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 | Given a sequence of positive terms $\{a_n\}$ whose sum of the first $n$ terms is $S_n$, for $n \geqslant 2$, we have $(a_n - S_{n-1})^2 = S_n S_{n-1}$, and $a_1 = 1$. Let $b_n = \log_2 \dfrac{a_{n+1}}{3}$, find the minimum value of $\dfrac{b_1 + b_2 + \ldots + b_n + 34}{n+1}$. | 9 | 126 | 1 |
math | If \(\log_{3n} 675 \sqrt{3} = \log_{n} 75\), determine the value of \(n^{5}\). | 5625 | 38 | 4 |
math | If the terminal sides of angles α and β are perpendicular, determine the relationship between α and β. | \beta = k \cdot 360^\circ + \alpha \pm 90^\circ | 20 | 22 |
math | A regular hexagon $ABCDEF$ has its center at $G$. Each of the vertices and the center are to be associated with one of the digits $1$ through $7$, with each digit used once, in such a way that the sums of the numbers on the lines $AGC$, $BGD$, and $CGE$ are all equal. In how many ways can this be done? | 144 | 83 | 3 |
math | Let \( n \) be a positive integer. When \( n > 100 \), what are the first two decimal places of the fractional part of \( \sqrt{n^2 + 3n + 1} \)? | 49 | 48 | 2 |
math | Find the probability that heads will appear an even number of times in an experiment where:
a) a fair coin is tossed $n$ times;
b) a coin with the probability of heads being $p$ ($0 < p < 1$) is tossed $n$ times. | \frac{1 + (1 - 2p)^n}{2} | 58 | 16 |
math | Let $a_n =\sum_{d|n} \frac{1}{2^{d+ \frac{n}{d}}}$ . In other words, $a_n$ is the sum of $\frac{1}{2^{d+ \frac{n}{d}}}$ over all divisors $d$ of $n$ .
Find $$ \frac{\sum_{k=1} ^{\infty}ka_k}{\sum_{k=1}^{\infty} a_k} =\frac{a_1 + 2a_2 + 3a_3 + ....}{a_1 + a_2 + a_3 +....} $$ | 4 | 151 | 1 |
math | Luna writes down all possible six-digit numbers that contain each of the digits 1, 2, 3, 4, 5, and 0 exactly once. What is the smallest number in Luna's list that is divisible by 5? | 123450 | 52 | 6 |
math | Using the systematic sampling method to select 32 people for a questionnaire survey from 960 people, determine the number of people among the 32 whose numbers fall within the interval [200, 480]. | 10 | 48 | 2 |
math | Given that $\overrightarrow{a}$ and $\overrightarrow{b}$ are two mutually perpendicular unit vectors on a plane, and vector $\overrightarrow{c}$ satisfies $\overrightarrow{c} + \overrightarrow{a} = λ(\overrightarrow{c} + \overrightarrow{b}), (λ \in \mathbb{R})$, find the minimum value of $|\overrightarrow{c}|$. | \frac{\sqrt{2}}{2} | 86 | 10 |
math | Given that there are 3 white balls and 2 red balls in bag A, and 2 white balls and 2 red balls in bag B, two balls are randomly taken from bag A and placed into bag B, then two balls are randomly taken from bag B, determine the probability that two white balls are taken from bag A, given that two white balls are taken ... | \frac{18}{37} | 79 | 9 |
math | $a,b,c,d,e$ are equal to $1,2,3,4,5$ in some order, such that no two of $a,b,c,d,e$ are equal to the same integer. Given that $b \leq d, c \geq a,a \leq e,b \geq e,$ and that $d\neq5,$ determine the value of $a^b+c^d+e.$ | 628 | 103 | 3 |
math | Form a square with sides of length $5$ , triangular pieces from the four coreners are removed to form a regular octagonn. Find the area **removed** to the nearest integer. | 4 | 40 | 1 |
math | In $\triangle ABC$, $A$, $B$, $C$ correspond to sides $a$, $b$, $c$, and it is known that $acosC + \frac{1}{2}c = b$.
(I) Find angle $A$;
(II) If $b=4$, $c=6$, find the values of $cosB$ and $cos(A+2B)$. | cos(A+2B) = -\frac{11}{14} | 86 | 17 |
math | Athletes A and B cannot run the first and fourth legs, respectively. Calculate the number of different possible sequences for the 4x100 relay race. | 14 | 33 | 2 |
math | Let \(AOB\) be a 60-degree angle. For any point \(P\) in the interior of \(\angle AOB\), let \(A'\) and \(B'\) be the feet of the perpendiculars from \(P\) to \(AO\) and \(BO\) respectively. Denote by \(r\) and \(s\) the distances \(OP\) and \(A'B'\). Find all possible pairs of real numbers \((r, s)\). | \left( r, \frac{r\sqrt{3}}{2} \right) | 96 | 20 |
math | Determine all functions $ f: \mathbb{N} \rightarrow \mathbb{N}$ which satisfy:
$ f(x\plus{}f(y))\equal{}f(x)\plus{}y$ for all $ x,y \in \mathbb{N}$ . | f(x) = x | 62 | 6 |
math | Compute $\arccos (\sin 3)$, where all functions are in radians. | 3 - \frac{\pi}{2} | 18 | 9 |
math | Let $a$ and $b$ be positive integers for which $ab - 8a + 7b = 637$. What is the minimal possible value of $|a - b|$? | 3 | 43 | 1 |
math | Evaluate the expression $\frac{(0.5)^{3.5}}{(0.05)^3}$.
A) $158.11\sqrt{5}$
B) $316.23\sqrt{5}$
C) $632.46\sqrt{5}$
D) $948.69\sqrt{5}$
E) $500\sqrt{5}$ | 316.23\sqrt{5} | 95 | 11 |
math | Matthew rolled a normal die 24 times. All numbers from 1 to 6 came up at least once. The number 1 came up more times than any other number. Matthew added up all the numbers. The total he obtained was the largest one possible. What total did he obtain?
A) 83
B) 84
C) 89
D) 90
E) 100 | 90 | 90 | 2 |
math | Three planes intersect pairwise and their three lines of intersection are mutually parallel, then determine the total number of parts the space is divided into. | 8 | 27 | 1 |
math | Given the parabola $y^{2}=4x$ whose directrix intersects with the hyperbola $\frac {x^{2}}{a^{2}}- \frac {y^{2}}{b^{2}}=1$ ($a > 0,b > 0$) at points $A$ and $B$, and point $F$ is the focus of the parabola. If $\triangle FAB$ is a right-angled triangle, then the range of the eccentricity of the hyperbola is \_\_\_\_\_\_... | (\sqrt {5},+\infty) | 117 | 9 |
math | There are 5 students and 2 teachers to be seated in a row for a group photo. The teachers cannot sit at either end and must sit together. How many different seating arrangements are possible? | 960 | 40 | 3 |
math |
Compute the definite integral:
$$
\int_{1 / 24}^{1 / 3} \frac{5 \sqrt{x+1}}{(x+1)^{2} \sqrt{x}} \, dx
$$ | 3 | 50 | 1 |
math | Mr. A owns a house valued at $12000. He sells it to Mr. B at a 15% loss, and then Mr. B sells it back to Mr. A at a 20% gain. Calculate the result of the two transactions. | 240 | 58 | 3 |
math | Let \( a, b, c \) be real numbers such that \( a + b + c = 1 \). Find the set of all possible values of \( ab + ac + bc \). | (-\infty, \frac{1}{2}] | 41 | 12 |
math | A plot of land is shaped as a trapezoid and is represented on a map using a scale where 1 cm equals 3 miles. The shorter base of the trapezoid is 12 cm, the longer base is 18 cm, and the height is 15 cm. Given that one square mile is equal to 640 acres, calculate the actual size of the plot in acres. | 1,296,000 \text{ acres} | 87 | 14 |
math | Given the function f(x) = cos²x + $\sqrt{3}$sinxcosx, where $x \in (0, \frac{\pi}{2})$, determine the interval where f(x) is monotonically increasing. | (0, \frac{\pi}{6}) | 50 | 10 |
math | Given the function $f(x) = \sin(2x - \frac{\pi}{6})$, if the graph of this function is translated to the left by $\frac{\pi}{6}$ units to obtain the graph of $y=g(x)$, determine the interval where the function $g(x)$ is monotonically increasing. | \left[-\frac{\pi}{3}, \frac{\pi}{6}\right] | 69 | 19 |
math | Is there a real number $m$ such that the equations $x^2 + mx + 2 = 0$ and $x^2 + 2x + m = 0$ have exactly one common real root? If such a real number exists, find this $m$ and the common real root of the two equations; if it does not exist, explain why. | 1 | 78 | 1 |
math | Given the area of $\triangle ABC$ is $S$, and $3 \overrightarrow{AB} \cdot \overrightarrow{AC} = 2S$.
$(1)$ Find the value of $\sin A$;
$(2)$ If $C= \frac{\pi}{4}$ and $\overrightarrow{AB} \cdot \overrightarrow{AC} = 16$, find $AC$. | 8 | 86 | 1 |
math | Given a sequence $\{a_n\}$ with a sum of the first $n$ terms denoted as $S_n$, and the point $(n, S_n)$ lies on the graph of the function $y=2^{x+1}-2$.
(I) Find the general term formula for the sequence $\{a_n\}$;
(II) Consider a sequence $\{b_n\}$ satisfying: $b_1=0$, $b_{n+1}+b_n=a_n$, find the sum formula of the f... | \lambda \in (1, +\infty) | 179 | 12 |
math | Let $a$ be a real constant, and $y=f(x)$ is an odd function defined on $\mathbb{R}$, and when $x < 0$, $f(x)=9x+\frac{a^2}{x}+7$. If $f(x) \geqslant a+1$ holds for all $x \geqslant 0$, then the range of values for $a$ is \_\_\_\_. | (-\infty, -\frac{8}{7}] | 96 | 13 |
math | Imagine you need to choose a license plate with 3 characters where the first character is a letter, the second character can either be a letter or a digit, and the last character must be a digit. Additionally, no two characters can be the same. How many different license plates can be created under these conditions? | 8190 | 63 | 4 |
math | Given a pentagon $ABCDE$, side $\overline{AB}$ is extended past $B$ to $A'$ so that $A'B = AB$. Likewise, points $B'$, $C'$, $D'$, and $E'$ are constructed by extending $\overline{BC}$, $\overline{CD}$, $\overline{DE}$, and $\overline{EA}$ such that $B'C = BC$, $C'D = CD$, $D'E = DE$, and $E'A = EA$ respectively. Now, ... | \left(\frac{1}{31}, \frac{2}{31}, \frac{4}{31}, \frac{8}{31}, \frac{16}{31}\right) | 260 | 45 |
math | Determine all composite positive integers \( n \) with the following property: If \( 1 = d_1 < d_2 < \ldots < d_k = n \) are all the positive divisors of \( n \), then
\[ \left(d_2 - d_1\right) : \left(d_3 - d_2\right) : \cdots : \left(d_k - d_{k-1}\right) = 1 : 2 : \cdots : (k-1). \]
(Walther Janous) | 6 | 119 | 1 |
math | Each of the six houses on one side of a street is connected by cable lines with each of the eight houses on the opposite side. How many pairwise intersections do the shadows of these cables form on the surface of the street, assuming no three of them intersect at a single point? Assume that the light producing these sh... | 420 | 68 | 3 |
math | The distance from the focus of the parabola $y^{2}=4x$ to the asymptotes of the hyperbola $x^{2}- \frac {y^{2}}{4}=1$ is $\frac {2 \sqrt {5}}{5}$. | \frac{2 \sqrt{5}}{5} | 58 | 12 |
math | The roots of the quadratic equation $x^{2}-1=3$ are ____. | x_{1} = 2, x_{2} = -2 | 18 | 15 |
math | Given that Casper can buy either 10 pieces of orange candy, 16 pieces of yellow candy, 18 pieces of maroon candy, or n pieces of violet candy, where the price of each piece of violet candy is 18 cents. Find the smallest possible value of n. | 40 | 62 | 2 |
math | Point $O$ lies on plane $\alpha$, and $A$, $B$, and $C$ are three non-collinear points on plane $\alpha$. The moving point $P$ on plane $\alpha$ satisfies $\overrightarrow{OP} = \overrightarrow{OA} + \lambda (\overrightarrow{AB} + \overrightarrow{AC})$. When the value of $\lambda$ makes $\overrightarrow{PA} \cdot (\ove... | \frac{1}{4} | 115 | 7 |
math | Two concentric circles share the same center with radii $15$ meters and $25$ meters respectively. A bee starts at point $A$ on the smaller circle, moves along an eighth of the outer circle's circumference, then goes directly towards the center, touches the center, and directly returns back to point $A$. Calculate the t... | \frac{25\pi}{4} + 50 | 77 | 14 |
math | A hyperbola and an ellipse share the same foci. The ellipse is given by the equation $\frac{y^2}{40} + \frac{x^2}{15} = 1$. Point $P(3, 4)$ lies on the asymptote of the hyperbola. Find the standard equation of the hyperbola and its eccentricity. | \frac{5}{4} | 78 | 7 |
math | What integer is closest to the value of $\sqrt[4]{15^4 + 10^4}$? | 15 | 25 | 2 |
math | Given that for a hyperbola $a=5$ and $c=7$, determine the standard equation of this hyperbola. | \frac {x^{2}}{25}- \frac {y^{2}}{24} = 1 \text{ or } \frac {y^{2}}{25}- \frac {x^{2}}{24} = 1 | 28 | 55 |
math | Given that $\frac{\cos 2\alpha}{\sqrt{2}\sin\left(\alpha+\frac{\pi}{4}\right)}=\frac{\sqrt{5}}{2}$, find the value of $\tan\alpha+\frac{1}{\tan\alpha}$. | -8 | 60 | 2 |
math | Given that point P is a moving point on the ellipse $\frac{x^{2}}{9}$$+ \frac{y^{2}}{5}$$=1, and F$_1$, F$_2$ are the two foci of the ellipse. Determine the maximum value of sin∠F$_1$PF$_2$. | \frac{4\sqrt{5}}{9} | 69 | 12 |
math | If $30^a = 4$ and $30^b = 9,$ then find $18^{(1 - a - b)/(2(1 - b))}.$ | \frac{5}{6} | 41 | 7 |
math | In parallelogram $ABCD$, $AD=1$, $\angle BAD=60^{\circ}$, and $E$ is the midpoint of $CD$. If $\overrightarrow{AD} \cdot \overrightarrow{EB}=2$, then the length of $AB$ is \_\_\_\_\_. | 12 | 66 | 2 |
math | Let $p_1 = 2012$ and $p_n = 2012^{p_{n-1}}$ for $n > 1$ . Find the largest integer $k$ such that $p_{2012}- p_{2011}$ is divisible by $2011^k$ . | 1 | 84 | 1 |
math | Let circle C be defined by the equation $(x-3)^2 + (y-5)^2 = 5$. A line $l$ passes through the center of circle C and intersects the circle at points A and B, and intersects the y-axis at point P. If point A is exactly the midpoint of segment BP, then the equation of line $l$ is __________. | 2x + y - 11 = 0 | 79 | 11 |
math | Suppose a three-digit positive integer is of the form "a₁a₂a₃", where a₁ > a₂ and a₂ < a₃. Such numbers are called concave numbers (e.g., 102, 312, 989, etc.). Determine the number of concave numbers in all three-digit positive integers. | 285 | 76 | 3 |
math | Given the set $\{-10, -7, -5, 0, 4, 6, 9\}$, find the minimum possible product of three different numbers from this set. | -540 | 41 | 4 |
math | If the four-digit number \(5ab4\) is a perfect square, then \(a + b =\) | 9 | 22 | 1 |
math | Let $(1+x)^8 = a + a_1x + \ldots + a_8x^8$, determine the number of odd numbers among the coefficients $a, a_1, \ldots, a_8$. | 2 | 49 | 1 |
math | A rectangular solid has the following properties:
- If the length decreases by 2 cm, and the width and height remain unchanged, the volume decreases by 48 cubic cm.
- If the width increases by 3 cm, and the length and height remain unchanged, the volume increases by 99 cubic cm.
- If the height increases by 4 cm, and ... | 290 | 108 | 3 |
math | The solution set of the inequality $\sqrt{x+3} > 3-x$ is \_\_\_\_\_\_. | (1, +\infty) | 25 | 8 |
math | Given the equation $\frac{x^2}{10-t}+\frac{y^2}{t-4}=1$, find the range of values for $t$ such that this equation represents an ellipse. | (4,7) \cup (7,10) | 43 | 13 |
math | On the side AB of square ABCD, an equilateral triangle AKB is constructed (externally). Find the radius of the circumscribed circle around triangle CKD if $\mathrm{AB}=1$. | 1 | 42 | 1 |
math | The domain of the function $y=\dfrac{\mathbf{\lg (x+1)}}{\mathbf{x-2}}$ is $(-\infty, -1) \cup (2, +\infty)$. | (-1,2)\cup(2,+\infty) | 49 | 13 |
math | Let $a, b, c$ be integers not all the same with $a, b, c\ge 4$ that satisfy $$ 4abc = (a + 3) (b + 3) (c + 3). $$ Find the numerical value of $a + b + c$ . | 16 | 72 | 2 |
math | Given the equation $(x - 2\cos\theta)^2 + (y - 2\sin\theta)^2 = 1$ $(0 \leq \theta \leq 2\pi)$, find the range of values of $\theta$ for which every solution pair $(x, y)$ satisfies the inequality $x \leq y$. | [\frac{5\pi}{12}, \frac{13\pi}{12}] | 75 | 21 |
math | Given that $S_{n}$ is the sum of the first $n$ terms of the sequence $\{a_n\}$, with $a_{1} < 2$, $a_n > 0$, and $6S_{n}= a_{n}^{2}+3a_{n}+2$ for $n \in \mathbb{N^*}$.
$(1)$ Find the general formula for the sequence $\{a_n\}$.
$(2)$ If for every $n \in \mathbb{N^*}$, $b_n=(-1)^{n} a_{n}^{2}$, find the sum of the fi... | T_{2n} = 18n^2-3n | 167 | 15 |
math | Carl discovers a cave with three types of rocks: $6$-pound rocks worth $$16$ each, $3$-pound rocks worth $$9$ each, and $2$-pound rocks worth $$3$ each. There are at least $15$ of each type. He can carry a maximum of $20$ pounds and no more than $5$ rocks in total. What is the maximum value, in dollars, of the rocks he... | 52 | 129 | 2 |
math | For the function $f\left(x\right)=\sin |x|+|\cos x|$, the following four statements are made:<br/>①$f\left(x\right)$ is an even function;<br/>②$f\left(x\right)$ is monotonically decreasing on the interval $(\frac{π}{2},π)$;<br/>③The range of $f\left(x\right)$ on the interval $(-\frac{π}{2},\frac{π}{2})$ is $[1,\sqrt{2}... | ①③④ | 180 | 6 |
math | In square ABCD, an isosceles triangle AEF is inscribed; point E lies on side BC, point F lies on side CD, and AE = AF. The tangent of angle AEF is 3. Find the cosine of angle FAD. | \frac{2\sqrt{5}}{5} | 54 | 12 |
math | Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that
\[f(f(x) + y) = f(x) - f(f(y) + f(-x)) + x\]for all real numbers $x$ and $y.$
Determine the number of possible values of $f(-2),$ and the sum of all possible values of $f(-2).$ Return the product of these two quantities. | 2 | 97 | 1 |
math | Given sets $A=\{x|x^{2}-7x+6\leqslant 0\}$ and $B=\{x|x^{2}-2x+1-m^{2}\leqslant 0, m \gt 0\}$.
$(1)$ If $m=1$, find $A\cap B$;
$(2)$ If $x\in A$ is a sufficient and necessary condition for $x\in B$, find the range of values for $m$. | [5,+\infty) | 108 | 7 |
math | Given $f(x)=x^{2}-2x-\ln (x+1)^{2}$.
(1) Find the interval(s) where $f(x)$ is monotonically increasing.
(2) If the function $F(x)=f(x)-x^{2}+3x+a$ has only one zero in the interval $[- \frac {1}{2},2]$, find the range of values for the real number $a$. | a=2\ln 2-1 | 94 | 9 |
math | In a factor tree, each value is the product of the two values below it, unless a value is a prime number already. Determine the value of \( X \) on the factor tree shown:
[asy]
draw((-1,-.3)--(0,0)--(1,-.3),linewidth(1));
draw((-2,-1.3)--(-1.5,-.8)--(-1,-1.3),linewidth(1));
draw((1,-1.3)--(1.5,-.8)--(2,-1.3),linewidth... | 11025 | 325 | 5 |
math | 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\% | 152 | 4 |
math | Let \( p(x) \) be a polynomial of degree \( 3n \) such that
\[
\begin{array}{c}
p(0)=p(3)=\cdots=p(3n)=2, \\
p(1)=p(4)=\cdots=p(3n-2)=1, \\
p(2)=p(5)=\cdots=p(3n-1)=0,
\end{array}
\]
and \( p(3n+1)=730 \). Find \( n \). | n = 4 | 116 | 4 |
math | In the Cartesian coordinate system $xoy$, the left branch of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \ (a > 0, b > 0)$ intersects with the parabola with focus $F$ described by $x^2 = 2py \ (p > 0)$ at points $M$ and $N$. If $|MF| + |NF| = 4|OF|$, then the eccentricity of that hyperbola is ______. | \frac{\sqrt{6}}{2} | 119 | 10 |
math | Laura added two three-digit positive integers. All six digits in these numbers are different. Laura's sum is a three-digit number $S$. What is the smallest possible value for the sum of the digits of $S$? | 4 | 45 | 1 |
math | In triangle ABC, medians BE and CD intersect at G. The midpoint of BD is F. Find the value of x if the area of triangle GFD is x times the area of triangle ABC. | \frac{1}{6} | 41 | 7 |
math | Find the number of permutations $x_1, x_2, x_3, x_4, x_5$ of numbers $1, 2, 3, 4, 5$ such that the sum of five products \[x_1x_2x_3 + x_2x_3x_4 + x_3x_4x_5 + x_4x_5x_1 + x_5x_1x_2\] is divisible by $3$. | 80 | 109 | 2 |
math | Given that three individuals, A, B, and C, are going to take a certain test, the probabilities of them meeting the standard are 0.8, 0.6, and 0.5, respectively. What is the probability that all three individuals meet the standard? | 0.24 | 58 | 4 |
math | Two lines $p$ and $q$ are parallel, and line $r$ intersects them. If the measure of $\angle A$ is $\frac{1}{5}$ the measure of $\angle B$. What is the degree measure of $\angle D$?
[asy]
size(100);
defaultpen(linewidth(0.7)+fontsize(9));
path p = (-1.35,0.72)--(0.45,0.72), q = (-1,0)--(1,0), r = (-0.67,1.09)--(0.27,-... | 30^\circ | 298 | 4 |
math | Alli rolls a standard $8$-sided die twice. What is the probability of rolling integers that differ by $3$ on her first two rolls? Express your answer as a common fraction. | \frac{1}{8} | 41 | 7 |
math | John draws a regular ten-pointed star in the sand, each with 10 outward-pointing and 10 inward-pointing points. He places one of twenty different sea shells at each of these 20 points. How many ways can he place the shells, if reflections and rotations of the arrangement are considered equivalent? | 19! | 66 | 3 |
math | Find all the solutions to
\[\sqrt[3]{2 - x} + \sqrt{x - 1} = 1.\]Enter all the solutions, separated by commas. | 1,2,10 | 38 | 6 |
math | Two concurrent forces $F_1=(\log_2, \log_2)$ and $F_2=(\log_5, \log_2)$ act on an object $M$, causing a displacement $s=(2\log_5, 1)$. Calculate the work $W$ done by the concurrent forces on the object $M$. | 2 | 74 | 1 |
math | $\triangle ABC$ and $\triangle ABD$ are inscribed in a circle of radius $r$ such that $AB$ is a chord within the circle, and $C$ and $D$ are on opposite arcs divided by $AB$. It's given that segment $CD$ is always perpendicular to $AB$. Points $C$ and $D$ do not coincide with $A$ or $B$. If $s = AC + AD$, then for all ... | 8r^2 | 114 | 4 |
math | In a geometric sequence $\{a_{n}\}$, $a_{7}=8a_{4}$, and $\frac{1}{2}a_{2}$, $a_{3}-4$, $a_{4}-12$ form an arithmetic sequence.
$(1)$ Find the general formula for $\{a_{n}\}$.
$(2)$ Let ${b}_{n}=(-1)^{n}\log_{2}{a}_{n}$. The sum of the first $n$ terms of the sequence $\{b_{n}\}$ is $T_{n}$. Find the value of $k$ su... | k=40 \text{ or } 37 | 143 | 12 |
math | Simplify the expression: $\frac{3^{n+5} - 3(3^n)}{3(3^{n+4})}$. Express your answer as a common fraction. | \frac{80}{27} | 40 | 9 |
math | In a Cartesian coordinate system, it is known that the distance from point $P\left(m-4,2m+7\right)$ to the two coordinate axes is equal. Find the value of $m$. | m = -11 \text{ or } m = -1 | 44 | 14 |
math | Let $a = \pi/2008$. Find the smallest positive integer $n$ such that \[2[\cos(a)\sin(a) + \cos(4a)\sin(2a) + \cos(9a)\sin(3a) + \cdots + \cos(n^2a)\sin(na)]\] is an integer. | 251 | 78 | 3 |
math | Right triangle $XYZ$ (hypotenuse $\overline{XZ}$) is inscribed in equilateral triangle $LMN,$ as shown. If $LZ = 4$ and $ZM = NX = 3,$ compute $NY.$ | 3 | 53 | 1 |
math | Given F<sub>1</sub> and F<sub>2</sub> are the left and right foci respectively of the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ (where $a > b > 0$) with semi-focal length $c$, the line $x = -\frac{a^2}{c}$ intersects the x-axis at point N, satisfying $\overrightarrow{F_{1}F_{2}} = 2\overrightarrow{NF_{1}}$, and $... | x = -1 | 255 | 4 |
math | Evaluate $\dfrac {\cos 20^{\circ}}{\cos 35^{\circ} \sqrt {1 - \sin 20^{\circ}}}$. | \sqrt{2} | 37 | 5 |
math | Given the parabola $y^2 = 8x$ with focus $F$, a line passing through $F$ at an angle of $60^\circ$ intersects the parabola at points $A$ and $B$ (with point $A$ in the first quadrant), and intersects its directrix at point $C$. Calculate the ratio of the areas of triangles $AOC$ and $BOF$, which is $\frac {S_{△AOC}}{S_... | \frac {S_{△AOC}}{S_{△BOF}} = \frac { \frac {1}{2}\cdot 2\cdot 8 \sqrt {3}}{ \frac {1}{2}\cdot 2\cdot \frac {4 \sqrt {3}}{3}} = 6 | 107 | 67 |
math | The coordinates of the focus of the parabola $x^{2}=-y$ are $\left( 0,-\dfrac{1}{4} \right)$. | \left(0, -\frac{1}{4}\right) | 37 | 15 |
math | Two circles with radii 1 and 2 have a common center \( O \). The area of the shaded region is three times smaller than the area of the larger circle. Find the angle \( \angle AOB \). | \frac{8\pi}{9} | 46 | 9 |
math | Given point \( O \) inside triangle \( \triangle ABC \), such that \( \overrightarrow{OA} + 2 \overrightarrow{OB} + 3 \overrightarrow{OC} = \overrightarrow{0} \), calculate the ratio of the area of \( \triangle ABC \) to the area of \( \triangle AOC \). | 3 | 74 | 1 |
math | Given the geometric sequence $\{a_n\}$, where $a_n > 0$ and $a_7 = \frac{\sqrt{2}}{2}$, find the minimum value of $\frac{1}{a_3} + \frac{2}{a_{11}}$. | 4 | 62 | 1 |
math | Two sides of a triangle are $8 \mathrm{dm}$ and $5 \mathrm{dm}$; the angle opposite to the first side is twice as large as the angle opposite to the second side. What is the length of the third side of the triangle? | 7.8 | 54 | 3 |
math | Given the functions $f(x) = \ln x$, $g(x) = \frac{1}{2}x^2 - 2x$, for $x > 2$, if $k(x - 2) < xf(x) + 2g'(x) + 3$ always holds, the maximum integer value of $k$ is ____. | k = 5 | 76 | 4 |
math | Given the function $f(x)=4\sin\left(8x-\frac{\pi}{9}\right)$, $x\in \left[0,+\infty \right)$, determine the initial phase of this harmonic motion. | -\frac{π}{9} | 50 | 7 |
math | Given that $(a+1)x - 1 - \ln x \leqslant 0$ holds for any $x \in [\frac{1}{2}, 2]$, find the maximum value of $a$. | 1 - 2\ln 2 | 48 | 8 |
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