Unnamed: 0
int64 0
40.3k
| problem
stringlengths 10
5.15k
| ground_truth
stringlengths 1
1.22k
| solved_percentage
float64 0
100
|
|---|---|---|---|
18,700 |
Given that p, q, r, and s are integers in the set {0, 1, 2, 3, 4}, calculate the number of ordered quadruples (p, q, r, s) such that p·s + q·r is odd.
|
168
| 18.75 |
18,701 |
A square $ABCD$ with a side length of $2$ is rotated around $BC$ to form a cylinder. Find the surface area of the cylinder.
|
16\pi
| 47.65625 |
18,702 |
Two non-collinear vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ are given, and $\overrightarrow{a}+2\overrightarrow{b}$ is perpendicular to $2\overrightarrow{a}-\overrightarrow{b}$, $\overrightarrow{a}-\overrightarrow{b}$ is perpendicular to $\overrightarrow{a}$. The cosine of the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is __________.
|
\frac{\sqrt{10}}{5}
| 86.71875 |
18,703 |
Find the number of 0-1 binary sequences formed by six 0's and six 1's such that no three 0's are together. For example, 110010100101 is such a sequence but 101011000101 and 110101100001 are not.
|
357
| 73.4375 |
18,704 |
(1) Find the value of $\cos\frac{5\pi}{3}$.
(2) Given that $\frac{\sin\alpha + 2\cos\alpha}{5\cos\alpha - \sin\alpha} = \frac{5}{16}$, find the value of $\tan\alpha$.
(3) Given that $\sin\theta = \frac{1}{3}$ and $\theta \in (0, \frac{\pi}{2})$, find the value of $\tan 2\theta$.
(4) In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. Moreover, $a\sin A\cos C + c\sin A\cos A = \frac{1}{3}c$, $D$ is the midpoint of $AC$, and $\cos B = \frac{2\sqrt{5}}{5}$, $BD = \sqrt{26}$. Find the length of the shortest side of triangle $ABC$.
|
2\sqrt{2}
| 1.5625 |
18,705 |
Let $AB$ be a diameter of a circle and let $C$ be a point on the segement $AB$ such that $AC : CB = 6 : 7$ . Let $D$ be a point on the circle such that $DC$ is perpendicular to $AB$ . Let $DE$ be the diameter through $D$ . If $[XYZ]$ denotes the area of the triangle $XYZ$ , find $[ABD]/[CDE]$ to the nearest integer.
|
13
| 18.75 |
18,706 |
Let $ABC$ be a triangle (right in $B$ ) inscribed in a semi-circumference of diameter $AC=10$ . Determine the distance of the vertice $B$ to the side $AC$ if the median corresponding to the hypotenuse is the geometric mean of the sides of the triangle.
|
5/2
| 17.96875 |
18,707 |
The second hand on the clock is 6 cm long. How far in centimeters does the tip of this second hand travel during a period of 15 minutes? Express your answer in terms of $\pi$.
|
180\pi
| 92.1875 |
18,708 |
If physical education is not the first class, and Chinese class is not adjacent to physics class, calculate the total number of different scheduling arrangements for five subjects - mathematics, physics, history, Chinese, and physical education - on Tuesday morning.
|
48
| 16.40625 |
18,709 |
Suppose $xy-5x+2y=30$ , where $x$ and $y$ are positive integers. Find the sum of all possible values of $x$
|
31
| 43.75 |
18,710 |
A cinema has 21 rows of seats, with 26 seats in each row. How many seats are there in total in this cinema?
|
546
| 100 |
18,711 |
Given that $x$ and $y$ satisfy the equation $x^2 + y^2 - 4x - 6y + 12 = 0$, find the minimum value of $x^2 + y^2$.
|
14 - 2\sqrt{13}
| 95.3125 |
18,712 |
Convert the following expressions between different bases: $110011_{(2)} = \_{(10)} = \_{(5)}$
|
51_{(10)} = 201_{(5)}
| 50 |
18,713 |
Determine how many non-similar regular 500-pointed stars exist, given that a regular $n$-pointed star adheres to the rules set in the original problem description.
|
100
| 48.4375 |
18,714 |
The line \(y = -\frac{1}{2}x + 8\) crosses the \(x\)-axis at \(P\) and the \(y\)-axis at \(Q\). Point \(T(r, s)\) is on line segment \(PQ\). If the area of \(\triangle POQ\) is twice the area of \(\triangle TOP\), then what is the value of \(r+s\)?
|
12
| 95.3125 |
18,715 |
Given that $(a+1)x - 1 - \ln x \leqslant 0$ holds for any $x \in [\frac{1}{2}, 2]$, find the maximum value of $a$.
|
1 - 2\ln 2
| 28.90625 |
18,716 |
Given the ellipse $C:\frac{{x}^{2}}{4}+\frac{{y}^{2}}{3}=1$, let ${F}_{1}$ and ${F}_{2}$ be its left and right foci, respectively. A line $l$ passing through point ${F}_{2}$ with a slope of $1$ intersects ellipse $C$ at two distinct points $M$ and $N$. Calculate the area of triangle $MN{F}_{1}$.
|
\frac{12\sqrt{2}}{7}
| 22.65625 |
18,717 |
If the complex number $z=(2-i)(a-i)$ (where $i$ is the imaginary unit) is a pure imaginary number, then the value of the real number $a$ is \_\_\_\_\_\_.
|
\frac{1}{2}
| 83.59375 |
18,718 |
In the geometric sequence $\{a_n\}$, if $a_n > a_{n+1}$, and $a_7 \cdot a_{14} = 6, a_4 + a_{17} = 5$, calculate $\frac{a_5}{a_{18}}$.
|
\frac{3}{2}
| 7.03125 |
18,719 |
Given the function $f(x)= \begin{cases} a+\ln x,x > 0 \\ g(x)-x,x < 0\\ \end{cases}$, which is an odd function, and $g(-e)=0$, find the value of $a$.
|
-1-e
| 58.59375 |
18,720 |
For how many integer values of \( n \) between 1 and 999 inclusive does the decimal representation of \( \frac{n}{1000} \) terminate?
|
999
| 92.96875 |
18,721 |
In square $ABCD$ with side length $2$ , let $M$ be the midpoint of $AB$ . Let $N$ be a point on $AD$ such that $AN = 2ND$ . Let point $P$ be the intersection of segment $MN$ and diagonal $AC$ . Find the area of triangle $BPM$ .
*Proposed by Jacob Xu*
|
2/7
| 63.28125 |
18,722 |
Given that in rhombus $ABCD$, the diagonal $AC$ is twice as long as diagonal $BD$, and point $E$ is on segment $AC$ such that line segment $BE$ bisects $\angle ABC$, determine the ratio of the area of triangle $ABE$ to the area of rhombus $ABCD$.
|
\frac{1}{4}
| 71.09375 |
18,723 |
Given the function $f(x)=2\sin (wx+\varphi+ \frac {\pi}{3})+1$ ($|\varphi| < \frac {\pi}{2},w > 0$) is an even function, and the distance between two adjacent axes of symmetry of the function $f(x)$ is $\frac {\pi}{2}$.
$(1)$ Find the value of $f( \frac {\pi}{8})$.
$(2)$ When $x\in(- \frac {\pi}{2}, \frac {3\pi}{2})$, find the sum of the real roots of the equation $f(x)= \frac {5}{4}$.
|
2\pi
| 71.09375 |
18,724 |
A rectangular box $Q$ is inscribed in a sphere of radius $s$. The surface area of $Q$ is 576, and the sum of the lengths of its 12 edges is 168. Determine the radius $s$.
|
3\sqrt{33}
| 43.75 |
18,725 |
Given a configuration of four unit squares arranged in a 2x2 grid, find the area of triangle $\triangle ABC$, where $A$ is the midpoint of the top side of the top-left square, $B$ is the bottom-right corner of the bottom-right square, and $C$ is the midpoint of the right side of the bottom-right square.
|
0.375
| 12.5 |
18,726 |
Find the smallest constant \(C\) such that
\[ x^2 + y^2 + z^2 + 1 \ge C(x + y + z) \]
for all real numbers \(x, y,\) and \(z.\)
|
\sqrt{\frac{4}{3}}
| 0 |
18,727 |
Anh traveled 75 miles on the interstate and 15 miles on a mountain pass. The speed on the interstate was four times the speed on the mountain pass. If Anh spent 45 minutes driving on the mountain pass, determine the total time of his journey in minutes.
|
101.25
| 53.90625 |
18,728 |
Joshua rolls two dice and records the product of the numbers face up. The probability that this product is composite can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$ . Compute $m+n$ .
*Proposed by Nathan Xiong*
|
65
| 44.53125 |
18,729 |
The gravitational force that Earth exerts on an object is inversely proportional to the square of the distance between the center of the Earth and the object. When Alice is on the surface of Earth, 6,000 miles from the center, the gravitational force is 400 Newtons. What is the gravitational force (in Newtons) that the Earth exerts on her when she's standing on a space station, 360,000 miles from the center of the earth? Express your answer as a fraction.
|
\frac{1}{9}
| 28.90625 |
18,730 |
From 5 students, 4 are to be selected to participate in competitions in four subjects: mathematics, physics, chemistry, and biology, with each competition having only 1 participant. If student A does not participate in the biology competition, then the total number of different selection schemes is \_\_\_\_\_\_.
|
96
| 94.53125 |
18,731 |
In $\triangle ABC$, if $a= \sqrt {5}$, $b= \sqrt {15}$, $A=30^{\circ}$, then $c=$ \_\_\_\_\_\_.
|
2 \sqrt {5}
| 0 |
18,732 |
Given a pyramid $P-ABC$ where $PA=PB=2PC=2$, and $\triangle ABC$ is an equilateral triangle with side length $\sqrt{3}$, the radius of the circumscribed sphere of the pyramid $P-ABC$ is _______.
|
\dfrac{\sqrt{5}}{2}
| 3.90625 |
18,733 |
Sandy and Sam each selected a positive integer less than 250. Sandy's number is a multiple of 15, and Sam's number is a multiple of 20. What is the probability that they selected the same number? Express your answer as a common fraction.
|
\frac{1}{48}
| 84.375 |
18,734 |
In a regular tetrahedron embedded in 3-dimensional space, the centers of the four faces are the vertices of a smaller tetrahedron. If the vertices of the larger tetrahedron are located on the surface of a sphere of radius \(r\), find the ratio of the volume of the smaller tetrahedron to that of the larger tetrahedron. Express your answer as a simplified fraction.
|
\frac{1}{27}
| 78.90625 |
18,735 |
What is the greatest common factor of 120, 180, and 300?
|
60
| 100 |
18,736 |
The expression \(\frac{3}{10}+\frac{3}{100}+\frac{3}{1000}\) is equal to:
|
0.333
| 50 |
18,737 |
In the cube ABCD-A<sub>1</sub>B<sub>1</sub>C<sub>1</sub>D<sub>1</sub>, the angle formed by the skew lines A<sub>1</sub>B and AC is \_\_\_\_\_\_°; the angle formed by the line A<sub>1</sub>B and the plane A<sub>1</sub>B<sub>1</sub>CD is \_\_\_\_\_\_\_\_\_°.
|
30
| 17.1875 |
18,738 |
The ellipse $5x^2 - ky^2 = 5$ has one of its foci at $(0, 2)$. Find the value of $k$.
|
-1
| 22.65625 |
18,739 |
Given an ellipse $C$: $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$ passing through the point $(0,4)$, with an eccentricity of $\frac{3}{5}$
1. Find the equation of $C$;
2. Find the length of the line segment intercepted by $C$ on the line passing through point $(3,0)$ with a slope of $\frac{4}{5}$.
|
\frac{41}{5}
| 16.40625 |
18,740 |
Crestview's school has expanded its official colors to include blue along with the original purple and gold. The students need to design a flag using three solid-colored horizontal stripes. Using one, two, or all three of the school colors, how many different flags are possible if adjacent stripes may be of the same color?
|
27
| 65.625 |
18,741 |
A cylinder has a radius of 5 cm and a height of 12 cm. What is the longest segment, in centimeters, that would fit inside the cylinder?
|
\sqrt{244}
| 0 |
18,742 |
Let $M = 39 \cdot 48 \cdot 77 \cdot 150$. Calculate the ratio of the sum of the odd divisors of $M$ to the sum of the even divisors of $M$.
|
\frac{1}{62}
| 20.3125 |
18,743 |
In parallelogram $EFGH$, point $Q$ is on $\overline{EF}$ such that $\frac {EQ}{EF} = \frac {23}{1005}$, and point $R$ is on $\overline{EH}$ such that $\frac {ER}{EH} = \frac {23}{2011}$. Let $S$ be the point of intersection of $\overline{EG}$ and $\overline{QR}$. Find $\frac {EG}{ES}$.
|
131
| 41.40625 |
18,744 |
Given that $|\vec{a}|=4$, and $\vec{e}$ is a unit vector. When the angle between $\vec{a}$ and $\vec{e}$ is $\frac{2\pi}{3}$, the projection of $\vec{a} + \vec{e}$ on $\vec{a} - \vec{e}$ is ______.
|
\frac{5\sqrt{21}}{7}
| 82.03125 |
18,745 |
In the Cartesian coordinate system xOy, curve $C_1: x^2+y^2=1$ is given. Taking the origin O of the Cartesian coordinate system xOy as the pole and the positive half-axis of x as the polar axis, a polar coordinate system is established with the same unit length. It is known that the line $l: \rho(2\cos\theta-\sin\theta)=6$.
(1) After stretching all the x-coordinates and y-coordinates of points on curve $C_1$ by $\sqrt{3}$ and 2 times respectively, curve $C_2$ is obtained. Please write down the Cartesian equation of line $l$ and the parametric equation of curve $C_2$;
(2) Find a point P on curve $C_2$ such that the distance from point P to line $l$ is maximized, and calculate this maximum value.
|
2\sqrt{5}
| 41.40625 |
18,746 |
A palindrome is an integer that reads the same forward and backward, such as 1221. What percent of the palindromes between 1000 and 2000 contain at least one digit 7?
|
10\%
| 22.65625 |
18,747 |
Given the function $f(x)=\sin x\cos x-\sqrt{3}\cos^{2}x$.
(1) Find the smallest positive period of $f(x)$;
(2) Find the maximum and minimum values of $f(x)$ when $x\in[0,\frac{\pi }{2}]$.
|
-\sqrt{3}
| 82.8125 |
18,748 |
Compute the sum of the squares of the first 10 base-6 numbers and express your answer in base 6. Specifically, find $1_6^2 + 2_6^2 + 3_6^2 + \cdots + 10_6^2$.
|
231_6
| 7.03125 |
18,749 |
For any real number $x$ , we let $\lfloor x \rfloor$ be the unique integer $n$ such that $n \leq x < n+1$ . For example. $\lfloor 31.415 \rfloor = 31$ . Compute \[2020^{2021} - \left\lfloor\frac{2020^{2021}}{2021} \right \rfloor (2021).\]
*2021 CCA Math Bonanza Team Round #3*
|
2020
| 87.5 |
18,750 |
Convert the decimal number 2011 to a base-7 number.
|
5602_7
| 0.78125 |
18,751 |
In an arithmetic sequence $\{a_n\}$ with a non-zero common difference, $a_1$, $a_2$, and $a_5$ form a geometric sequence, and the sum of the first $10$ terms of this sequence is $100$. The sum of the first $n$ terms of the sequence $\{b_n\}$ is $S_n$, and it satisfies $S_n=2b_n-1$.
$(I)$ Find the general formula for the sequences $\{a_n\}$ and $\{b_n\}$;
$(II)$ Let $C_n=a_n+\log_{\sqrt{2}} b_n$. The sum of the first $n$ terms of the sequence $\{C_n\}$ is $T_n$. If the sequence $\{d_n\}$ is an arithmetic sequence, and $d_n= \frac{T_n}{n+c}$, where $c\neq 0$.
$(i)$ Find the non-zero constant $C$;
$(ii)$ If $f(n)=\frac{d_n}{(n+36)d_{n+1}}$ $(n\in \mathbb{N}^*)$, find the maximum value of the term in the sequence $\{f(n)\}$.
|
\frac{1}{49}
| 26.5625 |
18,752 |
Determine the minimum value of the function $$y = \frac {4x^{2}+2x+5}{x^{2}+x+1}$$ for \(x > 1\).
|
\frac{16 - 2\sqrt{7}}{3}
| 3.125 |
18,753 |
In the Cartesian coordinate system, O is the origin. Given vector $\overrightarrow{a}=(2,1)$, point $A(1,0)$, and point $B(\cos\theta,t)$,
(1) If $\overrightarrow{a} \parallel \overrightarrow{AB}$ and $|\overrightarrow{AB}| = \sqrt{5}|\overrightarrow{OA}|$, find the coordinates of vector $\overrightarrow{OB}$.
(2) If $\overrightarrow{a} \perp \overrightarrow{AB}$, find the minimum value of $y=\cos^2\theta-\cos\theta+\left(\frac{t}{4}\right)^2$.
|
-\frac{1}{5}
| 82.03125 |
18,754 |
Eyes are the windows of the soul. In order to protect students' eyesight, Qihang High School conducts eye examinations for students every semester. The table below shows the results of the right eye vision examination for 39 students in a certain class at the school. In this set of vision data, the median is ______.
| Vision | $4.0$ | $4.1$ | $4.2$ | $4.3$ | $4.4$ | $4.5$ | $4.6$ | $4.7$ | $4.8$ | $4.9$ | $5.0$ |
|--------|-------|-------|-------|-------|-------|-------|-------|-------|-------|-------|-------|
| Number | $1$ | $2$ | $6$ | $3$ | $3$ | $4$ | $1$ | $2$ | $5$ | $7$ | $5$ |
|
4.6
| 12.5 |
18,755 |
The sum of all of the digits of the integers from 1 to 2008 is:
|
28054
| 7.03125 |
18,756 |
Given an ellipse C: $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)$ with its right focus F, upper vertex B, the line BF intersects C at another point A, and the projection of point A on the x-axis is A1. O is the origin, and if $\overrightarrow{BO}=2\overrightarrow{A_{1}A}$, determine the eccentricity of C.
|
\frac{\sqrt{3}}{3}
| 35.9375 |
18,757 |
A $10$ by $4$ rectangle has the same center as a circle of radius $5$. Calculate the area of the region common to both the rectangle and the circle.
A) $40 + 2\pi$
B) $36 + 4\pi$
C) $40 + 4\pi$
D) $44 + 4\pi$
E) $48 + 2\pi$
|
40 + 4\pi
| 34.375 |
18,758 |
In the rectangular coordinate system xOy, the parametric equations of the curve C are $$\begin{cases} x=1+2\cos\theta, \\ y= \sqrt {3}+2\sin\theta\end{cases}$$ (where θ is the parameter). 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 coordinate equation of the straight line l₁ is $$θ=α(0＜α＜ \frac {π}{2})$$. Rotate the straight line l₁ counterclockwise around the pole O by $$\frac {π}{3}$$ units to obtain the straight line l₂.
1. Find the polar coordinate equations of C and l₂.
2. Suppose the straight line l₁ and the curve C intersect at O, A two points, and the straight line l₂ and the curve C intersect at O, B two points. Find the maximum value of |OA|+|OB|.
|
4 \sqrt {3}
| 0 |
18,759 |
In triangle $\triangle ABC$, the sides corresponding to angles $A$, $B$, and $C$ are $a$, $b$, $c$ respectively, and $a\sin B=-\sqrt{3}b\cos A$.
$(1)$ Find the measure of angle $A$;
$(2)$ If $b=4$ and the area of $\triangle ABC$ is $S=2\sqrt{3}$, find the perimeter of $\triangle ABC$.
|
6 + 2\sqrt{7}
| 75.78125 |
18,760 |
A workshop has fewer than $60$ employees. When these employees are grouped in teams of $8$, $5$ employees remain without a team. When arranged in teams of $6$, $3$ are left without a team. How many employees are there in the workshop?
|
45
| 3.125 |
18,761 |
$(1)$ If $A=\frac{a-1}{a+2}•\frac{{a}^{2}-4}{{a}^{2}-2a+1}÷\frac{1}{a-1}$, simplify $A$;<br/>$(2)$ If $a$ satisfies $a^{2}-a=0$, find the value of $A$.
|
-2
| 9.375 |
18,762 |
A point $P$ is randomly placed in the interior of right triangle $ABC$, with coordinates $A(0,6)$, $B(9,0)$, and $C(0,0)$. What is the probability that the area of triangle $APC$ is more than a third of the area of triangle $ABC$?
|
\frac{1}{3}
| 5.46875 |
18,763 |
Remove all positive integers that are divisible by 7, and arrange the remaining numbers in ascending order to form a sequence $\{a_n\}$. Calculate the value of $a_{100}$.
|
116
| 7.8125 |
18,764 |
A right circular cone is cut into five pieces by four planes parallel to its base, each piece having equal height. Determine the ratio of the volume of the second-largest piece to the volume of the largest piece.
|
\frac{37}{61}
| 28.90625 |
18,765 |
To address the threat of AIDS to humanity, now three research institutes, designated as Institute A, Institute B, and Institute C, are independently developing an AIDS vaccine. The probabilities of successfully developing a vaccine are respectively $\frac{1}{2}$, $\frac{1}{3}$, and $\frac{1}{4}$. Calculate:
(1) The probability that exactly one research institute is successful in developing the vaccine;
(2) In order to achieve at least a $\frac{99}{100}$ probability that the vaccine is successfully developed (meaning at least one research institute is successful), how many institutes similar to Institute B are minimally required? (Reference data: $\lg 2=0.3010$, $\lg 3=0.4771$)
|
12
| 98.4375 |
18,766 |
For $-1<r<1$, let $T(r)$ denote the sum of the geometric series \[20 + 10r + 10r^2 + 10r^3 + \cdots.\] Let $b$ between $-1$ and $1$ satisfy $T(b)T(-b)=5040$. Find $T(b)+T(-b)$.
|
504
| 22.65625 |
18,767 |
Given vectors $\overrightarrow {a}$ and $\overrightarrow {b}$, the magnitude of $\overrightarrow {a}$ is the positive root of the equation x^2+x-2=0, $|\overrightarrow {b}|= \sqrt {2}$, and $(\overrightarrow {a}- \overrightarrow {b})\cdot \overrightarrow {a}=0$, find the angle between $\overrightarrow {a}$ and $\overrightarrow {b}$.
|
\frac{\pi}{4}
| 53.90625 |
18,768 |
Given the function $f(x)=\sin \frac {πx}{6}$, and the set $M={0,1,2,3,4,5,6,7,8}$, calculate the probability that $f(m)f(n)=0$ for randomly selected different elements $m$, $n$ from $M$.
|
\frac{5}{12}
| 46.875 |
18,769 |
Roll two dice consecutively. Let the number on the first die be $m$, and the number on the second die be $n$. Calculate the probability that:
(1) $m+n=7$;
(2) $m=n$;
(3) The point $P(m,n)$ is inside the circle $x^2+y^2=16$.
|
\frac{2}{9}
| 69.53125 |
18,770 |
All positive integers whose digits add up to 12 are listed in increasing order: $39, 48, 57, ...$. What is the twelfth number in that list?
|
165
| 0.78125 |
18,771 |
Let $S_n$ be the sum of the first $n$ terms of an arithmetic sequence $\{a_n\}$ with distinct terms, given that $a_3a_5=3a_7$, and $S_3=9$.
$(1)$ Find the general formula for the sequence $\{a_n\}$.
$(2)$ Let $T_n$ be the sum of the first $n$ terms of the sequence $\left\{ \frac{1}{a_na_{n+1}} \right\}$, find the maximum value of $\frac{T_n}{a_{n+1}}$.
|
\frac{1}{16}
| 50.78125 |
18,772 |
The operation $ \diamond $ is defined for positive integers $a$ and $b$ such that $a \diamond b = a^2 - b$. Determine how many positive integers $x$ exist such that $20 \diamond x$ is a perfect square.
|
19
| 41.40625 |
18,773 |
Given that Jackie has $40$ thin rods, one of each integer length from $1 \text{ cm}$ through $40 \text{ cm}$, with rods of lengths $5 \text{ cm}$, $12 \text{ cm}$, and $20 \text{ cm}$ already placed on a table, find the number of the remaining rods that she can choose as the fourth rod to form a quadrilateral with positive area.
|
30
| 55.46875 |
18,774 |
A numerical sequence is defined by the conditions: \( a_{1} = 1 \), \( a_{n+1} = a_{n} + \left\lfloor \sqrt{a_{n}} \right\rfloor \).
How many perfect squares are there among the first terms of this sequence that do not exceed \( 1{,}000{,}000 \)?
|
10
| 98.4375 |
18,775 |
Given a quadratic function $f\left(x\right)=ax^{2}+bx+c$, where $f\left(0\right)=1$, $f\left(1\right)=0$, and $f\left(x\right)\geqslant 0$ for all real numbers $x$. <br/>$(1)$ Find the analytical expression of the function $f\left(x\right)$; <br/>$(2)$ If the maximum value of the function $g\left(x\right)=f\left(x\right)+2\left(1-m\right)x$ over $x\in \left[-2,5\right]$ is $13$, determine the value of the real number $m$.
|
m = 2
| 73.4375 |
18,776 |
The relevant departments want to understand the popularization of knowledge about the prevention of H1N1 influenza in schools, so they designed a questionnaire with 10 questions and conducted a survey in various schools. Two classes, A and B, from a certain middle school were randomly selected, with 5 students from each class participating in the survey. The scores of the 5 students in class A were: 5, 8, 9, 9, 9; the scores of the 5 students in class B were: 6, 7, 8, 9, 10.
(Ⅰ) Please estimate which class, A or B, has more stable questionnaire scores;
(Ⅱ) If we consider the scores of the 5 students in class B as a population and use a simple random sampling method to draw a sample with a size of 2, calculate the probability that the absolute difference between the sample mean and the population mean is not less than 1.
|
\dfrac{2}{5}
| 4.6875 |
18,777 |
Given data: $2$, $5$, $7$, $9$, $11$, $8$, $7$, $8$, $10$, the $80$th percentile is ______.
|
10
| 61.71875 |
18,778 |
Given the function $f(x)=\sin (2x+ \frac {\pi}{3})$.
$(1)$ If $x\in(- \frac {\pi}{6},0]$, find the minimum value of $4f(x)+ \frac {1}{f(x)}$ and determine the value of $x$ at this point;
$(2)$ If $(a\in(- \frac {\pi}{2},0),f( \frac {a}{2}+ \frac {\pi}{3})= \frac { \sqrt {5}}{5})$, find the value of $f(a)$.
|
\frac {3 \sqrt {3}-4}{10}
| 0 |
18,779 |
Given the function g(n) = log<sub>27</sub>n if log<sub>27</sub>n is rational, and 0 otherwise, find the value of the sum from n=1 to 7290 of g(n).
|
12
| 32.03125 |
18,780 |
The Greater Fourteen Basketball League has two divisions, each containing seven teams. Each team plays each of the other teams in its own division twice and every team in the other division twice. How many league games are scheduled?
|
182
| 79.6875 |
18,781 |
Given that six balls are numbered 1, 2, 3, 4, 5, and 6, and the requirement that ball number 1 must be adjacent to ball number 2, and ball number 5 must not be adjacent to ball number 6, calculate the total number of different arrangements.
|
144
| 77.34375 |
18,782 |
Given the function $f(x)=2\ln (3x)+8x+1$, find the value of $\underset{\Delta x\to 0}{{\lim }}\,\dfrac{f(1-2\Delta x)-f(1)}{\Delta x}$.
|
-20
| 89.0625 |
18,783 |
Find the value of $x$ such that it is the mean, median, and mode of the $9$ data values $-10, -5, x, x, 0, 15, 20, 25, 30$.
A) $\frac{70}{7}$
B) $\frac{75}{7}$
C) $\frac{80}{7}$
D) $10$
E) $15$
|
\frac{75}{7}
| 1.5625 |
18,784 |
The coefficient of $x^2$ in the expansion of $(1+2x)^5$ is __________.
|
40
| 96.875 |
18,785 |
Given the function \( y = \frac{1}{2}\left(x^{2}-100x+196+\left|x^{2}-100x+196\right|\right) \), what is the sum of the function values when the variable \( x \) takes on the 100 natural numbers \( 1, 2, 3, \ldots, 100 \)?
|
390
| 35.15625 |
18,786 |
Determine the constant term in the expansion of $(x-2+ \frac {1}{x})^{4}$.
|
70
| 53.90625 |
18,787 |
The square \( STUV \) is formed by a square bounded by 4 equal rectangles. The perimeter of each rectangle is \( 40 \text{ cm} \). What is the area, in \( \text{cm}^2 \), of the square \( STUV \)?
(a) 400
(b) 200
(c) 160
(d) 100
(e) 80
|
400
| 49.21875 |
18,788 |
In $\triangle ABC$, $A=30^{\circ}$, $2 \overrightarrow{AB}\cdot \overrightarrow{AC}=3 \overrightarrow{BC}^{2}$, find the cosine value of the largest angle in $\triangle ABC$.
|
-\frac{1}{2}
| 44.53125 |
18,789 |
Suppose $a_1, a_2, a_3, \dots$ is an increasing arithmetic progression of positive integers. Given that $a_3 = 13$ , compute the maximum possible value of \[ a_{a_1} + a_{a_2} + a_{a_3} + a_{a_4} + a_{a_5}. \]*Proposed by Evan Chen*
|
365
| 57.03125 |
18,790 |
The rapid growth of the Wuhan Economic and Technological Development Zone's economy cannot be separated from the support of the industrial industry. In 2022, the total industrial output value of the entire area reached 3462.23 billion yuan, ranking first in the city. Express the number 3462.23 in scientific notation as ______.
|
3.46223 \times 10^{3}
| 0 |
18,791 |
The parabola with the equation \( y = 25 - x^2 \) intersects the \( x \)-axis at points \( A \) and \( B \).
(a) Determine the length of \( AB \).
(b) Rectangle \( ABCD \) is formed such that \( C \) and \( D \) are below the \( x \)-axis and \( BD = 26 \). Determine the length of \( BC \).
(c) If \( CD \) is extended in both directions, it meets the parabola at points \( E \) and \( F \). Determine the length of \( EF \).
|
14
| 21.09375 |
18,792 |
Al-Karhi's rule for approximating the square root. If \(a^{2}\) is the largest square contained in the given number \(N\), and \(r\) is the remainder, then
$$
\sqrt{N}=\sqrt{a^{2}+r}=a+\frac{r}{2a+1}, \text{ if } r<2a+1
$$
Explain how Al-Karhi might have derived this rule. Estimate the error by calculating \(\sqrt{415}\) in the usual way to an accuracy of \(0.001\).
|
20.366
| 66.40625 |
18,793 |
Calculate the value of the expression \[(5^{1003}+6^{1004})^2-(5^{1003}-6^{1004})^2\] and express it in the form of $k\cdot30^{1003}$ for some positive integer $k$.
|
24
| 92.1875 |
18,794 |
Let $a$ and $b$ be integers such that $ab = 144.$ Find the minimum value of $a + b.$
|
-145
| 92.96875 |
18,795 |
Simplify first, then evaluate: $(1-\frac{1}{a+1})÷\frac{a^2-2a+1}{a^2-1}$, where $a=-2$.
|
\frac{2}{3}
| 71.09375 |
18,796 |
Let $\mathcal{P}$ be the parabola in the plane determined by the equation $y = 2x^2$. Suppose a circle $\mathcal{C}$ intersects $\mathcal{P}$ at four distinct points. If three of these points are $(-7, 98)$, $(-1, 2)$, and $(6, 72)$, find the sum of the distances from the vertex of $\mathcal{P}$ to all four intersection points.
|
\sqrt{9653} + \sqrt{5} + \sqrt{5220} + \sqrt{68}
| 14.84375 |
18,797 |
What is the sum of the odd positive integers less than 50?
|
625
| 100 |
18,798 |
Given the function $f(x)= \begin{cases}x-1,0 < x\leqslant 2 \\ -1,-2\leqslant x\leqslant 0 \end{cases}$, and $g(x)=f(x)+ax$, where $x\in[-2,2]$, if $g(x)$ is an even function, find the value of the real number $a$.
|
-\dfrac{1}{2}
| 92.1875 |
18,799 |
Calculate:<br/>$(1)2(\sqrt{3}-\sqrt{5})+3(\sqrt{3}+\sqrt{5})$;<br/>$(2)-{1}^{2}-|1-\sqrt{3}|+\sqrt[3]{8}-(-3)×\sqrt{9}$.
|
11 - \sqrt{3}
| 91.40625 |
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