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 | Given the function \( f(x)=\cos x + m \left(x+\frac{\pi}{2}\right) \sin x \) where \( m \leqslant 1 \):
(1) Discuss the number of zeros of \( f(x) \) in the interval \( (-\pi, 0) \).
(2) If there exists \( t > 0 \) such that \( |f(x)| < -2x - \pi \) holds for \( x \in \left(-\frac{\pi}{2} - t, -\frac{\pi}{2}\right) \... | m = -1 | 143 | 4 |
math | If $1998$ is written as a product of two positive integers whose difference is as small as possible, calculate the difference between these two integers. | 17 | 32 | 2 |
math | The derivative of the function $f(x) = (x+1)(x^2-x+1)$ is calculated. | 3x^2 | 25 | 4 |
math | Given that \( F_{1} \) and \( F_{2} \) are the left and right foci of the hyperbola \( C: \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 \, (a > 0, b > 0) \), and that the circle with diameter \( F_{1}F_{2} \) intersects the hyperbola \( C \) at point \( P \) in the second quadrant, if the eccentricity of the hyperbola ... | \frac{4}{5} | 144 | 7 |
math | Simplify and evaluate
(Ⅰ) Evaluate \\( \dfrac{ \sqrt{3}\sin (- \dfrac{20}{3}\pi)}{\tan \dfrac{11}{3}\pi}-\cos \dfrac{13}{4}\pi\cdot\tan (- \dfrac{35}{4}\pi) \).
(Ⅱ) Evaluate: \\( \dfrac{\sqrt{1-2\sin {10}^{\circ }\cos {10}^{\circ }}}{\cos {10}^{\circ }-\sqrt{1-{\cos }^{2}{170}^{\circ }}} \)
(Ⅲ) If \\( \sin \theta, \... | - \dfrac{ \sqrt{7}}{2} | 225 | 13 |
math | The smallest positive integer n that satisfies √n - √(n-1) < 0.01 | 2501 | 24 | 4 |
math | Given the ellipse $C$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a > b > 0)$ with its right focus at $F(1,0)$, and the endpoints of the minor axis are $B_{1}$, $B_{2}$, and $\overrightarrow{FB_{1}} \cdot \overrightarrow{FB_{2}}=-a$.
1. Find the equation of the ellipse $C$;
2. A line $l$ passing through point $F$ with... | (0, \frac{1}{4}) | 193 | 10 |
math | Consider a sequence $y_1, y_2, y_3, \dots$ defined by
\begin{align*}
y_1 &= \sqrt[3]{5}, \\
y_2 &= (\sqrt[3]{5})^{\sqrt[3]{5}},
\end{align*}
and in general,
\[y_n = (y_{n - 1})^{\sqrt[3]{5}}\]
for $n > 1.$ What is the smallest value of $n$ for which $y_n$ is an integer? | 4 | 117 | 1 |
math | If the "mean reciprocal" of the first n terms of the sequence $\{a_n\}$ is $\frac{1}{2n-1}$, determine the general term formula of the sequence $\{a_n\}$. | 4n-3 | 47 | 4 |
math | Given a complex number $z = (a^{2} - 7a + 12) + (a^{2} - 5a + 6)i$ where $a \in \mathbb{R}$, for what value(s) of $a$ is $z$ a real number? For what value(s) of $a$ is $z$ an imaginary number? For what value(s) of $a$ is $z$ a pure imaginary number? | a = 4 \text{ for } z \text{ to be pure imaginary.} | 100 | 19 |
math | 50 schoolchildren and their parents are going on a tour to Nizhny Novgorod, some of whom drive cars. Each car can accommodate 6 people, including the driver. What is the minimum number of parents that need to be invited on the tour? | 10 | 55 | 2 |
math | Graph the set of points on the coordinate plane whose coordinates satisfy the equation \(4 x^{2} y^{2}=4 x y+3\). | y = \frac{3/2}{x} \quad \text{and} \quad y = \frac{-1/2}{x} | 31 | 31 |
math | Given $a \in \mathbb{R}$, let $f(x) = \ln x - ax$.
(I) Find the interval(s) where $f(x)$ is strictly increasing.
(II) Let $F(x) = f(x) + ax^2 + ax$. Determine if $F(x)$ has any extreme values (maximum or minimum). If they exist, find them and provide a reason. | \ln \sqrt{-\frac{1}{2a}} - \frac{1}{2} | 86 | 21 |
math | Given an arithmetic sequence with the first term $a$ and common difference $d$, determine the conditions under which it contains negative terms and only a finite number of negative terms. | a<0, d>0 | 35 | 7 |
math | It is known that point $M$ is a point on the parabola $y^2 = 4x$, and $F$ is the focus of the parabola. Point $A$ is on the circle $C$: $(x-4)^2 + (y-1)^2 = 1$. Find the minimum value of $|MA| + |MF|$. | 4 | 80 | 1 |
math | Express $1.\overline{024}$ as a reduced fraction, given that $0.\overline{008}$ is $\frac{1}{125}$. | \frac{128}{125} | 39 | 11 |
math | Given that point E is on the ellipse C: $$\frac {x^{2}}{a^{2}}$$ + $$\frac {y^{2}}{b^{2}}$$ = 1 (a > b > 0), a circle with center E is tangent to the x-axis at the right focus F₂ of the ellipse C, and intersects the y-axis at points A and B. Also, ∆ABE is an equilateral triangle with side length 2.
(I) Find the equati... | \frac {18}{5} | 185 | 8 |
math | If the digit 2 is placed after a three-digit number whose hundreds' digit is $a$, tens' digit is $b$, and units' digit is $c$, calculate the resulting four-digit number. | 1000a + 100b + 10c + 2 | 42 | 19 |
math | The graph of \[\frac{(x-2)^2}{a^2} + \frac{(y-3)^2}{b^2} = 1\] has its foci at $(2,\pm 5),$ while the graph of \[\frac{(x-2)^2}{a^2} - \frac{(y-3)^2}{b^2} = 1\] has its foci at $(\pm 7,3).$ Compute the value of $|ab|$. | \sqrt{\frac{609}{4}} | 108 | 11 |
math | Given the original prices of a coat, a hat, and a pair of gloves are $120, $30, and $50 respectively, calculate the percent of the total original prices that is the total amount saved when the coat is purchased at a 20% discount, the hat at a 40% discount, and the gloves at a 30% discount. | 25.5\% | 80 | 6 |
math | Given the sequence $\{a\_n\}$ that satisfies: $a\_1=1$, $a\_{n+1}+a\_n=(\frac{1}{3})^{n}$, $n∈N^{*}$, find $\lim\limits_{n→∞}a_{2n}$ $\_\_\_\_\_\_\_\_$. | - \frac{3}{4} | 77 | 8 |
math | In the plane rectangular coordinate system $xOy$, the line $l$ passes through the origin $O$ with an inclination angle of $\theta _{0}$. The parametric equations of the curve $C$ are $\left\{\begin{array}{l}{x=1+cosα}\\{y=\sqrt{3}+sinα}\end{array}\right.$ ($\alpha$ is the parameter), with the coordinate origin as the p... | (2\sqrt{3}, 4) | 195 | 10 |
math | Let $p:\exists x\in \left[-1,3\right]$, $x^{2}-3x-a \gt 0$. If $\neg p$ is a false proposition, then the range of real number $a$ is ______. | (-\infty, 4) | 53 | 8 |
math | Given a four-digit number \(\overline{abcd}\), when divided by 2, 3, 4, 5, 6, and 7, the remainders are all different and none of them are 0. Find the minimum value of \(\overline{abcd}\). | 1259 | 62 | 4 |
math | Given the function $f(x) = 2^x - 3x$, the number of zeros of the function $f(x)$ is ______. | 1 | 31 | 1 |
math | Consider two 101-digit numbers: $707,070,707,...,070,707$ and $606,060,606,...,060,606$. Find the sum of the units digit and the ten-thousand's digit of their product. | 6 | 72 | 1 |
math | The tangent line to the curve $f(x)=e^{x}$ at $x=0$ is tangent to the curve $g(x)=ax^{2}-a$ ($a\neq 0$). The equation of the line that passes through the tangent point and is perpendicular to this tangent line is __________. | x+y+1=0 | 66 | 6 |
math | $|5x^2-\tfrac25|\le|x-8|$ if and only if $x$ is in the interval $[a, b]$ . There are relatively prime positive integers $m$ and $n$ so that $b -a =\tfrac{m}{n}$ . Find $m + n$ . | 18 | 82 | 2 |
math | The graph of the function $y= \sqrt {3}\sin 2x-\cos 2x$ can be obtained by shifting the graph of the function $y=2\sin (2x+ \frac {\pi}{6})$ to the right by at least \_\_\_\_\_\_ units. | \frac {\pi}{6} | 66 | 7 |
math | Express the data "2684 billion" in scientific notation. | 2.684\times 10^{11} | 14 | 14 |
math | Find all positive integer pairs $(a, b)$ such that $\frac{2^{a+b}+1}{2^a+2^b+1}$ is an integer.**General**
Find all positive integer triples $(t, a, b)$ such that $\frac{t^{a+b}+1}{t^a+t^b+1}$ is an integer. | (2, 1, 1) | 86 | 9 |
math | Let \( p \) be a prime number. If there exists a positive integer \( n \) such that \( p \) divides \( n^{2} + 7n + 23 \), then the minimum value of \( p \) is ______. | 11 | 54 | 2 |
math | If two lines $p$ and $q$ have equations $y = -2x + 8$ and $y = -3x + 9$, what is the probability that a point randomly selected in the 1st quadrant and below $p$ will fall between $p$ and $q$? | 0.16 | 64 | 4 |
math | Calculate \(\operatorname{tg} \alpha\) if \(3 \operatorname{tg} \alpha - \sin \alpha + 4 \cos \alpha = 12\). | 4 | 40 | 1 |
math | The Tigers beat the Sharks 3 out of 5 times they initially played. Then, they played $N$ more games, and the Sharks ended up winning more than 90% of all the games played. Find the minimum possible value for $N$. | 26 | 53 | 2 |
math | In the equation $x + 3y = 3$, if $y$ is expressed in terms of $x$, then $x = \:$_____; if $x$ is expressed in terms of $y$, then $y = \:$_____. | \frac{3 - x}{3} | 53 | 9 |
math | Given the function $f(x)=\begin{cases} & \dfrac{1}{2}\sqrt{x^{2}+1},x\geqslant 0, \\ & -\ln (1-x),x < 0, \\ \end{cases}$ if the function $F(x)=f(x)-kx$ has exactly two zeros, then the range of $k$ is \_\_\_\_\_\_\_\_. | (\dfrac {1}{2},1) | 93 | 10 |
math | If \(a\), \(b\), \(c\), \(d\), \(e\), and \(f\) are integers for which \(1728x^3+64 = (ax^2 + bx + c)(dx^2 + ex + f)\) for all \(x\) values, determine the value of \(a^2+b^2+c^2+d^2+e^2+f^2\). | 23456 | 92 | 5 |
math | A point $P$ is chosen in the interior of $\triangle ABC$ such that when lines are drawn through $P$ parallel to the sides of $\triangle ABC$ , the resulting smaller triangles $t_{1}$ , $t_{2}$ , and $t_{3}$ in the figure, have areas $4$ , $9$ , and $49$ , respectively. Find the area of $\triangle ABC$ . [asy] size(200)... | 144 | 347 | 3 |
math | Let $f$ be a mapping from point set $A$ to point set $B$, such that for any $(x,y) \in A$, we have $f(x,y) = (y-x, y+x)$. Given a sequence of points in set $A$, $P_n(a_n, b_n)$ where $n \in \mathbb{N^{*}}$, and $P_{n+1}(a_{n+1}, b_{n+1}) = f(a_n, b_n)$ for each $n$. The point $P_1$ is given by $(0,2)$. Find the length ... | 2^{1007} | 155 | 7 |
math | How many ordered pairs of integers \((x, y)\) satisfy the equation
\[
x^{2} + y^{2} = 2(x + y) + xy?
\] | 6 | 40 | 1 |
math | The equation of a circle with center at $(1, -1)$ and radius $2$ is $(x-1)^2+(y+1)^2=4$. | (x-1)^2 + (y+1)^2 = 4 | 35 | 15 |
math | What is the area enclosed by the quadrilateral with vertices at $(6,1)$, $(1,6)$, $(4,3)$, and $(8,8)$? | 9 | 37 | 1 |
math | How many positive numbers are there among the first 100 terms of the sequence: $\sin 1^{\circ}, \sin 10^{\circ}, \sin 100^{\circ}, \sin 1000^{\circ}, \ldots ?$ | 3 | 61 | 1 |
math | What is the probability that two individuals, Person A and Person B, entering a subway station with three automatic ticket gates labeled \\(A\\), \\(B\\), and \\(C\\), select the same ticket gate? | \dfrac{1}{3} | 46 | 7 |
math | Suppose that $n$ is a positive integer such that in base $7$, then $n$ can be expressed as $\overline{ABC}_7$, and in base $11$, then $n$ can be expressed as $\overline{CBA}_{11}$. Find the largest possible value of $n$ in base $10$. | 247 | 74 | 3 |
math | If the price of a product increased from 5.00 reais to 5.55 reais, what was the percentage increase? | 11\% | 30 | 4 |
math | Given the curves $C\_1$ and $C\_2$, where $C\_1$: $\begin{cases} x = -4 + \cos t \\ y = 3 + \sin t \end{cases}$ ($t$ is a parameter) and $C\_2$: $\begin{cases} x = 6\cos \theta \\ y = 2\sin \theta \end{cases}$ ($\theta$ is a parameter).
1. Find the equations of $C\_1$ and $C\_2$ in the standard form and explain what t... | [1 + \sqrt{2}, +\infty) | 348 | 13 |
math | Jo adds up all the positive integers from 1 to 200. Meanwhile, Alex rounds every integer to its nearest multiple of 5 (rounding 2.5 up) before adding these values from 1 to 200. What is the positive difference between Jo's sum and Alex's sum? | 0 | 65 | 1 |
math | Given the function $f(x)=ax^{2}-(2a+1)x+a+1$.
$(1)$ If $a=2$, solve the inequality $f(x) \geqslant 0$ with respect to $x$;
$(2)$ If for $a \in [-2,2]$, $f(x) < 0$ always holds, find the range of the real number $x$. | (1, \dfrac{3}{2}) | 90 | 11 |
math | Consider the parabola $C$: $y^{2}=4x$ with focus $F$. The line $l$ passing through $F$ intersects $C$ at points $A$ and $B$. Given point $M(-1,2)$, if $\overrightarrow{MA} \cdot \overrightarrow{MB}=0$, then the slope of line $l$ is $k=$\_\_\_\_\_\_. | k=1 | 90 | 3 |
math | Given that $F_1$ and $F_2$ are the left and right foci of the ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, and point $P\left( 1, \frac{3}{2} \right)$ lies on it, with $PF_1 + PF_2 = 4$.
$(1)$ Find the standard equation of the ellipse $C$.
$(2)$ Let circle $O$ have $F_1$ and $F_2$ as its diameter. L... | \pm \frac{\sqrt{2}}{2} | 203 | 12 |
math | Determine the sum of the interior numbers from the eighth, ninth, and tenth rows of Pascal's Triangle, and then calculate the total sum of these sums. | 890 | 32 | 3 |
math | Given that $A$ is a moving point on the curve $C: 4x^{2}-y+1=0$, and there is a fixed point $M(-2,0)$. If $\overrightarrow{AT}=2\overrightarrow{TM}$, find the equation of the trajectory of the moving point $T$. | 4(3x+4)^{2}-3y+1=0 | 69 | 16 |
math | Consider an equilateral triangle $ABC$ with side length $3$. Right triangle $CBD$ is constructed outwardly on side $BC$ of triangle $ABC$ such that $CB = BD$ and $BCD$ is a right angle at $B$. Find $\sin^2\left(\angle CAD\right)$.
A) $\frac{3}{4}$
B) $\frac{1}{4}$
C) $\frac{1}{2}$
D) $\frac{\sqrt{2}}{2}$
E) $\f... | \frac{1}{2} | 122 | 7 |
math | Find the expanded form of the expression $(4x + 3)(2x - 7) + x$.
A) $8x^2 - 21x - 21$
B) $8x^2 - 22x - 21$
C) $8x^2 - 28x - 21$
D) $8x^2 - 21x - 14$ | 8x^2 - 21x - 21 | 92 | 13 |
math | First, a number \( a \) is randomly selected from the set \(\{1,2,3, \cdots, 99,100\}\), then a number \( b \) is randomly selected from the same set. Calculate the probability that the last digit of \(3^{a} + 7^{b}\) is 8. | \frac{3}{16} | 76 | 8 |
math | Determine the value of the expression $\sin 15^{\circ}\sin 105^{\circ}-\cos 15^{\circ}\cos 105^{\circ}$. | \frac{1}{2} | 44 | 7 |
math | Given the function $f(x) = \sqrt{x+3} + \frac{1}{x+2}$,
(1) Find the domain of the function;
(2) Find the value of $f(-3), f(\frac{2}{3})$. | \frac{8\sqrt{33} + 9}{24} | 56 | 17 |
math | Given \( f(x) + g(x) = \sqrt{\frac{1 + \cos 2x}{1 - \sin x}} \) for \( x \in \left( -\frac{\pi}{2}, \frac{\pi}{2} \right) \), where \( f(x) \) is an odd function and \( g(x) \) is an even function, determine the value of \( [f(x)]^2 - [g(x)]^2 \). | -2 \cos x | 102 | 5 |
math | Define the sequence $(y_n)$ by $y_1 = 150$ and $y_k = y_{k-1}^2 + 2y_{k-1}$ for all $k \geq 2$. Compute
\[
\frac{1}{y_1 + 1} + \frac{1}{y_2 + 1} + \frac{1}{y_3 + 1} + \dotsb.
\] | \frac{1}{151} | 99 | 9 |
math | If the power function $y=(m^{2}-2m-2)x^{-4m-2}$ is a decreasing function on $x \in (0,+\infty)$, then the value of the real number $m$ is \_\_\_\_\_\_. | m = 3 | 57 | 4 |
math | Given a quadratic function \( f(x) = ax^2 + bx + c \) where \( a \), \( b \), and \( c \) are real numbers and \( a \neq 0 \). The following conditions hold: (1) \( f(-1) = 0 \), (2) For any \( x \in \mathbb{R} \), \( x \leq f(x) \leq \frac{1}{2}(x^2 + 1) \). Find the value of \( a \). | a = \frac{1}{4} | 115 | 9 |
math | A sequence consists of $1500$ terms. Each term after the first is $2$ larger than the previous term. The sum of the $1500$ terms is $12000$. When every third term is added up starting with the first term, what is the sum? | 3000 | 64 | 4 |
math | A spherical soap bubble lands on a horizontal wet surface and forms a hemisphere. The volume of the hemisphere is known to be $36\pi$ cm³. Find the radius of the original bubble. | 3 \text{ cm} | 41 | 6 |
math | Suppose three whole numbers in ascending order have pairwise sums of 18, 23, and 27, respectively. Find the middle number. | 11 | 32 | 2 |
math | Given the digits 1, 2, 3, 7, 8, 9, find the smallest sum of two 3-digit numbers that can be obtained by placing each of these digits in one of the six boxes in the given addition problem, with the condition that each number must contain one digit from 1, 2, 3 and one digit from 7, 8, 9. | 417 | 85 | 3 |
math | Find the solution set for the following inequalities:
\\((1)2x^{2}+x-3 < 0\\)；
\\((2)x(9-x) > 0\\). | (0,9) | 41 | 5 |
math | What is the total number of digits used when the first 3002 positive even integers are written? | 11456 | 22 | 5 |
math | There are five distinct nonzero natural numbers; the smallest one is 7. If one of them is decreased by 20, and the other four numbers are each increased by 5, the resulting set of numbers remains the same. What is the sum of these five numbers? | 85 | 56 | 2 |
math | A bug starts at a vertex of a square. On each move, it randomly selects one of the three vertices where it is not currently located, and crawls along an edge of the square to that vertex. Given that the probability that the bug moves to its starting vertex on its eighth move is $m/n$, where $m$ and $n$ are relatively p... | 2734 | 83 | 4 |
math | Solve the inequality \(\left(\sqrt{x^{3}+2 x-58}+5\right)\left|x^{3}-7 x^{2}+13 x-3\right| \leqslant 0\). | x = 2 + \sqrt{3} | 53 | 10 |
math | Points $A$, $B$, $C$, $D$, and $E$ are located in 3-dimensional space with $AB= BC= CD= DE= EA= 2$ and $\angle ABC = \angle CDE = \angle
DEA = 90^\circ$. The plane of triangle $ABC$ is parallel to $\overline{DE}$. What is the area of triangle $BDE$? | 2 | 90 | 1 |
math | Given the function $f\left( x \right)=x\left( {{e}^{x}}+1 \right)$,
(1) Find the equation of the tangent line to the graph of the function $y=f\left( x \right)$ at the point $\left(0,f\left(0\right)\right)$;
(2) If the function $g\left( x \right)=f\left( x \right)-a{{e}^{x}}-x$, find the maximum value of the function... | g\left( 2 \right)=\left( 2-a \right){e}^{2} | 134 | 23 |
math | Let $a, b, c$ be three non-zero integers. It is known that the sums $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}$ and $\frac{b}{a}+\frac{c}{b}+\frac{a}{c}$ are integers. Find these sums. | 3 \text{ or } -3 | 76 | 8 |
math | Given that the exponential function $y=a^x$ is an increasing function on the real number line $\mathbb{R}$, the range of $a$ is __. | a>1 | 36 | 3 |
math | Find the equation of the circle that passes through point $A(3,2)$, has its center on the line $y=2x$, and is tangent to the line $y=2x+5$. | (x-2)^{2}+(y-4)^{2}=5 \text{ or } (x- \dfrac {4}{5})^{2}+(y- \dfrac {8}{5})^{2}=5 | 44 | 50 |
math | Let \\(\alpha\\) be an acute angle. If \\(\sin \left(\alpha+ \frac {\pi}{6}\right)= \frac {3}{5}\\), then \\(\cos \left(2\alpha- \frac {\pi}{6}\right)=\\) ______. | \frac {24}{25} | 61 | 9 |
math | In \\(\triangle ABC\\), let the sides opposite to angles \\(A\\), \\(B\\), and \\(C\\) be \\(a\\), \\(b\\), and \\(c\\) respectively. Let vector \\( \overrightarrow{m}=(\cos A+ \sqrt {2},\sin A)\\) and vector \\( \overrightarrow{n}=(-\sin A,\cos A)\\). If \\(| \overrightarrow{m}+ \overrightarrow{n}|=2\\),
\\((1)\\) fin... | 16 | 169 | 2 |
math | Given $a > 0$, $b > 0$, and $2a + b = 1$, find the maximum value of $$2 \sqrt {ab} - 4a^{2} - b^{2}.$$ | \frac{\sqrt{2} - 1}{2} | 48 | 13 |
math | Olivia's Omelette Oasis offers a range of omelettes that include various fillings: cheese, ham, mushrooms, peppers, onions, tomatoes, spinach, and olives. A customer can choose omelette egg base ranging from one to four eggs, and any combination of fillings. How many different kinds of omelettes can be ordered? | 1024 | 73 | 4 |
math | Calculate the limit of the numerical sequence:
$$
\lim _{n \rightarrow \infty}\left(\frac{3 n^{2}-6 n+7}{3 n^{2}+20 n-1}\right)^{-n+1}
$$ | e^{\frac{26}{3}} | 55 | 10 |
math | Determine the area of the smallest region enclosed by $y = |x|$ and $x^2 + y^2 = 9$. | \frac{9\pi}{4} | 29 | 9 |
math | The number of integers between 208 and 2008 ending with 1 is: | 180 | 21 | 3 |
math | How many numbers can you get by multiplying two or more distinct members of the set $\{1,2,4,7,13\}$ together? | 11 | 32 | 2 |
math | Given vectors $\overrightarrow{a}=(\sin \theta,\cos \theta-2\sin \theta)$ and $\overrightarrow{b}=(1,2)$, where $0 < \theta < \pi$.
$(1)$ If $\overrightarrow{a}\parallel \overrightarrow{b}$, find the value of $\sin \theta\cdot\cos \theta$;
$(2)$ If $|\overrightarrow{a}|=|\overrightarrow{b}|$, find the value of $\theta$... | \frac {3\pi}{4} | 108 | 9 |
math | Find the minimum value of
\[(13 - x)(11 - x)(13 + x)(11 + x) + 1000.\] | 424 | 36 | 3 |
math | Given the function $y=\sin (2x+\frac{π}{3})$, determine the horizontal shift required to obtain this graph from the graph of the function $y=\sin 2x$. | \frac{\pi}{6} | 41 | 7 |
math | Given \( f(x) = \left\{ \begin{array}{ll} x + \frac{1}{2}, & x \in \left[0, \frac{1}{2}\right) \\ 2(1-x), & x \in \left[\frac{1}{2}, 1\right] \end{array} \right. \), where \( f_{1}(x) = f(x) \) and \( f_{n}(x) = f(f_{n-1}(x)) \), find \( f_{27} \left( \frac{1}{5} \right) \). | \frac{4}{5} | 135 | 7 |
math | If the function $f(x)$ is an even function defined on $\mathbb{R}$, and it is monotonically decreasing on $(-\infty, 0]$, and $f(1) = 0$, then the range of $x$ for which $f(x) < 0$ is. | (-1, 1) | 67 | 6 |
math | If three unit vectors $\overrightarrow{a}$, $\overrightarrow{b}$, and $\overrightarrow{c}$ on the plane satisfy $|\overrightarrow{a}\cdot \overrightarrow{b}|=\frac{1}{2}$ and $|\overrightarrow{a}\cdot \overrightarrow{c}|=\frac{\sqrt{3}}{2}$, then the set of all possible values of $\overrightarrow{b}\cdot \overrightarro... | \left\{-\frac{\sqrt{3}}{2}, 0, \frac{\sqrt{3}}{2}\right\} | 100 | 30 |
math | Given $31 \cdot 36$ satisfies $0 \leq x \leq y$ and $\sqrt{1992}=\sqrt{x}+\sqrt{y}$, find the number of different integer pairs $(x, y)$. | 2 | 54 | 1 |
math | Evaluate the complex number sum $15 e^{3 \pi i/13} + 15 e^{24 \pi i/26}$ and express as $r e^{i \theta}.$ Enter the ordered pair $(r, \theta).$ | \left(30 \cos \left(\frac{9 \pi}{26}\right), \frac{15 \pi}{26}\right) | 56 | 34 |
math | Let $ a$, $ b$, $ c$, $ x$, $ y$, and $ z$ be real numbers satisfying:
\begin{align*}
17x + b y + c z &= 0 \\
a x + 29 y + c z &= 0 \\
a x + b y + 53 z &= 0.
\end{align*}
Assume that $ a \ne 17$ and $ x \ne 0$. Find the value of:
\[ \frac{a}{a - 17} + \frac{b}{b - 29} + \frac{c}{c - 53} \, ?\] | 1 | 151 | 1 |
math | Find the intercept on the x-axis of the line that is perpendicular to 4x + 3y - 7 = 0 and forms a triangle with the coordinate axes having an area of 6. | 4 | 42 | 1 |
math | In the decimal representation of the even number \( M \), only the digits \( 0, 2, 4, 5, 7, \) and \( 9 \) participate, and digits may repeat. It is known that the sum of the digits of the number \( 2M \) is 31, and the sum of the digits of the number \( M / 2 \) is 28. What values can the sum of the digits of the numb... | 29 | 112 | 2 |
math | Given $g(x)=mx+2$ and $f(x)=x^{2}-2x$, if for all $x\_1\in[-1,2]$ there exists an $x\_0\in[-1,2]$ such that $g(x\_1)=f(x\_0)$ holds true, then the range of values for $m$ is _____. | [-1, \frac{1}{2}] | 77 | 10 |
math | In triangle \(ABC\) with the ratio of the sides \(AB: AC = 5: 2\), the angle bisector of \(\angle BAC\) intersects side \(BC\) at point \(L\). Find the length of segment \(AL\) if the length of the vector \(2 \cdot \overrightarrow{AB} + 5 \cdot \overrightarrow{AC}\) is 2016. | 288 | 88 | 3 |
math | Determine the constant $k$ such that the quadratic equation $5kx^2 + 30x + 10 = 0$ has exactly one solution. Find this solution. | -\frac{2}{3} | 40 | 7 |
math | Using the six digits 0, 1, 2, 3, 4, 5:
(1) How many four-digit even numbers with no repeated digits can be formed?
(2) How many five-digit numbers that are multiples of 5 with no repeated digits can be formed? | 216 | 63 | 3 |
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