problem stringlengths 14 4.09k | solution stringlengths 1 802k | task_type stringclasses 3
values | problem_tokens int64 5 895 |
|---|---|---|---|
Convert the binary number 111.11 to decimal. | 7.75 | math | 14 |
Emily has a deck of cards that she arranges in a specific pattern repeating after every 17 cards: $$A, 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K, A, 2, 3, 4, A, 2, \cdots.$$ What is the 68th card in this sequence? | 4 | math | 91 |
Given real numbers $a$ and $b$, with $1 < a < b$, find the absolute difference between the average and the median of the four numbers $1$, $a+1$, $2a+b$, and $a+b+1$. | \left| \frac{1}{4} \right| | math | 52 |
Given $x_1$ and $x_2$ are the two roots of the quadratic equation $x^2 - 5x - 3 = 0$, find:
(1) $x_1^2 + x_2^2$
(2) $\frac{1}{x_1} - \frac{1}{x_2}$. | \frac{1}{x_1} - \frac{1}{x_2} = \frac{ \sqrt{37}}{3} | math | 78 |
Given a set of data $1$, $x$, $5$, $7$ with a unique mode and a median of $6$, find the average value of the set. | 5 | math | 36 |
Given $A=5\sqrt{2x+1}$, $B=3\sqrt{x+3}$, $C=\sqrt{10x+3y}$, where $A$ and $B$ are the simplest quadratic surds, and $A+B=C$, find the value of $\sqrt{2y-x^2}$. | 14 | math | 73 |
Consider the polynomial equations
\[90x^4 + ax^3 + bx^2 + cx + 15 = 0\]
and
\[15x^5 + dx^4 + ex^3 + fx^2 + gx + 90 = 0\].
These equations have a common rational root \( p \) which is not an integer and is negative. Determine \( p \). | -\frac{1}{3} | math | 86 |
Simplify and then evaluate the expression: $(x+2+\frac{4}{{x-2}})\div \frac{{{x^3}}}{{{x^2}-4x+4}}$, where $x$ is a suitable non-negative integer that satisfies the condition $x\leqslant 2$. | -1 | math | 66 |
The Euler family has four girls each aged $6$, two boys aged $10$ each, and a girl aged $16$. What is the mean (average) of the ages of the children? | \frac{60}{7} | math | 42 |
A right-angled isosceles triangle $ABP$ with $AB = BP = 4$ inches is placed inside a square $AXYZ$ with side length $8$ inches so that point $B$ is on side $AX$ of the square. The triangle is rotated clockwise about the midpoint of each side ($AB$, then $BP$, etc.), continuing this process along the sides of the square... | 12\pi\sqrt{2} | math | 156 |
Two chords of a circle, each equal to 12 and 16, are drawn through the ends of the diameter and intersect on the circumference. Find the distances from the center of the circle to these chords. | 8 \text{ and } 6 | math | 44 |
Find the eccentricity of the hyperbola $\frac{x^2}{4} - y^2 = 1$. | \frac{\sqrt{5}}{2} | math | 25 |
Given the fraction $\frac{3}{x-2}$, determine the range of values for $x$. | x\neq 2 | math | 22 |
Given that Xiao Wang bought $1000$ shares of a certain company's fund at $35$ yuan per share, and the price changes of the fund from Monday to Friday are as follows: $+4.5$, $+4$, $-1$, $-2.5$, $-6$ yuan, determine the price per share at the close of trading on Friday. | 34 \, \text{yuan} | math | 81 |
Given $ a_{i} \in \left\{0,1,2,3,4\right\}$ for every $ 0\le i\le 9$ and $6 \sum _{i = 0}^{9}a_{i} 5^{i} \equiv 1\, \, \left(mod\, 5^{10} \right)$ , find the value of $ a_{9} $. | 4 | math | 95 |
1. Given that the eccentricity of an ellipse is $\frac{\sqrt{7}}{4}$, and the distance from one endpoint of the minor axis to the right focus is 4, find the standard equation of the ellipse.
2. Given that a hyperbola passes through point A(6, -5), and one focus is (-6, 0), find the standard equation of the hyperbola. | \frac{x^2}{16} - \frac{y^2}{20} = 1 | math | 86 |
A pedestrian left point $A$ for a walk at a speed of $v$ km/h. After he had walked 6 km away from $A$, a cyclist left $A$ following him at a speed that was 9 km/h faster than the pedestrian's speed. When the cyclist caught up with the pedestrian, they turned back and returned together to $A$ at a speed of 4 km/h. At wh... | 6 \text{ km/h} | math | 101 |
Sector $OAB$ is a third of a circle of radius 5 cm. A circle is drawn inside this sector, tangent at three points. Find the number of centimeters in the radius of the inscribed circle. Express your answer in simplest radical form. | \frac{5(\sqrt{3}-1)}{2} \text{ centimeters} | math | 53 |
In quadrilateral $ABCD$, the diagonals $AC$ and $BD$ intersect at $O$. Given that $OB = 4$, $OD = 6$, $OA = 8$, $OC = 3$, and $AB = 6$, determine the length of $AD$. | \sqrt{166} | math | 62 |
Calculate the sum of the series $2 + 4 + 8 + 16 + 32 + \cdots + 512 + 1000$. | 2022 | math | 38 |
In the triangular pyramid $P-ABC$, $PA\bot $ plane $ABC$, $\triangle ABC$ is an isosceles triangle, where $AB=BC=2$, $\angle ABC=120{}^\circ $, and $PA=4$. The surface area of the circumscribed sphere of the triangular pyramid $P-ABC$ is __________. | 32\pi | math | 78 |
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} | math | 115 |
The yearly changes in the population census of a town for four consecutive years are, respectively, 25% increase, 25% increase, 25% decrease, 25% decrease. The net change over the four years, to the nearest percent, is: | -12 | math | 57 |
Given the quadratic equation $x^2 - 2px + (p^2 - 4) = 0$, find the difference between the larger root and the smaller root. | 4 | math | 37 |
Given the parabola $y=x^2-3mx+m+n$, for all real numbers $m$, find the conditions that $n$ must satisfy to ensure that the parabola intersects the x-axis. | n \leq -\frac{1}{9} | math | 44 |
(1) Given that $\alpha$ is an angle in the second quadrant and $\sin \alpha = \frac{3}{5}$, find the value of $\frac{1 + \sin \alpha + \cos \alpha + 2 \sin \alpha \cos \alpha}{1 + \sin \alpha + \cos \alpha}$.
(2) Given that $\alpha$ is an angle in the second quadrant, simplify the expression: $\cos \alpha \sqrt{\frac{... | \cos \alpha \sqrt{\frac{1 - \sin \alpha}{1 + \sin \alpha}} + \sin \alpha \sqrt{\frac{1 - \cos \alpha}{1 + \cos \alpha}} = \sin \alpha - \cos \alpha | math | 140 |
The number $(\sqrt{5} + 2)^{1997}$ rounded to 100 digits after the decimal point is calculated to find the 100th digit after the decimal point. | 0 | math | 45 |
Given a sequence $\{a_{n}\}$ that satisfies ${a_1}=2,(n-1){a_n}+n{a_{n-1}}=0$ for $n≥2, n∈{N^*}$. Find:<br/>
$(1)$ The general formula for the sequence $\{a_{n}\}$;<br/>
$(2)$ Let $S_{n}$ be the sum of the first $n$ terms of the sequence $\{a_{n}\}$, find $S_{2023}$. | 2024 | math | 115 |
Given that vector $\overrightarrow{a}\cdot(\overrightarrow{a}+2\overrightarrow{b})=0$, $|\overrightarrow{a}|=2$, $|\overrightarrow{b}|=2$, find the angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$. | \frac{2\pi}{3} | math | 66 |
Given a number X randomly selected in the interval [-2,3], determine the probability that X is less than or equal to 1. | \dfrac{3}{5} | math | 28 |
A teacher is assigning grades to a class of 12 students. The teacher gives out A's, B's, C's, and D's. Calculate the number of different ways in which the teacher can assign grades to all his students. | 16777216 | math | 49 |
Given an ellipse with its focus on the x-axis: $\frac{{x}^{2}}{4}+\frac{{y}^{2}}{m}=1$, and eccentricity $\frac{1}{2}$.<br/>$(1)$ Find the value of the real number $m$;<br/>$(2)$ A line $l$ passing through the point $P(0,2)$ intersects the ellipse at points $A$ and $B$, with the midpoint of line segment $AB$ being $M$.... | y = -x + 2 | math | 142 |
Let the function $f(x) = e^x(x^3 - 3x + 3) - ae^x - x$. If the inequality $f(x) \leq 0$ has a solution, then find the minimum value of the real number $a$. | 1 - \frac{1}{e} | math | 58 |
Find all values of the parameter \(a\), for each of which there exists a number \(b\) such that the system
\[
\left\{
\begin{array}{l}
x^{2}+y^{2}+2 a(a+y-x)=49 \\
y=15 \cos (x-b)-8 \sin (x-b)
\end{array}
\right.
\]
has at least one solution \((x, y)\). | a \in [-24, 24] | math | 97 |
Given that line $l$ passes through the point $(-3, 0)$ and is perpendicular to the line $2x - y - 3 = 0$, determine the equation of line $l$. | x + 2y + 3 = 0 | math | 43 |
What integer is closest to the value of $\sqrt[4]{15^4 + 10^4}$? | 15 | math | 25 |
Given the parametric equations of curve $C_1$ are $\begin{cases}x= \sqrt{t} \\ y= \dfrac{ \sqrt{3t}}{3} \end{cases}$ (where $t$ is the parameter). Establishing a polar coordinate system with the origin as the pole and the positive half-axis of $x$ as the polar axis, the polar equation of curve $C_2$ is $\rho=2$. Find t... | \left( \sqrt{3},1\right) | math | 117 |
The function \( f(x) \) is defined on the set of real numbers and satisfies the equation:
\[ f(x) - \frac{1}{2} f\left(\frac{x}{2}\right) = x^2 \]
Find \( f(x) \). | \frac{8}{7} x^{2} | math | 57 |
Given that $\cos \left( \frac{5\pi}{12}-\theta \right)= \frac{1}{3}$, find the value of $\sin \left( \frac{\pi}{12}+\theta \right)$. | \frac{1}{3} | math | 53 |
Given $\overrightarrow{a}=(\cos α,\sin α)$ and $\overrightarrow{b}=(\cos β,\sin β)(0 < α < \dfrac {π}{2}\,,\,- \dfrac {π}{2} < β < 0)$ with $|\overrightarrow{a}- \overrightarrow{b}|= \dfrac {2 \sqrt {5}}{5}$,
(I) find the value of $\cos (α-β)$;
(II) if $\cos β= \dfrac {12}{13}$, find the value of $\cos α$. | \dfrac {56}{65} | math | 129 |
Let $g(x)$ be a function defined piecewise as follows:
\[ g(x) = \left\{
\begin{array}{cl}
-x & \text{if } x\leq 0, \\
2x - 45 & \text{if } x>0.
\end{array}
\right. \]
If $a$ is negative, find $a$ such that $g(g(g(11))) = g(g(g(a)))$. | a = -34 | math | 99 |
One pen costs $2. Calculate the cost of 10 pens. | \$20 | math | 15 |
Find the equation of line $l$ that passes through point $A(-1,1)$ and is parallel to the line $x+3y+4=0$. | x+3y-2=0 | math | 35 |
Nathan and his two younger twin sisters' ages multiply to 72. Find the sum of their three ages. | 14 | math | 24 |
Find the values of $x$ and $y$ such that $\sqrt{4 - 5x + y} = 9$. | y = 77 + 5x | math | 28 |
Let $p, q, r$, and $s$ be positive real numbers such that
\[
\begin{array}{c@{\hspace{3pt}}c@{\hspace{3pt}}c@{\hspace{3pt}}c@{\hspace{3pt}}c}
p^2+q^2&=&r^2+s^2&=&2500,\\
pr&=&qs&=&1200.
\end{array}
\]
Compute the value of $\lfloor T \rfloor$, where $T=p+q+r+s$. | 140 | math | 127 |
Given that a line passes through point $P(2,0)$, and the chord length intercepted by the circle $(x-3)^{2}+(y-2)^{2}=4$ is $2 \sqrt {3}$, what is the equation of this line? | x=2 \text{ and } 3x-4y-6=0 | math | 58 |
The quadratic $2x^2 - 28x + 50$ can be written in the form $(x+b)^2+c$, where $b$ and $c$ are constants. What is $b+c$? | -55 | math | 48 |
For the number $25$, the first operation is defined as $2^{3}+5^{3}=133$, the second operation as $1^{3}+3^{3}+3^{3}=55$. If this process is repeated, the number obtained after the $2016$th operation is ______. | 250 | math | 71 |
Humanity discovers a system with 11 habitable planets, 5 of which are "Earth-like" and 6 are "Mars-like." Earth-like planets require 2 units of colonization each, and Mars-like planets need 1 unit each. If humanity has 14 total units available for colonization, in how many different ways can they occupy these planets? | 20 | math | 75 |
Several different positive integers are written on a blackboard. The product of the smallest two of them is 16. The product of the largest two of them is 225. What is the sum of all the integers written on the blackboard? | 44 | math | 52 |
Given positive numbers \(a, b, c, x, y, z\) that satisfy the equations \(cy + bz = a\), \(az + cx = b\), and \(bx + ay = c\), find the minimum value of the function
\[ f(x, y, z) = \frac{x^2}{1 + x} + \frac{y^2}{1 + y} + \frac{z^2}{1 + z}. \] | 1/2 | math | 98 |
Given a regular nonagon \(N\), with \(O\) as the center of its circumcircle, \(PQ\) and \(QR\) as adjacent sides of \(N\). Let \(A\) be the midpoint of \(PQ\), and \(B\) be the midpoint of the radius perpendicular to \(QR\). Find the angle between \(AO\) and \(AB\). | 30^{\circ} | math | 78 |
In the Cartesian coordinate plane $(xOy)$, point $P$ is a moving point on the graph of the function $f(x) = \ln x (x \geqslant 1)$. The tangent line $l$ to the graph at point $P$ intersects the $x$-axis at point $M$. The line perpendicular to $l$ passing through point $P$ intersects the $x$-axis at point $N$. Let $t$ b... | t_{\text{max}} = \frac{1}{2}\left(2e + \frac{1}{e} - e\right) = \frac{e^2 + 1}{2e} | math | 122 |
A smooth ball with a radius of 1 cm was dipped in red paint and set between two absolutely smooth concentric spheres with radii of 4 cm and 6 cm, respectively (the ball is outside the smaller sphere but inside the larger one). Upon contact with both spheres, the ball leaves a red mark. During its movement, the ball tra... | 83.25 | math | 129 |
Lottery. (For 7th grade, 3 points) It so happened that Absent-Minded Scientist has only 20 rubles left, but he needs to buy a bus ticket to get home. The bus ticket costs 45 rubles. Nearby the bus stop, instant lottery tickets are sold for exactly 10 rubles each. With a probability of $p = 0.1$, a ticket contains a win... | 0.19 | math | 141 |
A company named KayakVentures started the year by manufacturing 5 kayaks in January. Each following month, they tripled the production from the previous month. How many total kayaks did KayakVentures manufacture by the end of April? | 200 | math | 50 |
Let $ABCD$ be a rectangle where $AB = 2 \times AD$. Point $E$ is the midpoint of side $\overline{AB}$, and $\overline{DE}$ meets diagonal $\overline{AC}$ at point $F$. The area of quadrilateral $BFED$ is $50$. Calculate the area of rectangle $ABCD$. | 300 | math | 77 |
A bag contains four pieces of paper, each labeled with one of the digits $1$, $2$, $3$, or $5$, with no repeats. Three of these pieces are drawn, one at a time without replacement, to construct a three-digit number. Calculate the probability that the three-digit number is a multiple of $3$ and also an odd number. | \frac{1}{4} | math | 74 |
Right-angled triangle $ABP$, where angle $B$ is $90^\circ$ and sides $AB = 2$ inches and $BP = 4$ inches, is placed inside square $AXYZ$ with a side of length $6$ inches so that $B$ is on side $AX$. The triangle is rotated clockwise about $B$, then $P$, then $A$, and so on along the sides of the square until $P$ return... | 24\pi | math | 151 |
Classes A and B participated in the same subject exam, with 50 students in Class A and 40 students in Class B. The average score of Class A is 76 with a variance of 96, while the average score of Class B is 85 with a variance of 60. What is the variance of the scores of all 90 students from Classes A and B combined? | 100 | math | 85 |
If a number is randomly selected from the interval $(0, \frac{1}{2})$, calculate the probability that the selected number is less than $\frac{1}{3}$. | \frac{2}{3} | math | 38 |
Given an arithmetic sequence {a_n} with the first term a_1 and common difference d, the sum of its first n terms is denoted by S_n. If the line y = a_1x + m has two intersections with the circle (x-2)^2 + y^2 = 1 that are symmetric with respect to the line x + y + d = 0, find the value of S_n. | -n^2 + 2n | math | 88 |
The function $f(x)=A\sin \omega x (\omega > 0)$, for any $x$, satisfies $f\left( x- \frac{1}{2} \right)=f\left( x+ \frac{1}{2} \right)$, and $f\left( -\frac{1}{4} \right)=-a$. Determine the value of $f\left( \frac{9}{4} \right)$. | \sqrt{2}a\times \frac{\sqrt{2}}{2}=a | math | 99 |
Given that a random variable $\xi$ is normally distributed as $\xi \sim N(0, \sigma^2)$, and $P(-2 < \xi \le 0) = 0.4$, calculate the probability $P(\xi > 2)$. | 0.1 | math | 57 |
Given an infinite geometric sequence $\{a_n\}$ with the sum of its terms denoted as $S_n$, the first term $a_1=1$, and the common ratio $a- \frac{3}{2}$, and $\lim_{n\rightarrow \infty}S_n=a$, find the value of $a$. | 2 | math | 71 |
The terms of the sequence $(b_i)$ defined by $b_{n + 2} = \frac {b_n + 2017} {1 + b_{n + 1}}$ for $n \ge 1$ are positive integers. Find the minimum possible value of $b_1 + b_2$. | 2018 | math | 70 |
Let $f(x)$ be an odd function on $\mathbb{R}$ , such that $f(x)=x^2$ when $x\ge 0$ . Knowing that for all $x\in [a,a+2]$ , the inequality $f(x+a)\ge 2f(x)$ holds, find the range of real number $a$ . | a \in [\sqrt{2}, +\infty) | math | 86 |
Given a class with 4 morning lessons and 2 afternoon lessons, and the requirement of scheduling Mathematics in the morning and Physical Education in the afternoon, determine the total number of different ways to arrange the schedule of 6 lessons: Chinese, Mathematics, Politics, English, Physical Education, and Art. | 192 | math | 60 |
Determine all pairs of integers \((x, y)\) such that \(9xy - x^2 - 8y^2 = 2005\). | (63, 58), (-63, -58), (459, 58), (-459, -58) | math | 36 |
(1) Given that the domain of the function $y=f(x)$ is $[-1,2]$, find the domain of the function $y=f(1-x^{2})$.
(2) Given that the domain of the function $y=f(2x-3)$ is $(-2,1]$, find the domain of the function $y=f(x)$. | (-7,-1] | math | 79 |
A triangle has an area of $40$, one side of length $12$, and the median to that side of length $10$. Let $\theta$ be the acute angle formed by that side and the median. What is $\cos{\theta}$?
A) $\frac{1}{3}$
B) $\frac{2}{3}$
C) $\frac{\sqrt{5}}{3}$
D) $\frac{3}{2}$
E) $\frac{\sqrt{2}}{2}$ | \frac{\sqrt{5}}{3} | math | 109 |
Given the equation x<sub>1</sub>+x<sub>2</sub>+x<sub>3</sub>+x<sub>4</sub>=8, where x<sub>2</sub>=2, calculate the number of positive integer solutions to the equation. | 10 | math | 59 |
Knowing that \(\cos ^{6} x + \sin ^{6} x = a\), find \(\cos ^{4} x + \sin ^{4} x\). | \frac{1+2a}{3} | math | 41 |
Alex has 6 pieces of paper, each with a different math problem. In how many ways can he distribute these problems to his 12 friends, given that each friend can receive more than one problem? | 2,\!985,\!984 | math | 42 |
In triangle $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given $\overrightarrow{m}=(a,2b-c)$, $\overrightarrow{n}=(\cos A,\cos C)$, and $\overrightarrow{m}$ parallel to $\overrightarrow{n}$.
$(1)$ Find the measure of angle $A$.
$(2)$ If $b+c=7$ and the area of $\t... | 12 | math | 125 |
If the function $f(x)=2x^{3}-3mx^{2}+6x$ is increasing on the interval $(2,+\infty)$, determine the range of the real number $m$. | (-\infty, \dfrac {5}{2}] | math | 44 |
Given \(\log _{4}(x+2y) + \log _{4}(x-2y) = 1\), find the minimum value of \(|x| - |y|\). | \sqrt{3} | math | 44 |
How many four-digit numbers satisfy the property that the second digit is the average of the first and the third digits? | 450 | math | 23 |
Given $p$: $0 < x < 2$, $q$: $x < a$, if $p$ is a sufficient but not necessary condition for $q$, then the range of the real number $a$ is. | [2, +\infty) | math | 47 |
Calculate: $(1\frac{2}{3})^{4}\times (-\frac{3}{5})^{5}$. | -\frac{3}{5} | math | 27 |
The sum of two factors of the polynomial $x^{5n} + x^n + 1$ when $n = 1$ and $x = 2$ is. | 12 | math | 37 |
Given the coordinates of $A$, $B$, and $C$ are $(4,6)$, $(3,0)$, and $(k,0)$ respectively, find the value of $k$ that makes $\overline{AC} + \overline{BC}$ as small as possible. | 3 | math | 62 |
A circle has a radius of 3 units. There are many line segments of length 4 units that are tangent to the circle at their midpoints. Find the area of the region consisting of all such line segments.
A) $3\pi$
B) $5\pi$
C) $4\pi$
D) $7\pi$
E) $6\pi$ | 4\pi | math | 79 |
In tetrahedron \(ABCD\), it is known that:
\[AB = AC = 3, \quad BD = BC = 4,\]
with \(BD \perp\) plane \(ABC\).
Find the circumradius of the tetrahedron \(ABCD\). | 2 | math | 60 |
If the inequality $(a-2)x^{2}+2(a-2)x-4 < 0$ holds true for all $a \in \mathbb{R}$, then the range of values for $x$ is $\_\_\_\_\_\_\_\_\_$. | \{-2,0\} | math | 59 |
Given that Four times Dick's age plus twice Tom's age equals three times Harry's age, and Three times the square of Harry's age is equal to twice the square of Dick's age added to four times the square of Tom's age, and their respective ages are relatively prime to each other, find the sum of the cubes of their ages. | 349 | math | 70 |
Find the point \( M^{\prime} \) that is symmetric to the point \( M \) with respect to the line.
$$
\begin{aligned}
& M(0, -3, -2) \\
& \frac{x-1}{1} = \frac{y+1.5}{-1} = \frac{z}{1}
\end{aligned}
$$ | M' = (1, 1, 1) | math | 82 |
Given the lines $l_1: ax+2y-1=0$ and $l_2: x+by-3=0$, where the angle of inclination of $l_1$ is $\frac{\pi}{4}$, find the value of $a$. If $l_1$ is perpendicular to $l_2$, find the value of $b$. If $l_1$ is parallel to $l_2$, find the distance between the two lines. | \frac{7\sqrt{2}}{4} | math | 101 |
Let \(a\), \(b\), and \(c\) be three positive real numbers such that \(abc \geq 1\). Show that:
\[
\frac{1}{2+a}+\frac{1}{2+b}+\frac{1}{2+c} \leq 1
\]
Determine the case of equality. | a = b = c = 1 | math | 73 |
(1) Calculate the value of $0.0081^{ \frac {1}{4}}+(4^{- \frac {3}{4}})^2+(\sqrt {8})^{- \frac {4}{3}}-16^{0.75}$;
(2) Given $log_{32}9=p$, $log_{27}25=q$, express $lg5$ in terms of $p$ and $q$. | \frac {15pq}{15pq+4} | math | 96 |
Given the function $f(x) = \cos^{2}x - \sin^{2}x + \frac{1}{2}, x \in (0, \pi)$.
1. Find the interval where $f(x)$ is monotonically increasing.
2. Let $\triangle ABC$ be an acute triangle with side $a = \sqrt{19}$ opposite to angle $A$ and side $b = 5$ opposite to angle $B$. If $f(A) = 0$, find the area of $\triangle A... | \frac{15\sqrt{3}}{4} | math | 114 |
George wants to buy pencils for his class. He has $\$9.30$ and each pencil costs $\$1.05$, tax included. If he buys more than 8 pencils, he gets a 10% discount on the total cost. How many pencils can George buy at most? | 9 | math | 63 |
The following table shows several pairs of values of the independent variable $x$ and the function $y$ of a quadratic function:<br/>
| $x$ | $\ldots $ | $0$ | $1$ | $\frac{3}{2}$ | $2$ | $\ldots $ |
|-------|-----------|-----|-----|---------------|-----|-----------|
| $y$ | $\ldots $ | $-\frac{7}{2}$ | $\frac{1}{2... | y=-2\left(x-\frac{3}{2}\right)^2+1 | math | 176 |
Suppose that $f$ is a function on the interval $[1,3]$ such that $-1\le f(x)\le 1$ for all $x$ and $\displaystyle \int_1^3f(x)\,dx=0.$ How large can $\displaystyle\int_1^3\frac{f(x)}x\,dx$ be? | \ln\left(\frac{4}{3}\right) | math | 92 |
Given the vectors $\overrightarrow{m}=(2\sin \omega x, \cos ^{2}\omega x-\sin ^{2}\omega x)$ and $\overrightarrow{n}=( \sqrt {3}\cos \omega x,1)$, where $\omega > 0$ and $x\in R$. If the minimum positive period of the function $f(x)= \overrightarrow{m}\cdot \overrightarrow{n}$ is $\pi$,
(I) Find the value of $\omega$.... | -\frac{3}{2} | math | 163 |
For a certain hyperbola,
\[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1,\]
where \(a \neq b\), the angle between the asymptotes is \(90^\circ\). Find \(\frac{a}{b}\). | 1 | math | 68 |
The store owner bought 2000 pens at $0.15 each and plans to sell them at $0.30 each, calculate the number of pens he needs to sell to make a profit of exactly $150. | 1000 | math | 50 |
Three people, A, B, and C, depart from location $A$ to location $B$. Person A departs at 8:00, person B at 8:20, and person C at 8:30. They all travel at the same speed. Ten minutes after person C departs, the distance of person A to location $B$ is exactly half the distance of person B to location $B$, and at this mom... | 2418 | math | 125 |
$(1)$ Calculate: $|10-(-6)|=\_\_\_\_\_\_$.
$(2)$ If $|m-3|=5$, find the value of $m$.
$(3)$ Given $|m-4|+|m+2|=6$, list all integers $m$ that satisfy the condition. | -2, -1, 0, 1, 2, 3, 4 | math | 71 |
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