Unnamed: 0
int64 0
40.3k
| problem
stringlengths 10
5.15k
| ground_truth
stringlengths 1
1.22k
| solved_percentage
float64 0
100
|
|---|---|---|---|
20,000 |
A right triangular prism $ABC-A_{1}B_{1}C_{1}$ has all its vertices on the surface of a sphere. Given that $AB=3$, $AC=5$, $BC=7$, and $AA_{1}=2$, find the surface area of the sphere.
|
\frac{208\pi}{3}
| 60.15625 |
20,001 |
Let $a$ and $b$ be even integers such that $ab = 144$. Find the minimum value of $a + b$.
|
-74
| 3.90625 |
20,002 |
What common fraction (that is, a fraction reduced to its lowest terms) is equivalent to $0.6\overline{41}$?
|
\frac{127}{198}
| 69.53125 |
20,003 |
A right triangle has sides of lengths 5 cm and 11 cm. Calculate the length of the remaining side if the side of length 5 cm is a leg of the triangle. Provide your answer as an exact value and as a decimal rounded to two decimal places.
|
9.80
| 2.34375 |
20,004 |
In the sequence ${a_{n}}$, $a_{n+1}=\begin{cases} 2a_{n}\left(a_{n} < \frac{1}{2}\right) \\ 2a_{n}-1\left(a_{n}\geqslant \frac{1}{2}\right) \end{cases}$, if $a_{1}=\frac{4}{5}$, then the value of $a_{20}$ is $\_\_\_\_\_\_$.
|
\frac{2}{5}
| 13.28125 |
20,005 |
A spiral staircase turns $180^\circ$ as it rises 8 feet. The radius of the staircase is 4 feet. What is the number of feet in the length of the handrail? Express your answer as a decimal to the nearest tenth.
|
14.9
| 4.6875 |
20,006 |
Given that $F_1$ and $F_2$ are the left and right foci of the ellipse $E$: $x^2 + \frac{y^2}{b^2} = 1 (0 < b < 1)$, and the line $l$ passing through $F_1$ intersects $E$ at points $A$ and $B$. If the sequence $|AF_2|, |AB|, |BF_2|$ forms an arithmetic progression, then:
(1) Find $|AB|$;
(2) If the slope of line $l$ is $1$, find the value of $b$.
|
\frac{\sqrt{2}}{2}
| 29.6875 |
20,007 |
The fractional equation $\dfrac{x-5}{x+2}=\dfrac{m}{x+2}$ has a root, determine the value of $m$.
|
-7
| 7.03125 |
20,008 |
Given that the sum of the first $n$ terms of the arithmetic sequence ${a_n}$ is $S_n$, if $a_2=0$, $S_3+S_4=6$, then the value of $a_5+a_6$ is $\_\_\_\_\_\_$.
|
21
| 67.1875 |
20,009 |
What is the reciprocal of $\frac{3}{4} + \frac{4}{5}$?
A) $\frac{31}{20}$
B) $\frac{20}{31}$
C) $\frac{19}{20}$
D) $\frac{20}{19}$
|
\frac{20}{31}
| 100 |
20,010 |
A digital clock displays time in a 24-hour format (from 00:00 to 23:59). Find the largest possible sum of the digits in this time display.
|
19
| 17.96875 |
20,011 |
Given the point $P(-\sqrt{3}, y)$ is on the terminal side of angle $\alpha$ and $\sin\alpha = \frac{\sqrt{13}}{13}$, find the value of $y$.
|
\frac{1}{2}
| 90.625 |
20,012 |
You have a square with vertices at $(2,1)$, $(5,1)$, $(2,4)$, and $(5,4)$. A line joining $(2,1)$ and $(5,3)$ divides the square into two regions. What fraction of the area of the square is above this line?
|
\frac{2}{3}
| 35.9375 |
20,013 |
Someone collected data relating the average temperature x (℃) during the Spring Festival to the sales y (ten thousand yuan) of a certain heating product. The data pairs (x, y) are as follows: (-2, 20), (-3, 23), (-5, 27), (-6, 30). Based on the data, using linear regression, the linear regression equation between sales y and average temperature x is found to be $y=bx+a$ with a coefficient $b=-2.4$. Predict the sales amount when the average temperature is -8℃.
|
34.4
| 7.8125 |
20,014 |
Suppose \( g(x) \) is a rational function such that \( 4g\left(\frac{1}{x}\right) + \frac{3g(x)}{x} = x^3 \) for \( x \neq 0 \). Find \( g(-3) \).
|
-\frac{6565}{189}
| 36.71875 |
20,015 |
A line $l$ is tangent to the circle $x^{2}+y^{2}=1$ and the sum of its intercepts on the two coordinate axes is equal to $\sqrt{3}$. Find the area of the triangle formed by line $l$ and the two coordinate axes.
|
\frac{3}{2}
| 8.59375 |
20,016 |
Let's call a number palindromic if it reads the same left to right as it does right to left. For example, the number 12321 is palindromic.
a) Write down any five-digit palindromic number that is divisible by 5.
b) How many five-digit palindromic numbers are there that are divisible by 5?
|
100
| 90.625 |
20,017 |
In the complex plane, $z,$ $z^2,$ $z^3$ represent, in some order, three vertices of a non-degenerate equilateral triangle. Determine all possible perimeters of the triangle.
|
3\sqrt{3}
| 42.1875 |
20,018 |
4. Suppose that $\overline{A2021B}$ is a six-digit integer divisible by $9$ . Find the maximum possible value of $A \cdot B$ .
5. In an arbitrary triangle, two distinct segments are drawn from each vertex to the opposite side. What is the minimum possible number of intersection points between these segments?
6. Suppose that $a$ and $b$ are positive integers such that $\frac{a}{b-20}$ and $\frac{b+21}{a}$ are positive integers. Find the maximum possible value of $a + b$ .
|
143
| 0 |
20,019 |
In the Cartesian coordinate system $xOy$, a polar coordinate system is established with the origin $O$ as the pole and the positive half-axis of the x-axis as the polar axis. It is known that the point $P(\sqrt {2}, \frac {7\pi}{4})$ lies on the line $l: \rho\cos\theta +2\rho\sin\theta +a=0$ ($a\in\mathbb{R}$).
(Ⅰ) Find the Cartesian equation of line $l$.
(Ⅱ) If point $A$ lies on the line $l$, and point $B$ lies on the curve $C: \begin{cases} x=t \\ y=\frac{1}{4}t^2 \end{cases}$ (where $t$ is a parameter), find the minimum value of $|AB|$.
|
\frac{\sqrt{5}}{10}
| 43.75 |
20,020 |
A sign painter paints individual numerals for a row of 100 houses. The houses are numbered with consecutive integers from 1 to 100. How many 9s are painted, and what is the total count of all digits used in painting?
|
192
| 35.15625 |
20,021 |
Given a set of four-ordered real number pairs \((a, b, c, d)\), where \(a, b, c, d \in \{0, 1, 2, 3\}\) and \(a, b, c, d\) can be the same, calculate how many such pairs exist so that \(ad - bc\) is odd.
|
96
| 81.25 |
20,022 |
Let S<sub>n</sub> be the sum of the first n terms of the arithmetic sequence {a<sub>n</sub>}, given that a<sub>7</sub> = 5 and S<sub>5</sub> = -55.
1. Find S<sub>n</sub>.
2. Let b<sub>n</sub> = $$\frac {S_{n}}{n}$$, find the sum of the first 19 terms, T<sub>19</sub>, of the sequence { $$\frac {1}{b_{n}b_{n+1}}$$}.
|
-\frac {1}{19}
| 68.75 |
20,023 |
Let $P(x) = x^3 + ax^2 + bx + 1$ be a polynomial with real coefficients and three real roots $\rho_1$ , $\rho_2$ , $\rho_3$ such that $|\rho_1| < |\rho_2| < |\rho_3|$ . Let $A$ be the point where the graph of $P(x)$ intersects $yy'$ and the point $B(\rho_1, 0)$ , $C(\rho_2, 0)$ , $D(\rho_3, 0)$ . If the circumcircle of $\vartriangle ABD$ intersects $yy'$ for a second time at $E$ , find the minimum value of the length of the segment $EC$ and the polynomials for which this is attained.
*Brazitikos Silouanos, Greece*
|
\sqrt{2}
| 9.375 |
20,024 |
The distances from three points lying in a horizontal plane to the base of a television tower are 800 m, 700 m, and 500 m, respectively. From each of these three points, the tower is visible (from the base to the top) at certain angles, with the sum of these three angles being $90^{\circ}$.
A) Find the height of the television tower (in meters).
B) Round the answer to the nearest whole number of meters.
|
374
| 64.84375 |
20,025 |
Consider the system of equations
\[
8x - 6y = c,
\]
\[
12y - 18x = d.
\]
If this system has a solution \((x, y)\) where both \(x\) and \(y\) are nonzero, find the value of \(\frac{c}{d}\), assuming \(d\) is nonzero.
|
-\frac{4}{9}
| 19.53125 |
20,026 |
Given points $P(-2,-3)$ and $Q(5, 3)$ in the $xy$-plane; point $R(2,m)$ is taken so that $PR+RQ$ is minimized. Determine the value of $m$.
A) $\frac{3}{5}$
B) $\frac{2}{5}$
C) $\frac{3}{7}$
D) $\frac{1}{5}$
|
\frac{3}{7}
| 13.28125 |
20,027 |
Given the function \\(f(x) = x^2 + 2ax + 4\\) and the interval \\([-3,5]\\), calculate the probability that the function has no real roots.
|
\dfrac{1}{2}
| 99.21875 |
20,028 |
Let \( x \) be a real number with the property that \( x+\frac{1}{x} = 4 \). Define \( S_m = x^m + \frac{1}{x^m} \). Determine the value of \( S_6 \).
|
2702
| 82.03125 |
20,029 |
Let \( r, s, \) and \( t \) be the roots of the equation \( x^3 - 15x^2 + 13x - 6 = 0 \). Find the value of \( \frac{r}{\frac{1}{r}+st} + \frac{s}{\frac{1}{s}+tr} + \frac{t}{\frac{1}{t}+rs} \).
|
\frac{199}{7}
| 85.15625 |
20,030 |
Given $cos(x+\frac{π}{6})+sin(\frac{2π}{3}+x)=\frac{1}{2}$, calculate $sin(2x-\frac{π}{6})$.
|
\frac{7}{8}
| 25.78125 |
20,031 |
By using equations, recurring decimals can be converted into fractions. For example, when converting $0.\overline{3}$ into a fraction, we can let $0.\overline{3} = x$. From $0.\overline{3} = 0.333\ldots$, we know that $10x = 3.333\ldots$. Therefore, $10x = 3 + 0.\overline{3}$. So, $10x = 3 + x$. Solving this equation, we get $x = \frac{1}{3}$, which means $0.\overline{3} = \frac{1}{3}$.
$(1)$ Convert $0.\overline{4}\overline{5}$ into a fraction, fill in the blanks below:
Let $0.\overline{4}\overline{5} = x$. From $0.\overline{4}\overline{5} = 0.4545\ldots$, we have $100x = 45.4545\ldots$.
So, $100x = 45 + 0.\overline{4}\overline{5}$. Therefore, ______. Solving this equation, we get $x = \_\_\_\_\_\_$.
$(2)$ Convert $0.2\overline{4}\overline{5}$ into a fraction.
|
\frac{27}{110}
| 67.1875 |
20,032 |
From the $7$ integers from $2$ to $8$, randomly select $2$ different numbers. The probability that these $2$ numbers are coprime is ______.
|
\frac{2}{3}
| 38.28125 |
20,033 |
In the Cartesian coordinate plane $(xOy)$, two acute angles $\alpha$ and $\beta$ are formed with the non-negative semi-axis of $Ox$ as the initial side. Their terminal sides intersect the unit circle at points $A$ and $B$ respectively. The vertical coordinates of $A$ and $B$ are $\frac{\sqrt{5}}{5}$ and $\frac{3\sqrt{10}}{10}$ respectively.
1. Find $\alpha - \beta$.
2. Find the value of $\cos(2\alpha - \beta)$.
|
\frac{3\sqrt{10}}{10}
| 89.0625 |
20,034 |
The stem-and-leaf plot shows the duration of songs (in minutes and seconds) played during a concert by a band. There are 15 songs listed in the plot. In the stem-and-leaf plot, $3 \ 15$ represents $3$ minutes, $15$ seconds, which is the same as $195$ seconds. Find the median duration of the songs. Express your answer in seconds.
\begin{tabular}{c|ccccc}
1&30&45&50&&\\
2&10&20&30&35&50\\
3&00&15&15&30&45\\
\end{tabular}
|
170
| 64.0625 |
20,035 |
In triangle ABC, the sides opposite to angles A, B, and C are a, b, and c respectively. Given that 2(tanA + tanB) = $\frac{\text{tanA}}{\text{cosB}} + \frac{\text{tanB}}{\text{cosA}}$.
(1) Find the value of $\frac{a+b}{c}$;
(2) If c = 2 and C = $\frac{\pi}{3}$, find the area of triangle ABC.
|
\sqrt{3}
| 87.5 |
20,036 |
Given \(0 \leq x_0 < 1\), let
\[
x_n = \left\{ \begin{array}{ll}
2x_{n-1} & \text{if } 2x_{n-1} < 1 \\
2x_{n-1} - 1 & \text{if } 2x_{n-1} \geq 1
\end{array} \right.
\]
for all integers \(n > 0\). Determine the number of initial values of \(x_0\) that satisfy \(x_0 = x_6\).
|
64
| 27.34375 |
20,037 |
Find the ratio of $AE:EC$ in $\triangle ABC$ given that $AB=6$, $BC=8$, $AC=10$, and $E$ is on $\overline{AC}$ with $BE=6$.
|
\frac{18}{7}
| 29.6875 |
20,038 |
The perimeter of triangle \(ABC\) is 1. Circle \(\omega\) is tangent to side \(BC\), the extension of side \(AB\) at point \(P\), and the extension of side \(AC\) at point \(Q\). A line passing through the midpoints of \(AB\) and \(AC\) intersects the circumcircle of triangle \(APQ\) at points \(X\) and \(Y\). Find the length of segment \(XY\).
|
0.5
| 60.9375 |
20,039 |
In a country there are $15$ cities, some pairs of which are connected by a single two-way airline of a company. There are $3$ companies and if any of them cancels all its flights, then it would still be possible to reach every city from every other city using the other two companies. At least how many two-way airlines are there?
|
21
| 13.28125 |
20,040 |
Express $361_9 + 4C5_{13}$ as a base 10 integer, where $C$ denotes the digit whose value is 12 in base 13.
|
1135
| 23.4375 |
20,041 |
The bases of an isosceles trapezoid are in the ratio 3:2. A circle is constructed on the larger base as its diameter, and this circle intersects the smaller base such that the segment cut off on the smaller base is equal to half of the smaller base. In what ratio does the circle divide the non-parallel sides of the trapezoid?
|
1:2
| 8.59375 |
20,042 |
Calculate both the product and the sum of the least common multiple (LCM) and the greatest common divisor (GCD) of $12$ and $15$.
|
63
| 69.53125 |
20,043 |
In triangle ABC, the angles A, B, and C are represented by vectors AB and BC with an angle θ between them. Given that the dot product of AB and BC is 6, and that $6(2-\sqrt{3})\leq|\overrightarrow{AB}||\overrightarrow{BC}|\sin(\pi-\theta)\leq6\sqrt{3}$.
(I) Find the value of $\tan 15^\circ$ and the range of values for θ.
(II) Find the maximum value of the function $f(\theta)=\frac{1-\sqrt{2}\cos(2\theta-\frac{\pi}{4})}{\sin\theta}$.
|
\sqrt{3}-1
| 35.15625 |
20,044 |
Given regular hexagon $ABCDEF$ , compute the probability that a randomly chosen point inside the hexagon is inside triangle $PQR$ , where $P$ is the midpoint of $AB$ , $Q$ is the midpoint of $CD$ , and $R$ is the midpoint of $EF$ .
|
\frac{3}{8}
| 0 |
20,045 |
Given that $\{a_{n}\}$ is a geometric sequence, and $a_{3}$ and $a_{7}$ are two roots of the equation $x^{2}+4x+1=0$, calculate the value of $a_{5}$.
|
-1
| 43.75 |
20,046 |
Given a function \\(f(x)\\) defined on \\(\mathbb{R}\\) that satisfies: the graph of \\(y=f(x-1)\\) is symmetric about the point \\((1,0)\\), and when \\(x \geqslant 0\\), it always holds that \\(f(x+2)=f(x)\\). When \\(x \in [0,2)\\), \\(f(x)=e^{x}-1\\), where \\(e\\) is the base of the natural logarithm, evaluate \\(f(2016)+f(-2017)\\).
|
1-e
| 71.875 |
20,047 |
Given that $|\overrightarrow{a}|=1$, $|\overrightarrow{b}|=\sqrt{2}$, and $\overrightarrow{a} \bot (\overrightarrow{a}-\overrightarrow{b})$, find the projection of vector $\overrightarrow{a}$ in the direction of vector $\overrightarrow{b}$.
|
\frac{\sqrt{2}}{2}
| 45.3125 |
20,048 |
A TV station broadcasts 5 advertisements in a row, among which there are 3 different commercial advertisements and 2 different World Expo promotional advertisements. The last advertisement broadcasted is a World Expo promotional advertisement, and the methods in which the 2 World Expo promotional advertisements are not broadcasted consecutively are $\boxed{36}$.
|
36
| 68.75 |
20,049 |
Evaluate the expression $3 - (-3)^{-\frac{2}{3}}$.
|
3 - \frac{1}{\sqrt[3]{9}}
| 22.65625 |
20,050 |
Given $x^{2}-5x-2006=0$, evaluate the algebraic expression $\dfrac {(x-2)^{3}-(x-1)^{2}+1}{x-2}$.
|
2010
| 62.5 |
20,051 |
In a certain region, the rate of taxation is half the amount of the income in thousands: that is, $\frac{x}{2}\%$ tax rate for an income of $x$ thousand dollars. What income, in dollars, will yield the highest take-home pay?
|
100000
| 87.5 |
20,052 |
There are 120 five-digit numbers formed by the digits 1, 2, 3, 4, 5, arranged in descending order. The 95th number is ______.
|
21354
| 38.28125 |
20,053 |
Given that the polar coordinate equation of curve C is ρ - 4cosθ = 0, establish a rectangular coordinate system with the pole as the origin and the polar axis as the positive semi-axis. Line l passes through point M(3, 0) with a slope angle of $\frac{\pi}{6}$.
(I) Find the rectangular coordinate equation of curve C and the parametric equation of line l;
(II) If line l intersects curve C at points A and B, find $\frac{1}{|MA|} + \frac{1}{|MB|}$.
|
\frac{\sqrt{15}}{3}
| 91.40625 |
20,054 |
The sequence $\{a_n\}$ satisfies $a_{n+1}+(-1)^n a_n = 2n-1$. Find the sum of the first $80$ terms of $\{a_n\}$.
|
3240
| 5.46875 |
20,055 |
Xiao Ming observed a faucet that was continuously dripping water due to damage. In order to investigate the waste caused by the water leakage, Xiao Ming placed a graduated cylinder under the faucet to collect water and recorded the total amount of water in the cylinder every minute. However, due to a delay in starting the timer, there was already a small amount of water in the cylinder at the beginning. Therefore, he obtained a set of data as shown in the table below:
| Time $t$ (minutes) | $1$ | $2$ | $3$ | $4$ | $5$ | $\ldots$ |
|--------------------|-----|-----|-----|-----|-----|---------|
| Total water amount $y$ (milliliters) | $7$ | $12$ | $17$ | $22$ | $27$ | $\ldots$ |
$(1)$ Investigation: Based on the data in the table above, determine the function relationship between the total water amount $y$ and time $t$. Find the expression of $y$ in terms of $t$.
$(2)$ Application:
1. Estimate the total water amount in the cylinder when Xiao Ming measures it at the $20$th minute.
2. A person drinks approximately $1500$ milliliters of water per day. Estimate how many days the water leaked from this faucet in a month (assuming $30$ days) can supply drinking water for one person.
|
144
| 13.28125 |
20,056 |
An entire floor is tiled with blue and white tiles. The floor has a repeated tiling pattern that forms every $8 \times 8$ square. Each of the four corners of this square features an asymmetrical arrangement of tiles where the bottom left $4 \times 4$ segment within each $8 \times 8$ square consists of blue tiles except for a $2 \times 2$ section of white tiles at its center. What fraction of the floor is made up of blue tiles?
A) $\frac{1}{2}$
B) $\frac{5}{8}$
C) $\frac{3}{4}$
D) $\frac{7}{8}$
E) $\frac{2}{3}$
|
\frac{3}{4}
| 4.6875 |
20,057 |
The numbers 2, 3, 5, 7, 11, 13, 17, 19 are arranged in a multiplication table, with four along the top and the other four down the left. The multiplication table is completed and the sum of the sixteen entries is tabulated. What is the largest possible sum of the sixteen entries?
\[
\begin{array}{c||c|c|c|c|}
\times & a & b & c & d \\ \hline \hline
e & & & & \\ \hline
f & & & & \\ \hline
g & & & & \\ \hline
h & & & & \\ \hline
\end{array}
\]
|
1482
| 16.40625 |
20,058 |
Given the function $f(x) = \sqrt{3}\sin 2x - 2\cos^2x + 1$.
(1) Find the range of the function $f(x)$ in the interval $\left[-\frac{\pi}{12}, \frac{\pi}{2}\right)$.
(2) Let $\alpha, \beta \in \left(0, \frac{\pi}{2}\right)$, $f\left(\frac{1}{2}\alpha + \frac{\pi}{12}\right) = \frac{10}{13}$, $f\left(\frac{1}{2}\beta + \frac{\pi}{3}\right) = \frac{6}{5}$. Find the value of $\sin(\alpha - \beta)$.
|
-\frac{33}{65}
| 87.5 |
20,059 |
The number $24!$ has many positive integer divisors. What is the probability that a divisor randomly chosen from these is odd?
|
\frac{1}{23}
| 71.09375 |
20,060 |
What is the constant term of the expansion of $\left(5x + \dfrac{1}{3x}\right)^8$?
|
\frac{43750}{81}
| 84.375 |
20,061 |
Let \\( \{a_n\} \\) be a sequence with the sum of the first \\( n \\) terms denoted as \\( S_n \\). If \\( S_2 = 4 \\) and \\( a_{n+1} = 2S_n + 1 \\) where \\( n \in \mathbb{N}^* \\), find the values of \\( a_1 \\) and \\( S_5 \\).
|
121
| 65.625 |
20,062 |
Compute $({11011_{(2)}} - {101_{(2)}} = )$\_\_\_\_\_\_\_\_\_\_$(.$ (represented in binary)
|
10110_{(2)}
| 33.59375 |
20,063 |
Convert $BD4_{16}$ to base 4.
|
233110_4
| 38.28125 |
20,064 |
A $6$ -inch-wide rectangle is rotated $90$ degrees about one of its corners, sweeping out an area of $45\pi$ square inches, excluding the area enclosed by the rectangle in its starting position. Find the rectangle’s length in inches.
|
12
| 26.5625 |
20,065 |
Given a random variable $X\sim N(2, \sigma ^{2})$, $P(X\leqslant 0)=0.15$, calculate $P(2\leqslant X\leqslant 4)$.
|
0.35
| 80.46875 |
20,066 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $a^{2}+b^{2}+4 \sqrt {2}=c^{2}$ and $ab=4$, find the minimum value of $\frac {\sin C}{\tan ^{2}A\cdot \sin 2B}$.
|
\frac {3 \sqrt {2}}{2}+2
| 0 |
20,067 |
Inside an isosceles triangle $\mathrm{ABC}$ with equal sides $\mathrm{AB} = \mathrm{BC}$ and an angle of 80 degrees at vertex $\mathrm{B}$, a point $\mathrm{M}$ is taken such that the angle $\mathrm{MAC}$ is 10 degrees and the angle $\mathrm{MCA}$ is 30 degrees. Find the measure of the angle $\mathrm{AMB}$.
|
70
| 76.5625 |
20,068 |
Given real numbers \(a, b, c\), the polynomial
$$
g(x) = x^{3} + a x^{2} + x + 10
$$
has three distinct roots, and these three roots are also roots of the polynomial
$$
f(x) = x^{4} + x^{3} + b x^{2} + 100 x + c.
$$
Find the value of \(f(1)\).
|
-7007
| 60.9375 |
20,069 |
(1) Use the Euclidean algorithm to find the greatest common divisor (GCD) of 117 and 182, and verify it using the method of successive subtraction.
(2) Use the Horner's method to calculate the value of the polynomial \\(f(x)=1-9x+8x^{2}-4x^{4}+5x^{5}+3x^{6}\\) at \\(x=-1\\).
|
12
| 67.1875 |
20,070 |
Given the function $f(x)=|2x-9|-|x-5|$.<br/>$(1)$ Find the solution set of the inequality $f(x)\geqslant 2x-1$;<br/>$(2)$ The minimum value of the function $y=f(x)+3|x-5|$ is $m$. For positive real numbers $a$ and $b$ satisfying $\frac{1}{a}+\frac{3}{b}=m$, find the minimum value of $a+3b$.
|
16
| 78.125 |
20,071 |
If \(a\), \(b\), and \(c\) are distinct positive integers such that \(abc = 16\), then the largest possible value of \(a^b - b^c + c^a\) is:
|
263
| 37.5 |
20,072 |
Given that the sequence $\{a\_n\}$ satisfies $\frac{1}{a_{n+1}} - \frac{1}{a_n} = d (n \in \mathbb{N}^*, d$ is a constant), the sequence $\{\frac{1}{b_n}\}$ is a harmonic sequence, and $b_1 + b_2 + b_3 + ... + b_9 = 90$, find the value of $b_4 + b_6$.
|
20
| 85.9375 |
20,073 |
If the eccentricity of the conic section \(C\): \(x^{2}+my^{2}=1\) is \(2\), determine the value of \(m\).
|
-\dfrac {1}{3}
| 83.59375 |
20,074 |
A school selects 4 teachers from 8 to teach in 4 remote areas, with one teacher per area. Among them, A and B cannot go together, and A and C must either both go or both not go. Derive the total number of different dispatch plans.
|
600
| 48.4375 |
20,075 |
Determine the number of six-letter words where the first and last letters are the same, and the second and fifth letters are also the same.
|
456976
| 60.15625 |
20,076 |
Given that $|\overrightarrow{a}|=4, |\overrightarrow{b}|=8$, and the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is $\frac{2\pi}{3}$.
(I) Find $|\overrightarrow{a}+\overrightarrow{b}|$;
(II) Find the value of $k$ such that $(\overrightarrow{a}+2\overrightarrow{b}) \perp (k\overrightarrow{a}-\overrightarrow{b})$.
|
-7
| 78.125 |
20,077 |
How many positive, three-digit integers contain at least one $4$ as a digit but do not contain a $6$ as a digit?
|
200
| 75 |
20,078 |
Given the line $y=kx+b$ is a tangent to the curve $f\left(x\right)=\ln x+2$ and also a tangent to the curve $g\left(x\right)=\ln \left(x+1\right)$, determine the value of $k-b$.
|
1 + \ln 2
| 75 |
20,079 |
Find the smallest possible value of \(x\) in the simplified form \(x=\frac{a+b\sqrt{c}}{d}\) if \(\frac{7x}{8}-1=\frac{4}{x}\), where \(a, b, c,\) and \(d\) are integers. What is \(\frac{acd}{b}\)?
|
-105
| 82.03125 |
20,080 |
Given a point $P(x,y)$ moving on the ellipse $\frac{x^{2}}{4} + \frac{y^{2}}{3} = 1$, let $d = \sqrt{x^{2} + y^{2} + 4y + 4} - \frac{x}{2}$. Find the minimum value of $d$.
A) $\sqrt{5} - 2$
B) $2\sqrt{2} - 1$
C) $\sqrt{5} - 1$
D) $\sqrt{6} - 1$
|
2\sqrt{2} - 1
| 7.03125 |
20,081 |
In a large box of ribbons, $\frac{1}{4}$ are yellow, $\frac{1}{3}$ are purple, $\frac{1}{6}$ are orange, and the remaining 40 ribbons are black. How many of the ribbons are purple?
|
53
| 75 |
20,082 |
In the Cartesian coordinate system, it is known that the terminal side of an angle $\alpha$ with the origin as the vertex and the non-negative half-axis of the $x$-axis as the initial side passes through the point $(-3,-4)$.
$(1)$ Find the value of $\frac{sin\alpha}{tan\alpha}$;
$(2)$ Find the value of $\frac{sin(\alpha+\frac{\pi}{2})\cdot cos(\frac{9\pi}{2}-\alpha)\cdot tan(2\pi-\alpha)\cdot cos(-\frac{3\pi}{2}+\alpha)}{sin(2\pi-\alpha)\cdot tan(-\alpha-\pi)\cdot sin(\pi+\alpha)}$.
|
\frac{3}{5}
| 45.3125 |
20,083 |
The graph of the function $f(x)=3 \sqrt {2}\cos (x+\varphi)+\sin x$, where $x\in \mathbb{R}$ and $\varphi\in\left(- \frac {\pi}{2}, \frac {\pi}{2}\right)$, passes through the point $\left( \frac {\pi}{2},4\right)$. Find the minimum value of $f(x)$.
|
-5
| 87.5 |
20,084 |
Given the polar equation of a line is $ρ\sin(θ+ \frac{π}{4})= \frac{\sqrt{2}}{2}$, and the parametric equation of the circle $M$ is $\begin{cases} x = 2\cosθ \\ y = -2 + 2\sinθ \end{cases}$, where $θ$ is the parameter.
(I) Convert the line's polar equation into a Cartesian coordinate equation;
(II) Determine the minimum distance from a point on the circle $M$ to the line.
|
\frac{3\sqrt{2}}{2} - 2
| 13.28125 |
20,085 |
$P(x)=ax^2+bx+c$ has exactly $1$ different real root where $a,b,c$ are real numbers. If $P(P(P(x)))$ has exactly $3$ different real roots, what is the minimum possible value of $abc$ ?
|
-2
| 28.90625 |
20,086 |
Tsrutsuna starts in the bottom left cell of a 7 × 7 square table, while Tsuna is in the upper right cell. The center cell of the table contains cheese. Tsrutsuna wants to reach Tsuna and bring a piece of cheese with him. From a cell Tsrutsuna can only move to the right or the top neighboring cell. Determine the number of different paths Tsrutsuna can take from the lower left cell to the upper right cell, such that he passes through the center cell.
*Proposed by Giorgi Arabidze, Georgia*
|
400
| 85.15625 |
20,087 |
Let $m$ be a real number where $m > 0$. If for any $x \in (1, +\infty)$, the inequality $2e^{2mx} - \frac{ln x}{m} ≥ 0$ always holds, then find the minimum value of the real number $m$.
|
\frac{1}{2e}
| 17.96875 |
20,088 |
What is the greatest integer less than 150 for which the greatest common divisor of that integer and 18 is 6?
|
138
| 85.9375 |
20,089 |
Given that the three sides of triangle $\triangle ABC$ are $a$, $a+3$, and $a+6$, and the largest angle is twice the smallest angle, calculate the cosine value of the smallest angle.
|
\frac{3}{4}
| 27.34375 |
20,090 |
Let $a=x^3-3x^2$, then the coefficient of the $x^2$ term in the expansion of $(a-x)^6$ is $\boxed{-192}$.
|
-192
| 69.53125 |
20,091 |
In the rectangular coordinate system $(xOy)$, the parametric equation of line $l$ is given by $ \begin{cases} x=-\frac{1}{2}t \\ y=2+\frac{\sqrt{3}}{2}t \end{cases} (t\text{ is the parameter})$, and a circle $C$ with polar coordinate equation $\rho=4\cos\theta$ is established with the origin $O$ as the pole and the positive half of the $x$-axis as the polar axis. Let $M$ be any point on circle $C$, and connect $OM$ and extend it to $Q$ such that $|OM|=|MQ|$.
(I) Find the rectangular coordinate equation of the trajectory of point $Q$;
(II) If line $l$ intersects the trajectory of point $Q$ at points $A$ and $B$, and the rectangular coordinates of point $P$ are $(0,2)$, find the value of $|PA|+|PB|$.
|
4+2\sqrt{3}
| 28.90625 |
20,092 |
Johny's father tells him: "I am twice as old as you will be seven years from the time I was thrice as old as you were". What is Johny's age?
|
14
| 27.34375 |
20,093 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. Given vectors $\overrightarrow{m} = (b+c, a^2 + bc)$ and $\overrightarrow{n} = (b+c, -1)$ with $\overrightarrow{m} \cdot \overrightarrow{n} = 0$.
(1) Find the size of angle $A$;
(2) If $a = \sqrt{3}$, find the maximum area of $\triangle ABC$.
|
\frac{\sqrt{3}}{4}
| 87.5 |
20,094 |
When the set of natural numbers is listed in ascending order, what is the smallest prime number that occurs after a sequence of six consecutive positive integers, all of which are nonprime?
|
37
| 0 |
20,095 |
Given that $({x}^{2}+1){(2x+1)}^{9}={a}_{0}+{a}_{1}(x+2)+{a}_{2}{(x+2)}^{2}+\cdots +{a}_{11}{(x+2)}^{11}$, find the value of $({a}_{0}+{a}_{1}+{a}_{2}+\cdots +{a}_{11})$.
|
-2
| 90.625 |
20,096 |
A regular hexagon has an area of $150\sqrt{3}$ cm². If each side of the hexagon is decreased by 3 cm, by how many square centimeters is the area decreased?
|
76.5\sqrt{3}
| 8.59375 |
20,097 |
Given the quadratic function \( y = x^{2} - \frac{2n+1}{n(n+2)} x + \frac{n+1}{n(n+2)^{2}} \), the length of the segment intercepted on the \( x \)-axis is \( d_n \). Find the value of \( \sum_{n=1}^{100} d_n \).
|
\frac{7625}{10302}
| 6.25 |
20,098 |
Given that $α \in (0, \frac{π}{3})$ satisfies the equation $\sqrt{6} \sin α + \sqrt{2} \cos α = \sqrt{3}$, find the values of:
1. $\cos (α + \frac{π}{6})$
2. $\cos (2α + \frac{π}{12})$
|
\frac{\sqrt{30} + \sqrt{2}}{8}
| 0 |
20,099 |
Choose $4-4$: Parameter Equation Lecture. In the plane rectangular coordinate system $xOy$, with $O$ as the pole and the non-negative half-axis of $x$ as the polar axis, establish a polar coordinate system. The polar coordinates of point $P$ are $(2\sqrt{3}, \dfrac{\pi}{6})$. The polar coordinate equation of curve $C$ is $\rho ^{2}+2\sqrt{3}\rho \sin \theta = 1$.
(Ⅰ) Write down the rectangular coordinates of point $P$ and the general equation of curve $C.
(Ⅱ) If $Q$ is a moving point on $C$, find the minimum value of the distance from the midpoint $M$ of $PQ$ to the line $l: \left\{\begin{array}{l}x=3+2t\\y=-2+t\end{array}\right.$ (where $t$ is a parameter).
|
\dfrac{11\sqrt{5}}{10} - 1
| 3.125 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.