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22,100 | Without using any tables, find the exact value of the product:
\[ P = \cos \frac{\pi}{15} \cos \frac{2\pi}{15} \cos \frac{3\pi}{15} \cos \frac{4\pi}{15} \cos \frac{5\pi}{15} \cos \frac{6\pi}{15} \cos \frac{7\pi}{15}. \] | 1/128 | 57.03125 |
22,101 | Given a large circle with a radius of 11 and small circles with a radius of 1, determine the maximum number of small circles that can be placed inside the large circle, such that each small circle is internally tangent to the large circle and the small circles do not overlap. | 31 | 74.21875 |
22,102 | From the numbers $1, 2, 3, 4, 5$, 3 numbers are randomly drawn (with replacement) to form a three-digit number. What is the probability that the sum of its digits equals 9? | $\frac{19}{125}$ | 0 |
22,103 | Among the following functions, identify which pairs represent the same function.
1. $f(x) = |x|, g(x) = \sqrt{x^2}$;
2. $f(x) = \sqrt{x^2}, g(x) = (\sqrt{x})^2$;
3. $f(x) = \frac{x^2 - 1}{x - 1}, g(x) = x + 1$;
4. $f(x) = \sqrt{x + 1} \cdot \sqrt{x - 1}, g(x) = \sqrt{x^2 - 1}$. | (1) | 0 |
22,104 | What is the difference between the maximum value and the minimum value of the sum $a_1 + 2a_2 + 3a_3 + 4a_4 + 5a_5$ where $\{a_1,a_2,a_3,a_4,a_5\} = \{1,2,3,4,5\}$ ? | 20 | 91.40625 |
22,105 | What percent of the positive integers less than or equal to $120$ have no remainders when divided by $6$? | 16.67\% | 96.09375 |
22,106 | Given that $l$ is the incenter of $\triangle ABC$, with $AC=2$, $BC=3$, and $AB=4$. If $\overrightarrow{AI}=x \overrightarrow{AB}+y \overrightarrow{AC}$, then $x+y=$ ______. | \frac {2}{3} | 72.65625 |
22,107 | Let $\triangle XYZ$ be a right triangle with $Y$ as the right angle. A circle with diameter $YZ$ intersects side $XZ$ at $W$. If $XW = 3$ and $YW = 9$, find the length of $WZ$. | 27 | 41.40625 |
22,108 | Given the function $f(x)=kx+b$, whose graph intersects the $x$ and $y$ axes at points A and B respectively, with $\overrightarrow{AB}=2\overrightarrow{i}+2\overrightarrow{j}$ ($\overrightarrow{i}$, $\overrightarrow{j}$ are unit vectors in the positive direction of the $x$ and $y$ axes). The function $g(x)=x^{2}-x-6$ is also given.
1. Find the values of $k$ and $b$.
2. When $x$ satisfies $f(x) > g(x)$, find the minimum value of the function $\frac{g(x)+1}{f(x)}$. | -3 | 56.25 |
22,109 | The difference between the maximum and minimum values of the function $f(x)= \frac{2}{x-1}$ on the interval $[-2,0]$ is $\boxed{\frac{8}{3}}$. | \frac{4}{3} | 40.625 |
22,110 | Given an odd function $f(x)$ defined on $\mathbb{R}$ that satisfies $f(2-x) - f(x) = 0$, and $f(-1) = 1$, calculate the value of $f(1) + f(2) + f(3) + \ldots + f(2010)$. | -1 | 61.71875 |
22,111 | Consider the arithmetic sequence defined by the set $\{2, 5, 8, 11, 14, 17, 20\}$. Determine the total number of different integers that can be expressed as the sum of three distinct members of this set. | 13 | 93.75 |
22,112 | In a geometric progression with a common ratio of 4, denoted as $\{b_n\}$, where $T_n$ represents the product of the first $n$ terms of $\{b_n\}$, the fractions $\frac{T_{20}}{T_{10}}$, $\frac{T_{30}}{T_{20}}$, and $\frac{T_{40}}{T_{30}}$ form another geometric sequence with a common ratio of $4^{100}$. Analogously, for an arithmetic sequence $\{a_n\}$ with a common difference of 3, if $S_n$ denotes the sum of the first $n$ terms of $\{a_n\}$, then _____ also form an arithmetic sequence, with a common difference of _____. | 300 | 8.59375 |
22,113 | In a right triangle JKL, the hypotenuse KL measures 13 units, and side JK measures 5 units. Determine $\tan L$ and $\sin L$. | \frac{5}{13} | 25.78125 |
22,114 | A positive integer whose digits are the same when read forwards or backwards is called a palindrome. An example of a palindrome is 13931. What is the sum of the digits of the next palindrome greater than 13931? | 10 | 60.9375 |
22,115 | Two students, A and B, each choose 2 out of 6 extracurricular reading materials. Calculate the number of ways in which the two students choose extracurricular reading materials such that they have exactly 1 material in common. | 60 | 0 |
22,116 | In the rectangular coordinate system on the plane, establish a polar coordinate system with the coordinate origin as the pole and the positive semi-axis of the $x$-axis as the polar axis. The polar coordinates of point $A$ are $\left( 4\sqrt{2}, \frac{\pi}{4} \right)$, and the polar equation of line $l$ is $\rho \cos \left( \theta - \frac{\pi}{4} \right) = a$, which passes through point $A$. The parametric equations of curve $C_1$ are given by $\begin{cases} x = 2 \cos \theta \\ y = \sqrt{3} \sin \theta \end{cases}$ ($\theta$ is the parameter).
(1) Find the maximum and minimum distances from points on curve $C_1$ to line $l$.
(2) Line $l_1$, which is parallel to line $l$ and passes through point $B(-2, 2)$, intersects curve $C_1$ at points $M$ and $N$. Compute $|BM| \cdot |BN|$. | \frac{32}{7} | 3.125 |
22,117 | The three sides of a triangle are 25, 39, and 40. Find the diameter of its circumscribed circle. | \frac{125}{3} | 0 |
22,118 | Given the function $y=f(x)$ that satisfies $f(-x)=-f(x)$ and $f(1+x)=f(1-x)$ for $x \in [-1,1]$ with $f(x)=x^{3}$, find the value of $f(2015)$. | -1 | 81.25 |
22,119 | Given that $a > 1$ and $b > 0$, and $a + 2b = 2$, find the minimum value of $\frac{2}{a - 1} + \frac{a}{b}$. | 4(1 + \sqrt{2}) | 0 |
22,120 | Given the function $f(x) = x^{3} + ax^{2} - 2x + 1$ has an extremum at $x=1$.
$(1)$ Find the value of $a$;
$(2)$ Determine the monotonic intervals and extremum of $f(x)$. | -\frac{1}{2} | 67.96875 |
22,121 | Given the parametric equation of curve $C\_1$ as $\begin{cases} x=a\cos \theta \\ y=b\sin \theta \end{cases}$ $(a > b > 0, \theta$ is the parameter$)$, and the point $M(1, \frac{ \sqrt{3}}{2})$ on curve $C\_1$ corresponds to the parameter $\theta= \frac{ \pi}{3}$. Establish a polar coordinate system with the origin $O$ as the pole and the positive half of the $x$-axis as the polar axis. The polar coordinate equation of curve $C\_2$ is $ρ=2\sin θ$.
1. Write the polar coordinate equation of curve $C\_1$ and the rectangular coordinate equation of curve $C\_2$;
2. Given points $M\_1$ and $M\_2$ with polar coordinates $(1, \frac{ \pi}{2})$ and $(2,0)$, respectively. The line $M\_1M\_2$ intersects curve $C\_2$ at points $P$ and $Q$. The ray $OP$ intersects curve $C\_1$ at point $A$, and the ray $OQ$ intersects curve $C\_1$ at point $B$. Find the value of $\frac{1}{|OA|^{2}}+ \frac{1}{|OB|^{2}}$. | \frac{5}{4} | 14.0625 |
22,122 |
A school has 1200 students, and each student participates in exactly \( k \) clubs. It is known that any group of 23 students all participate in at least one club in common, but no club includes all 1200 students. Find the minimum possible value of \( k \). | 23 | 20.3125 |
22,123 | Given the "ratio arithmetic sequence" $\{a_{n}\}$ with $a_{1}=a_{2}=1$, $a_{3}=3$, determine the value of $\frac{{a_{2019}}}{{a_{2017}}}$. | 4\times 2017^{2}-1 | 0 |
22,124 | Consider a large square divided into a grid of \(5 \times 5\) smaller squares, each with side length \(1\) unit. A shaded region within the large square is formed by connecting the centers of four smaller squares, creating a smaller square inside. Calculate the ratio of the area of the shaded smaller square to the area of the large square. | \frac{2}{25} | 21.875 |
22,125 | Given a triangular pyramid $D-ABC$ with all four vertices lying on the surface of a sphere $O$, if $DC\bot $ plane $ABC$, $\angle ACB=60^{\circ}$, $AB=3\sqrt{2}$, and $DC=2\sqrt{3}$, calculate the surface area of sphere $O$. | 36\pi | 84.375 |
22,126 | Five consecutive two-digit positive integers, each less than 40, are not prime. What is the largest of these five integers? | 36 | 3.125 |
22,127 | Let the sum of the first $n$ terms of an arithmetic sequence $\{a_n\}$ be $S_n$. If $a_5=3$ and $S_{10}=40$, then the minimum value of $nS_n$ is . | -32 | 38.28125 |
22,128 | Let $x_1,$ $x_2,$ $x_3,$ $x_4$ be the roots of the polynomial $f(x) = x^4 - x^3 + x^2 + 1$. Define $g(x) = x^2 - 3$. Find the product:
\[ g(x_1) g(x_2) g(x_3) g(x_4). \] | 142 | 10.9375 |
22,129 | A right circular cone with a base radius $r$ and height $h$ lies on its side on a flat table. As the cone rolls on the surface of the table without slipping, the point where the cone's base connects with the table traces a circular arc centered at the vertex of the cone. The cone first returns to its original position after making $20$ complete rotations. The value of $h/r$ in simplest form can be expressed as $\lambda\sqrt{k}$, where $\lambda$ and $k$ are positive integers, and $k$ is not divisible by the square of any prime. Find $\lambda + k$. | 400 | 80.46875 |
22,130 | The sequence $b_1, b_2, b_3, \dots$ satisfies $b_1 = 25,$ $b_{12} = 125,$ and for all $n \ge 3,$ $b_n$ is the arithmetic mean of the first $n - 1$ terms. Find $b_2.$ | 225 | 78.90625 |
22,131 | Two cards are dealt from a standard deck of 52 cards. What is the probability that the first card dealt is a $\clubsuit$ and the second card dealt is a $\heartsuit$? | \frac{13}{204} | 100 |
22,132 | Calculate the probability that in a family where there is already one child who is a boy, the next child will also be a boy. | 1/3 | 0.78125 |
22,133 | In $\triangle ABC$, the lengths of the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $(2b-c)\cos A=a\cos C$.
(1) Find the measure of angle $A$;
(2) If $a=3$ and $b=2c$, find the area of $\triangle ABC$. | \frac{3\sqrt{3}}{2} | 96.875 |
22,134 | How many ways are there to arrange the letters of the word $\text{C}_1\text{O}_1\text{M}_1\text{M}_2\text{U}_1\text{N}_1\text{I}_1\text{T}_1$, in which the two M's are considered different? | 40320 | 66.40625 |
22,135 | In Mr. Lee's classroom, there are six more boys than girls among a total of 36 students. What is the ratio of the number of boys to the number of girls? | \frac{7}{5} | 53.90625 |
22,136 | In a square diagram divided into 64 smaller equilateral triangular sections, shading follows a pattern where every alternate horizontal row of triangles is filled. If this pattern begins from the first row at the bottom (considering it as filled), what fraction of the triangle would be shaded in such a 8x8 triangular-section diagram?
A) $\frac{1}{3}$
B) $\frac{1}{2}$
C) $\frac{2}{3}$
D) $\frac{3}{4}$
E) $\frac{1}{4}$ | \frac{1}{2} | 94.53125 |
22,137 | If there exists a line $l$ that is a tangent to the curve $y=x^{2}$ and also a tangent to the curve $y=a\ln x$, then the maximum value of the real number $a$ is ____. | 2e | 45.3125 |
22,138 | Given a circle of radius 3, there are many line segments of length 4 that are tangent to the circle at their midpoints. Find the area of the region consisting of all such line segments.
A) $9\pi$
B) $\pi$
C) $4\pi$
D) $13\pi$
E) $16\pi$ | 4\pi | 40.625 |
22,139 | Given four points $P, A, B, C$ on a sphere, if $PA$, $PB$, $PC$ are mutually perpendicular and $PA=PB=PC=1$, calculate the surface area of this sphere. | 3\pi | 82.8125 |
22,140 | When someone is at home, the probability of a phone call being answered at the first ring is 0.1, at the second ring is 0.2, at the third ring is 0.4, and at the fourth ring is 0.1. Calculate the probability that the phone call is answered within the first four rings. | 0.8 | 96.09375 |
22,141 | All positive integers whose digits add up to 14 are listed in increasing order: $59, 68, 77, ...$. What is the fifteenth number in that list? | 266 | 30.46875 |
22,142 | Given that $\sin x-2\cos x= \sqrt {5}$, find the value of $\tan x$. | -\dfrac{1}{2} | 85.15625 |
22,143 | Each of two boxes contains four chips numbered $1$, $2$, $3$, and $4$. Calculate the probability that the product of the numbers on the two chips is a multiple of $4$. | \frac{1}{2} | 31.25 |
22,144 | If the function $f(x) = \tan(2x - \frac{\pi}{6})$, then the smallest positive period of $f(x)$ is \_\_\_\_\_\_; $f\left(\frac{\pi}{8}\right)=$ \_\_\_\_\_\_. | 2 - \sqrt{3} | 87.5 |
22,145 | Water is the source of life and one of the indispensable important material resources for human survival and development. In order to better manage water quality and protect the environment, the Municipal Sewage Treatment Office plans to purchase 10 sewage treatment equipment in advance. There are two models, $A$ and $B$, with their prices and sewage treatment capacities as shown in the table below:<br/>
| | $A$ model | $B$ model |
|----------|-----------|-----------|
| Price (million yuan) | $12$ | $10$ |
| Sewage treatment capacity (tons/month) | $240$ | $200$ |
$(1)$ In order to save expenses, the Municipal Sewage Treatment Office plans to purchase sewage treatment equipment with a budget not exceeding $105$ million yuan. How many purchasing plans do you think are possible?<br/>
$(2)$ Under the condition in $(1)$, if the monthly sewage treatment volume must not be less than $2040$ tons, to save money, please help the Municipal Sewage Treatment Office choose the most cost-effective plan. | 102 | 30.46875 |
22,146 | Find all real numbers \( x \) such that
\[
\frac{16^x + 25^x}{20^x + 32^x} = \frac{9}{8}.
\] | x = 0 | 2.34375 |
22,147 | Given the sequence $\\_a{n}\_$, where $\_a{n}>0$, $\_a{1}=1$, and $\_a{n+2}=\frac{1}{a{n}+1}$, and it is known that $\_a{6}=a{2}$, find the value of $\_a{2016}+a{3}=\_\_\_\_\_\_$. | \frac{\sqrt{5}}{2} | 58.59375 |
22,148 | China has become the world's largest electric vehicle market. Electric vehicles have significant advantages over traditional vehicles in ensuring energy security and improving air quality. After comparing a certain electric vehicle with a certain fuel vehicle, it was found that the average charging cost per kilometer for electric vehicles is $0.6$ yuan less than the average refueling cost per kilometer for fuel vehicles. If the charging cost and refueling cost are both $300$ yuan, the total distance that the electric vehicle can travel is 4 times that of the fuel vehicle. Let the average charging cost per kilometer for this electric vehicle be $x$ yuan.
$(1)$ When the charging cost is $300$ yuan, the total distance this electric vehicle can travel is ______ kilometers. (Express using an algebraic expression with $x$)
$(2)$ Please calculate the average travel cost per kilometer for these two vehicles separately.
$(3)$ If the other annual costs for the fuel vehicle and electric vehicle are $4800$ yuan and $7800$ yuan respectively, in what range of annual mileage is the annual cost of buying an electric vehicle lower? (Annual cost $=$ annual travel cost $+$ annual other costs) | 5000 | 23.4375 |
22,149 | (1) Given $\cos \alpha =\frac{\sqrt{5}}{3}, \alpha \in \left(-\frac{\pi }{2},0\right)$, find $\sin (\pi -\alpha)$;
(2) Given $\cos \left(\theta+ \frac{\pi}{4}\right)= \frac{4}{5}, \theta \in \left(0, \frac{\pi}{2}\right)$, find $\cos \left(\frac{\pi }{4}-\theta \right)$. | \frac{3}{5} | 90.625 |
22,150 | 500 × 3986 × 0.3986 × 5 = ? | 0.25 \times 3986^2 | 0 |
22,151 | In the triangle below, find $XY$. Triangle $XYZ$ is a right triangle with $XZ = 18$ and $Z$ as the right angle. Angle $Y = 60^\circ$.
[asy]
unitsize(1inch);
pair P,Q,R;
P = (0,0);
Q= (1,0);
R = (0.5,sqrt(3)/2);
draw (P--Q--R--P,linewidth(0.9));
draw(rightanglemark(R,P,Q,3));
label("$X$",P,S);
label("$Y$",Q,S);
label("$Z$",R,N);
label("$18$",(P+R)/2,W);
label("$60^\circ$",(0.9,0),N);
[/asy] | 36 | 53.90625 |
22,152 | Determine the probability that a 4 × 4 square grid becomes a single uniform color (all white or all black) after rotation. | \frac{1}{32768} | 21.875 |
22,153 | Given $A = 30^\circ$ and $B = 60^\circ$, calculate the value of $(1+\tan A)(1+\tan B)$. | 2 + \frac{4\sqrt{3}}{3} | 10.9375 |
22,154 | Point P is located on side AB of triangle ABC. What is the probability that the area of triangle PBC is less than or equal to 1/3 of the area of triangle ABC. | \frac{1}{3} | 38.28125 |
22,155 | Let \(b_n = 7^n + 9^n\). Determine the remainder when \(b_{86}\) is divided by \(50\). | 40 | 55.46875 |
22,156 | Calculate \(3^5 \cdot 6^5\). | 1,889,568 | 0 |
22,157 | Three concentric circles with radii 5 meters, 10 meters, and 15 meters, form the paths along which an ant travels moving from one point to another symmetrically. The ant starts at a point on the smallest circle, moves radially outward to the third circle, follows a path on each circle, and includes a diameter walk on the smallest circle. How far does the ant travel in total?
A) $\frac{50\pi}{3} + 15$
B) $\frac{55\pi}{3} + 25$
C) $\frac{60\pi}{3} + 30$
D) $\frac{65\pi}{3} + 20$
E) $\frac{70\pi}{3} + 35$ | \frac{65\pi}{3} + 20 | 3.125 |
22,158 | In the expansion of $(x-y)^{8}(x+y)$, the coefficient of $x^{7}y^{2}$ is ____. | 20 | 85.15625 |
22,159 | Let \( Q \) be the product of the first \( 50 \) positive even integers. Find the largest integer \( k \) such that \( Q \) is divisible by \( 2^k \). | 97 | 90.625 |
22,160 | In an arithmetic sequence $\{a_n\}$, it is known that $a_1 + a_3 = 0$ and $a_2 + a_4 = -2$. Find the sum of the first 10 terms of the sequence $\left\{ \frac{a_n}{2^{n-1}} \right\}$. | \frac{5}{256} | 1.5625 |
22,161 | Given the function $f(x)=2x^{3}+ax^{2}+bx+1$, it reaches an extreme value of $-6$ at $x=1$.
(1) Find the values of the real numbers $a$ and $b$.
(2) Find the maximum and minimum values of the function $f(x)$ on the interval $[−2,2]$. | -6 | 36.71875 |
22,162 | Determine the volume of the solid formed by the set of vectors $\mathbf{v}$ such that
\[\mathbf{v} \cdot \mathbf{v} = \mathbf{v} \cdot \begin{pmatrix} -6 \\ 18 \\ 12 \end{pmatrix}.\] | \frac{4}{3} \pi \cdot 126 \sqrt{126} | 0 |
22,163 | In the provided polygon, each side is perpendicular to its adjacent sides, and all 36 of the sides are congruent. The perimeter of the polygon is 72. Inside, the polygon is divided into rectangles instead of squares. Find the area of the polygon. | 72 | 8.59375 |
22,164 | Given $f(x)=\sin 2x+\cos 2x$.
$(1)$ Find the period and the interval of monotonic increase of $f(x)$.
$(2)$ Find the maximum and minimum values of the function $f(x)$ on $[0,\frac{π}{2}]$. | -1 | 82.8125 |
22,165 | What common fraction (that is, a fraction reduced to its lowest terms) is equivalent to $0.4\overline{13}$? | \frac{409}{990} | 86.71875 |
22,166 | If one vertex of an ellipse and its two foci form the vertices of an equilateral triangle, then find the eccentricity $e$ of the ellipse. | \dfrac{1}{2} | 39.0625 |
22,167 | Calculate the arithmetic mean of 17, 29, 45, and 64. | 38.75 | 2.34375 |
22,168 | In right triangle $ABC$, $\sin A = \frac{3}{5}$ and $\sin B = 1$. Find $\sin C$. | \frac{4}{5} | 73.4375 |
22,169 | Points $P$, $Q$, $R$, and $S$ are in space such that each of $\overline{SP}$, $\overline{SQ}$, and $\overline{SR}$ is perpendicular to the other two. If $SP = SQ = 12$ and $SR = 7$, then what is the volume of pyramid $SPQR$? | 168 | 90.625 |
22,170 | In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are denoted as $a$, $b$, $c$ respectively, and it is given that $c\cos B + b\cos C = 3a\cos B$.
$(1)$ Find the value of $\cos B$;
$(2)$ If $\overrightarrow{BA} \cdot \overrightarrow{BC} = 2$, find the minimum value of $b$. | 2\sqrt{2} | 81.25 |
22,171 | Calculate the expression $\left(100 - \left(5000 - 500\right)\right) \times \left(5000 - \left(500 - 100\right)\right)$. | -20240000 | 53.90625 |
22,172 | In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively. Given that $a + b = 6$, $c = 2$, and $\cos C = \frac{7}{9}$:
1. Find the values of $a$ and $b$.
2. Calculate the area of triangle $ABC$. | 2\sqrt{2} | 84.375 |
22,173 | Shift the graph of the function $f(x) = 2\sin(2x + \frac{\pi}{4})$ to the right by $\varphi (\varphi > 0)$ units, then shrink the x-coordinate of each point on the graph to half of its original value (the y-coordinate remains unchanged), and make the resulting graph symmetric about the line $x = \frac{\pi}{4}$. Determine the minimum value of $\varphi$. | \frac{3}{8}\pi | 0 |
22,174 | The measures of angles \( X \) and \( Y \) are both positive, integer numbers of degrees. The measure of angle \( X \) is a multiple of the measure of angle \( Y \), and angles \( X \) and \( Y \) are supplementary angles. How many measures are possible for angle \( X \)? | 17 | 39.84375 |
22,175 | Given a sequence $\{a_{n}\}$ where $a_{1}=1$ and $a_{n+1}=\left\{\begin{array}{l}{{a}_{n}+1, n \text{ is odd}}\\{{a}_{n}+2, n \text{ is even}}\end{array}\right.$
$(1)$ Let $b_{n}=a_{2n}$, write down $b_{1}$ and $b_{2}$, and find the general formula for the sequence $\{b_{n}\}$.
$(2)$ Find the sum of the first $20$ terms of the sequence $\{a_{n}\}$. | 300 | 79.6875 |
22,176 | Let the integer part and decimal part of $2+\sqrt{6}$ be $x$ and $y$ respectively. Find the values of $x$, $y$, and the square root of $x-1$. | \sqrt{3} | 15.625 |
22,177 | In the Cartesian coordinate system $xOy$, point $F$ is a focus of the ellipse $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, and point $B_1(0, -\sqrt{3})$ is a vertex of $C$, $\angle OFB_1 = \frac{\pi}{3}$.
$(1)$ Find the standard equation of $C$;
$(2)$ If point $M(x_0, y_0)$ is on $C$, then point $N(\frac{x_0}{a}, \frac{y_0}{b})$ is called an "ellipse point" of point $M$. The line $l$: $y = kx + m$ intersects $C$ at points $A$ and $B$, and the "ellipse points" of $A$ and $B$ are $P$ and $Q$ respectively. If the circle with diameter $PQ$ passes through point $O$, find the area of $\triangle AOB$. | \sqrt{3} | 11.71875 |
22,178 | In the rectangular coordinate system $(xOy)$, a line $l_{1}$ is given by the equation $y = \tan \alpha \cdot x \ (0 \leqslant \alpha < \pi, \alpha \neq \frac{\pi}{2})$, and a parabola $C$ is given by the parametric equations $\begin{cases} x = t^{2} \\ y = -2t \end{cases} \ (t \text{ is a parameter})$. Establish a polar coordinate system with the origin $O$ as the pole and the non-negative semi-axis of the $x$-axis as the polar axis.
1. Find the polar equations of the line $l_{1}$ and the parabola $C$.
2. If the line $l_{1}$ intersects the parabola $C$ at point $A$ (distinct from the origin $O$), draw a line $l_{2}$ passing through the origin and perpendicular to $l_{1}$. The line $l_{2}$ intersects the parabola $C$ at point $B$ (distinct from the origin $O$). Find the minimum value of the area of triangle $OAB$. | 16 | 84.375 |
22,179 | On a Cartesian coordinate plane, points \((4,-1)\) and \((-1, 3)\) are adjacent corners on a square. Calculate the area of this square. | 41 | 95.3125 |
22,180 | Let \( n \in \mathbf{Z}_{+} \). When \( n > 100 \), the first two digits of the decimal part of \( \sqrt{n^{2}+3n+1} \) are ______. | 50 | 32.8125 |
22,181 | Let $S_{n}$ be the sum of the first $n$ terms of the sequence $\{a_{n}\}$, $a_{2}=5$, $S_{n+1}=S_{n}+a_{n}+4$; $\{b_{n}\}$ is a geometric sequence, $b_{2}=9$, $b_{1}+b_{3}=30$, with a common ratio $q \gt 1$.
$(1)$ Find the general formulas for sequences $\{a_{n}\}$ and $\{b_{n}\}$;
$(2)$ Let all terms of sequences $\{a_{n}\}$ and $\{b_{n}\}$ form sets $A$ and $B$ respectively. Arrange the elements of $A\cup B$ in ascending order to form a new sequence $\{c_{n}\}$. Find $T_{20}=c_{1}+c_{2}+c_{3}+\cdots +c_{20}$. | 660 | 1.5625 |
22,182 | Determine the exact value of the series
\[\frac{1}{3 + 1} + \frac{2}{3^2 + 1} + \frac{4}{3^4 + 1} + \frac{8}{3^8 + 1} + \frac{16}{3^{16} + 1} + \dotsb.\] | \frac{1}{2} | 67.96875 |
22,183 | Given $x= \frac {\pi}{12}$ is a symmetry axis of the function $f(x)= \sqrt {3}\sin(2x+\varphi)+\cos(2x+\varphi)$ $(0<\varphi<\pi)$, after shifting the graph of function $f(x)$ to the right by $\frac {3\pi}{4}$ units, find the minimum value of the resulting function $g(x)$ on the interval $\left[-\frac {\pi}{4}, \frac {\pi}{6}\right]$. | -1 | 10.15625 |
22,184 | Given three points in space: A(0,1,5), B(1,5,0), and C(5,0,1), if the vector $\vec{a}=(x,y,z)$ is perpendicular to both $\overrightarrow{AB}$ and $\overrightarrow{AC}$, and the magnitude of vector $\vec{a}$ is $\sqrt{15}$, then find the value of $x^2y^2z^2$. | 125 | 60.9375 |
22,185 | Given the space vectors $\overrightarrow{a}, \overrightarrow{b}$ that satisfy: $(\overrightarrow{a}+ \overrightarrow{b})\perp(2 \overrightarrow{a}- \overrightarrow{b})$, $(\overrightarrow{a}-2 \overrightarrow{b})\perp(2 \overrightarrow{a}+ \overrightarrow{b})$, find $\cos < \overrightarrow{a}, \overrightarrow{b} >$. | -\frac{\sqrt{10}}{10} | 21.09375 |
22,186 | If real numbers \( x \) and \( y \) satisfy the relation \( xy - x - y = 1 \), calculate the minimum value of \( x^{2} + y^{2} \). | 6 - 4\sqrt{2} | 12.5 |
22,187 | Given that a square $S_1$ has an area of $25$, the area of the square $S_3$ constructed by bisecting the sides of $S_2$ is formed by the points of bisection of $S_2$. | 6.25 | 33.59375 |
22,188 | In $\bigtriangleup ABC$, $AB = 75$, and $AC = 100$. A circle with center $A$ and radius $AB$ intersects $\overline{BC}$ at points $B$ and $X$. It is known that $\overline{BX}$ and $\overline{CX}$ have integer lengths. Calculate the length of segment $BC$. | 125 | 15.625 |
22,189 | Given that $\dfrac {\pi}{4} < \alpha < \dfrac {3\pi}{4}$ and $0 < \beta < \dfrac {\pi}{4}$, with $\cos \left( \dfrac {\pi}{4}+\alpha \right)=- \dfrac {3}{5}$ and $\sin \left( \dfrac {3\pi}{4}+\beta \right)= \dfrac {5}{13}$, find the value of $\sin(\alpha+\beta)$. | \dfrac {63}{65} | 39.0625 |
22,190 | Given that $x > 0$ and $y > 0$, and $\frac{4}{x} + \frac{3}{y} = 1$.
(I) Find the minimum value of $xy$ and the values of $x$ and $y$ when the minimum value is obtained.
(II) Find the minimum value of $x + y$ and the values of $x$ and $y$ when the minimum value is obtained. | 7 + 4\sqrt{3} | 50.78125 |
22,191 | Let $\mathcal{T}$ be the set $\lbrace1,2,3,\ldots,12\rbrace$. Let $m$ be the number of sets of two non-empty disjoint subsets of $\mathcal{T}$. Calculate the remainder when $m$ is divided by $1000$. | 625 | 0 |
22,192 | Given the function $y = x - 5$, let $x = 0.5, 1, 1.5, 2, 2.5, 3, 3.5, 4, 4.5, 5$, we can obtain 10 points on the graph of the function. Randomly select two points $P(a, b)$ and $Q(m, n)$ from these 10 points. What is the probability that $P$ and $Q$ lie on the same inverse proportion function graph? | \frac{4}{45} | 29.6875 |
22,193 | In the Cartesian coordinate plane $xOy$, a circle with center $C(1,1)$ is tangent to the $x$-axis and $y$-axis at points $A$ and $B$, respectively. Points $M$ and $N$ lie on the line segments $OA$ and $OB$, respectively. If $MN$ is tangent to circle $C$, find the minimum value of $|MN|$. | 2\sqrt{2} - 2 | 6.25 |
22,194 | Given the function $f(x)=\frac{\ln x}{x+1}$.
(1) Find the equation of the tangent line to the curve $y=f(x)$ at the point $(1,f(1))$;
(2) For $t < 0$, and $x > 0$ with $x\neq 1$, the inequality $f(x)-\frac{t}{x} > \frac{\ln x}{x-1}$ holds. Find the maximum value of the real number $t$. | -1 | 53.90625 |
22,195 | Given a hyperbola with left and right foci at $F_1$ and $F_2$ respectively, a chord $AB$ on the left branch passing through $F_1$ with a length of 5. If $2a=8$, determine the perimeter of $\triangle ABF_2$. | 26 | 66.40625 |
22,196 | Toss a fair coin, with the probability of landing heads or tails being $\frac {1}{2}$ each. Construct the sequence $\{a_n\}$ such that
$$
a_n=
\begin{cases}
1 & \text{if the } n\text{-th toss is heads,} \\
-1 & \text{if the } n\text{-th toss is tails.}
\end{cases}
$$
Define $S_n =a_1+a_2+…+a_n$. Find the probability that $S_2 \neq 0$ and $S_8 = 2$. | \frac {13}{128} | 7.03125 |
22,197 | **
Consider a set of numbers $\{1, 2, 3, 4, 5, 6, 7\}$. Two different natural numbers are selected at random from this set. What is the probability that the greatest common divisor (gcd) of these two numbers is one? Express your answer as a common fraction.
** | \frac{17}{21} | 0.78125 |
22,198 | The ages of Jo, her daughter, and her grandson are all even numbers. The product of their three ages is 2024. How old is Jo? | 46 | 39.84375 |
22,199 | Given the function $f(x)=f'(1)e^{x-1}-f(0)x+\frac{1}{2}x^{2}(f′(x) \text{ is } f(x))$'s derivative, where $e$ is the base of the natural logarithm, and $g(x)=\frac{1}{2}x^{2}+ax+b(a\in\mathbb{R}, b\in\mathbb{R})$
(I) Find the analytical expression and extreme values of $f(x)$;
(II) If $f(x)\geqslant g(x)$, find the maximum value of $\frac{b(a+1)}{2}$. | \frac{e}{4} | 28.125 |
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