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40.3k
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| ground_truth
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100
|
|---|---|---|---|
18,500 |
Using $600$ cards, $200$ of them having written the number $5$ , $200$ having a $2$ , and the other $200$ having a $1$ , a student wants to create groups of cards such that the sum of the card numbers in each group is $9$ . What is the maximum amount of groups that the student may create?
|
100
| 24.21875 |
18,501 |
Among the following 4 propositions, the correct one is:
(1) If a solid's three views are completely identical, then the solid is a cube;
(2) If a solid's front view and top view are both rectangles, then the solid is a cuboid;
(3) If a solid's three views are all rectangles, then the solid is a cuboid;
(4) If a solid's front view and left view are both isosceles trapezoids, then the solid is a frustum.
|
(3)
| 0 |
18,502 |
Given two circles C<sub>1</sub>: $x^{2}+y^{2}-x+y-2=0$ and C<sub>2</sub>: $x^{2}+y^{2}=5$, determine the positional relationship between the two circles; if they intersect, find the equation of the common chord and the length of the common chord.
|
\sqrt{2}
| 32.8125 |
18,503 |
How many positive, three-digit integers contain at least one $4$ as a digit but do not contain a $6$ as a digit?
|
200
| 73.4375 |
18,504 |
The average age of 8 people in a room is 25 years. A 20-year-old person leaves the room. Calculate the average age of the seven remaining people.
|
\frac{180}{7}
| 3.90625 |
18,505 |
Lingling and Mingming were racing. Within 5 minutes, Lingling ran 380.5 meters, and Mingming ran 405.9 meters. How many more meters did Mingming run than Lingling?
|
25.4
| 100 |
18,506 |
Given the function $f(x) = (2 - a)(x - 1) - 2 \ln x (a \in \mathbb{R})$.
(1) If the tangent line of the curve $g(x) = f(x) + x$ at the point $(1, g(1))$ passes through the point $(0, 2)$, find the monotonically decreasing interval of the function $g(x)$;
(2) If the function $y = f(x)$ has no zeros in the interval $(0, \frac{1}{2})$, find the minimum value of the real number $a$.
|
2 - 4 \ln 2
| 69.53125 |
18,507 |
In triangle $ABC$, $BC = 1$ unit and $\measuredangle BAC = 40^\circ$, $\measuredangle ABC = 90^\circ$, hence $\measuredangle ACB = 50^\circ$. Point $D$ is midway on side $AB$, and point $E$ is the midpoint of side $AC$. If $\measuredangle CDE = 50^\circ$, compute the area of triangle $ABC$ plus twice the area of triangle $CED$.
A) $\frac{1}{16}$
B) $\frac{3}{16}$
C) $\frac{4}{16}$
D) $\frac{5}{16}$
E) $\frac{7}{16}$
|
\frac{5}{16}
| 14.84375 |
18,508 |
Determine how many hours it will take Carl to mow the lawn, given that the lawn measures 120 feet by 100 feet, the mower's swath is 30 inches wide with an overlap of 6 inches, and Carl walks at a rate of 4000 feet per hour.
|
1.5
| 67.96875 |
18,509 |
Given a mall with four categories of food: grains, vegetable oils, animal products, and fruits and vegetables, with 40, 10, 20, and 20 varieties, respectively, calculate the total sample size if 6 types of animal products are sampled.
|
27
| 87.5 |
18,510 |
Given a geometric sequence $\{a_n\}$ with $a_1=1$, $0<q<\frac{1}{2}$, and for any positive integer $k$, $a_k - (a_{k+1}+a_{k+2})$ is still an element of the sequence, find the common ratio $q$.
|
\sqrt{2} - 1
| 18.75 |
18,511 |
Given that the angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is $60^{\circ}$, $|\overrightarrow{a}|=1$, and $|\overrightarrow{b}|=2$,
(1) Find $(2\overrightarrow{a}-\overrightarrow{b})\cdot\overrightarrow{a}$;
(2) Find $|2\overrightarrow{a}+\overrightarrow{b}|$.
|
2\sqrt{3}
| 96.875 |
18,512 |
Identical red balls and three identical black balls are arranged in a row, numbered from left to right as 1, 2, 3, 4, 5, 6. Calculate the number of arrangements where the sum of the numbers of the red balls is less than the sum of the numbers of the black balls.
|
10
| 72.65625 |
18,513 |
Mrs. Thompson teaches science to 20 students. She noticed that when she had recorded the grades for 19 of these students, their average was 76. After including Oliver's grade, the new average grade for all 20 students became 78. Calculate Oliver's score on the test.
|
116
| 14.84375 |
18,514 |
If four distinct positive integers $a$, $b$, $c$, $d$ satisfy $\left(6-a\right)\left(6-b\right)\left(6-c\right)\left(6-d\right)=9$, then $a+b+c+d$ is ______.
|
24
| 85.9375 |
18,515 |
Triangle $XYZ$ has vertices $X(1, 9)$, $Y(3, 1)$, and $Z(9, 1)$. A line through $Y$ cuts the area of $\triangle XYZ$ in half. Find the sum of the slope and the $y$-intercept of this line.
|
-3
| 32.03125 |
18,516 |
Two circles with radii of 4 and 5 are externally tangent to each other and are both circumscribed by a third circle. Find the area of the shaded region outside these two smaller circles but within the larger circle. Express your answer in terms of $\pi$. Assume the configuration of tangency and containment is similar to the original problem, with no additional objects obstructing.
|
40\pi
| 26.5625 |
18,517 |
In trapezoid $ABCD$ with $AD\parallel BC$ , $AB=6$ , $AD=9$ , and $BD=12$ . If $\angle ABD=\angle DCB$ , find the perimeter of the trapezoid.
|
39
| 3.125 |
18,518 |
When three standard dice are tossed, the numbers $a, b, c$ are obtained. Find the probability that $abc = 72$.
|
\frac{1}{24}
| 32.03125 |
18,519 |
Arrange the positive odd numbers as shown in the pattern below. What is the 5th number from the left in the 21st row?
$$
1 \\
3 \quad 5 \quad 7 \\
9 \quad 11 \quad 13 \quad 15 \quad 17 \\
19 \quad 21 \quad 23 \quad 25 \quad 27 \quad 29 \quad 31 \\
\ldots \quad \quad \quad \ldots \quad \quad \quad \ldots
$$
|
809
| 56.25 |
18,520 |
Given that 50 students are selected from 2013 students using a two-step process, first eliminating 13 students through simple random sampling and then selecting 50 from the remaining 2000 using systematic sampling, determine the probability of each person being selected.
|
\frac{50}{2013}
| 85.15625 |
18,521 |
Given the vertices of a triangle A(0, 5), B(1, -2), C(-6, m), and the midpoint of BC is D, when the slope of line AD is 1, find the value of m and the length of AD.
|
\frac{5\sqrt{2}}{2}
| 54.6875 |
18,522 |
Simplify first, then evaluate: \\((x+2)^{2}-4x(x+1)\\), where \\(x= \sqrt {2}\\).
|
-2
| 90.625 |
18,523 |
Determine the area of the region bounded by the graph of \[x^2+y^2 = 6|x-y| + 6|x+y|\].
A) 54
B) 63
C) 72
D) 81
E) 90
|
72
| 26.5625 |
18,524 |
A construction team has 6 projects (A, B, C, D, E, F) that need to be completed separately. Project B must start after project A is completed, project C must start after project B is completed, and project D must immediately follow project C. Determine the number of different ways to schedule these 6 projects.
|
20
| 2.34375 |
18,525 |
Is there a real number $a$ such that the function $y=\sin^2x+a\cos x+ \frac{5}{8}a- \frac{3}{2}$ has a maximum value of $1$ on the closed interval $\left[0,\frac{\pi}{2}\right]$? If it exists, find the corresponding value of $a$. If not, explain why.
|
\frac{3}{2}
| 35.9375 |
18,526 |
In the plane rectangular coordinate system $xOy$, the parameter equation of the line $l$ is $\left\{{\begin{array}{l}{x=3-\frac{{\sqrt{3}}}{2}t,}\\{y=\sqrt{3}-\frac{1}{2}t}\end{array}}\right.$ (where $t$ is the parameter). Establish a polar coordinate system with the origin $O$ as the pole and the positive half-axis of the $x$-axis as the polar axis. The polar coordinate equation of the curve $C$ is $ρ=2sin({θ+\frac{π}{6}})$.
$(1)$ Find the general equation of the line $l$ and the rectangular coordinate equation of the curve $C$;
$(2)$ If the polar coordinates of point $P$ are $({2\sqrt{3},\frac{π}{6}})$, the line $l$ intersects the curve $C$ at points $A$ and $B$. Find the value of $\frac{1}{{|{PA}|}}+\frac{1}{{|{PB}|}}$.
|
\frac{\sqrt{3}}{2}
| 40.625 |
18,527 |
A right circular cylinder with radius 3 is inscribed in a hemisphere with radius 7 so that its bases are parallel to the base of the hemisphere. What is the height of this cylinder?
|
\sqrt{40}
| 0 |
18,528 |
Given the equation csa\sin (+ \dfrac {\pi}{6})+siasn(a- \dfrac {\pi}{3})=, evaluate the expression.
|
\dfrac{1}{2}
| 8.59375 |
18,529 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively, where $C$ is an obtuse angle. Given $\cos(A + B - C) = \frac{1}{4}$, $a = 2$, and $\frac{\sin(B + A)}{\sin A} = 2$.
(1) Find the value of $\cos C$;
(2) Find the length of $b$.
|
\sqrt{6}
| 77.34375 |
18,530 |
For any real number $x$, the symbol $\lfloor x \rfloor$ represents the integer part of $x$, that is, $\lfloor x \rfloor$ is the largest integer not exceeding $x$. Calculate the value of $\lfloor \log_{2}1 \rfloor + \lfloor \log_{2}2 \rfloor + \lfloor \log_{2}3 \rfloor + \lfloor \log_{2}4 \rfloor + \ldots + \lfloor \log_{2}1024 \rfloor$.
|
8204
| 86.71875 |
18,531 |
In the Cartesian coordinate system, with the origin as the pole and the positive half-axis of the x-axis as the polar axis, a polar coordinate system is established. The polar equation of line $l$ is $\rho\cos(\theta+ \frac{\pi}{4})= \frac{\sqrt{2}}{2}$, and the parametric equation of curve $C$ is $\begin{cases} x=5+\cos\theta \\ y=\sin\theta \end{cases}$ (where $\theta$ is the parameter).
(1) Find the Cartesian equation of line $l$ and the ordinary equation of curve $C$;
(2) Curve $C$ intersects the x-axis at points $A$ and $B$, with the x-coordinate of point $A$ being less than that of point $B$. Let $P$ be a moving point on line $l$. Find the minimum value of the perimeter of $\triangle PAB$.
|
\sqrt{34}+2
| 5.46875 |
18,532 |
Two 24-sided dice have the following configurations: 5 purple sides, 8 blue sides, 10 red sides, and 1 gold side. What is the probability that when both dice are rolled, they will show the same color?
|
\dfrac{95}{288}
| 80.46875 |
18,533 |
Given the hyperbola $C$: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ ($a > 0$, $b > 0$), with the circle centered at the right focus $F$ of $C$ ($c$, $0$) and with radius $a$ intersects one of the asymptotes of $C$ at points $A$ and $B$. If $|AB| = \frac{2}{3}c$, determine the eccentricity of the hyperbola $C$.
|
\frac{3\sqrt{5}}{5}
| 80.46875 |
18,534 |
A kite has sides $15$ units and $20$ units meeting at a right angle, and its diagonals are $24$ units and $x$ units. Find the area of the kite.
|
216
| 45.3125 |
18,535 |
Add 47.2189 to 34.0076 and round to the nearest hundredth.
|
81.23
| 94.53125 |
18,536 |
Given that $z$ is a complex number such that $z+\frac{1}{z}=2\cos 5^\circ$, find $z^{1500}+\frac{1}{z^{1500}}$.
|
-\sqrt{3}
| 0 |
18,537 |
When two fair 8-sided dice (labeled from 1 to 8) are tossed, the numbers \(a\) and \(b\) are obtained. What is the probability that the two-digit number \(ab\) (where \(a\) and \(b\) are digits) and both \(a\) and \(b\) are divisible by 4?
|
\frac{1}{16}
| 96.875 |
18,538 |
A rectangular board of 12 columns and 12 rows has squares numbered beginning in the upper left corner and moving left to right so row one is numbered 1 through 12, row two is 13 through 24, and so on. Determine which number of the form $n^2$ is the first to ensure that at least one shaded square is in each of the 12 columns.
|
144
| 28.90625 |
18,539 |
Let $ ABCD$ be a quadrilateral in which $ AB$ is parallel to $ CD$ and perpendicular to $ AD; AB \equal{} 3CD;$ and the area of the quadrilateral is $ 4$ . if a circle can be drawn touching all the four sides of the quadrilateral, find its radius.
|
\frac{\sqrt{3}}{2}
| 16.40625 |
18,540 |
A point M on the parabola $y=-4x^2$ is at a distance of 1 from the focus. What is the y-coordinate of point M?
|
-\frac{15}{16}
| 35.15625 |
18,541 |
Find the largest three-digit integer starting with 8 that is divisible by each of its distinct, non-zero digits except for 7.
|
864
| 5.46875 |
18,542 |
In $\triangle ABC$, if $\angle A=60^{\circ}$, $\angle C=45^{\circ}$, and $b=4$, then the smallest side of this triangle is $\_\_\_\_\_\_\_.$
|
4\sqrt{3}-4
| 42.96875 |
18,543 |
Let $a$, $b$, and $c$ be positive integers such that $a + b + c = 30$ and $\gcd(a,b) + \gcd(b,c) + \gcd(c,a) = 11$. Determine the sum of all possible distinct values of $a^2 + b^2 + c^2$.
A) 302
B) 318
C) 620
D) 391
E) 419
|
620
| 8.59375 |
18,544 |
Given the ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \left( a > b > 0 \right)$ has an eccentricity of $\frac{\sqrt{2}}{2}$, and the distance from one endpoint of the minor axis to the right focus is $\sqrt{2}$. The line $y = x + m$ intersects the ellipse $C$ at points $A$ and $B$.
$(1)$ Find the equation of the ellipse $C$;
$(2)$ As the real number $m$ varies, find the maximum value of $|AB|$;
$(3)$ Find the maximum value of the area of $\Delta ABO$.
|
\frac{\sqrt{2}}{2}
| 41.40625 |
18,545 |
In the Cartesian coordinate system $xOy$, with the origin as the pole and the positive half-axis of the $x$-axis as the polar axis, the polar coordinate equation of the curve $C_{1}$ is $\rho \cos \theta = 4$.
$(1)$ Let $M$ be a moving point on the curve $C_{1}$, point $P$ lies on the line segment $OM$, and satisfies $|OP| \cdot |OM| = 16$. Find the rectangular coordinate equation of the locus $C_{2}$ of point $P$.
$(2)$ Suppose the polar coordinates of point $A$ are $({2, \frac{π}{3}})$, point $B$ lies on the curve $C_{2}$. Find the maximum value of the area of $\triangle OAB$.
|
2 + \sqrt{3}
| 24.21875 |
18,546 |
Let *Revolution* $(x) = x^3 +Ux^2 +Sx + A$ , where $U$ , $S$ , and $A$ are all integers and $U +S + A +1 = 1773$ . Given that *Revolution* has exactly two distinct nonzero integer roots $G$ and $B$ , find the minimum value of $|GB|$ .
*Proposed by Jacob Xu*
<details><summary>Solution</summary>*Solution.* $\boxed{392}$ Notice that $U + S + A + 1$ is just *Revolution* $(1)$ so *Revolution* $(1) = 1773$ . Since $G$ and $B$ are integer roots we write *Revolution* $(X) = (X-G)^2(X-B)$ without loss of generality. So Revolution $(1) = (1-G)^2(1-B) = 1773$ . $1773$ can be factored as $32 \cdot 197$ , so to minimize $|GB|$ we set $1-G = 3$ and $1-B = 197$ . We get that $G = -2$ and $B = -196$ so $|GB| = \boxed{392}$ .</details>
|
392
| 91.40625 |
18,547 |
Find the common ratio of the infinite geometric series: $$\frac{7}{8} - \frac{5}{12} + \frac{35}{144} - \dots$$
|
-\frac{10}{21}
| 89.84375 |
18,548 |
The function $f(x)=\log_{a}(1-x)+\log_{a}(x+3)$ where $0 < a < 1$.
$(1)$ Find the zeros of the function $f(x)$.
$(2)$ If the minimum value of the function $f(x)$ is $-2$, find the value of $a$.
|
\frac{1}{2}
| 96.09375 |
18,549 |
Define $\lfloor x \rfloor$ as the largest integer less than or equal to $x$ . Define $\{x \} = x - \lfloor x \rfloor$ . For example, $\{ 3 \} = 3-3 = 0$ , $\{ \pi \} = \pi - 3$ , and $\{ - \pi \} = 4-\pi$ . If $\{n\} + \{ 3n\} = 1.4$ , then find the sum of all possible values of $100\{n\}$ .
*Proposed by Isabella Grabski*
|
145
| 12.5 |
18,550 |
Given a complex number $z = (a^2 - 4) + (a + 2)i$ where $a \in \mathbb{R}$:
(Ⅰ) If $z$ is a pure imaginary number, find the value of the real number $a$;
(Ⅱ) If the point corresponding to $z$ in the complex plane lies on the line $x + 2y + 1 = 0$, find the value of the real number $a$.
|
a = -1
| 79.6875 |
18,551 |
What is the value of $\sqrt{2 \cdot 4! \cdot 4!}$ expressed as a positive integer?
|
24\sqrt{2}
| 3.125 |
18,552 |
Find the number of real solutions of the equation
\[\frac{x}{50} = \cos x.\]
|
31
| 25.78125 |
18,553 |
Consider the line $18x + 9y = 162$ forming a triangle with the coordinate axes. Calculate the sum of the lengths of the altitudes of this triangle.
A) 21.21
B) 42.43
C) 63.64
D) 84.85
E) 105.06
|
42.43
| 63.28125 |
18,554 |
Given the lines $l_1: ax+2y-1=0$ and $l_2: x+by-3=0$, where the angle of inclination of $l_1$ is $\frac{\pi}{4}$, find the value of $a$. If $l_1$ is perpendicular to $l_2$, find the value of $b$. If $l_1$ is parallel to $l_2$, find the distance between the two lines.
|
\frac{7\sqrt{2}}{4}
| 72.65625 |
18,555 |
Given real numbers $x$ and $y$ satisfying $x^{2}+y^{2}-4x-2y-4=0$, find the maximum value of $x-y$.
|
1+3\sqrt{2}
| 88.28125 |
18,556 |
Given the right focus $F$ and the right directrix $l$ of the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, with eccentricity $e = \frac{\sqrt{5}}{5}$. Draw $AM \perp l$ through the vertex $A(0, b)$, with $M$ as the foot of the perpendicular. Find the slope of the line $FM$.
|
\frac{1}{2}
| 55.46875 |
18,557 |
Given the diameter $d=\sqrt[3]{\dfrac{16}{9}V}$, find the volume $V$ of the sphere with a radius of $\dfrac{1}{3}$.
|
\frac{1}{6}
| 34.375 |
18,558 |
In the Cartesian coordinate system $xOy$, the terminal side of angle $\theta$ with $Ox$ as the initial side passes through the point $\left( \frac{3}{5}, \frac{4}{5} \right)$, then $\sin \theta=$ ______, $\tan 2\theta=$ ______.
|
-\frac{24}{7}
| 95.3125 |
18,559 |
Using the digits 0, 1, 2, 3, 4, 5 to form numbers without repeating any digit. Calculate:
(1) How many six-digit numbers can be formed?
(2) How many three-digit numbers can be formed that contain at least one even number?
(3) How many three-digit numbers can be formed that are divisible by 3?
|
40
| 10.15625 |
18,560 |
Given the state income tax rate is $q\%$ for the first $\$30000$ of yearly income plus $(q + 1)\%$ for any amount above $\$30000$, and Samantha's state income tax amounts to $(q + 0.5)\%$ of her total annual income, determine Samantha's annual income.
|
60000
| 60.15625 |
18,561 |
What three-digit positive integer is one more than a multiple of 3, 4, 5, 6, and 7?
|
421
| 86.71875 |
18,562 |
What is the 10th term of an arithmetic sequence of 20 terms with the first term being 7 and the last term being 67?
|
\frac{673}{19}
| 97.65625 |
18,563 |
Given the sequence 2008, 2009, 1, -2008, -2009, ..., where each term from the second term onward is equal to the sum of the term before it and the term after it, calculate the sum of the first 2013 terms of this sequence.
|
4018
| 92.96875 |
18,564 |
Given that the ratio of the length, width, and height of a rectangular prism is $4: 3: 2$, and that a plane cuts through the prism to form a hexagonal cross-section (as shown in the diagram), with the minimum perimeter of such hexagons being 36, find the surface area of the rectangular prism.
|
208
| 32.8125 |
18,565 |
How many four-digit numbers starting with the digit $2$ and having exactly three identical digits are there?
|
27
| 5.46875 |
18,566 |
Given the point \( P \) on the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \left(a>b>0, c=\sqrt{a^{2}-b^{2}}\right)\), and the equation of the line \( l \) is \(x=-\frac{a^{2}}{c}\), and the coordinate of the point \( F \) is \((-c, 0)\). Draw \( PQ \perp l \) at point \( Q \). If the points \( P \), \( Q \), and \( F \) form an isosceles triangle, what is the eccentricity of the ellipse?
|
\frac{\sqrt{2}}{2}
| 28.125 |
18,567 |
Six small equilateral triangular corrals, each with a side length of 1 unit, are enclosed by a fence. The exact amount of fencing that enclosed these six corrals is reused to form one large equilateral triangular corral. What is the ratio of the total area of the six small corrals to the area of the new large corral? Express your answer as a common fraction.
|
\frac{1}{6}
| 97.65625 |
18,568 |
Simplify first, then evaluate: $[\left(2x-y\right)^{2}-\left(y+2x\right)\left(y-2x\right)]\div ({-\frac{1}{2}x})$, where $x=\left(\pi -3\right)^{0}$ and $y={({-\frac{1}{3}})^{-1}}$.
|
-40
| 59.375 |
18,569 |
Given $x=1-2a$, $y=3a-4$.
$(1)$ Given that the arithmetic square root of $x$ is $3$, find the value of $a$;
$(2)$ If $x$ and $y$ are both square roots of the same number, find this number.
|
25
| 81.25 |
18,570 |
Cars A and B simultaneously depart from locations $A$ and $B$, and travel uniformly towards each other. When car A has traveled 12 km past the midpoint of $A$ and $B$, the two cars meet. If car A departs 10 minutes later than car B, they meet exactly at the midpoint of $A$ and $B$. When car A reaches location $B$, car B is still 20 km away from location $A$. What is the distance between locations $A$ and $B$ in kilometers?
|
120
| 47.65625 |
18,571 |
Given the function $f(x)=\sin \omega x\cos \omega x- \sqrt{3}\cos^2\omega x+ \frac{\sqrt{3}}{2} (\omega > 0)$, the two adjacent axes of symmetry of its graph are $\frac{\pi}{2}$.
$(1)$ Find the equation of the axis of symmetry for the function $y=f(x)$.
$(2)$ If the zeros of the function $y=f(x)- \frac{1}{3}$ in the interval $(0,\pi)$ are $x_{1}$ and $x_{2}$, find the value of $\cos (x_{1}-x_{2})$.
|
\frac{1}{3}
| 48.4375 |
18,572 |
Three balls numbered 1, 2, and 3 are placed in a bag. A ball is drawn, the number recorded, and then the ball is returned. This process is repeated three times. Calculate the probability that the sum of the three recorded numbers is less than 8.
|
\frac{23}{27}
| 25.78125 |
18,573 |
Given the derivative of the function $f(x)$ is $f'(x) = 2 + \sin x$, and $f(0) = -1$. The sequence $\{a_n\}$ is an arithmetic sequence with a common difference of $\frac{\pi}{4}$. If $f(a_2) + f(a_3) + f(a_4) = 3\pi$, calculate $\frac{a_{2016}}{a_{2}}$.
|
2015
| 60.15625 |
18,574 |
The volume of a given sphere is \( 72\pi \) cubic inches. Calculate the surface area of the sphere in terms of \( \pi \).
|
36\pi 2^{2/3}
| 0 |
18,575 |
If the square roots of a positive number are $a+2$ and $2a-11$, find the positive number.
|
225
| 28.90625 |
18,576 |
What is the least five-digit positive integer which is congruent to 7 (mod 21)?
|
10,003
| 0 |
18,577 |
About 40% of students in a certain school are nearsighted, and about 30% of the students in the school use their phones for more than 2 hours per day, with a nearsighted rate of about 50% among these students. If a student who uses their phone for no more than 2 hours per day is randomly selected from the school, calculate the probability that the student is nearsighted.
|
\frac{5}{14}
| 71.09375 |
18,578 |
In the polar coordinate system, the equation of circle C is $\rho=4\sqrt{2}\cos(\theta-\frac{\pi}{4})$. A Cartesian coordinate system is established with the pole as the origin and the positive x-axis as the polar axis. The parametric equation of line $l$ is $\begin{cases} x=t+1 \\ y=t-1 \end{cases}$ (where $t$ is the parameter).
(1) Find the Cartesian equation of circle C and the standard equation of line $l$.
(2) Suppose line $l$ intersects circle C at points A and B. Find the area of triangle $\triangle ABC$.
|
2\sqrt{3}
| 47.65625 |
18,579 |
A store received a large container of milk. The salesperson has a balance scale that lacks weights (milk bottles can be placed on the scale), and there are 3 identical milk bottles, two of which are empty, and one has 1 liter of milk. How can exactly 85 liters of milk be measured into one bottle using the balance scale no more than 8 times (assuming the capacity of the milk bottles exceeds 85 liters)?
|
85
| 17.1875 |
18,580 |
Find \( g(2021) \) if for any real numbers \( x, y \) the following equation holds:
\[ g(x-y) = g(x) + g(y) - 2022(x + y) \]
|
4086462
| 94.53125 |
18,581 |
A rectangular prism has vertices at the corners and edges joining them similarly to a cube. The prism dimensions differ along each axis; therefore, no two adjoining sides are of the same length. If one side has a length ratio of 2:3 with another, and there are three dimensions under consideration, compute how many total diagonals (both face diagonals that lie within the surfaces and space diagonals that span the entire prism) exist.
|
16
| 55.46875 |
18,582 |
Given the expansion of the binomial $({x+\frac{a}{{\sqrt{x}}}})^n$ where $n\in{N^*}$, in the expansion, ___, ___. Given the following conditions: ① the ratio of the binomial coefficients of the second term to the third term is $1:4$; ② the sum of all coefficients is $512$; ③ the $7$th term is a constant term. Choose two appropriate conditions from the above three conditions to fill in the blanks above, and complete the following questions.
$(1)$ Find the value of the real number $a$ and the term with the largest binomial coefficient in the expansion;
$(2)$ Find the constant term in the expansion of $({x\sqrt{x}-1}){({x+\frac{a}{{\sqrt{x}}}})^n}$.
|
-48
| 25 |
18,583 |
Given that the product of the digits of a 3-digit positive integer equals 36, calculate the number of such integers.
|
21
| 36.71875 |
18,584 |
If $N$ is represented as $11000_2$ in binary, what is the binary representation of the integer that comes immediately before $N$?
|
$10111_2$
| 0 |
18,585 |
The graph of the function $y=\sin(\omega x+ \frac {5\pi}{6})$ where $0<\omega<\pi$ intersects with the coordinate axes at points closest to the origin, which are $(0, \frac {1}{2})$ and $( \frac {1}{2}, 0)$. Determine the axis of symmetry of this graph closest to the y-axis.
|
-1
| 57.8125 |
18,586 |
Find the value of \(m + n\) where \(m\) and \(n\) are integers defined as follows: The positive difference between the roots of the quadratic equation \(5x^2 - 9x - 14 = 0\) can be expressed as \(\frac{\sqrt{m}}{n}\), with \(m\) not divisible by the square of any prime number.
|
366
| 88.28125 |
18,587 |
Calculate: $\left(-2\right)^{0}-3\tan 30^{\circ}-|\sqrt{3}-2|$.
|
-1
| 91.40625 |
18,588 |
Several people completed the task of planting 2013 trees, with each person planting the same number of trees. If 5 people do not participate in the planting, the remaining people each need to plant 2 more trees but still cannot complete the task. However, if each person plants 3 more trees, they can exceed the task. How many people participated in the planting?
|
61
| 57.8125 |
18,589 |
How many squares of side at least $8$ have their four vertices in the set $H$, where $H$ is defined by the points $(x,y)$ with integer coordinates, $-8 \le x \le 8$ and $-8 \le y \le 8$?
|
285
| 8.59375 |
18,590 |
Given $M=\{1,2,x\}$, we call the set $M$, where $1$, $2$, $x$ are elements of set $M$. The elements in the set have definiteness (such as $x$ must exist), distinctiveness (such as $x\neq 1, x\neq 2$), and unorderedness (i.e., changing the order of elements does not change the set). If set $N=\{x,1,2\}$, we say $M=N$. It is known that set $A=\{2,0,x\}$, set $B=\{\frac{1}{x},|x|,\frac{y}{x}\}$, and if $A=B$, then the value of $x-y$ is ______.
|
\frac{1}{2}
| 32.8125 |
18,591 |
Calculate the value of \[\cot(\cot^{-1}5 + \cot^{-1}11 + \cot^{-1}17 + \cot^{-1}23).\]
|
\frac{97}{40}
| 0 |
18,592 |
Given that $\overrightarrow{e}$ is a unit vector, $|\overrightarrow{a}|=4$, and the angle between $\overrightarrow{a}$ and $\overrightarrow{e}$ is $\frac{2}{3}π$, find the projection of $\overrightarrow{a}$ onto $\overrightarrow{e}$.
|
-2
| 35.9375 |
18,593 |
A certain shopping mall's sales volume (unit: ten thousand pieces) for five consecutive years is as shown in the table below:
| $x$(year) | 1 | 2 | 3 | 4 | 5 |
|-----------|---|---|---|---|---|
| $y$(sales volume) | 5 | 5 | 6 | 7 | 7 |
$(1)$ Find the linear regression equation $\hat{y}=\hat{b}x+\hat{a}$ for the sales volume $y$ and the corresponding year $x$;
$(2)$ If two sets of data are randomly selected from the five sets, find the probability that the two selected sets are exactly the data from two consecutive years.
Note: In the linear regression equation $y=bx+a$, $\hat{b}=\frac{\sum_{i=1}^{n}{{x}_{i}{y}_{i}-n\overline{x}\overline{y}}}{\sum_{i=1}^{n}{{x}_{i}^{2}-n\overline{{x}}^{2}}}$, $\hat{a}=\overline{y}-\hat{b}\overline{x}$. Where $\overline{x}$ and $\overline{y}$ are sample averages, and the linear regression equation can also be written as $\hat{y}=\hat{b}x+\hat{a}$.
|
\frac{2}{5}
| 74.21875 |
18,594 |
In the rectangular coordinate system (xOy), a pole coordinate system is established with O as the pole and the positive semi-axis of x as the polar axis. The shortest distance between a point on the curve C: ρcosθ - ρsinθ = 1 and a point on the curve M: x = -2 + cosφ, y = 1 + sinφ (φ is a parameter) can be calculated.
|
2\sqrt{2}-1
| 56.25 |
18,595 |
If $\displaystyle\prod_{i=6}^{2021} (1-\tan^2((2^i)^\circ))$ can be written in the form $a^b$ for positive integers $a,b$ with $a$ squarefree, find $a+b$ .
*Proposed by Deyuan Li and Andrew Milas*
|
2018
| 37.5 |
18,596 |
A market survey shows that the price $f(t)$ and sales volume $g(t)$ of a particular product in Oriental Department Store over the past month (calculated based on 30 days) approximately satisfy the functions $f(t)=100(1+ \frac {1}{t})$ and $g(t)= \begin{cases} 100+t & 1\leqslant t < 25,t\in N \\ 150-t & 25\leqslant t\leqslant 30,t\in N \end{cases}$, respectively.
(1) Find the daily sales revenue $W(t)$ of the product in terms of time $t (1\leqslant t\leqslant 30,t\in N)$;
(2) Calculate the maximum and minimum daily sales revenue $W(t)$.
|
12100
| 13.28125 |
18,597 |
On Arbor Day, a class at a certain school divided into 10 small groups to participate in tree planting activities. The number of trees planted by the 10 groups is shown in the table below:
| Number of Trees Planted | 5 | 6 | 7 |
|--------------------------|-----|-----|-----|
| Number of Groups | 3 | 4 | 3 |
The variance of the number of trees planted by these 10 groups is ______.
|
0.6
| 93.75 |
18,598 |
Cagney can frost a cupcake every 25 seconds and Lacey can frost a cupcake every 35 seconds. They work together for 7 minutes, but there is a 1-minute period where only Cagney is frosting because Lacey takes a break. What is the number of cupcakes they can frost in these 7 minutes?
|
26
| 28.125 |
18,599 |
Given a line segment of length $6$ is divided into three line segments of lengths that are positive integers, calculate the probability that these three line segments can form a triangle.
|
\frac {1}{10}
| 41.40625 |
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