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 | Person A and person B each guess a riddle in each guessing activity. If one person guesses correctly and the other person guesses incorrectly, the person who guessed correctly wins; otherwise, it is a tie. It is known that in each activity, the probabilities of A and B guessing correctly are $\frac{5}{6}$ and $\frac{3}... | \frac{7}{27} | 131 | 8 |
math | What is the lowest prime number that is thirteen more than a cube? | 229 | 14 | 3 |
math | The bacteria in a laboratory culture grow at a rate such that their number quadruples every 12 hours. If there are initially 200 bacteria in the culture, calculate how many hours it will take until there are exactly 819,200 bacteria. Also, determine how many bacteria there will be after 24 hours. | 3,200 | 71 | 5 |
math | The side length of square $A$ is 48 cm. The side length of square $B$ is 60 cm. What is the ratio of the area of square $A$ to the area of square $B$, and what is the ratio of the perimeter of square $A$ to the perimeter of square $B$? Express your answer as a common fraction. | \frac{4}{5} | 78 | 7 |
math | Given the binomial coefficients $C_{10}^{r+1}$ and $C_{10}^{17-r}$, determine the number of possible values for their sum. | 2 | 39 | 1 |
math | The left and right foci of the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \left( a > b > 0 \right)$ are $F_1$ and $F_2$, respectively. $B$ is the bottom vertex of the ellipse, $A$ is the right vertex of the ellipse, and point $M$ is on the ellipse such that $MF_2 \perp x$-axis. The origin is $O$, and $OM \parallel ... | (0, \frac{\pi}{2}] | 169 | 10 |
math | In the Cartesian coordinate system xOy, with the origin as the pole O and the positive x-axis as the polar axis, the polar equation of circle C is $\rho=4\sqrt{2}\cos(\theta+ \frac{\pi}{4})$.
(1) Convert the polar equation of circle C into a Cartesian coordinate equation;
(2) A line l with a slope of 1 passes through p... | \frac{\sqrt{6}}{2} | 126 | 10 |
math | Given the function $f(x) = \log_{\frac{1}{2}} \sqrt{-x^2 + 2x + 8}$.
(1) Find the domain of $f(x)$;
(2) Find the range of $f(x)$. | \left[\log_{\frac{1}{2}}3, +\infty\right) | 57 | 21 |
math | Given that the domain of the function $f(x)$ is $\mathbb{R}$, and for all $x \in \mathbb{R}$, $f(-x) = f(x)$ holds.
1. If $x \geq 0$, $f(x) = (\frac{1}{2})^x$, find the solution set of the inequality $f(x) > \frac{1}{4}$.
2. If $f(x+1)$ is an even function, and when $x \in [0, 1]$, $f(x) = 2^x$, find the analytical exp... | 2^{x - 2015} | 153 | 10 |
math | What is the remainder when $(x + 2)^{2023}$ is divided by $x^2 + x + 1$? | 1 | 31 | 1 |
math | The sequence $\left\{a_{n}\right\}$ satisfies $a_{1}=\frac{1}{2}, a_{n+1}=a_{n}^{2}+a_{n}, n \in \mathbf{N}^{*}, b_{n}=\frac{1}{1+a_{n}}$.
Given:
$$
S_{n}=b_{1}+b_{2}+\cdots+b_{n}, P_{n}=b_{1} b_{2} \cdots b_{n},
$$
find the value of $2 P_{n}+S_{n}$. | 2 | 133 | 1 |
math | In a new logo design, a square of side length 24 inches contains four larger circles and one smaller circle. Each of the larger circles is tangent to two sides of the square and two other larger circles. The smaller circle is placed at the center and is tangent to all four larger circles. Calculate the total shaded are... | 576 - 153\pi | 69 | 10 |
math | The equation $x^{2} = x\sin x + \cos x$ has how many real solutions? | 2 | 23 | 1 |
math | Determine all positive integers $n$ such that the number $n(n+2)(n+4)$ has at most $15$ positive divisors. | 1, 2, 3, 4, 5, 7, 9 | 39 | 19 |
math | Three candles can burn for 30, 40, and 50 minutes, respectively (but are not ignited simultaneously). It is known that the three candles are burning simultaneously for 10 minutes, and only one candle is burning for 20 minutes. How long are exactly two candles burning simultaneously? | 35 | 64 | 2 |
math | Let $A=\{1,a-1\}$, $B=\{-1,2a-3,1-2a\}$. If $A\subseteq B$, then solve for the value of $a$. | 0 | 46 | 1 |
math | Given the function $f(x) = 4x^2 - 4ax + a^2 - 2a + 2$ has a maximum value of 3 on the interval $[0, 2]$. Find the value of the real number $a$. | 1 + \sqrt{2} | 57 | 7 |
math | Find the number of different arrangements for a class to select 6 people to participate in two volunteer activities, with each activity accommodating no more than 4 people. | 50 | 32 | 2 |
math | Given an geometric sequence $\{a_{n}\}$ with the sum of the first $n$ terms denoted as $S_{n}$. It is known that $a_{3}=\frac{1}{8}$, and the sequence formed by $S_{2}+\frac{1}{16}$, $S_{3}$, $S_{4}$ is an arithmetic sequence. Additionally, the sequence $\{b_{n}\}$ satisfies $b_{n}=8n$.
(I) Find the general term formu... | 16 - \frac{16+8n}{2^{n}} | 157 | 16 |
math | Find the number of ordered pairs $(x,y)$ of real numbers such that
\[9^{x^2 + y} + 9^{x + y^2} = 1.\] | 1 | 40 | 1 |
math | Compute the following expression:
\[ 2(1+2(1+2(1+2(1+2(1+2(1+2(1+2))))))) \] | 510 | 40 | 3 |
math | A rectangular prism has six faces, twelve edges, and eight vertices. A segment, such as $x$, which joins two vertices not joined by an edge, is called a diagonal. Considering the rectanglular prism has dimensions such that the length, height, and width are all different, calculate how many total diagonals it has and ho... | 16 | 79 | 2 |
math | A magazine printed photos of three celebrities along with three photos of the celebrities as babies. The baby pictures did not identify the celebrities. Readers were asked to match each celebrity with the correct baby pictures. What is the probability that a reader guessing at random will match all three correctly? | \frac{1}{6} | 55 | 7 |
math | Given a point $P$ on the right branch of the hyperbola $x^2-y^2=a^2$ ($a>0$), with $A_1$ and $A_2$ being the left and right vertices of the hyperbola, respectively, and $\angle A_2PA_1 = 2\angle PA_1A_2$, find the measure of $\angle PA_1A_2$. | \frac{\pi}{8} | 92 | 7 |
math | Find maximal positive integer $p$ such that $5^7$ is sum of $p$ consecutive positive integers | 125 | 29 | 3 |
math | At Joe's Fruit Stand, 5 bananas cost as much as 3 apples, and 8 apples cost as much as 5 oranges. How many oranges cost as much as 25 bananas? | 9 | 41 | 1 |
math | Among 10 products, there are 8 qualified products and 2 unqualified products. Two products are drawn without replacement from these 10 products, one at a time. If it is known that one of the draws is a qualified product, calculate the probability that the other draw is also a qualified product. | \frac{7}{11} | 64 | 8 |
math | $\bigcirc \bigcirc \div \square=14 \cdots 2$, how many ways are there to fill the square? | 4 | 29 | 1 |
math | Given the function \( f(x) = x^2 - a x + a - 1 \) where \( a \in \mathbb{R} \), if for any \( a \in (0, 4) \), there exists \( x_0 \in [0, 2] \) such that \( t \leq \left| f(x_0) \right| \) holds, then the range of \( t \) is . | 1 | 97 | 1 |
math | Three circles with radii $2$, $3$, and $4$ are mutually externally tangent. What is the area of the triangle determined by their points of tangency?
A) $\frac{6}{5}$
B) $\frac{10}{3}$
C) $\frac{8}{3}$
D) 4 | \frac{8}{3} | 70 | 7 |
math | Find all functions $f:\mathbb{R} \to \mathbb{R}$ such that for any two real numbers $x,y$ holds
$$f(xf(y)+2y)=f(xy)+xf(y)+f(f(y)).$$ | f(x) = 2x \text{ and } f(x) = 0 | 52 | 18 |
math | When studying the correlation between two variables, it is observed that the sample points in the scatter plot are concentrated around a certain exponential curve $y=e^{bx+a}$. Let $z=\ln y$, and the linear regression equation is obtained as $\hat{z} =0.25x-2.58$. Determine the regression equation for this model. | y = e^{0.25x - 2.58} | 74 | 16 |
math | The largest number by which the expression $n^4 - n^2$ is divisible for all possible integral values of $n$. | 12 | 27 | 2 |
math | Let $\mathbf{a} = \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 2 \\ 0 \\ -1 \end{pmatrix}.$ Find the vector $\mathbf{v}$ that satisfies $\mathbf{v} \times \mathbf{a} = \mathbf{b} \times \mathbf{a}$ and $\mathbf{v} \times \mathbf{b} = \mathbf{a} \times \mathbf{b}.$ | \begin{pmatrix} 3 \\ 1 \\ -1 \end{pmatrix} | 129 | 20 |
math | If one of the 13 provinces or territories is chosen at random, calculate the probability that it joined Canadian Confederation between 1890 and 1969. | \frac{4}{13} | 37 | 8 |
math | Let $m > n$ be positive integers such that $3(3mn - 2)^2 - 2(3m -3n)^2 = 2019$ . Find $3m + n$ .
| 46 | 52 | 2 |
math | 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} | 113 | 13 |
math | Let $p, q, r, s, t, u, v, w$ be distinct elements in the set $\{-8, -6, -4, -1, 1, 3, 5, 14\}$. What is the minimum possible value of $(p+q+r+s)^2 + (t+u+v+w)^2$? | 10 | 78 | 2 |
math | Given an arithmetic sequence $\{a_{n}\}$ with the sum of the first $n$ terms as $S_{n}$, $a_{6} \lt 0$, and $a_{4}+a_{9} \gt 0$, determine the largest value of $n$ that satisfies the inequality $S_{n} \lt 0$. | 11 | 76 | 2 |
math | Determine the value of \(\frac{c}{b}\) when the quadratic \(x^2 - 2100x - 8400\) is expressed in the form \((x+b)^2 + c\), where \(b\) and \(c\) are constants. | 1058 | 61 | 4 |
math | Let numbers $x$ and $y$ be chosen independently at random from the intervals $[0, \pi]$ and $[-\frac{\pi}{2}, \frac{\pi}{2}]$, respectively. Define $P(\alpha)$ as the probability that
\[\cos^2{x} + \cos^2{y} < \alpha\]
where $\alpha$ is a constant with $1 < \alpha \leq 2$. Find the maximum value of $P(\alpha)$.
A) $\fr... | \frac{\pi}{2} | 146 | 7 |
math | A coordinate system and parametric equations (4-4):
In the rectangular coordinate system $xoy$, the curve $C_1$ is defined by the parametric equations: $\begin{cases} x = \cos\alpha \\ y = \sin^2\alpha \end{cases}$ $(\alpha$ is the parameter$)$, and in the polar coordinate system with the origin $o$ as the pole and the... | \sqrt{2} - 1 | 232 | 8 |
math | A student, Leo, needs to complete homework assignments to earn grades. For the first four grades, he needs to complete one assignment per grade. For the next seven grades, he requires three assignments per grade. For all subsequent grades, the number of homework assignments per grade increases by two for each set of fo... | 153 | 125 | 3 |
math | In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $b\sin B-a\sin A= \frac{1}{2}a\sin C$ and the area of $\triangle ABC$ is $a^{2}\sin B$, then $\cos B=$ ______. | \frac{3}{4} | 80 | 7 |
math | Let the function $g$ take nonnegative integers to real numbers, defined by $g(1) = 2$, and
\[g(m + n) + g(m - n) = 2[g(m) + g(n)]\]
for all nonnegative integers $m \ge n$. Find the value of $g(10)$. | 200 | 73 | 3 |
math | A plane is drawn through a side of the lower base of a cube, dividing the volume of the cube in the ratio \( m: n \), measured from the lower base. Find the angle between this plane and the plane of the base, given that \( m \leq n \). | \alpha = \arctan \left(\frac{2m}{m + n}\right) | 59 | 21 |
math | Let the function $f(x) = ax^2 + b$ ($a \neq 0$). If $\int_{0}^{3} f(x) \, dx = 3f(x_0)$, then $x_0 = \_\_\_\_\_\_$. | \pm \sqrt{3} | 61 | 7 |
math | Three people, Jia, Yi, and Bing, participated in a competition and they took the top 3 places (with no ties). Jia said: "I am first", Yi said: "I am not first", and Bing said: "I am not third". Only one of them is telling the truth. If the rankings of Jia, Yi, and Bing are respectively $A, B, C$, then the three-digit n... | 312 | 103 | 3 |
math | Let $n$ be an even positive integer. Alice and Bob play the following game. Before the start of the game, Alice chooses a set $S$ containing $m$ integers and announces it to Bob. The players then alternate turns, with Bob going first, choosing $i\in\{1,2,\dots, n\}$ that has not been chosen and setting the valu... | m = 2^{\frac{n}{2}} | 209 | 11 |
math | Given the quadratic equation in $x$: $x^{2}-4x+2k=0$.
$(1)$ If the equation has real roots, find the range of real values for $k$.
$(2)$ If $k$ is the largest integer that satisfies $(1)$, and one of the roots of the equation $x^{2}-4x+2k=0$ is a root of the quadratic equation $x^{2}-2mx+3m-1=0$, find the value of ... | x = 4 | 118 | 4 |
math | Let \( g : \mathbb{R} \to \mathbb{R} \) be a function such that
\[ g(xg(y) - x) = xy - g(x) \] for all \( x, y \).
Let \( m \) be the number of possible values of \( g(-2) \), and let \( t \) be the sum of all possible values of \( g(-2) \). Find \( m \times t \). | 0 | 99 | 1 |
math | Consider a regular tetrahedron with vertices \(A\), \(B\), \(C\), and \(D\), where each edge of the tetrahedron is of length 1. Let \(P\) be a point on edge \(AB\) such that \(AP = t \times AB\) and \(Q\) be a point on edge \(CD\) such that \(CQ = s \times CD\), where \(0 \leq t, s \leq 1\). Find the least possible dis... | \frac{\sqrt{2}}{2} | 118 | 10 |
math | Real numbers \( x, y, \) and \( z \) satisfy the equation:
\[ 3(x + y + z) = x^2 + y^2 + z^2. \]
Let \( N \) be the maximum value of \( xy + xz + yz \), and let \( n \) be the minimum value of \( xy + xz + yz \). Find \( N + 5n \). | 27 | 92 | 2 |
math | Thirty-two 5-inch wide square posts are evenly spaced with 4 feet between adjacent posts to enclose a square field. What is the outer perimeter, in feet, of the fence? | 125\frac{1}{3} \text{ feet} | 38 | 15 |
math | If the complex number $z$ corresponds to a point in the second quadrant of the complex plane, and $|z|=2$, then $z$ is equal to ______. (Write down one answer) | -1 + \sqrt{3}i | 42 | 9 |
math | A piece of fabric shrinks by $\frac{1}{18}$ of its length and by $\frac{1}{14}$ of its width during washing. What length of the fabric needs to be taken in order to have $221 \mathrm{~m}^{2}$ after washing, if its width was 875 mm before washing? | 288 \, \text{m} | 75 | 10 |
math | In her first $10$ basketball games, Rachel scored $9, 5, 7, 4, 8, 6, 2, 3, 5,$ and $6$ points. In her next game, she scored fewer than $10$ points and her points-per-game average for these $11$ games was an integer. In the following game, she again scored fewer than $10$ points and her average for the $12$ games was al... | \textbf{(A)}\ 0 | 151 | 9 |
math | Mady has an infinite number of balls and boxes available to her. The empty boxes, each capable of holding sixteen balls, are arranged in a row from left to right. At the first step, she places a ball in the first box (the leftmost box) of the row. At each subsequent step, she places a ball in the first box of the row t... | 30 | 136 | 2 |
math | Given a sequence $\{a_n\}$ with the sum of its first $n$ terms as $S_n$, and it satisfies the equation $3S_n = 2a_n + 1$:
(1) Find the general term formula for the sequence $\{a_n\}$;
(2) Let another sequence $\{b_n\}$ be defined by $b_n = (n + 1)a_n$, find the sum of the first $n$ terms of the sequence $\{b_n\}$, deno... | \frac{4}{9} - \frac{3n+4}{9} \cdot (-2)^n | 115 | 24 |
math | Given a "za field" with bases of 10 bu and 20 bu, and a height of 10 bu, and a "gui field" with a base of 8 bu and a height of 5 bu, calculate the probability that a tea tree planted randomly in the "za field" is exactly planted in the "gui field". | \frac{2}{15} | 73 | 8 |
math | Gretchen has ten socks, two of each color: red, blue, green, yellow, and purple. She randomly draws five socks. What is the probability that she has exactly two pairs of socks with the same color? | \frac{5}{42} | 46 | 8 |
math | Given a set of data: 10, 10, x, 8, where the median is equal to the mean, find the median of this data set. | 10 | 36 | 2 |
math | Given the function $f(x) = x^3 - 3x$,
(1) Determine the intervals of monotonicity for $f(x)$.
(2) Find the maximum and minimum values of $f(x)$ on the interval $[-3, 2]$. | -18 | 59 | 3 |
math | Given that the equation \(2x^{2} + kx - 2k + 1 = 0\) has two real roots whose sum of squares is \(\frac{29}{4}\), determine the value of \(k\). | 3 \text{ or } -11 | 51 | 9 |
math | A city uses a lottery system for assigning car permits, with 300,000 people participating in the lottery and 30,000 permits available each month.
1. If those who win the lottery each month exit the lottery, and those who do not win continue in the following month's lottery, with an additional 30,000 new participants ad... | 10 | 181 | 2 |
math | Given the functions $f$ that satisfy $f(x+6) + f(x-6) = f(x)$ for all real $x$, determine the least common positive period $p$ for all such functions. | 36 | 44 | 2 |
math | Given a rectangular pool measuring 20 m by 8 m, with a 1 m wide walkway around the outside, calculate the area of the walkway. | 60 \, \text{m}^2 | 34 | 11 |
math | Two cards are chosen at random from a standard 52-card deck. What is the probability that both cards are numbers (2 through 10) totaling to 12? | \frac{35}{663} | 38 | 10 |
math | Determine the largest prime number less than 5000 of the form \( a^n - 1 \), where \( a \) and \( n \) are positive integers, and \( n \) is greater than 1. | 127 | 49 | 3 |
math | Given the function $f(x) = ax^3 + b\sin x + 4$ ($a, b \in \mathbb{R}$), and $f(\lg(\log_2 10)) = 5$, then $f(\lg(\lg 2)) =$ | 3 | 61 | 1 |
math | In the rectangular coordinate system $(xOy)$, there is a line $l_{1}$: $x=-2$, and a curve $C$: $\begin{cases} x=2\cos \theta \\ y=2+2\sin \theta \end{cases}(\theta$ is a parameter$)$. Establish a polar coordinate system with the coordinate origin $O$ as the pole and the positive half of the $x$-axis as the polar axis.... | (-2 \sqrt {2}, \frac {\pi}{4}) | 207 | 14 |
math | The volume of a refrigerator is approximately 150 what units? | 150\,\text{Liters} | 14 | 10 |
math | Last year, Ms. Jane Doe received an inheritance, and paid $25\%$ in federal taxes on the inheritance, and then paid $15\%$ of what remained in state taxes. She paid a total of $\textdollar15000$ for both taxes. Calculate the value of her inheritance. | 41379 | 68 | 5 |
math | If the foci of the ellipse $\dfrac{x^{2}}{a^{2}}+ \dfrac{y^{2}}{b^{2}}=1$ are on the $x$-axis, and a tangent line to the circle $x^{2}+y^{2}=4$ passing through point $C(2,1)$ intersects the circle at points $A$ and $B$, such that line $AB$ exactly passes through the right focus and the top vertex of the ellipse, then t... | \dfrac {x^{2}}{20}+ \dfrac {y^{2}}{16}=1 | 116 | 25 |
math | Find three different polynomials \(P(x)\) with real coefficients such that \(P(x^2 + 1) = P(x)^2 + 1\) for all real \(x\). | P(x) = x, P(x) = x^2 + 1, P(x) = x^4 + 2x^2 + 2 | 40 | 33 |
math | A residential building has a construction cost of 250 yuan per square meter. Considering a useful life of 50 years and an annual interest rate of 5%, what monthly rent per square meter is required to recoup the entire investment? | 1.14 | 50 | 4 |
math | Three congruent circles of radius $2$ are drawn in the plane so that each circle passes through the centers of the other two circles. The region common to all three circles has a boundary consisting of three congruent circular arcs. Let $K$ be the area of the triangle whose vertices are the midpoints of those arcs.... | 300 | 111 | 3 |
math | Let $f(x)$ be a differentiable function satisfying $\lim_{x\rightarrow 0} \frac{f(1)-f(1+2x)}{2x} = 1$. Find the slope of the tangent line to the curve $y=f(x)$ at the point $(1, f(1))$. | -1 | 68 | 2 |
math | A point $(x, y)$ is randomly picked from inside the rectangle with vertices $(0,0)$, $(6,0)$, $(6,3)$, and $(0,3)$. What is the probability that $2x < y$? | \frac{1}{8} | 53 | 7 |
math | Calculate $52430_{7} - 4320_{8}$ in base 10. | 10652 | 25 | 5 |
math | Two right circular cones with vertices facing down as shown in the figure below contain the same amount of liquid. The radii of the tops of the liquid surfaces are $3$ cm and $6$ cm. Into each cone is dropped a spherical marble of radius $1$ cm, which sinks to the bottom and is completely submerged without spilling any... | 4:1 | 99 | 3 |
math | Find the number of permutations $(a_1, a_2, a_3, a_4, a_5, a_6, a_7)$ of $(1,2,3,4,5,6,7)$ that satisfy
\[\frac{a_1 + 1}{2} \cdot \frac{a_2 + 2}{2} \cdot \frac{a_3 + 3}{2} \cdot \frac{a_4 + 4}{2} \cdot \frac{a_5 + 5}{2} \cdot \frac{a_6 + 6}{2} \cdot \frac{a_7 + 7}{2} > 7!.\] | 5039 | 156 | 4 |
math | Given the containers A, B, and C, where A is $\tfrac{4}{5}$ full, B is initially empty and becomes $\tfrac{3}{5}$ full after pouring from A, and C is initially empty and becomes $\tfrac{3}{4}$ full after pouring from B, determine the ratio of the volume of A to the volume of C. | \frac{15}{16} | 78 | 9 |
math | The inclination angle of the line $y = x + m$ is __________. | \frac{\pi}{4} | 17 | 7 |
math | (1) Given that the domain of the function $y=f(x)$ is $[-1,2]$, find the domain of the function $y=f(1-x^{2})$.
(2) Given that the domain of the function $y=f(2x-3)$ is $(-2,1]$, find the domain of the function $y=f(x)$. | (-7,-1] | 79 | 5 |
math | Let $V$ be a rectangular prism with integer side lengths. The largest face has area 240 and the smallest face has area 48. A third face has area $x$, where $x$ is not equal to 48 or 240. What is the sum of all possible values of $x$? | 260 | 70 | 3 |
math | A book with 73 pages numbered 1 to 73 has its pages renumbered in reverse, from 73 to 1. How many pages do the new page number and old page number share the same units digit? | 15 | 49 | 2 |
math | In the diagram below, $\|\overrightarrow{OA}\| = 1,$ $\|\overrightarrow{OB}\| = 1,$ and $\|\overrightarrow{OC}\| = \sqrt{2}.$ Also, $\tan \angle AOC = 7$ and $\angle BOC = 45^\circ.$
[asy]
unitsize(2 cm);
pair A, B, C, O;
A = (1,0);
B = (-0.6,0.8);
C = (0.2,1.4);
O = (0,0);
draw(O--A,Arrow(6));
draw(O--B,Arrow(6))... | \left( \frac{5}{4}, \frac{7}{4} \right) | 234 | 20 |
math | Quadrilateral $PQRS$ is inscribed in a circle with segment $PR$ as a diameter of the circle. If $m\angle RPS = 60^\circ$ and $m\angle QPR = 30^\circ$, determine the ratio of the area of $PQRS$ to the area of the circle. Express your answer as a common fraction in simplest radical form in terms of $\pi$ as $\frac{a+\sqr... | 7 | 132 | 1 |
math | A rectangular box has a volume of $4608$ cubic inches and a surface area of $1824$ square inches. The sum of the lengths of its $12$ edges is $216$ inches. What is the volume of the box if its length, width, and height were each increased by two inches? | 6656 \text{ cubic inches} | 71 | 10 |
math | Given Ellen has a $3 \times 7$ index card. If she shortens the length of one side of this card by $2$ inches and the other side by $1$ inch, the card would have an area of $10$ square inches. Find the area of the card in square inches if instead she shortens the length of the first side by $1$ inch and the second side ... | 10 | 90 | 2 |
math | When $0.73\overline{864}$ is expressed as a fraction in the form $\frac{y}{999900}$, what is the value of $y$? | 737910 | 44 | 6 |
math | For all $x \in (0, +\infty)$, the inequality $(2x - 2a + \ln \frac{x}{a})(-2x^{2} + ax + 5) \leq 0$ always holds. Determine the range of values for the real number $a$. | \left\{ \sqrt{5} \right\} | 66 | 13 |
math | A certain shopping mall is conducting a trial sale of a type of clothing with a cost of $60 per item. It is stipulated that during the trial sale period, the selling price must not be lower than the cost price, and the profit must not exceed 40%. It was found during the trial sale that the sales volume $y$ (in units) a... | 70 \leq x \leq 84 | 203 | 12 |
math | If $2^a + 2^b = 5^c + 5^d$, where $a, b, c, d$ are integers and at most how many of them can be negative?
A) 4
B) 3
C) 2
D) 1
E) 0 | \textbf{(E)}\ 0 | 68 | 9 |
math | Given the vectors $\overrightarrow{a}=(\cos x, \sin x)$, $\overrightarrow{b}=(\cos x + 2\sqrt{3}, \sin x)$, and $\overrightarrow{c}=(0, 1)$, where $x \in \mathbb{R}$:
1. If $\overrightarrow{a} \perp \overrightarrow{c}$, find the value of $\cos 2x$.
2. If the function $f(x) = \overrightarrow{a} \cdot (\overrightarrow{b}... | 5 | 149 | 1 |
math | If point \( P \) is the circumcenter of \(\triangle ABC\) and \(\overrightarrow{PA} + \overrightarrow{PB} + \lambda \overrightarrow{PC} = \mathbf{0}\), where \(\angle C = 120^\circ\), then find the value of the real number \(\lambda\). | -1 | 75 | 2 |
math | Given an arithmetic sequence {a<sub>n</sub>} with a common difference d > 0, and a<sub>2</sub> is the geometric mean of a<sub>1</sub> and a<sub>4</sub>. Let $$b_{n}=a_{2^{n}}$$. If for any n ∈ N*, we have $$\frac {1}{b_{1}}+ \frac {1}{b_{2}}+…+ \frac {1}{b_{n}}<3$$, find the range of d. | [\frac {1}{3}, +\infty) | 117 | 12 |
math | Given 60 bottles of mineral water, numbered from 1 to 60, determine the possible sequence of individual numbers drawn if a systematic sampling method is used to draw 6 bottles for inspection. | 3, 13, 23, 33, 43, 53 | 41 | 21 |
math | Let $S_n$ be the sum of the first $n$ terms of an arithmetic sequence $\{a_n\}$, with $S_{10}=16$ and $S_{100}-S_{90}=24$. Find $S_{100}$. | 200 | 61 | 3 |
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