\overline{XZ}$ such that $YJ = JL = LM = MO$. What is the ratio of the area of trapezoid $NOXZ$ to the area of triangle $XYZ$? | \frac{7}{16} | 108 | 8 |
math | Given 6 parking spaces in a row and 3 cars that need to be parked such that no two cars are next to each other, calculate the number of different parking methods. | 24 | 36 | 2 |
math | 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 | 43 | 2 |
math | A sequence of three real numbers forms an arithmetic progression with a first term of 5. If 3 is added to the second term and 15 is added to the third term, the three resulting numbers form a geometric progression. What is the largest possible value for the first term of the geometric progression? | 5 | 62 | 1 |
math | Find the volume of a cube if the distance from its space diagonal to a non-intersecting edge is $d$. | 2d^3 \sqrt{2} | 24 | 9 |
math | Given $\sin(\alpha - \beta) = \frac{1}{3}$ and $\cos \alpha \sin \beta = \frac{1}{6}$, find $\cos(2\alpha + 2\beta)$. | \frac{1}{9} | 49 | 7 |
math | A block of iron solidifies from molten iron, and its volume reduces by $\frac{1}{34}$. Then, if this block of iron melts back into molten iron (with no loss in volume), by how much does its volume increase? | \frac{1}{33} | 53 | 8 |
math | Given the matrix $$M= \begin{bmatrix} 1 & 2 \\ a & 1\end{bmatrix}$$ with one of its eigenvalues being λ=3, and its corresponding eigenvector being $$\overrightarrow {α}= \begin{bmatrix} \overset{1}{1}\end{bmatrix}$$, find the equation of the curve l<sub>2</sub> resulting from the transformation of the line l<sub>1</sub... | x+1=0 | 116 | 5 |
math | A square sheet of paper has each of its four corners folded into smaller isosceles right triangles which are then cut out. If the side length $AB$ of the original square is $16$ units, and each corner triangle is made from folds such that the legs of the triangles are equal to half the side length of smaller squares fo... | 32 | 85 | 2 |
math | Jack, Jill, and John play a game in which each randomly picks and then replaces a card from a standard 52 card deck, until a spade card is drawn. What is the probability that Jill draws the spade? (Jack, Jill, and John draw in that order, and the game repeats if no spade is drawn.) | 12/37 | 70 | 5 |
math | Given that the right focus of the ellipse $C$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a > b > 0)$ is $(\sqrt{5},0)$, and the eccentricity is $\frac{\sqrt{5}}{3}$.
(1) Find the standard equation of the ellipse $C$;
(2) If the moving point $P(x_{0},y_{0})$ is outside the ellipse $C$, and the two tangents from point ... | x^{2}+y^{2}=13 | 140 | 11 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.