problem stringlengths 14 4.09k | solution stringlengths 1 802k | task_type stringclasses 3
values | problem_tokens int64 5 895 |
|---|---|---|---|
If $0 \le \theta \le 4 \pi,$ find all values of $\theta$ which satisfy
\[\log_{\frac{1}{2} \sin 2 \theta} \sin \theta = \frac{1}{2}.\]Enter all the solutions, separated by commas. | \frac{\pi}{4}, \frac{9 \pi}{4} | math | 65 |
During the epidemic prevention and control period, people should wear masks when going out to protect themselves from being infected with the novel coronavirus. A certain pharmacy purchased a certain number of packs of disposable medical masks for $4000$ yuan, which were quickly sold out. The store then purchased a sec... | 3 | math | 208 |
Let the function $f(x) = \begin{cases} e^{x-1}, & x < 1 \\ x^{\frac{1}{3}}, & x \geq 1 \end{cases}$. Determine the range of $x$ for which $f(x) \leq 2$. | (-\infty, 8] | math | 66 |
Define $a*b = 2 \times \left\{ \frac{a}{2} \right\} + 3 \times \left\{ \frac{a+b}{6} \right\}$, where the symbol $\{x\}$ represents the fractional part of $x$, such as $\{2.016\} = 0.016$. Then, calculate $1.4*3.2$ if the answer is represented as a decimal. | 3.7 | math | 102 |
Given $y^2 = 16x$ and point $A(1, 2)$, with $P$ being a point on the parabola and $F$ the focus of the parabola, find the minimum value of $|PF| + |PA|$. | 5 | math | 60 |
The sequence \( a_{n} \) is given as follows:
\[ a_{1} = 1, \quad a_{n+1} = a_{n} + \frac{2 a_{n}}{n}, \quad \text{for} \, n \geq 1. \text{ Find } a_{200} \] | 20100 | math | 77 |
Given Brenda received $\frac{4}{15}$ of the total votes and this equals 48 votes, and Colby received 35 votes, determine the total number of votes cast in the election. | 180 | math | 43 |
Given the ratio of the surface areas of two spheres is 1:3, find the ratio of their volumes. | 1:3\sqrt{3} | math | 23 |
A $200\times 420\times 480$ rectangular solid is made by gluing together $1\times 1\times 1$ cubes. An internal diagonal of this solid passes through the interiors of how many of the $1\times 1\times 1$ cubes? | 1000 | math | 67 |
A circle is tangent to the lines $5x - 2y = 20$ and $5x - 2y = -40.$ The center of the circle lies on the line $3x + y = 0.$ Find the center of the circle. | \left(-\frac{10}{11}, \frac{30}{11}\right) | math | 57 |
Five students sign up for basketball, chess, and environmental clubs, with each student limited to joining one of them. Among them, Xiao Bin definitely will not join the chess club, Xiao Cong definitely will not join the basketball club, and Xiao Hao definitely will not join the environmental club. Determine the total ... | 72 | math | 66 |
Given that Vertex E of right triangle ABE (where ∠AEB = 90°) is in the interior of unit square ABCD. Let R be the region consisting of all points inside ABCD and outside triangle ABE whose distance from AD is between 1/4 and 1/2. What is the area of R? | \frac{1}{8} | math | 71 |
Alex writes down all the five-digit numbers that contain each of the digits 1, 2, 6, 7, and 8 exactly once. What is the smallest number in Alex's list that is divisible by 6? | 12678 | math | 48 |
Solve the equations:<br/>$(1)\left\{\begin{array}{l}2x-y=3\\ x+y=-12\end{array}\right.$;<br/>$(2)\frac{2}{1-x}+1=\frac{x}{1+x}$. | x = -3 | math | 60 |
Determine the interval of monotonic increase for the function $y = \frac{1}{2}\sin x + \frac{\sqrt{3}}{2}\cos x$ where $x \in [0, \frac{\pi}{2}]$. | \left[0, \frac{\pi}{6}\right] | math | 53 |
Given that the common ratio of the geometric sequence $\{a_{n}\}$ is $q$ with $|q|\neq 1$, and $a_{1}=-1$, find the value of $m$ if $a_{m}=a_{1} \cdot a_{2} \cdot a_{3} \cdot a_{4} \cdot a_{5}$. | 11 | math | 81 |
Given the function $f(x)=x^{3}$, find the solution set of the inequality $f(2x)+f(x-1) < 0$. | (-\infty, \frac {1}{3}) | math | 34 |
Given that $\text{1 mile} = \text{10 furlongs}$ and $\text{1 furlong} = \text{50 rods}$, find the number of rods in one mile. | 500 | math | 45 |
The number of intersection points between the graph of the function $y=f(x)$ and the y-axis is at most one. | 1 | math | 25 |
Find the smallest positive real number \( c \) such that for all nonnegative real numbers \( x \) and \( y \),
\[
\sqrt{xy} + c \sqrt{|x - y|} \ge \frac{x + y}{2}.
\] | \frac{1}{2} | math | 56 |
(1) Given the power function $f(x) = (-2m^2+m+2)x^{-2m+1}$ is an even function, find the expression for $f(x)$;
(2) Given $x+x^{-1}=3$ ($x>1$), find the value of $x^2-x^{-2}$. | 3\sqrt{5} | math | 73 |
A manufacturer plans to hold a promotional event in 2010. After investigation and calculation, the annual sales volume of the product (i.e., the factory's annual output $x$ in ten thousand units) and the annual promotional expenses $m$ (in ten thousand yuan) ($m\geqslant 0$) satisfy $x=3- \dfrac {k}{m+1}$ ($k$ is a con... | 3 | math | 260 |
Given that point P lies on the line $x-2y-1=0$ and point Q lies on the line $x-2y+3=0$, if the midpoint of line segment PQ is $M(x_0, y_0)$ and $y_0 > -x_0 + 2$, find the range of values for $\sqrt{(x_0-4)^2 + y_0^2}$. | [\sqrt{5}, +\infty) | math | 91 |
Eight women of different heights are at a conference. Their heights in centimeters are 150, 155, 160, 165, 170, 175, 180, and 185. Each woman decides to only shake hands with women whose height difference is 10 centimeters or less. How many handshakes take place? | 7 | math | 86 |
In the scatter plot of a sample data group $((x\_1, y\_1), (x\_2, y\_2), ..., (x\_6, y\_6))$, if all sample points $(x\_i, y\_i) (i=1,2,...,6)$ oscillate near the curve $y=bx^2−\frac{1}{3}$. Given that $\sum\limits^{6 }\_{i\_=1 } x\_i=11$, $\sum\limits^{6 }\_{i\_=1 } y\_i=13$, and $\sum\limits^{6 }\_{i\_=1 } x^2\_i=21$... | b=\frac{5}{7} | math | 169 |
Given that for any $n\in \mathbb{N}^{*}$, the function $f(x)=x^{2}+2x$ has a tangent line with slope $a_{n}$ at the point $(n,f(n))$. The sequence $\{b_{n}\}$ is a geometric sequence with a common ratio greater than $0$, where $b_{1}=3$ and $b_{3}-b_{2}=18$.
$(1)$ Find the general formulas for sequences $\{a_{n}\}$ a... | \frac{2315}{11} | math | 183 |
It is known that the cost price of a certain product is 40 yuan per item, and the selling price is 60 yuan per item, with a weekly sales volume of 300 items. Market research shows that for every 1 yuan increase in price, 10 fewer items will be sold each week. At what price should the product be set to maximize the stor... | 6250 | math | 82 |
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 | math | 136 |
Find the positive integer $n$ such that the least common multiple of $n$ and $n - 30$ is $n + 1320$ . | 165 | math | 43 |
On the \(xy\)-plane, let \(S\) denote the region consisting of all points \((x, y)\) for which
\[
\left|x+\frac{1}{2} y\right| \leq 10, \quad |x| \leq 10, \quad \text{and} \quad |y| \leq 10.
\]
The largest circle centered at \((0,0)\) that can be fitted in the region \(S\) has area \(k \pi\). Find the value of \(k\). | 80 | math | 123 |
Let \( p(x) = x^2 - x + 1 \). Let \(\alpha\) be a root of \( p(p(p(p(x)))) \). Find the value of
\[
(p(\alpha) - 1) p(\alpha) p(p(\alpha)) p(p(p(\alpha)))
\] | -1 | math | 67 |
Given the proposition q: The inequality $-3^x \leq a$ holds true for all positive real numbers $x$, find the range of the real number $a$. | [-1, +\infty) | math | 37 |
Calculate the tax rate K using the formula $K=(1-7x)^{n}$ $(n\in\mathbb{N^{*}})$. Given that the sum of the binomial coefficients in the expanded form of K is 64, calculate the value of K when $x=\frac{3}{700}$ using the rounding method. If the value is accurate to 0.01, what is the digit in the hundredths place? | 3 | math | 97 |
For how many positive integer values of $N$ does the expression $\dfrac{48}{N+3}$ result in an integer? | 7 | math | 29 |
The table shows a set of equations, and based on it, a student conjectured that $S_{2n-1} = (2n-1)(an^2 + bn + c)$. The teacher confirmed that the conjecture was correct. Determine the value of $a - b + c$.
The equations are given as follows:
$$
S_1 = 1, \\
S_2 = 2 + 3 = 5, \\
S_3 = 4 + 5 + 6 = 15, \\
S_4 = 7 + 8 + 9 +... | a - b + c = 5 | math | 169 |
The heights of the two trees are in the ratio $3:4$ and the difference in their heights is $16$ feet, calculate the height of the taller tree. | 64 | math | 36 |
In a class of 40 students, it is known that $\frac{1}{2}$ of the students have a dog, $\frac{2}{5}$ have a cat, 8 students have a different type of pet, and 7 students have no pets at all. The Venn diagram details are as follows: 12 students only have dogs, 3 students have both dogs and other pets but no cats, and 11 s... | 5 | math | 107 |
Three distinct vertices of a regular tetrahedron are chosen at random. Determine the probability that the plane determined by these three vertices contains points inside the tetrahedron. | 0 | math | 35 |
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} | math | 129 |
Given that $\{a\_n\}$ is an arithmetic sequence with a common difference of $d$, and the sum of its first $n$ terms is $S\_n$, where $S\_4 = 2S\_2 + 4$.
1. Find the value of the common difference $d$;
2. If for any $n \in \mathbb{N}^*$, the inequality $S\_n \geq S\_8$ holds, find the range of possible values for $a\_1... | [-8, -7] | math | 111 |
Determine the probability that the yellow ball ends up in a higher-numbered bin than the blue ball. | \frac{2}{5} | math | 21 |
A car travels due east at $\frac 23$ mile per minute on a long, straight road. At the same time, a circular storm, whose radius is $51$ miles, moves southeast at $\frac 12\sqrt{2}$ mile per minute. At time $t=0$, the center of the storm is $110$ miles due north of the car. At time $t=t_1$ minutes, the car enters the st... | 198 | math | 131 |
Given sets $P=\{x|3a-10\leqslant x \lt 2a+1\}$ and $Q=\{x||2x-3|\leqslant 7\}$.
$(1)$ When $a=2$, find $P\cap (\complement _{R}Q)$;
$(2)$ If "$x\in P$" is a necessary but not sufficient condition for "$x\in Q$", find the range of real numbers $a$. | (2, \frac{8}{3}] | math | 108 |
What is the base six product of the numbers $132_{6}$ and $14_{6}$? | 2332_6 | math | 24 |
Given that $a \in (0, 2)$, and $\tan a = 2$, find the value of $\cos \left( a - \frac{\pi}{4} \right)$. | \frac{3\sqrt{10}}{10} | math | 43 |
Compute the integer $n > 3$ for which
\[\log_{10} (n - 3)! + \log_{10} (n - 1)! + 3 = 2 \log_{10} n!.\] | 6 | math | 54 |
In the trapezoid \(ABCD\) with bases \(AD = 20\) and \(BC = 10\), circles constructed using sides \(AB\), \(BC\), and \(CD\) as diameters intersect at one point. The length of the diagonal \(AC\) is 18. Find the length of \(BD\). | 24 | math | 73 |
In the diagram, line $AB$ is parallel to line $EF$ and $AFC$ is a straight line. Points $D$, $E$ are on line $AB$, and $G$ is on $EF$. Given that $\angle AFC = 180^\circ$, $\angle AFE = 100^\circ$, and $\angle EFA = 60^\circ$, determine the value of $x,$ which is $\angle DGC$.
[asy]
draw((0,0)--(-.5,5)--(8,5)--(6.5,0)-... | 80^\circ | math | 352 |
Two men and two women are lined up in a row. The number of ways they can be arranged alternately by gender. | 8 | math | 25 |
The difference between the two natural numbers that add up to $20,000$ and one of which, when multiplied by $7$, results in the other number. | 15,000 | math | 36 |
A certain clothing supermarket purchased several pieces of children's clothing at a unit price of $30$ yuan. The pricing department stipulates that the selling price should not be less than $30$ yuan per item, nor exceed $60$ yuan per item. After a period of sales, it was found that when the selling price was $60$ yuan... | 1950 \text{ yuan} | math | 209 |
Determine the number of ordered pairs $(m, n)$ that satisfy $m$ and $n \in \{-1,0,1,2,3\}$, and the equation $mx^2 + 2x + n = 0$ has real solutions. | 17 | math | 57 |
Given a sequence $\left\{x_{n}\right\}$ that satisfies $x_{n+1}=x_{n}-x_{n-1}$ for $n \geqslant 2$, with $x_{1}=a$ and $x_{2}=b$. Let $S_{n}= x_{1}+x_{2}+\cdots+x_{n}$. Determine $S_{100}$ and $x_{100}$. | -a, \quad 2b - a | math | 100 |
Find all values of \(k\) such that the quadratic expression
\[x^2 - (k - 4) x - k + 7 > 0\]
is true for all \(x\). | (-2, 6) | math | 43 |
In a group of 40 students, 20 play soccer, 19 play volleyball, and 15 play exactly one of these two sports. How many students do not play soccer or volleyball?
(a) 7
(b) 5
(c) 13
(d) 9
(e) 10 | 13 | math | 69 |
If line $l$ passes through point $P(0, 1)$ and is tangent to the circle $x^2+y^2=1$, then the equation of line $l$ is. | y=1 | math | 42 |
After two price reductions, the retail price of a certain product dropped from 800 yuan to 578 yuan. Calculate the average percentage decrease per reduction. | 15\% | math | 34 |
The projection of $\begin{pmatrix} 2 \\ -3 \\ 1 \end{pmatrix}$ onto a certain vector $\mathbf{v}$ is $\begin{pmatrix} -2 \\ 3 \\ -1.5 \end{pmatrix}$. Find the projection of $\begin{pmatrix} 4 \\ 1 \\ -3 \end{pmatrix}$ onto $\mathbf{v}$. | \begin{pmatrix} \frac{1}{15.25} \\ -\frac{1.5}{15.25} \\ \frac{0.75}{15.25} \end{pmatrix} | math | 89 |
For how many values of $c$ in the interval $[0, 500]$ does the equation \[5 \lfloor x \rfloor + 3 \lceil x \rceil = c\] have a solution for $x$? | 126 | math | 55 |
A laptop is originally priced at $1200$ dollars and is put on sale for $30\%$ off. If a $12\%$ tax was added to the sale price, calculate the total selling price (in dollars) of the laptop. | 940.8 | math | 56 |
Find all positive solutions (\(x_{1}>0, x_{2}>0, x_{3}>0, x_{4}>0, x_{5}>0\)) of the system of equations
$$
\left\{\begin{array}{l}
x_{1}+x_{2}=x_{3}^{2} \\
x_{2}+x_{3}=x_{4}^{2} \\
x_{3}+x_{4}=x_{5}^{2} \\
x_{4}+x_{5}=x_{1}^{2} \\
x_{5}+x_{1}=x_{2}^{2}
\end{array}\right.
$$ | x_1 = x_2 = x_3 = x_4 = x_5 = 2 | math | 147 |
Given $S=(x-a)^2+(\ln x-a)^2$ ($a\in \mathbb{R}$), calculate the minimum value of $S$. | \frac{1}{2} | math | 34 |
Solve the systems of equations:
a) $\left\{\begin{array}{l}x^{2}-3 x y-4 y^{2}=0, \\ x^{3}+y^{3}=65 ;\end{array}\right.$
b) $\left\{\begin{array}{l}x^{2}+2 y^{2}=17, \\ 2 x y-x^{2}=3\end{array}\right.$ | (3, 2), (-3, -2), \left(\frac{\sqrt{3}}{3}, \frac{5\sqrt{3}}{3}\right), \left(-\frac{\sqrt{3}}{3}, -\frac{5\sqrt{3}}{3}\right) | math | 97 |
If $x\log_{3}4=1$, then $x=$ ______; $4^{x}+4^{-x}=$ ______. | \dfrac{10}{3} | math | 31 |
The sum of the dimensions of a rectangular prism is the sum of the number of edges, corners, and faces, where the dimensions are 2 units by 3 units by 4 units. Calculate the resulting sum. | 26 | math | 44 |
Given that $\{a_n\}$ is an arithmetic sequence with a non-zero common difference, and $\{b_n\}$ is a geometric sequence, where $a_1=2$, $b_1=1$, $a_2=b_2$, $2a_4=b_3$, and there exist constants $\alpha$ and $\beta$ such that $a_n=\log_{\alpha}b_n+\beta$ holds for every positive integer $n$, then $\alpha^{\beta}=$ ? | 4 | math | 107 |
Given that the line $x-y-4=0$ intersects the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ ($a>b>0$) at points A and B, and the x-coordinate of the midpoint of A and B is 3, find the eccentricity of the ellipse. | \sqrt{\frac{2}{3}} | math | 77 |
As shown in Figure 1, a line segment of length 1 is divided into two parts, $x$ and $y$. Then, the segment of length $x$ is bent into a semicircular arc $ACB$, and the segment of length $y$ is folded into a rectangle $ABDE$ along three sides $(BD, DE, EA)$, forming a closed "curved shape" $ACBDEA$. What is the maximum ... | \frac{1}{2(\pi+4)} | math | 100 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $a=\frac{\sqrt{3}}{2}b$, $B=C$.
(I) Find the value of $\cos B$;
(II) Let $f(x)=\sin(2x+B)$, find the value of $f(\frac{\pi}{6})$. | \frac{3+\sqrt{13}}{8} | math | 94 |
In triangle $ABC$, $BC = 30 \sqrt{2}$ and $\angle C = 45^\circ$. Let the perpendicular bisector of $BC$ intersect $BC$ and $AC$ at $D$ and $E$, respectively. Find the length of $DE$. | 15 | math | 61 |
Express with formulas:<br/>$(1)$ The difference between the reciprocal of $x$ and $y$ ($x\neq 0$);<br/>$(2)$ The difference between the square of the sum of $a$ and $b$ and the product of $a$ and $b$. | (a+b)^2 - ab | math | 64 |
Suppose $a$ and $b$ are positive integers such that $\gcd(a,b)$ is divisible by exactly $7$ distinct primes and $\mathop{\text{lcm}}[a,b]$ is divisible by exactly $28$ distinct primes.
If $a$ has fewer distinct prime factors than $b$, then $a$ has at most how many distinct prime factors? | 17 | math | 78 |
Given that $F\_1$ and $F\_2$ are the left and right foci of the ellipse $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ (where $a > b > 0$), the ellipse $C$ passes through the point $(-\sqrt{3}, 1)$ and shares a focus with the parabola $y^2 = -8x$.
1. Find the equation of the ellipse $C$.
2. The line $l$ passing through ... | \sqrt{6} | math | 160 |
Given the equation
\[ x_{1}^{2} + x_{2}^{2} = (x_{1} + x_{2})^{2} - 2 x_{1} x_{2} \]
and Vieta's formulas:
\[
\left\{
\begin{array}{l}
x_{1} + x_{2} = -(m+1) \\
x_{1} x_{2} = 2 m - 2
\end{array}
\right.
\]
where \( D = (m+3)^{2} \geq 0 \), find the value of \( x_{1}^{2} + x_{2}^{2} \). Then, determine \( y = (m-1)... | 4 \text{ at } m=1 | math | 178 |
The average rate of change of the functions $y=-x^{2}$, $y= \frac{1}{x}$, $y=2x+1$, $y= \sqrt{x}$ near $x=1$ (where $\Delta x$ is very small) is the largest for which function. | 2x+1 | math | 65 |
The rules for a race require that all runners start at $A$, touch any part of the 1500-meter wall, and stop at $B$. What is the number of meters in the minimum distance a participant must run? Express your answer to the nearest meter. Assume the distances from A to the nearest point on the wall is 400 meters, and from ... | 1803 | math | 92 |
Given proposition $p$: $|4x-3|\leqslant 1$ and proposition $q$: $x^{2}-(2a+1)x+a(a+1)\leqslant 0$. If $\neg p$ is a necessary but not sufficient condition for $\neg q$, then the range of values for the real number $a$ is _____. | [0, \frac{1}{2}] | math | 79 |
At the Galaxy Gym, a survey was conducted among its members. The average age of the female members was found to be 35 years old, while the average age of the male members was 45 years old. The average age of the entire membership was found to be 40 years old. Determine the ratio of the number of female to male members.... | 1 | math | 82 |
What is the largest $n$ for which it is possible to construct two bi-infinite sequences $A$ and $B$ such that any subsequence of $B$ of length $n$ is contained in $A$, $A$ has a period of 1995, and $B$ does not have this property (is either non-periodic or has a period of a different length)? | 1995 | math | 83 |
Given set A={-2, 0, 1, 3}, and set B={x|-x∈A, 1-x∉A}, find the set B. | \{-3, -1, 2\} | math | 37 |
If point $P(\cos \alpha, \sin \alpha)$ lies on the line $y=-2x$, then the value of $\sin 2\alpha + 2\cos 2\alpha$ is ____.
A) $- \frac {14}{5}$
B) $- \frac {7}{5}$
C) $-2$
D) $\frac {4}{5}$ | -2 | math | 87 |
Determine the minimum possible value of the sum
\[\frac{a}{3b} + \frac{b}{6c} + \frac{c}{9a},\]where $a,$ $b,$ and $c$ are positive real numbers. | \frac{3}{\sqrt[3]{162}} | math | 55 |
Given that $x$ and $y$ are positive real numbers, and they satisfy the equation $2x^{2}+8y^{2}+xy=2$, find the maximum value of $x+2y$. | \frac{4}{3} | math | 47 |
Find the minimum value of
\[2 \sqrt[4]{x} + \frac{1}{x}\] for \(x > 0.\) | 3 | math | 32 |
Given $f(α)=\frac{{\sin(π+α)}}{{\tan(π-α)}}$.
$(1)$ Find the value of $f({\frac{{5π}}{6}})$;
$(2)$ If $f({\frac{π}{3}-α})=\frac{2}{3}$, find the value of $\sin({α+\frac{π}{6}})$. | \frac{2}{3} | math | 89 |
Find two numbers such that their sum is 60, and the sum of their greatest common divisor and least common multiple is 84. | 24 \text{ and } 36 | math | 29 |
In a newly built road in a city, there are 12 street lamps. To save electricity without affecting normal lighting, three of them can be turned off. However, the lamps at both ends cannot be turned off, nor can two adjacent lamps be turned off. How many methods are there to turn off the lamps? | 56 | math | 65 |
In the polar coordinate system, the coordinates of point M are $$(3, \frac{\pi}{2})$$, and the equation of curve C is $$\rho = 2\sqrt{2}\sin\left(\theta + \frac{\pi}{4}\right)$$; taking the pole as the origin of the coordinate system and the positive half-axis of the x-axis as the polar axis, a line l with a slope of -... | \frac{3\sqrt{3}}{2} | math | 148 |
The distance from the origin to the line $4x+3y-12=0$ is __________. | \frac {12}{5} | math | 24 |
Given non-negative integers $a$ and $b$ satisfying $|a-b|+ab=1$, let $M=\{(a,b)\}$ be the set of all such pairs $(a,b)$. Find the number of elements in $M$. | 3 | math | 52 |
A small town has fewer than 6000 inhabitants. We know that there are $10\%$ more girls than boys among the children, and $15\%$ more men than women among the adults. There are $20\%$ more children than adults in the town.
How many people live in the town? | 3311 | math | 71 |
Calculate the expression $(-2)^4 + (-2)^3 + (-2)^2 + (-2)^1 + 2^1 + 2^2 + 2^3 + 2^4$. | 40 | math | 44 |
There are four balls in a bag, which are identical in shape and size. The balls are numbered $1$, $2$, $3$, and $4$ respectively.
(i) Calculate the probability that the product of the numbers of two randomly drawn balls from the bag is not greater than $4$.
(ii) Randomly draw a ball from the bag, its number is $m$, put... | P = \frac{5}{8} | math | 114 |
Three planes can divide space into at least parts, and at most parts. | 4; 8 | math | 19 |
The sequence $\left\{x_{n}\right\}$ is defined as follows: $x_{1}=\frac{1}{2}, x_{n+1}=x_{n}^{2}+x_{n}$. Find the integer part of the following sum: $\frac{1}{1+x_{1}}+\frac{1}{1+x_{2}}+\cdots+\frac{1}{1+x_{200}}$. | 1 | math | 94 |
Given a positive integer \( n \geq 3 \), for an \( n \)-dimensional real vector \( \left( x_1, x_2, \cdots, x_n \right) \), if every permutation \( y_1, y_2, \cdots, y_n \) of its elements satisfies \( \sum_{i=1}^{n-1} y_i y_{i+1} \geq -1 \), then the vector \( \left( x_1, x_2, \cdots, x_n \right) \) is called "shining... | -\frac{n-1}{2} | math | 185 |
In the polar coordinate system, the equation of curve C is $$\rho=4(\cos\theta+\sin\theta)- \frac {6}{\rho}$$, with the pole O as the origin and the positive half-axis of the x-axis as the polar axis, a Cartesian coordinate system is established.
(1) Find the parametric equation of curve C;
(2) In the Cartesian coordin... | (3, 3) | math | 118 |
How many different $5 \times 5$ arrays whose entries are 1's, -1's, and 0's have the property that the sum of the entries in each row is 0, the sum of the entries in each column is 0, and each row and column should contain exactly one 0? | 933120 | math | 66 |
Given two lines $l_1: x-2y+4=0$ and $l_2: x+y-2=0$ intersect at point P
(1) Find the coordinates of point P;
(2) Let line $l_3: 3x-4y+5=0$, find the equations of the lines that pass through point P and are parallel and perpendicular to line $l_3$, respectively. | 4x+3y-6=0 | math | 92 |
Let $a$ and $b$ be positive real numbers such that
\[\frac{1}{a + 3} + \frac{1}{b + 3} = \frac{1}{4}.\]Find the minimum value of $a + 3b.$ | 4 + 8 \sqrt{3} | math | 59 |
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