task_type stringclasses 4
values | problem stringlengths 14 5.23k | solution stringlengths 1 8.29k | problem_tokens int64 9 1.02k | solution_tokens int64 1 1.98k |
|---|---|---|---|---|
math | In the regular pyramid $S$-$ABCD$, $O$ is the projection of the apex on the base, $P$ is the midpoint of the lateral edge $SD$, and $SO = OD$. The angle formed by line $BC$ and plane $PAC$ is ______. | 30^\circ | 60 | 4 |
math | Given triangle $ABC$, if $\sin A + 2\sin B\cos C = 0$, find the maximum value of $\tan A$. | \frac{1}{\sqrt{3}} | 31 | 10 |
math | Given that in $\triangle ABC$, $C = 2A$, $\cos A = \frac{3}{4}$, and $2 \overrightarrow{BA} \cdot \overrightarrow{CB} = -27$.
(I) Find the value of $\cos B$;
(II) Find the perimeter of $\triangle ABC$. | 15 | 71 | 2 |
math | The function \(f(x) = 5x^2 - 15x - 2\) has a minimum value when x is negative. | -13.25 | 30 | 6 |
math | If $a^{m}=2$ and $a^{n}=3$, then $a^{m+n}=$______, $a^{m-2n}=$______. | \frac{2}{9} | 37 | 7 |
math | Let $s$ be the result of tripling both the base and exponent of $c^d$, where $d$ is a non-zero integer. If $s$ equals the product of $c^d$ by $y^d$, determine the value of $y$. | 27c^2 | 57 | 5 |
math | In how many distinct ways can I arrange my six keys on a keychain, if I want to put my house key next to my car key and my mailbox key next to my bike key? Two arrangements are not considered different if the keys are in the same order (or can be made to be in the same order without taking the keys off the chain--that ... | 3 | 80 | 1 |
math | For constants $a$ and $b$, let
\[f(x) = \left\{
\begin{array}{cl}
ax + b & \text{if } x < 1, \\
7 - 2x & \text{if } x \ge 1.
\end{array}
\right.\]
The function $f$ has the property that $f(f(x)) = x$ for all $x$. What is the value of $a + b$? | 3 | 101 | 1 |
math | John is deciding whether to go hiking tomorrow. If it rains, there is a 10% chance he will go hiking, but if it is sunny, there is a 90% chance he will go. The weather forecast predicts a 30% chance of rain. What is the probability that John will go hiking? Express your answer as a percent. | 66\% | 74 | 4 |
math | The interval that contains the root of the function $f(x) = x^5 + x - 3$ is to be found. | [1,2] | 28 | 5 |
math | Given a 3 cm by 3 cm by 3 cm cube, a 1 cm by 1 cm by 1 cm cube is cut from one corner and a 2 cm by 2 cm by 2 cm cube is cut from the opposite corner, calculate the surface area of the resulting solid. | 54 | 63 | 2 |
math | Given that $\frac{n}{p\_1+p\_2+...+p\_n}$ is the "harmonic mean" of $n$ positive numbers $p\_1$, $p\_2$, ..., $p\_n$, and the harmonic mean of the first $n$ terms of the sequence $\{a\_n\}$ is $\frac{1}{n}$, determine the value of $\frac{1}{a\_1a\_2} + \frac{1}{a\_2a\_3} + ... + \frac{1}{a\_10a\_11}$. | \frac{10}{21} | 125 | 9 |
math | Given the sets $A=\{x\mid ax-2=0\}$ and $B=\{x\mid x^2-3x+2=0\}$, and $A \subseteq B$, then the set $C$ consisting of the values of the real number $a$ is | \{1, 2\} | 63 | 8 |
math | Fifty slips are placed into a hat, each bearing a number from 1 to 10, with each number entered on five slips. Five slips are drawn from the hat at random and without replacement. Let $p$ be the probability that all five slips bear the same number. Let $q$ be the probability that two of the slips bear a number $a$ and ... | 450 | 99 | 3 |
math | At the World Meteorological Conference, each participant took turns announcing the average monthly temperature in their home city. Meanwhile, all others recorded the product of the temperature in their city and the temperature being announced. In total, 62 positive numbers and 48 negative numbers were recorded. What is... | 3 | 72 | 1 |
math | What is the largest 4-digit integer congruent to $7 \pmod{19}$? | 9982 | 21 | 4 |
math | Derive the formula \( I_{2}^{*}=\frac{1}{n} \sum_{i=1}^{n} \frac{\varphi\left(x_{i}\right)}{f\left(x_{i}\right)} \) for estimating the definite integral \( I=\int_{a}^{b} \varphi(x) \, \mathrm{d} x \), where \( f(x) \) is the density of an auxiliary random variable \( X \) in the integration interval \( (a, b) \); \( x... | I_{2}^{*} = \frac{1}{n} \sum_{i=1}^{n} \frac{\varphi(x_i)}{f(x_i)} | 153 | 38 |
math | In $\triangle ABC$, the sides opposite to angles A, B, and C are $a$, $b$, and $c$, respectively. If $$\frac{a^2}{bc} - \frac{c}{b} - \frac{b}{c} = \sqrt{3}$$, and the radius of the circumcircle of $\triangle ABC$ is 3, find $a$. | a = 3 | 84 | 4 |
math | Given that Alice and Bob shared a large pizza cut into 12 equally-sized slices, the cost of the plain pizza was $12, and there was an additional cost of $4 to add olives to three slices, find the difference in the amount Bob paid and the amount Alice paid, after Bob ate all the olive-topped slices and 3 slices of the p... | 0 | 87 | 1 |
math | Two lines passing through point \( M \), which lies outside the circle with center \( O \), touch the circle at points \( A \) and \( B \). Segment \( OM \) is divided in half by the circle. In what ratio is segment \( OM \) divided by line \( AB \)? | 1:3 | 63 | 3 |
math | For every $m$ and $k$ integers with $k$ odd and a divisor of 150, denote by $\left[\frac{m}{k}\right]$ the integer closest to $\frac{m}{k}$. For every odd integer $k$ that divides 150, let $P(k)$ be the probability that
\[\left[\frac{n}{k}\right] + \left[\frac{150 - n}{k}\right] = \left[\frac{150}{k}\right]\]
for an in... | \frac{1}{5} | 237 | 7 |
math | Vanya came up with a three-digit prime number in which all digits are different.
What digit can it end with if its last digit is equal to the sum of the first two digits? | 7 | 38 | 1 |
math | Given the function $f(x)=2\sin (2x+ \frac {\pi}{6})-1$ $(x\in\mathbb{R})$, find the maximum and minimum values of $f(x)$ in the interval $\left[0, \frac {\pi}{2}\right]$. | -2 | 64 | 2 |
math | Sequence $(a_n)$ is defined as $a_{n+1}-2a_n+a_{n-1}=7$ for every $n\geq 2$ , where $a_1 = 1, a_2=5$ . What is $a_{17}$ ? | 905 | 69 | 3 |
math | A circle of radius $15$ inches has its center at the vertex $C$ of an equilateral triangle $ABC$ and passes through the other two vertices. The side $BC$ extended through $C$ intersects the circle at point $E$. Find the number of degrees in angle $AEC$. | 90 | 63 | 2 |
math | Given a number $n\in\mathbb{Z}^+$ and let $S$ denotes the set $\{0,1,2,...,2n+1\}$ . Consider the function $f:\mathbb{Z}\times S\to [0,1]$ satisfying two following conditions simultaneously:
i) $f(x,0)=f(x,2n+1)=0\forall x\in\mathbb{Z}$ ;
ii) $f(x-1,y)+f(x+1,y)+f(x,y-1)+f(x,y+1)=1$ for all $x\in\mathbb{Z}$ ... | 2n + 1 | 273 | 5 |
math | In the exam, there were 25 problems of three types: easy ones worth 2 points each, medium ones worth 3 points each, and hard ones worth 5 points each. Correctly solved problems were rewarded with the specified points based on difficulty; otherwise, they received 0 points. The best possible total score on the exam was 8... | 40 | 107 | 2 |
math | Let $n$ be a positive integer such that the decimal representation of $n!$ ends in $k$ zeros and the decimal representation of $(3n)!$ ends in $4k$ zeros. Let $t$ denote the sum of the four smallest possible values of $n$. Find the sum of the digits of $t$. | 7 | 69 | 1 |
math | Given $x∈({0,\frac{π}{2}})$, and $ax \lt \sin x \lt bx$ always holds, find the minimum value of $b-a$. | 1-\frac{2}{π} | 39 | 8 |
math | Given that the function $f(x) = -x^3 + ax^2 - 4$ attains an extremum at $x = 2$, and if $m, n \in [-1,1]$, then the minimum value of $f(m) + f'(n)$ is ______. | -4 + (-9) = -13 | 64 | 10 |
math | \(\left(\frac{3}{5}\right)^{2 \log _{9}(x+1)} \cdot\left(\frac{125}{27}\right)^{\log _{v 27}(x-1)}=\frac{\log _{5} 27}{\log _{5} 243}\). | x = 2 | 77 | 4 |
math | Michael picks a random subset of the complex numbers \(\left\{1, \omega, \omega^{2}, \ldots, \omega^{2017}\right\}\) where \(\omega\) is a primitive \(2018^{\text {th }}\) root of unity and all subsets are equally likely to be chosen. If the sum of the elements in his subset is \(S\), what is the expected value of \(|S... | \frac{1009}{2} | 115 | 10 |
math | Find the remainder when the polynomial $x^{1002}$ is divided by the polynomial $(x^2 - 1)(x + 1).$ | 1 | 33 | 1 |
math | How many two digit numbers have exactly $4$ positive factors? $($ Here $1$ and the number $n$ are also considered as factors of $n. )$ | 31 | 45 | 2 |
math | Given the function $g(x)=x^{2}-2 (x \in \mathbb{R})$ and $f(x) = \begin{cases} g(x)+x+4, & x < g(x) \\ g(x)-x, & x \geq g(x) \end{cases}$, determine the range of $f(x)$. | [-2.25, 0] \cup (2, +\infty) | 76 | 19 |
math | Given the ellipse \(\frac{x^{2}}{5^{2}}+\frac{y^{2}}{4^{2}}=1\), a line passes through its left focus \(F_{1}\) and intersects the ellipse at points \(A\) and \(B\). Point \(D(a, \theta)\) is located to the right of \(F_{1}\). The lines \(AD\) and \(BD\) intersect the left directrix of the ellipse at points \(M\) and \... | a = 5 | 135 | 4 |
math | My friend thought of an integer between 10 and 19. To guess the number he thought of, I can ask him questions that he will answer with "yes" or "no": What is the minimum number of questions, and which questions should I ask to determine the number he thought of? | 3 | 62 | 1 |
math | In an isosceles right triangle \( \triangle ABC \), \( \angle A = 90^\circ \), \( AB = 1 \). \( D \) is the midpoint of \( BC \), \( E \) and \( F \) are two other points on \( BC \). \( M \) is the other intersection point of the circumcircles of \( \triangle ADE \) and \( \triangle ABF \); \( N \) is the other inters... | \sqrt{2} | 154 | 5 |
math | Find the sum of the thousands digit and the units digit of the product of the two 101-digit numbers 404040404...040404 and 707070707...070707. | 10 | 59 | 2 |
math | Observe the following equations:
$$1^{2}=1$$
$$1^{2}-2^{2}=-3$$
$$1^{2}-2^{2}+3^{2}=6$$
$$1^{2}-2^{2}+3^{2}-4^{2}=-10$$
$$\ldots$$
According to this pattern, the $n^{th}$ equation can be written as \_\_\_\_\_\_. | 1^{2}-2^{2}+3^{2}-\ldots+(-1)^{n-1}n^{2}=\frac{(-1)^{n+1}}{2}n(n+1) | 96 | 48 |
math | Let \( b = 1^{2} - 2^{2} + 3^{2} - 4^{2} + 5^{2} - \ldots - 2012^{2} + 2013^{2} \). Determine the remainder of \( b \) divided by 2015. | 1 | 73 | 1 |
math | Given a complex number $Z$ that satisfies the equation $Z(i-1)=2i$ (where $i$ is the imaginary unit), determine $\overset{.}{Z}$. | 1+i | 40 | 2 |
math | Let the function $f(x)=x^{3}+x$, where $x\in\mathbb{R}$. If for $0 < \theta < \frac{\pi}{2}$, the inequality $f(m\sin \theta)+f(1-m) > 0$ always holds, determine the range of the real number $m$. | (-\infty, 1] | 74 | 8 |
math | How many integers between $200$ and $250$ have three different digits in decreasing order? | 1 | 23 | 1 |
math | In a rectangle of size $3 \times 4$, 4 points are chosen. Find the smallest number $C$ such that the distance between some two of these points does not exceed $C$. | 2.5 | 41 | 3 |
math | \[\left\{\begin{array}{l}
3(2-\sqrt{x-y})^{-1}+10(2+\sqrt{x+y})^{-1}=5, \\
4(2-\sqrt{x-y})^{-1}-5(2+\sqrt{x+y})^{-1}=3.
\end{array}\right.\]
\[\left\{\begin{array}{l}
\sqrt[3]{x}+\sqrt[3]{y}=4, \\
x+y=28.
\end{array}\right.\] | (1, 27), (27, 1) | 112 | 14 |
math | Each corner cube is removed from this $3\text{ cm}\times 3\text{ cm}\times 3\text{ cm}$ cube. The surface area of the remaining figure is
[asy]
draw((2.7,3.99)--(0,3)--(0,0));
draw((3.7,3.99)--(1,3)--(1,0));
draw((4.7,3.99)--(2,3)--(2,0));
draw((5.7,3.99)--(3,3)--(3,0));
draw((0,0)--(3,0)--(5.7,0.99));
draw((0,1)--(3,1... | 54 | 324 | 2 |
math | The monotonic decreasing interval of the function $f\left(x\right)={\left( \frac{1}{3}\right)}^{-{x}^{2}-4x+3}$ is \_\_\_\_\_\_. | (-\infty, -2] | 49 | 8 |
math | Find the values of $x$ and $y$ such that $\sqrt{4 - 5x + y} = 9$. | y = 77 + 5x | 28 | 9 |
math | Let the function $f(x)$ be defined on $\mathbb{R}$ and satisfy $f(2-x) = f(2+x)$ and $f(7-x) = f(7+x)$. Also, in the closed interval $[0, 7]$, only $f(1) = f(3) = 0$. Determine the number of roots of the equation $f(x) = 0$ in the closed interval $[-2005, 2005]$. | 802 | 107 | 3 |
math | Given $\triangle BAD$ is right-angled at $B$, on $AD$ there is a point $C$ for which $AC=CD$ and $AB=BC$. Determine the magnitude of $\angle DAB$. | 60^\circ | 46 | 4 |
math | A train departs from point $A$ and travels towards point $B$ at a uniform speed. 11 minutes later, another train departs from point $B$ traveling towards point $A$ at a constant speed on the same route. After their meeting point, it takes the trains 20 minutes and 45 minutes to reach $B$ and $A$, respectively. In what ... | 9/5 | 97 | 3 |
math | Let $x$ and $y$ be positive real numbers. Find the minimum value of
\[
\frac{\sqrt{(x^2 + y^2)(4x^2 + 2y^2)}}{xy}.
\] | 2 + \sqrt{2} | 50 | 7 |
math | A positive integer will be called "sparkly" if its smallest positive divisor, other than 1, equals the total number of divisors (including 1). How many of the numbers \(2, 3, \ldots, 2003\) are sparkly? | 3 | 58 | 1 |
math | Given a quadratic equation in terms of $x$: $mx^{2}-4x+1=0$.
$(1)$ If $1$ is one of the roots of this equation, find the value of $m$;
$(2)$ If the quadratic equation $mx^{2}-4x+1=0$ has real roots, find the range of values for $m$. | m \leq 4 \text{ and } m \neq 0 | 80 | 17 |
math | Ann wants to build a $6$-step staircase using toothpicks. She already made a $4$-step staircase using $28$ toothpicks. The number of toothpicks added for each additional step increases by $3$ more than the previous step's increase. Calculate the total number of additional toothpicks Ann needs to complete the $6$-step s... | 33 | 80 | 2 |
math | Given that $\frac{a}{30-a}+\frac{b}{70-b}+\frac{c}{75-c}=9$, evaluate $\frac{6}{30-a}+\frac{14}{70-b}+\frac{15}{75-c}$. | 35 | 62 | 2 |
math | Given the variable $a$ and $θ∈R$, find the minimum value of $(a-2\cos θ)^{2}+(a-5 \sqrt {2}-2\sin θ)^{2}$. | 9 | 48 | 1 |
math | A rug is made with three different colors as shown. The areas of the three differently colored regions form an arithmetic progression. The inner rectangle is two feet wide, and each of the two shaded regions is $2$ feet wide on all four sides. Find the length in feet of the inner rectangle. | 4 | 60 | 1 |
math | If \(x\), \(y\), and \(z\) are positive real numbers with \(xy = 30\), \(xz = 60\), and \(yz = 90\), what is the value of \(x+y+z\)? | 11\sqrt{5} | 54 | 7 |
math | Let \( m = 2^{40}5^{24} \). How many positive integer divisors of \( m^2 \) are less than \( m \) but do not divide \( m \)? | 959 | 45 | 3 |
math | Given the set \( S = \{1, 2, \cdots, 2005\} \), and a subset \( A \subseteq S \) such that the sum of any two numbers in \( A \) is not divisible by 117, determine the maximum value of \( |A| \). | 1003 | 69 | 4 |
math | The value of 1 + 3^2 is | 10 | 11 | 2 |
math |
When I saw Eleonora, I found her very pretty. After a brief trivial conversation, I told her my age and asked how old she was. She answered:
- When you were as old as I am now, you were three times older than me. And when I will be three times older than I am now, together our ages will sum up to exactly a century.
... | 15 | 87 | 2 |
math | A regular octagon and a rectangle share a common side \( \overline{AD} \). Determine the degree measure of the exterior angle \( BAC \) formed at vertex \( A \) where \( \angle BAD \) is an angle of the octagon and \( \angle CAD \) is an angle of the rectangle. | 135^\circ | 68 | 5 |
math | Find the standard equations for the following curves:
(1) A parabola whose focus is the left vertex of the ellipse $\frac{x^2}{9} + \frac{y^2}{16} = 1$;
(2) A hyperbola that shares asymptotes with the hyperbola $\frac{y^2}{5} - \frac{x^2}{5} = 1$ and passes through the point $(1, \sqrt{3})$. | \frac{y^2}{2} - \frac{x^2}{2} = 1 | 101 | 21 |
math | 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... | \frac{\sqrt{3}}{2} | 243 | 10 |
math | Provide the proof process for the following inequalities:
\\(①\\) If \\(a\\), \\(b∈R\\), then \\(\dfrac{b}{a}+\dfrac{a}{b}\geqslant 2\sqrt{\dfrac{b}{a}\cdot \dfrac{a}{b}}=2\\);
\\(②\\) If \\(x > 0\\), then \\(\cos x+\dfrac{1}{\cos x}\geqslant 2\sqrt{\cos x\cdot \dfrac{1}{\cos x}}=2\\);
\\(③\\) If \\(x < 0\\), then \... | ①②③ | 304 | 6 |
math | Among the following statements, the correct ones are \_\_\_\_\_\_.
① The set of angles whose terminal sides fall on the y-axis is $$\{α|α= \frac {kπ}{2}, k∈Z\}$$;
② A center of symmetry for the graph of the function $$y=2\cos(x- \frac {π}{4})$$ is $$( \frac {3π}{4}, 0)$$;
③ The function y=tanx is increasing in the ... | ②④ | 173 | 4 |
math | The wristwatch is 5 minutes slow per hour; 5.5 hours ago, it was set to the correct time. It is currently 1 PM on a clock that shows the correct time. How many minutes will it take for the wristwatch to show 1 PM? | 30 | 57 | 2 |
math | Let $a, b, c, d$ be real numbers, none of which are equal to $-1,$ and let $\omega$ be a complex number such that $\omega^4 = 1$ and $\omega \neq 1.$ If
\[
\frac{1}{a + \omega} + \frac{1}{b + \omega} + \frac{1}{c + \omega} + \frac{1}{d + \omega} = \frac{4}{1 + \omega},
\]
then find
\[
\frac{1}{a + 1} + \frac{1}{b + 1} ... | 2 | 162 | 1 |
math | Three friends, Rowan, Sara, and Tim, are playing a monetary game. Each starts with $3. A bell rings every 20 seconds, and with each ring, any player with money chooses one of the other two players independently at random and gives them $1. The game continues for 2020 rounds. What is the probability that at the end of t... | \frac{1}{4} | 125 | 7 |
math | Given triangle $ABC$ with opposite sides $a$, $b$, and $c$ to angles $A$, $B$, and $C$ respectively, and given that $\sin C= \frac {56}{65}$, $\sin B= \frac {12}{13}$, and $b=3$, find the value of $c$. | \frac {14}{5} | 76 | 8 |
math | For each integer $1\le j\le 2017$ , let $S_j$ denote the set of integers $0\le i\le 2^{2017} - 1$ such that $\left\lfloor \frac{i}{2^{j-1}} \right\rfloor$ is an odd integer. Let $P$ be a polynomial such that
\[P\left(x_0, x_1, \ldots, x_{2^{2017} - 1}\right) = \prod_{1\le j\le 2017} \left(1 - \prod_{i\in S_j} ... | 2 | 257 | 1 |
math | **Background:** Define a new operation "$*$": the result of the operation takes the positive sign if the signs are the same, and negative if the signs are different. The value is the sum of their absolute values. When $0$ is operated with any number using "$*$", the result is the absolute value of that number. For exam... | 2a + 3 | 218 | 5 |
math | Rectangle $ABCD$ contains a point $Q$ such that the distances from $Q$ to the vertices $A$, $B$, $C$, and $D$ are $5$, $x$, $13$, and $12$ inches respectively. Determine the value of $x$.
**A)** $5$
**B)** $7$
**C)** $5\sqrt{2}$
**D)** $13$ | 5\sqrt{2} | 94 | 6 |
math | There exists a unique strictly increasing sequence of nonnegative integers $b_1 < b_2 < … < b_m$ such that $\frac{2^{225}+1}{2^{15}+1} = 2^{b_1} + 2^{b_2} + … + 2^{b_m}$, determine the value of $m$. | 8 | 80 | 1 |
math | Let $\{a_n\}$ be a decreasing geometric sequence, where $a_1+a_2=11$, $a_1⋅a_2=10$, and $\log{a_1}+\log{a_2}+\log{a_3}+...+\log{a_{10}}=$ | -35 | 70 | 3 |
math | In the geometric sequence {a_n}, it is known that $a_1 + a_2 + a_3 = 1$, and $a_4 + a_5 + a_6 = -2$. Find the sum of the first 15 terms of the sequence, $S_{15}$. | S_{15} = 11 | 66 | 9 |
math | A regular dodecahedron is projected orthogonally onto a plane, and its image is an $n$-sided polygon. What is the smallest possible value of $n$ ? | 6 | 41 | 1 |
math | Solve the systems of equations:
a) $\left\{\begin{array}{l}x^{2}-3 x y-4 y^{2}=0, \\ x^{3}+y^{3}=65 ;\end{array}\right.$
b) $\left\{\begin{array}{l}x^{2}+2 y^{2}=17, \\ 2 x y-x^{2}=3\end{array}\right.$ | (3, 2), (-3, -2), \left(\frac{\sqrt{3}}{3}, \frac{5\sqrt{3}}{3}\right), \left(-\frac{\sqrt{3}}{3}, -\frac{5\sqrt{3}}{3}\right) | 97 | 66 |
math | Given two lines $l_1: (k-3)x + (5-k)y + 1 = 0$ and $l_2: 2(k-3)x - 2y + 3 = 0$ are perpendicular, determine the value(s) of $k$. | 1 \text{ or } 4 | 60 | 8 |
math | Express 1.36 billion in scientific notation. | 1.36\times 10^{9} | 11 | 12 |
math | For real numbers \( x \) and \( y \) within the interval \([0, 12]\):
$$
xy = (12 - x)^2 (12 - y)^2
$$
What is the maximum value of the product \( xy \)? | 81 | 57 | 2 |
math | Given the function $f(x) = \sin\left(\frac{\pi}{6}x\right)\cos\left(\frac{\pi}{6}x\right)$, find the value of $f(1) + f(2) + \ldots + f(2018)$. | 0 | 65 | 1 |
math | Find the equation of the line that passes through the intersection point of the lines $x+y-3=0$ and $2x-y=0$, and is perpendicular to the line $2x+y-5=0$. | x - 2y + 3 = 0 | 46 | 11 |
math | Given $a\in R$, the functions are defined as follows: $f(x)=\frac{1}{3}{x^3}-\frac{1}{2}(a-1)x^{2}-ax-3$ and $g\left(x\right)=x-2\ln x$. Find:
1. When $a=1$, find the equation of the tangent line to the function $y=f\left(x\right)$ at the point $\left(3,f\left(3\right)\)$.
2. If the decreasing interval of the function... | (2-2\ln 2, 3-2\ln 3] | 189 | 18 |
math | Team A and Team B are competing in a table tennis match, with each team having three players. The players for Team A are \( A_1, A_2, A_3 \), and the players for Team B are \( B_1, B_2, B_3 \). According to past competition statistics, the win probabilities between players are as follows:
\[
\begin{array}{|c|c|c|}
\hl... | E(\xi) = \frac{22}{15}, \quad E(\eta) = \frac{23}{15} | 303 | 30 |
math | How many even numbers between $1000$ and $9999$ have all distinct digits? | 2296 | 23 | 4 |
math | Ben rolls 6 fair 12-sided dice, with each die numbered from 1 to 12. What is the probability that exactly three of the dice show a prime number? | \frac{857500}{2985984} | 38 | 18 |
math | Let the solution set of the inequality $x(x-a-1)<0$ (where $a \in \mathbb{R}$) be $M$, and the solution set of the inequality $x^2-2x-3\leq0$ be $N$.
(Ⅰ) When $a=1$, find the set $M$;
(Ⅱ) If $M \subseteq N$, find the range of the real number $a$. | [-2, 2] | 98 | 6 |
math | Shown below are rows 1, 2, and 3 of Pascal's triangle.
\[
\begin{array}{ccccccc}
& & 1 & & 1 & & \\
& 1 & & 2 & & 1 & \\
1 & & 3 & & 3 & & 1
\end{array}
\]
Let \((d_i), (e_i), (f_i)\) be the sequence, from left to right, of elements in the 2010th, 2011th, and 2012th rows, respectively, with the leftmost element occurr... | \frac{1}{2} | 193 | 7 |
math | For a positive integer \( n \), let \( x_n \) be the real root of the equation \( n x^{3} + 2 x - n = 0 \). Define \( a_n = \left[ (n+1) x_n \right] \) (where \( [x] \) denotes the greatest integer less than or equal to \( x \)) for \( n = 2, 3, \ldots \). Then find \( \frac{1}{1005} \left( a_2 + a_3 + a_4 + \cdots + a... | 2013 | 138 | 4 |
math | Given the function f(x) = e^x(-x + lnx + a), where e is the base of the natural logarithm, a is a constant, and a ≤ 1.
(I) Determine whether the function f(x) has an extreme value point in the interval (1, e), and explain the reason.
(II) If a = ln2, find the minimum integer value of k such that f(x) < k is always true... | 0 | 94 | 1 |
math | Given $\alpha \in (0, \pi)$, if $\sin \alpha + \cos \alpha = \frac{\sqrt{3}}{3}$, calculate the value of $\cos^2 \alpha - \sin^2 \alpha$. | -\frac{\sqrt{5}}{3} | 52 | 10 |
math | Given $f\left(x\right)=\frac{\sqrt{{x}^{2}+1}+x-1}{\sqrt{{x}^{2}+1}+x+1}+ax^{3}+bx-8$ and $f\left(-2\right)=10$, find $f\left(2\right)=\_\_\_\_\_\_$. | -26 | 84 | 3 |
math | A rectangular container measuring $8\, \text{cm}$ by $4\, \text{cm}$ by $10\, \text{cm}$ is filled with a liquid $Y$ that does not mix with water. When poured into a large body of water, it forms a circular film of thickness $0.05$ cm on the surface. Find the radius, in centimeters, of the resulting circular film. | \sqrt{\frac{6400}{\pi}} | 90 | 13 |
math | Given the set $A=\{x|x^2-3x+2\leq0\}$, the set $B=\{y|y=x^2-4x+a\}$, and the set $C=\{x|x^2-ax-4\leq0\}$. Proposition $p$: $A\cap B\neq\emptyset$; Proposition $q$: $A\cap C=A$.
(1) If proposition $p$ is false, find the range of the real number $a$;
(2) If both propositions $p$ and $q$ are true, find the range of th... | [0,6] | 142 | 5 |
math | Both $a$ and $b$ are positive integers, and $b > 1$. When $a^b$ is the greatest possible value less than 500, and additionally, $a + b$ is even, what is the sum of $a$ and $b$? | 24 | 61 | 2 |
math | Given $p$ is an odd prime and $a$ is an integer, find
$$
\mid\left\{(x, y) \mid x^2 + y^2 \equiv a \pmod{p}, x, y \in \{0, 1, \ldots, p-1\} \right\}\mid
$$ | p+1 | 76 | 3 |
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