Unnamed: 0
int64 0
40.3k
| problem
stringlengths 10
5.15k
| ground_truth
stringlengths 1
1.22k
| solved_percentage
float64 0
100
|
|---|---|---|---|
17,000 |
If the moving point $P$ is on the line $y=x+1$, and the moving point $Q$ is on the curve $x^{2}=-2y$, calculate the minimum value of $|PQ|$.
|
\frac{\sqrt{2}}{4}
| 1.5625 |
17,001 |
A solid is formed by rotating a triangle with sides of lengths 3, 4, and 5 around the line containing its shortest side. Find the surface area of this solid.
|
36\pi
| 42.1875 |
17,002 |
Let $P(x) = b_0 + b_1x + b_2x^2 + \dots + b_mx^m$ be a polynomial with integer coefficients, where $0 \le b_i < 5$ for all $0 \le i \le m$.
Given that $P(\sqrt{5})=23+19\sqrt{5}$, compute $P(3)$.
|
132
| 59.375 |
17,003 |
Calculate \(14 \cdot 31\) and \(\left\lfloor\frac{2+\sqrt{2}}{2}\right\rfloor + \left\lfloor\frac{3+\sqrt{3}}{3}\right\rfloor + \left\lfloor\frac{4+\sqrt{4}}{4}\right\rfloor + \cdots + \left\lfloor\frac{1989+\sqrt{1989}}{1989}\right\rfloor + \left\lfloor\frac{1990+\sqrt{1990}}{1990}\right\rfloor\).
|
1989
| 0.78125 |
17,004 |
How many ways are there to put 7 balls in 2 boxes if the balls are distinguishable but the boxes are not?
|
64
| 73.4375 |
17,005 |
In $\triangle ABC$, $b^{2}=ac$, and $a+c=3$, $\cos B= \frac{3}{4}$, then $\overset{→}{AB} \cdot \overset{→}{BC} =$______.
|
-\frac{3}{2}
| 91.40625 |
17,006 |
Tim plans a weeklong prank to repeatedly steal Nathan's fork during lunch. He involves different people each day:
- On Monday, he convinces Joe to do it.
- On Tuesday, either Betty or John could undertake the prank.
- On Wednesday, there are only three friends from whom he can seek help, as Joe, Betty, and John are not available.
- On Thursday, neither those involved earlier in the week nor Wednesday's helpers are willing to participate, but four new individuals are ready to help.
- On Friday, Tim decides he could either do it himself or get help from one previous assistant who has volunteered again.
How many different combinations of people could be involved in the prank over the week?
|
48
| 14.0625 |
17,007 |
What is the smallest positive integer $n$ such that $\frac{n}{n+103}$ is equal to a terminating decimal?
|
22
| 54.6875 |
17,008 |
Let \(a\), \(b\), \(c\), and \(d\) be positive integers with \(a < 3b\), \(b < 4c\), and \(c < 5d\). If \(d < 80\), find the largest possible value for \(a\).
|
4724
| 91.40625 |
17,009 |
If for any \( x \in \mathbf{R} \), the function \( f(x) \) satisfies the equation \( f(x+2009) = -f(x+2008) \), and \( f(2009) = -2009 \), determine the value of \( f(-1) \).
|
-2009
| 77.34375 |
17,010 |
A road of 1500 meters is being repaired. In the first week, $\frac{5}{17}$ of the total work was completed, and in the second week, $\frac{4}{17}$ was completed. What fraction of the total work was completed in these two weeks? And what fraction remains to complete the entire task?
|
\frac{8}{17}
| 94.53125 |
17,011 |
Real numbers between 0 and 1, inclusive, are chosen based on the outcome of flipping two fair coins. If two heads are flipped, then the chosen number is 0; if a head and a tail are flipped (in any order), the number is 0.5; if two tails are flipped, the number is 1. Another number is chosen independently in the same manner. Calculate the probability that the absolute difference between these two numbers, x and y, is greater than $\frac{1}{2}$.
|
\frac{1}{8}
| 15.625 |
17,012 |
Given functions $f(x)=xe^x$ and $g(x)=-\frac{lnx}{x}$, if $f(x_{1})=g(x_{2})=t\left( \gt 0\right)$, find the maximum value of $\frac{{x}_{1}}{{x}_{2}{e}^{t}}$.
|
\frac{1}{e}
| 50 |
17,013 |
Given a hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ ($a>0, b>0$) with a point C on it, a line passing through the center of the hyperbola intersects the hyperbola at points A and B. Let the slopes of the lines AC and BC be $k_1$ and $k_2$ respectively. Find the eccentricity of the hyperbola when $\frac{2}{k_1 k_2} + \ln{k_1} + \ln{k_2}$ is minimized.
|
\sqrt{3}
| 54.6875 |
17,014 |
Given a sequence of positive terms $\{a\_n\}$, with $a\_1=2$, $(a\_n+1)a_{n+2}=1$, and $a\_2=a\_6$, find the value of $a_{11}+a_{12}$.
|
\frac{1}{9}+\frac{\sqrt{5}}{2}
| 2.34375 |
17,015 |
Subtract $123.45$ from $567.89.$ Express the result as a decimal to the nearest hundredth.
|
444.44
| 92.96875 |
17,016 |
The value of \((4 + 44 + 444) \div 4\) is:
|
123
| 47.65625 |
17,017 |
In the sequence $\{a_n\}$, if $a_1=-2$ and for any $n\in\mathbb{N}^*$, $a_{n+1}=1+2a_n$, then the sum of the first $10$ terms of the sequence $\{a_n\}$ is ______.
|
-1033
| 39.84375 |
17,018 |
A class has prepared 6 programs to participate in the Xiamen No.1 Middle School Music Square event. The order of the programs has the following requirements: Programs A and B must be adjacent, and Programs C and D cannot be adjacent. How many possible arrangements of the program order are there for this event?
|
144
| 85.15625 |
17,019 |
5 students stand in a row for a photo, where students A and B must stand next to each other, and A cannot stand at either end. Calculate the total number of possible arrangements.
|
36
| 48.4375 |
17,020 |
Lead used to make twelve solid lead balls, each with a radius of 2 cm, is reused to form a single larger solid lead sphere. What is the radius of this larger sphere?
|
\sqrt[3]{96}
| 10.9375 |
17,021 |
Given $\sin\alpha= \frac {2 \sqrt {2}}{3}$, $\cos(\alpha+\beta)=- \frac {1}{3}$, and $\alpha, \beta\in(0, \frac {\pi}{2})$, determine the value of $\sin(\alpha-\beta)$.
|
\frac {10 \sqrt {2}}{27}
| 0 |
17,022 |
Cylinder $C$'s height is equal to the diameter of cylinder $D$ and cylinder $C$'s diameter is equal to the height $h$ of cylinder $D$. If the volume of cylinder $D$ is three times the volume of cylinder $C$, the volume of cylinder $D$ can be written as $M \pi h^3$ cubic units. Find the value of $M$.
|
\frac{9}{4}
| 46.09375 |
17,023 |
Find
\[
\cos \left( 4 \arccos \frac{2}{5} \right).
\]
|
-\frac{47}{625}
| 94.53125 |
17,024 |
Triangle $\triangle ABC$ has a right angle at $C$, $\angle A = 45^\circ$, and $AC=12$. Find the radius of the incircle of $\triangle ABC$.
|
6 - 3\sqrt{2}
| 0 |
17,025 |
In an opaque bag, there are four identical balls labeled with numbers $3$, $4$, $5$, and $6$ respectively. Outside the bag, there are two balls labeled with numbers $3$ and $6$. Determine the probability that a triangle with the drawn ball and the numbers on the two balls outside the bag forms an isosceles triangle.
|
\frac{1}{4}
| 69.53125 |
17,026 |
Simplify and write the result as a common fraction: $$\sqrt[4]{\sqrt[3]{\sqrt{\frac{1}{65536}}}}$$
|
\frac{1}{\sqrt[3]{4}}
| 33.59375 |
17,027 |
If the vector $\overrightarrow{a} = (x, y-1)$ is collinear with the vector $\overrightarrow{b} = (3, -2)$, then the minimum value of $z = \log_{2}(4^x + 8^y)$ is ______.
|
\frac{5}{2}
| 72.65625 |
17,028 |
Given two fixed points on the plane, \\(A(-2,0)\\) and \\(B(2,0)\\), and a moving point \\(T\\) satisfying \\(|TA|+|TB|=2 \sqrt {6}\\).
\\((\\)I\\()\\) Find the equation of the trajectory \\(E\\) of point \\(T\\);
\\((\\)II\\()\\) A line passing through point \\(B\\) and having the equation \\(y=k(x-2)\\) intersects the trajectory \\(E\\) at points \\(P\\) and \\(Q\\) \\((k\neq 0)\\). If \\(PQ\\)'s midpoint is \\(N\\) and \\(O\\) is the origin, the line \\(ON\\) intersects the line \\(x=3\\) at point \\(M\\). Find the maximum value of \\( \dfrac {|PQ|}{|MB|}\\).
|
\sqrt {3}
| 0 |
17,029 |
Let the probability of germination for each seed be 0.9, and 1000 seeds have been planted. For each seed that does not germinate, 2 more seeds need to be replanted. Let X be the number of replanted seeds. Calculate the expected value of X.
|
200
| 90.625 |
17,030 |
Given the arithmetic sequence $\left\{ a_n \right\}$ where each term is positive, the sum of the first $n$ terms is $S_n$. When $n \in N^*, n \geqslant 2$, it holds that $S_n = \frac{n}{n-1}\left( a_n^2 - a_1^2 \right)$. Find the value of $S_{20} - 2S_{10}$.
|
50
| 58.59375 |
17,031 |
If the price of a stamp is 45 cents, what is the maximum number of stamps that could be purchased with $50?
|
111
| 92.1875 |
17,032 |
In the polar coordinate system, let curve \(C_1\) be defined by \(\rho\sin^2\theta = 4\cos\theta\). Establish a Cartesian coordinate system \(xOy\) with the pole as the origin and the polar axis as the positive \(x\)-axis. The curve \(C_2\) is described by the parametric equations:
\[ \begin{cases}
x = 2 + \frac{1}{2}t \\
y = \frac{\sqrt{3}}{2}t
\end{cases} \]
where \(t\) is the parameter.
(1) Find the Cartesian equations for \(C_1\) and \(C_2\).
(2) If \(C_1\) and \(C_2\) intersect at points \(A\) and \(B\), and there is a fixed point \(P\) with coordinates \((2,0)\), find the value of \(|PA| \cdot |PB|\).
|
\frac{32}{3}
| 20.3125 |
17,033 |
Given that x > 0, y > 0, and x + 2y = 4, find the minimum value of $$\frac {(x+1)(2y+1)}{xy}$$.
|
\frac {9}{2}
| 32.03125 |
17,034 |
Given that the sum of the first $n$ terms of an arithmetic sequence $\{a\_n\}$ is $S\_n$, with $a\_2 = 4$ and $S\_{10} = 110$, find the minimum value of $\frac{S\_n + 64}{a\_n}$.
|
\frac{17}{2}
| 60.15625 |
17,035 |
Integers $x$ and $y$ with $x>y>0$ satisfy $x+y+xy=101$. What is $x$?
|
50
| 67.96875 |
17,036 |
Let $f : N \to N$ be a strictly increasing function such that $f(f(n))= 3n$ , for all $n \in N$ . Find $f(2010)$ .
Note: $N = \{0,1,2,...\}$
|
3015
| 52.34375 |
17,037 |
It is currently 3:00:00 PM, as shown on a 12-hour digital clock. In 300 hours, 55 minutes, and 30 seconds, what will the time be and what is the sum of the hours, minutes, and seconds?
|
88
| 28.125 |
17,038 |
A pyramid has a square base $ABCD$ and a vertex $E$. The area of square $ABCD$ is $256$, and the areas of $\triangle ABE$ and $\triangle CDE$ are $120$ and $136$, respectively. The distance from vertex $E$ to the midpoint of side $AB$ is $17$. What is the volume of the pyramid?
- **A)** $1024$
- **B)** $1200$
- **C)** $1280$
- **D)** $1536$
- **E)** $1600$
|
1280
| 56.25 |
17,039 |
Given $\overrightarrow{m}=(\sin \omega x,-1)$, $\overrightarrow{n}=(1,- \sqrt {3}\cos \omega x)$ where $x\in\mathbb{R}$, $\omega > 0$, and $f(x)= \overrightarrow{m}\cdot \overrightarrow{n}$, and the distance between a certain highest point and its adjacent lowest point on the graph of function $f(x)$ is $5$,
$(1)$ Find the interval of monotonic increase for the function $f(x)$;
$(2)$ If $f\left( \dfrac {3\theta}{\pi}\right)= \dfrac {6}{5}$ where $\theta\in\left(- \dfrac {5\pi}{6}, \dfrac {\pi}{6}\right)$, then find the value of $f\left( \dfrac {6\theta}{\pi}+1\right)$.
|
\dfrac {48}{25}
| 34.375 |
17,040 |
Let $ABC$ be triangle such that $|AB| = 5$ , $|BC| = 9$ and $|AC| = 8$ . The angle bisector of $\widehat{BCA}$ meets $BA$ at $X$ and the angle bisector of $\widehat{CAB}$ meets $BC$ at $Y$ . Let $Z$ be the intersection of lines $XY$ and $AC$ . What is $|AZ|$ ? $
\textbf{a)}\ \sqrt{104}
\qquad\textbf{b)}\ \sqrt{145}
\qquad\textbf{c)}\ \sqrt{89}
\qquad\textbf{d)}\ 9
\qquad\textbf{e)}\ 10
$
|
10
| 54.6875 |
17,041 |
When \( q(x) = Dx^4 + Ex^2 + Fx + 6 \) is divided by \( x - 2 \), the remainder is 14. Find the remainder when \( q(x) \) is divided by \( x + 2 \).
|
14
| 10.9375 |
17,042 |
Given the sequence ${a_n}$, $a_1=1$ and $a_n a_{n+1} + \sqrt{3}(a_n - a_{n+1}) + 1 = 0$. Determine the value of $a_{2016}$.
|
2 - \sqrt{3}
| 53.125 |
17,043 |
In the rectangular coordinate system $xOy$, the ordinary equation of curve $C_1$ is $x^2+y^2-2x=0$. Establish a polar coordinate system with the origin $O$ as the pole and the positive semi-axis of $x$ as the polar axis. The polar equation of curve $C_2$ is $\rho^{2}= \frac {3}{1+2\sin^{2}\theta }$.
(I) Find the parametric equation of $C_1$ and the rectangular equation of $C_2$;
(II) The ray $\theta= \frac {\pi}{3}(\rho\geq0)$ intersects $C_1$ at a point $A$ distinct from the pole, and $C_2$ at point $B$. Find $|AB|$.
|
\frac { \sqrt {30}}{5}-1
| 0 |
17,044 |
Two arithmetic sequences $\{a_{n}\}$ and $\{b_{n}\}$ have the sums of the first $n$ terms as $S_{n}$ and $T_{n}$, respectively. It is known that $\frac{{S}_{n}}{{T}_{n}}=\frac{7n+2}{n+3}$. Find $\frac{{a}_{7}}{{b}_{7}}$.
|
\frac{93}{16}
| 64.0625 |
17,045 |
Four steel balls, each with a radius of 1, are to be completely packed into a container shaped as a regular tetrahedron. What is the minimum height of the tetrahedron?
|
2 + \frac{2 \sqrt{6}}{3}
| 47.65625 |
17,046 |
a) Vanya flips a coin 3 times, and Tanya flips a coin 2 times. What is the probability that Vanya gets more heads than Tanya?
b) Vanya flips a coin $n+1$ times, and Tanya flips a coin $n$ times. What is the probability that Vanya gets more heads than Tanya?
|
\frac{1}{2}
| 87.5 |
17,047 |
Given the exponential function y=f(x) whose graph passes through the point $\left( \frac{1}{2}, \frac{\sqrt{2}}{2} \right)$, find the value of $\log_2 f(2)$.
|
-2
| 82.03125 |
17,048 |
Simplify first, then evaluate: $(1-\frac{2}{x+1})÷\frac{x^2-1}{2x+2}$, where $x=\pi ^{0}+1$.
|
\frac{2}{3}
| 89.0625 |
17,049 |
Mayuki walks once around a track shaped with straight sides and semicircular ends at a constant speed daily. The track has a width of \(4\) meters, and it takes her \(24\) seconds longer to walk around the outside edge than the inside edge. Determine Mayuki's speed in meters per second.
|
\frac{\pi}{3}
| 88.28125 |
17,050 |
In writing the integers from 20 through 199 inclusive, how many times is the digit 7 written?
|
38
| 1.5625 |
17,051 |
If 2023 were expressed as a sum of distinct powers of 2, what would be the least possible sum of the exponents of these powers?
|
48
| 29.6875 |
17,052 |
Let the base areas of two cylinders be $S_1$ and $S_2$, and their volumes be $\upsilon_1$ and $\upsilon_2$, respectively. If their lateral areas are equal, and $$\frac {S_{1}}{S_{2}}= \frac {16}{9},$$ then the value of $$\frac {\upsilon_{1}}{\upsilon_{2}}$$ is \_\_\_\_\_\_.
|
\frac {4}{3}
| 98.4375 |
17,053 |
Given that the random variable $\xi$ follows a normal distribution $N(1,4)$, if $p(\xi > 4)=0.1$, then $p(-2 \leqslant \xi \leqslant 4)=$ _____ .
|
0.8
| 69.53125 |
17,054 |
Define a function $g(x),$ for positive integer values of $x,$ by
\[
g(x) = \left\{
\begin{aligned}
\log_3 x & \quad \text{if } \log_3 x \text{is an integer} \\
1 + g(x + 1) & \quad \text{otherwise}.
\end{aligned}
\right.
\]
Compute $g(50)$.
|
35
| 7.03125 |
17,055 |
In the Cartesian coordinate system $xOy$, the parametric equation of curve $C$ is $\begin{cases} & x=2\sqrt{3}\cos a \\ & y=2\sin a \end{cases}$, where $a$ is a parameter and $a \in (0, \pi)$. In the polar coordinate system with the origin $O$ as the pole and the positive half axis of $x$ as the polar axis, the polar coordinates of point $P$ are $(4\sqrt{2}, \frac{\pi}{4})$, and the polar equation of line $l$ is $\rho \sin(\theta - \frac{\pi}{4}) + 5\sqrt{2} = 0$.
(1) Find the Cartesian equation of line $l$ and the general equation of curve $C$.
(2) Suppose $Q$ is a moving point on curve $C$, and $M$ is the midpoint of segment $PQ$. Find the maximum distance from point $M$ to the line $l$.
|
6\sqrt{2}
| 80.46875 |
17,056 |
The lengths of the three sides of a triangle are \( 10 \), \( y+5 \), and \( 3y-2 \). The perimeter of the triangle is \( 50 \). What is the length of the longest side of the triangle?
|
25.75
| 1.5625 |
17,057 |
In the expansion of $(\frac{3}{{x}^{2}}+x+2)^{5}$, the coefficient of the linear term in $x$ is ____.
|
200
| 39.0625 |
17,058 |
Given that angle DEF is a right angle and the sides of triangle DEF are the diameters of semicircles, the area of the semicircle on segment DE equals $18\pi$, and the arc of the semicircle on segment DF has length $10\pi$. Determine the radius of the semicircle on segment EF.
|
\sqrt{136}
| 0 |
17,059 |
A regular hexahedron with an edge length of $1$ is cut by planes passing through the common vertex of three edges and their respective midpoints. After removing the $8$ triangular pyramids, the volume of the remaining convex polyhedron is $\_\_\_\_\_\_$.
|
\frac{5}{6}
| 21.875 |
17,060 |
A sphere intersects the $xy$-plane in a circle centered at $(3,5,0)$ with a radius of 2. The sphere also intersects the $yz$-plane in a circle centered at $(0,5,-8),$ with radius $r.$ Find $r.$
|
\sqrt{59}
| 28.90625 |
17,061 |
Two people, A and B, participate in a general knowledge competition, with a total of 4 different questions, including 2 multiple-choice questions and 2 true/false questions. A and B each draw one question (without repetition).
$(1)$ What is the probability that A draws a multiple-choice question and B draws a true/false question?
$(2)$ What is the probability that at least one of A and B draws a multiple-choice question?
|
\frac{5}{6}
| 93.75 |
17,062 |
From the five points consisting of the four vertices and the center of a square, any two points are chosen. The probability that the distance between these two points is not less than the side length of the square is ______.
|
\frac{3}{5}
| 49.21875 |
17,063 |
Square \( ABCD \) has a side length of 12 inches. A segment \( AE \) is drawn where \( E \) is on side \( DC \) and \( DE \) is 5 inches long. The perpendicular bisector of \( AE \) intersects \( AE, AD, \) and \( BC \) at points \( M, P, \) and \( Q \) respectively. The ratio of the segments \( PM \) to \( MQ \) is:
|
5:19
| 1.5625 |
17,064 |
If the operation symbol "△" is defined as: $a \triangle b = a + b + ab - 1$, and the operation symbol "⊗" is defined as: $a \otimes b = a^2 - ab + b^2$, evaluate the value of $3 \triangle (2 \otimes 4)$.
|
50
| 88.28125 |
17,065 |
A convex polyhedron S has vertices U1, U2, …, Um, and 120 edges. This polyhedron is intersected by planes Q1, Q2, …, Qm, where each plane Qk intersects only those edges that are connected to vertex Uk. No two planes intersect within the volume or on the surface of S. As a result, m pyramids are formed along with a new polyhedron T. Determine the number of edges that polyhedron T now possesses.
|
360
| 7.03125 |
17,066 |
The perimeter of quadrilateral PQRS, made from two similar right-angled triangles PQR and PRS, is given that the length of PQ is 3, the length of QR is 4, and ∠PRQ = ∠PSR. Find the perimeter of PQRS.
|
22
| 6.25 |
17,067 |
Given $f(\alpha)= \dfrac {\sin (\alpha- \dfrac {5\pi}{2})\cos ( \dfrac {3\pi}{2}+\alpha)\tan (\pi-\alpha)}{\tan (-\alpha-\pi)\sin (\pi-\alpha)}$.
(1) Simplify $f(\alpha)$
(2) If $\cos (\alpha+ \dfrac {3\pi}{2})= \dfrac {1}{5}$ and $\alpha$ is an angle in the second quadrant, find the value of $f(\alpha)$.
|
\dfrac{2\sqrt{6}}{5}
| 59.375 |
17,068 |
Determine the smallest integral value of $n$ such that the quadratic equation
\[3x(nx+3)-2x^2-9=0\]
has no real roots.
A) -2
B) -1
C) 0
D) 1
|
-1
| 86.71875 |
17,069 |
Susie was given $\$1,500$ for her birthday. She decides to invest the money in a bank account that earns $12\%$ interest, compounded quarterly. How much total interest will Susie have earned 4 years later?
|
901.55
| 0 |
17,070 |
How many of the numbers
\[
a_1\cdot 5^1+a_2\cdot 5^2+a_3\cdot 5^3+a_4\cdot 5^4+a_5\cdot 5^5+a_6\cdot 5^6
\]
are negative if $a_1,a_2,a_3,a_4,a_5,a_6 \in \{-1,0,1 \}$ ?
|
364
| 51.5625 |
17,071 |
Let \( r(x) \) have a domain of \(\{ -2, -1, 0, 1 \}\) and a range of \(\{ 1, 3, 5, 7 \}\). Let \( s(x) \) have a domain of \(\{ 0, 1, 2, 3, 4, 5 \}\) and be defined by \( s(x) = 2x + 1 \). What is the sum of all possible values of \( s(r(x)) \)?
|
21
| 7.8125 |
17,072 |
For each positive integer $n$ , consider the highest common factor $h_n$ of the two numbers $n!+1$ and $(n+1)!$ . For $n<100$ , find the largest value of $h_n$ .
|
97
| 92.96875 |
17,073 |
Given real numbers $x$ and $y$ satisfy that three of the four numbers $x+y$, $x-y$, $\frac{x}{y}$, and $xy$ are equal, determine the value of $|y|-|x|$.
|
\frac{1}{2}
| 53.125 |
17,074 |
Given $f(x)=\cos x\cdot\ln x$, $f(x_{0})=f(x_{1})=0(x_{0}\neq x_{1})$, find the minimum value of $|x_{0}-x_{1}|$ ___.
|
\dfrac {\pi}{2}-1
| 89.0625 |
17,075 |
Using arithmetic operation signs, write the largest natural number using two twos.
|
22
| 58.59375 |
17,076 |
Given that a high school senior year has 12 classes, with exactly 8 classes to be proctored by their own homeroom teachers, find the number of different proctoring arrangements for the math exam.
|
4455
| 50.78125 |
17,077 |
Given non-zero vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfying $\overrightarrow{a}^{2}=(5 \overrightarrow{a}-4 \overrightarrow{b})\cdot \overrightarrow{b}$, find the minimum value of $\cos < \overrightarrow{a}, \overrightarrow{b} >$.
|
\frac {4}{5}
| 68.75 |
17,078 |
Ottó decided to assign a number to each pair \((x, y)\) and denote it as \((x \circ y)\). He wants the following relationships to hold:
a) \(x \circ y = y \circ x\)
b) \((x \circ y) \circ z = (x \circ z) \circ (y \circ z)\)
c) \((x \circ y) + z = (x + z) \circ (y + z)\).
What number should Ottó assign to the pair \((1975, 1976)\)?
|
1975.5
| 25.78125 |
17,079 |
What is the smallest positive integer $x$ that, when multiplied by $900$, produces a product that is a multiple of $1152$?
|
32
| 96.09375 |
17,080 |
Find the set of values for parameter \(a\) for which the sum of the cubes of the roots of the equation \(x^{2} + ax + a + 1 = 0\) is equal to 1.
|
-1
| 0 |
17,081 |
The equation of a line is given by $Ax+By=0$. If we choose two different numbers from the set $\{1, 2, 3, 4, 5\}$ to be the values of $A$ and $B$ each time, then the number of different lines that can be obtained is ．
|
18
| 7.03125 |
17,082 |
Consider the function $g(x) = \frac{ax+b}{cx+d}$, where $a$, $b$, $c$, and $d$ are nonzero real numbers. Assume $g(5) = 5$ and $g(25) = 25$, and it also satisfies $g(g(x)) = x$ for all values except $\frac{-d}{c}$. Find the unique number that is not in the range of $g$.
|
15
| 59.375 |
17,083 |
The sum of all digits used in the numbers 1, 2, 3, ..., 999 is .
|
13500
| 70.3125 |
17,084 |
Determine the value of the following expression, simplified as a fraction: $$1+\cfrac{3}{2+\cfrac{5}{6}}$$
|
\frac{35}{17}
| 60.15625 |
17,085 |
Given that $\alpha \in \left(\frac{\pi}{2}, \pi\right)$ and $\sin \alpha = \frac{4}{5}$, calculate the value of $\sin 2\alpha$.
|
-\frac{24}{25}
| 100 |
17,086 |
Triangle $PQR$ has side lengths $PQ=160, QR=300$, and $PR=240$. Lines $m_P, m_Q$, and $m_R$ are drawn parallel to $\overline{QR}, \overline{RP}$, and $\overline{PQ}$, respectively, such that the intersections of $m_P, m_Q$, and $m_R$ with the interior of $\triangle PQR$ are segments of lengths $75, 60$, and $20$, respectively. Find the perimeter of the triangle whose sides lie on lines $m_P, m_Q$, and $m_R$.
|
155
| 6.25 |
17,087 |
Sara lists the whole numbers from 1 to 50. Lucas copies Sara's numbers, replacing each occurrence of the digit '3' with the digit '2'. Calculate the difference between Sara's sum and Lucas's sum.
|
105
| 2.34375 |
17,088 |
Dana goes first and Carl's coin lands heads with probability $\frac{2}{7}$, and Dana's coin lands heads with probability $\frac{3}{8}$. Find the probability that Carl wins the game.
|
\frac{10}{31}
| 37.5 |
17,089 |
Given a line $l$ with an inclination angle of $\theta$, if $\cos\theta= \frac {4}{5}$, calculate the slope of this line.
|
\frac{3}{4}
| 76.5625 |
17,090 |
Given that $\frac{\sin \theta + \cos \theta}{\sin \theta - \cos \theta} = 2$, find the value of $\frac{\sin \theta}{\cos^{3} \theta} + \frac{\cos \theta}{\sin^{3} \theta}$.
|
\frac{820}{27}
| 92.1875 |
17,091 |
Add 24.567 to 38.924, then multiply the sum by 2.5, and round the result to the nearest hundredth.
|
158.73
| 4.6875 |
17,092 |
Find one fourth of 12.8, expressed as a simplified improper fraction and also as a mixed number.
|
3 \frac{1}{5}
| 33.59375 |
17,093 |
A bag contains 4 blue marbles and 6 green marbles. Three marbles are drawn one after another without replacement. What is the probability that the first marble is blue, the second one is green, and the third one is also green?
|
\frac{1}{6}
| 89.0625 |
17,094 |
In triangle $\triangle ABC$, $sin(A+\frac{π}{4})sin(B+\frac{π}{4})=cosAcosB$. Find:<br/>
$(1)$ the value of angle $C$;<br/>
$(2)$ if $AB=\sqrt{2}$, find the minimum value of $\overrightarrow{CA}•\overrightarrow{CB}$.
|
-\sqrt{2}+1
| 0.78125 |
17,095 |
A local government intends to encourage entrepreneurship by rewarding newly established small and micro enterprises with an annual output value between 500,000 and 5,000,000 RMB. The reward scheme follows these principles: The bonus amount $y$ (in ten thousand RMB) increases with the yearly output value $x$ (in ten thousand RMB), the bonus is no less than 700,000 RMB, and the bonus does not exceed 15% of the annual output value.
1. If an enterprise has an output value of 1,000,000 RMB and is eligible for a 90,000 RMB bonus, analyze whether the function $y=\log x + kx + 5$ (where $k$ is a constant) is in line with the government's reward requirements, and explain why (given $\log 2 \approx 0.3, \log 5 \approx 0.7$).
2. If the function $f(x) = \frac{15x - a}{x + 8}$ is adopted as the reward model, determine the minimum value of the positive integer $a$.
|
315
| 17.1875 |
17,096 |
Let $O$ be the origin. Determine the scalar $m$ such that for any points $A, B, C, D$, if the equation
\[4 \overrightarrow{OA} - 3 \overrightarrow{OB} + 6 \overrightarrow{OC} + m \overrightarrow{OD} = \mathbf{0}\]
holds, then the points $A, B, C$, and $D$ are coplanar.
|
-7
| 75 |
17,097 |
Convert -630° to radians.
|
-\frac{7\pi}{2}
| 80.46875 |
17,098 |
Six test scores have a mean of 85, a median of 88, and a mode of 90. The highest score exceeds the second highest by 5 points. Find the sum of the three highest scores.
|
275
| 12.5 |
17,099 |
What is the value of $x$ if $x=\frac{2023^2 - 2023 + 1}{2023}$?
|
2022 + \frac{1}{2023}
| 87.5 |
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