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 | In a rectangular parallelepiped, let dimensions be \( AB = 4 \), \( BC = 2 \), and \( CG = 3 \). Point \( M \) is the midpoint of edge \( \overline{FG} \), and \( E \) is a point on edge \( \overline{AB} \) such that \( AE = 1 \). Find the volume of the pyramid with base \( BCEF \) and apex \( M \). | 3 | 98 | 1 |
math | Each edge of a regular tetrahedron is given a stripe. The choice of which edge to stripe is made at random. What is the probability that there is at least one triangle face with all its edges striped? | \frac{1695}{4096} | 44 | 13 |
math | Senya has three straight sticks, each 24 centimeters long. Senya broke one of them into two pieces so that using the two pieces of the broken stick and the two whole sticks, he could form the outline of a right triangle. How many square centimeters is the area of this triangle? | 216 | 62 | 3 |
math | A conveyor system produces on average 85% of first-class products. How many products need to be sampled so that, with a probability of 0.997, the deviation of the frequency of first-class products from 0.85 in absolute magnitude does not exceed 0.01? | 11475 | 63 | 5 |
math | Given the ellipse $$E: \frac {x^{2}}{a^{2}}+y^{2}=1(a>1)$$, the line passing through points A(0, -1) and B(a, 0) has a distance from the origin of $$\frac { \sqrt {3}}{2}$$.
(Ⅰ) Find the equation of ellipse E;
(Ⅱ) The line $l$: $y=kx+1$ intersects ellipse E at points C and D. The circle with diameter CD passes thro... | y= \frac {1}{3}x+1 | 130 | 12 |
math | In triangle $XYZ$, where $XY = 7$, $YZ = 6$, and $ZX = 5$, determine the probability that a randomly selected point inside the triangle is closer to vertex $Z$ than to either $X$ or $Y$. | \frac{1}{4} | 53 | 7 |
math | Nine tiles are numbered $1, 2, 3, \ldots, 9,$ respectively. Each of three players randomly selects and keeps three of the tile, and sums those three values. The probability that all three players obtain an odd sum is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$ | 17 | 88 | 2 |
math | Find the equation of the line that passes through point A (3, 2) and is perpendicular to the line $4x+5y-8=0$. | 4y-5x+7=0 | 34 | 9 |
math | Laura's uncle made 60 cupcakes for a family gathering. Of these cupcakes, one-third contained blueberries, one-fourth contained sprinkles, half contained frosting, and one-fifth contained pecans. What is the largest possible number of cupcakes that had none of these ingredients? | 0 | 57 | 1 |
math | 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 | 107 | 1 |
math | Given points $F_{1}$ and $F_{2}$ are respectively the left and right foci of the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ ($a > 0, b > 0$), and $P$ is any point on the left branch of the hyperbola. If the minimum value of $\frac{|PF_{2}|^{2}}{|PF_{1}|}$ is $9a$, then calculate the eccentricity of the h... | 5 | 120 | 1 |
math | Outside the line $l$, there is a point $A$ at a distance of $7$ cm from the line $l$. $B$ is any point on the line $l$. The possible length of the line segment $AB$ is ____ cm, the reason is ____. | AB \geqslant 7 \text{ cm} | 58 | 13 |
math | The letters of the alphabet are assigned numeric values using an alternating, cyclic sequence of $\{-3, -1, 0, 1, 3\}$. Starting with A and ending with Z following the pattern:
$$
3, 1, 0, -1, -3, -1, 0, 1, 3, 1, 0, -1, -3, -1, 0, 1, \ldots
$$
Two complete cycles are displayed. The letter A has a value of $3$, B is $1$... | 5 | 152 | 1 |
math | What percent of $x$ is equal to $60\%$ of $30\%$ of $x$? | 18\% | 27 | 4 |
math | A square and a circle intersect so that each side of the square contains a chord of the circle equal in length to twice the radius of the circle. What is the ratio of the area of the square to the area of the circle? Express your answer as a common fraction in terms of $\pi$. | \frac{2}{\pi} | 60 | 8 |
math | A deck of forty cards consists of four $1$'s, four $2$'s,..., and four $10$'s. A matching pair (two cards with the same number) is removed from the deck. Given that these cards are not returned to the deck, let $m/n$ be the probability that two randomly selected cards also form a pair, where $m$ and $n$ are relatively ... | 758 | 97 | 3 |
math | Given that point A lies on the circle C: x² + y² = 1, a line is drawn from point A perpendicular to the y-axis, intersecting the y-axis at point B. Point P satisfies the equation $\overrightarrow {BP}=2 \overrightarrow {BA}$.
1. Find the trajectory equation of point P.
2. Let Q be a point on the line l: x = 3, and O t... | \frac{3}{2} | 116 | 7 |
math | Find the minimum value of
\[\sqrt{x^2 + (1 - x)^2} + \sqrt{(1 - x)^2 + (1 + x)^2}\]over all real numbers $x.$ | \sqrt{5} | 45 | 5 |
math | For every positive integer $n$ , define $S_n$ to be the sum
\[
S_n = \sum_{k = 1}^{2010} \left( \cos \frac{k! \, \pi}{2010} \right)^n .
\]
As $n$ approaches infinity, what value does $S_n$ approach? | 1944 | 87 | 4 |
math | Given the product of two positive integers $a$ and $b$ is $143$, where Alice mistakenly reversed the digits of the two-digit number $a$ to obtain this value, calculate the correct value of the product of $a$ and $b$. | 341 | 54 | 3 |
math | Given a function $f(x)$ defined on $\mathbb{R}$, and the graph of the function $y=f(x-3)$ is symmetric about the point $(3,0)$. When $x \geq 0$, $f(x)=x^2+2x$. If $f(2-a^2) > f(a)$, determine the range of the real number $a$. | (-2, 1) | 84 | 6 |
math | A car's brakes are applied suddenly, and it travels 35 feet in the first second after the brakes are applied. The distance it travels in each subsequent second is 10 feet less than the distance traveled in the previous second. Calculate the total distance the car travels from the time the brakes are applied until it co... | 5 | 83 | 1 |
math | Given that $\alpha$ and $\beta$ are acute angles, and $\tan\alpha= \frac {1}{7}$, $\cos(\alpha+\beta)= \frac {2 \sqrt {5}}{5}$, find the value of $\cos 2\beta$. | \frac{4}{5} | 58 | 7 |
math | Find the sum of the $2007$ roots of $(x-1)^{2007}+2(x-2)^{2006}+3(x-3)^{2005}+\cdots+2006(x-2006)^2+2007(x-2007)$.
| 2005 | 77 | 4 |
math | If the complex number $z=(1+i)(3-2i)$ (where $i$ is the imaginary unit), find the quadrant in which the point representing the complex number $z$ in the complex plane is located. | 1 | 46 | 1 |
math | Given vectors $\overrightarrow{a}, \overrightarrow{b}$ satisfy $|\overrightarrow{a}|=1$, $|\overrightarrow{a}+ \overrightarrow{b}|= \sqrt{7}$, $\overrightarrow{a}\cdot (\overrightarrow{b}- \overrightarrow{a})=-4$, find the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$. | \dfrac{5\pi}{6} | 86 | 9 |
math | Find the second-next perfect square after a perfect square number $x$. | x + 4\sqrt{x} + 4 | 14 | 11 |
math | Given the line $l$: $\begin{cases}x=1+ \frac{1}{2}t \\ y= \frac{ \sqrt{3}}{2}t\end{cases} (t$ is a parameter$)$ and the curve $C_{1}$: $\begin{cases}x=\cos \theta \\ y=\sin \theta\end{cases} (\theta$ is a parameter$)$:
(1) Assume that $l$ intersects $C_{1}$ at two points $A$ and $B$, find $|AB|$;
(2) If the horizonta... | \frac{ \sqrt{6}}{4}( \sqrt{2}-1) | 210 | 18 |
math | Let $S = \{1, 22, 333, \dots , 999999999\}$ . For how many pairs of integers $(a, b)$ where $a, b \in S$ and $a < b$ is it the case that $a$ divides $b$ ? | 14 | 82 | 2 |
math | To express 20 as a sum of distinct powers of 2, we would write $20 = 2^4 + 2^2$. The sum of the exponents of these powers is $4 + 2 = 6$. If 1562 were expressed as a sum of distinct powers of 2, what would be the least possible sum of the exponents of these powers? | 27 | 84 | 2 |
math | Consider the following trigonometric identities:
① $\tan(10^\circ)\tan(20^\circ) + \tan(20^\circ)\tan(60^\circ) + \tan(60^\circ)\tan(10^\circ) = 1$;
② $\tan(13^\circ)\tan(35^\circ) + \tan(35^\circ)\tan(42^\circ) + \tan(42^\circ)\tan(13^\circ) = 1$;
③ $\tan(5^\circ)\tan(100^\circ) + \tan(100^\circ)\tan(-15^\circ) ... | 1 | 268 | 1 |
math | A sphere with center $O$ has radius $6$. A triangle with sides of length $15, 15,$ and $24$ is situated in space so that each of its sides is tangent to the sphere. Find the distance between $O$ and the plane determined by the triangle. | 2\sqrt{5} | 62 | 6 |
math | Let \( Q(z) = z^8 - (8\sqrt{2} - 10)z^4 + (8\sqrt{2} - 11) \). What is the minimum perimeter among all the 8-sided polygons in the complex plane whose vertices are precisely the zeros of \( Q(z) \)? | 8\sqrt{2} | 69 | 6 |
math | Given the universal set $U=\{a_{1},a_{2},a_{3},a_{4}\}$, the set $A$ is a subset of $U$ with exactly two elements, and satisfies the following three conditions:
1. If $a_{1} \in A$, then $a_{2} \in A$;
2. If $a_{3} \notin A$, then $a_{2} \notin A$;
3. If $a_{3} \in A$, then $a_{4} \notin A$.
Then, the set $A=$ ______.... | \{a_{2},a_{3}\} | 135 | 11 |
math | If $2008=2^{a_{1}}+2^{a_{2}}+\cdots+2^{a_{n}}$, where $a_{1}, a_{2}, \cdots, a_{n}$ are distinct non-negative integers, then the relationship between $\sin \sum_{i=1}^{n} a_{i}$, $\cos \sum_{i=1}^{n} a_{i}$, and $\tan \sum_{i=1}^{n} a_{i}$ is ______. (Given that $\pi=3.141592 \cdots$) | \sin \sum_{i=1}^{n} a_{i} > \tan \sum_{i=1}^{n} a_{i} > \cos \sum_{i=1}^{n} a_{i} | 132 | 50 |
math | Li Yun is sitting by the window in a train moving at a speed of 60 km/h. He sees a freight train with 30 cars approaching from the opposite direction. When the head of the freight train passes the window, he starts timing, and he stops timing when the last car passes the window. The recorded time is 18 seconds. Given t... | 44 | 122 | 2 |
math | If $|a-1|=4$, $|-b|=|-7|$, and $|a+b|\neq a+b$, find the value of $2a+b$. | 3 \text{ or } -13 | 37 | 9 |
math | If $a-b=2$ and $b-c=-3$, calculate the value of $a-c$. | -1 | 22 | 2 |
math | Two vertical poles stand on sloped ground. The bottoms of the poles are 20 feet apart. One pole is 12 feet tall, and the other is 30 feet tall. The ground slopes upward between the poles, starting from the shorter pole and rising linearly at a rate of 1 foot height increase every 4 feet horizontal. How long, in feet, i... | \sqrt{569} | 97 | 7 |
math | A bag contains four pieces of paper, each labeled with one of the digits $1$, $2$, $3$, or $5$, with no repeats. Three of these pieces are drawn, one at a time without replacement, to construct a three-digit number. Calculate the probability that the three-digit number is a multiple of $3$ and also an odd number. | \frac{1}{4} | 74 | 7 |
math | Either increasing the radius or the height of a cylinder by seven inches will result in the same volume. The original height of the cylinder is three inches. What is the original radius in inches? | r = 7 | 38 | 4 |
math | Translate a parabola $M_1$: $y = ax^2 + c$ to get another parabola $M_2$. $M_2$ passes through the vertex $A$ of $M_1$, and the axis of symmetry of $M_2$ intersects parabolas $M_1$ and $M_2$ at points $B$ and $C$ respectively. If the coordinates of point $C$ are $(2, c-5)$, then the area of $\triangle ABC$ is _________... | 10 | 114 | 2 |
math | How many ordered pairs of integers \((a, b)\) satisfy the following inequalities?
\[
\begin{aligned}
a^2 + b^2 &< 25 \\
a^2 + b^2 &< 10a \\
a^2 + b^2 &< 10b
\end{aligned}
\] | 13 | 73 | 2 |
math | There is a box containing 7 blue and 5 red balls. What is the minimum number of balls that need to be drawn to ensure there are at least 2 blue and 1 red balls among them? | 8 \text{ balls} | 43 | 6 |
math | Suppose $a$ and $b$ are positive integers such that $a$ has $4$ factors and $b$ has $a$ factors. If $b$ is divisible by $a$, determine the smallest possible value of $b$. | 24 | 51 | 2 |
math | Given that the sum of the binomial coefficients of the expansion of $(x+ \frac {m}{x})^{n}$ is $256$
$(1)$ Find $n$;
$(2)$ If the constant term in the expansion is $\frac {35}{8}$, find the value of $m$;
$(3)$ If the coefficient of the largest term in the expansion is only for the $6$th and $7$th terms, find the ... | m=2 | 106 | 3 |
math | The graph of the function $f(x)=\frac{x}{x+a}$ is symmetric about the point $(1,1)$, and the function $g(x)=\log_{10}(10^x+1)+bx$ is even. Find the value of $a+b$. | -\frac{3}{2} | 60 | 7 |
math | Given a line $l$ passing through point $P(-2, 1)$.
(1) When the distance from line $l$ to points $B(-5, 4)$ and $C(3, 2)$ is equal, find the equation of line $l$;
(2) When the area of the triangle formed by line $l$ with the x-axis and y-axis is $\frac{1}{2}$, find the equation of line $l$. | x+y+1=0 \text{ or } x+4y-2=0 | 99 | 19 |
math | Consider this histogram of scores for 100 students taking a test:
- Each interval represents a distinct score range, and the students' scores are distributed over these intervals with provided frequencies. Determine the score interval that contains the median.
- The intervals and student counts are as follows:
- $85... | 70-74 | 200 | 5 |
math | Given a sequence $\{a_n\}$, where consecutive terms $a_n$ and $a_{n+1}$ are the roots of the equation $x^2 - nx + b_n = 0$, and $a_{10} = 7$, find $b_{17}$. | 66 | 63 | 2 |
math | "Xiao Mahu" opened an "Animal Restaurant". When calculating this week's turnover, he mistook the 3 in the hundreds place for an 8. When calculating the money spent on purchases, he mistook the 8 in the tens place for a 5. As a result, he calculated that he made a profit of 1320 yuan this week. Students, how much money ... | 850 | 100 | 3 |
math | Let $p,$ $q,$ and $r$ be real numbers, and let $A,$ $B,$ $C$ be points such that the midpoint of $\overline{BC}$ is $(p, p, 0),$ the midpoint of $\overline{AC}$ is $(0, q, q),$ and the midpoint of $\overline{AB}$ is $(r, 0, r).$ Find
\[\frac{AB^2 + AC^2 + BC^2}{p^2 + q^2 + r^2}.\] | 8 | 118 | 1 |
math | Given that the solution set of the inequality $ax^2+5x+b<0$ is $\{x|-3<x<2\}$, then the solution set of the inequality $bx^2+5x+a>0$ is \_\_\_\_\_\_. | \left(-\frac{1}{3}, \frac{1}{2}\right) | 57 | 19 |
math | The children went to the forest to pick mushrooms. If Anya gives half of her mushrooms to Vitya, all the children will have the same number of mushrooms. But if Anya instead gives all her mushrooms to Sasha, Sasha will have as many mushrooms as all the others combined. How many children went to pick mushrooms? | 6 | 66 | 1 |
math | In a table tennis tournament, each participant played against every other participant once. Each match was officiated by one referee. All referees officiated a different number of matches. Player Ivanov claims that all his matches were officiated by different referees. Players Petrov and Sidorov claim the same about th... | \text{No} | 73 | 5 |
math | Determine the number of significant digits in the measurement of the side of a square whose computed area is $2.3406$ square inches to the nearest ten-thousandth of a square inch. | 5 | 42 | 1 |
math | Find the positive integer $k$ such that the roots of $x^3 - 15x^2 + kx -1105$ are three distinct collinear points in the complex plane. | 271 | 47 | 3 |
math | Express $0.5\overline{023}$ as a common fraction. | \frac{1045}{1998} | 18 | 13 |
math | The numbers $1522$, $1689$, and $1754$ each are 4-digit numbers starting with $1$, and each has exactly two identical digits. How many such numbers are there if the identical digits should not be 1 or 5? | 126 | 59 | 3 |
math | Find the number of distinct numbers in the list
\[
\left\lfloor \frac{1^2}{500} \right\rfloor, \ \left\lfloor \frac{2^2}{500} \right\rfloor, \ \left\lfloor \frac{3^2}{500} \right\rfloor, \ \dots, \ \left\lfloor \frac{2000^2}{500} \right\rfloor.
\] | 8000 | 112 | 4 |
math | Given the function $f(x) = x^3 + ax$ has two extreme points on $\mathbb{R}$, the range of the real number $a$ is. | a < 0 | 37 | 4 |
math | Given vectors $\overrightarrow{a}=(\sin x,1)$ and $\overrightarrow{b}=(\cos x,2)$, where $x\in \mathbb{R}$, and the function $f(x)=\overrightarrow{a}\cdot \overrightarrow{b}$.
$(1)$ When $x\in \left[ -\frac{\pi}{12},\frac{\pi}{3} \right]$, find the maximum and minimum values of $|\overrightarrow{a}+\overrightarrow{b}|... | 7 | 183 | 1 |
math | Draw chord OA from the origin O on the circle $x^2+y^2-8x=0$.
(1) Find the equation of the trajectory of the midpoint M of chord OA;
(2) Extend OA to N such that $|OA|=|AN|$, find the equation of the trajectory of point N. | x^2+y^2-16x=0 | 69 | 12 |
math | The sum of the numerical coefficients in the expansion of the binomial $(x+y)^8$ is | 256 | 20 | 3 |
math | 1) \(\lim_{n \rightarrow \infty}\left(1 + \frac{a}{n}\right)^{n}\)
2) \(\lim_{x \rightarrow 0} \sqrt[x]{1 - 2x}\)
3) \(\lim_{t \rightarrow \infty}\left(\frac{t - 3}{t + 2}\right)^{2t + 1}\)
4) \(\lim_{x \rightarrow \frac{\pi}{4}} (\tan x)^{\tan(2x)}\) | e^{-1} | 117 | 4 |
math | The diagonals of rectangle $PQRS$ intersect at point $X$. If $PS = 6$ and $RS=8$, then what is $\sin \angle PXS$? | \frac{24}{25} | 40 | 9 |
math | A square is divided into four congruent rectangles, as shown. If the perimeter of each of these four rectangles is 40 inches, what is the perimeter of the square, in inches?
[asy]
draw((0,0)--(0,4)--(4,4)--(4,0)--cycle);
draw((0,2)--(4,2));
draw((2,0)--(2,4));
[/asy] | \frac{160}{3} \text{ inches} | 90 | 14 |
math | The house number. A person mentioned that his friend's house is located on a long street (where the houses on the side of the street with his friend's house are numbered consecutively: $1, 2, 3, \ldots$), and that the sum of the house numbers from the beginning of the street to his friend's house matches the sum of the... | 204 | 139 | 3 |
math | How many zeros are in the expansion of $999,\!999,\!999,\!998^2$? | 11 | 31 | 2 |
math |
Two people are flipping a coin: one flipped it 10 times, and the other 11 times.
What is the probability that the second person gets more heads than the first person? | \frac{1}{2} | 40 | 7 |
math | Find the maximum value of the real number $m$ such that the inequality $${[b - (a - 2)]^2} + {[\ln b - (a - 1)]^2} \geqslant {m^2} - m$$ holds for any $b > 0, a \in \mathbb{R}$. | 2 | 76 | 1 |
math | Given the sets $A = \{ x \mid -3 < x < 1 \}$ and $B = \{ x \mid \log_2|x| < 1 \}$, find the intersection $A \cap B$. | (-2, 0) \cup (0, 1) | 50 | 14 |
math | Ria has three counters marked 1, 5, and 11. She wants to place them side-by-side to make a four-digit number. How many different four-digit numbers can she make? | 4 | 42 | 1 |
math | Let $\triangle PQR$ be a right triangle such that $Q$ is a right angle. A circle with diameter $QR$ intersects side $PR$ at $S$. If $PS = 3$ and $QS = 9$, what is $RS$? | 27 | 56 | 2 |
math | In triangle $XYZ$, $XY = 5$, $XZ = 9$, $YZ = 11$, and point $W$ lies on $\overline{YZ}$ such that $\overline{XW}$ bisects $\angle YXZ$. Calculate $\cos \angle YXW$. | \frac{\sqrt{15}}{6} | 65 | 11 |
math | Given that a student's Chinese score (x) is higher than 85 points and their Mathematics score (y) is not less than 80 points, determine the system of inequalities that represents this situation. | \begin{cases}
x > 85 \\
y \geq 80
\end{cases} | 43 | 24 |
math | Given that the sequence $\{a_n\}$ is a non-zero arithmetic sequence, and $S_n$ is the sum of its first $n$ terms, with the condition that $a_n = \sqrt{S_{2n-1}}$ for $n \in \mathbb{N}^*$. If the inequality $\frac{\lambda}{a_n} \leqslant \frac{n+8}{n}$ holds for any $n \in \mathbb{N}^*$, then the maximum value of the re... | 9 | 120 | 1 |
math | In the picture, there is a grid consisting of 25 small equilateral triangles. How many rhombuses can be made from two adjacent small triangles? | 30 | 32 | 2 |
math | Suppose $z^{3}=2+2i$ , where $i=\sqrt{-1}$ . The product of all possible values of the real part of $z$ can be written in the form $\frac{p}{q}$ where $p$ and $q$ are relatively prime positive integers. Find $p+q$ . | 3 | 81 | 1 |
math | Given the sets $A = \{1, 2, 3\}$ and $B = \{x | x^2 - (a+1)x + a = 0, x \in \mathbb{R}\}$. If $A \cup B = A$, find the real number $a$. | 1, 2, 3 | 67 | 7 |
math | Given that points $C$ and $D$ are two moving points on the ellipse $\frac{x^2}{4} + y^2 = 1$, and point $M(0, 2)$. If $\overrightarrow{MD} = \lambda \overrightarrow{MC}$, then the range of values for the real number $\lambda$ is _______. | [\frac{1}{3}, 3] | 76 | 10 |
math | Find all continuous functions defined for all \( x \) that satisfy the equation \( f(x) = a^{x} f(x / 2) \), where \( a \) is a fixed positive number. | f(x) = C a^{2x} | 43 | 10 |
math | Given the function $f(x) = \sqrt{2}\sin(x + \frac{π}{4})$, where $x \in [0,π]$, determine the interval on which the function $f(x)$ is monotonically increasing. | [0, \frac{π}{4}] | 52 | 10 |
math | The sequence of numbers \( a_{1}, a_{2}, \ldots, a_{2016} \) is a geometric progression, and the sequence \( b_{1}, b_{2}, \ldots, b_{2016} \) is an arithmetic progression. It is known that among all quadratic trinomials \( P_{i}(x) = x^{2} + a_{i} x + b_{i} \) for \( i = 1, \ldots, 2016 \), only one quadratic trinomia... | 1, 2016 | 143 | 7 |
math | Find the number of positive integers $n \le 500$ such that $21n$ is a perfect square. | 4 | 27 | 1 |
math | The vectors $\mathbf{a},$ $\mathbf{b},$ and $\mathbf{c}$ satisfy $\|\mathbf{a}\| = 1,$ $\|\mathbf{b}\| = \sqrt{2},$ $\|\mathbf{c}\| = 3,$ and
\[\mathbf{a} \times (\mathbf{a} \times \mathbf{c}) + 2\mathbf{b} = \mathbf{0}.\]
If $\theta$ is the angle between $\mathbf{a}$ and $\mathbf{c},$ find all possible values of $\the... | 70.53^\circ \text{ or } 109.47^\circ | 139 | 21 |
math | The distance from the origin to the line $3x + 4y + 5 = 0$ can be calculated. | 1 | 26 | 1 |
math | In the rectangular coordinate system, the graph of the function \( y = \frac{1}{|x|} \) is denoted by \( \Gamma \). Let points \( P \) and \( Q \) on \( \Gamma \) satisfy the following conditions: \( P \) is in the first quadrant, \( Q \) is in the second quadrant, and the line segment \( P Q \) is tangent to the porti... | 4 | 118 | 1 |
math | The greatest common divisor of 2183 and 1947 is ______. | 59 | 19 | 2 |
math | Select 4 volunteers from a group of 5, including candidates A and B, to carry out four different tasks labeled A, B, C, and D, with each person responsible for one task. Given that neither A nor B can undertake task A, there are a total of ____ distinct task assignment schemes. | 72 | 63 | 2 |
math | What is the largest number, all of whose digits are either 2 or 3, and whose digits add up to $13$? | 33332 | 29 | 5 |
math | In a city, there are 57 bus routes. It is known that:
1. From any stop, you can reach any other stop without transferring.
2. For each pair of routes, there is exactly one stop where you can transfer from one route to the other.
3. Each route has at least three stops.
How many stops does each of the 57 routes have? | 8 | 79 | 1 |
math | Given the function $f(x)=2x^{2}-2ax+b$, with $f(-1)=-8$. It is known that $f(x) \geqslant f(-1)$ holds true for all $x \in R$. Let set $A={x|f(x) > 0}$, and set $B={x||x-t|\leqslant 1}$.
(I) Find $(C_{R}A)∪B$ when $t=1$.
(II) Assuming proposition $P$: $A∩B \neq$ empty set, if $\neg P$ is true, find the range of values ... | [-2,0] | 144 | 5 |
math | Given points A(-1, 2, 3) and B(0, 0, 5), find the condition that a point P(x, y, z) must satisfy to be equidistant from points A and B. | 2x-4y+4z=11 | 49 | 11 |
math | Given $f(x) = |x-3| + |x-4|$.
(1) If the solution set of the inequality $f(x) < a$ with respect to $x$ is not empty, find the range of values for the real number $a$;
(2) Solve the inequality: $f(x) \geqslant 7 + 7x - x^2$. | (-\infty, 0] \cup [7, +\infty) | 85 | 18 |
math | (1) Convert the decimal number $23$ to binary.
(2) If selecting $3$ representatives from four people named A, B, C, and D, the probability of A being selected is ______.
(3) Given real numbers $x$ and $y$ satisfy the system of inequalities:
\[
\begin{cases}
x - y - 2 \leqslant 0 \\
x + 2y - 5 \geqslant 0 \\
y - 2 \le... | \frac{8}{3} | 221 | 7 |
math | A landscaping team planned to have 6 workers green a 180 square meter area. Due to the addition of 2 workers during construction, the task was completed 3 hours ahead of schedule. Assuming each worker greens the same area per hour, calculate the greening area per worker per hour. Let the greening area per worker per ho... | \frac{180}{6x} - \frac{180}{(6+2)x} = 3 | 87 | 27 |
math | Given the line $l: ax + y + b = 0$ intersects with the circle $O: x^{2} + y^{2} = 4$ at points $A$ and $B$, and $M(\sqrt{3}, -1)$, and $\overrightarrow{OA} + \overrightarrow{OB} = \frac{2}{3} \overrightarrow{OM}$, then $\sqrt{3}ab = $ ______. | -4 | 96 | 2 |
math | Suppose the estimated $20$ billion dollar cost to send a person to the planet Mars is shared equally by the $250$ million people in the U.S. Then each person's share is | 80 | 42 | 2 |
math | In the triangle shown below, suppose $\cos R = \frac{3}{5}$. If RS = 10, what is the length of QS?
[asy]
pair Q,R,S;
S = (0,0);
Q = (sqrt(91),0);
R = (sqrt(91),-6);
draw(S--Q--R--S);
draw(rightanglemark(S,Q,R,20));
label("$S$",S,NW);
label("$Q$",Q,NE);
label("$R$",R,SE);
label("$10$",(R+S)/2,SW);
[/asy] | 8 | 130 | 1 |
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