problem stringlengths 14 4.09k | solution stringlengths 1 802k | task_type stringclasses 3
values | problem_tokens int64 5 895 |
|---|---|---|---|
Given the universal set $M=\{1,m,3+(m^{2}-5m-6)i\}$ and the set $N=\{x|x^{2}-2x-3=0\}$, if $M∩N=\{3\}$, find $M∪N$. | \{-1,1,3,3-12i\} | math | 62 |
There exists a sequence of numbers $a_{1}$, $a_{2}$, $a_{3}$, $\ldots$, $a_{n}$, $\ldots$ that satisfies the following conditions: $a_{1}=\frac{1}{2}$, $a_{n}=\frac{1}{1-{a}_{n-1}}$ (for $n\geqslant 2$ and $n$ is an integer). Find:<br/>
$(1) a_{2}=$______;<br/>
$(2) a_{1}+a_{2}+a_{3}+\ldots +a_{2023}=\_\_\_\_\_\_$. | 1011.5 | math | 149 |
Given two lines \( l_{1}: 2x + y = 3 \) and \( l_{2}: x + 4y = 2 \). The line \( l \) passes through the intersection point \( P \) of \( l_{1} \) and \( l_{2} \), as well as the origin. Find the equation of the line \( l \). | x - 10y = 0 | math | 81 |
In a crown, five diamonds are embedded at five positions, equally spaced around a circle. If there are three different colors of diamonds to choose from, how many different ways are there to embed the diamonds? Assume that the five positions are indistinguishable from each other. | 51 | math | 54 |
The probability of A not losing is $\dfrac{1}{3} + \dfrac{1}{2}$. | \dfrac{1}{6} | math | 25 |
Lines $p$ and $q$ are parallel. $m\angle E = 150^\circ$, and $m\angle G = 70^\circ$. What is the number of degrees in $m\angle F$?
[asy]
size(100); real h = 1.2; currentpen = fontsize(10pt);
draw(Label("$p$",Relative(1)),(0,0)--(1,0),E);
draw(Label("$q$",Relative(1)),(0,-h)--(1,-h),E);
draw((0,-h)--h/2*(cos(150*pi/180... | 110^\circ | math | 289 |
Let α be in the interval $(-\frac{\pi}{2}, \frac{\pi}{2})$, and given that $\sin\alpha = -\frac{3}{5}$, find the value of $\cos(-\alpha)$. | \frac{4}{5} | math | 51 |
Solve in $\mathbb{C}^{3}$ the system
$$
\left\{\begin{array}{l}
a^{2}+a b+c=0 \\
b^{2}+b c+a=0 \\
c^{2}+c a+b=0
\end{array}\right.
$$ | (0, 0, 0) \ \text{and} \ \left( -\frac{1}{2}, -\frac{1}{2}, -\frac{1}{2} \right) | math | 68 |
Given a sequence $\{a_n\}$ with $a_1=1$ and $a_{n+1}= \frac{2a_n}{a_n+2}$, find the value of $a_{10}$. | \frac{2}{11} | math | 49 |
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 | math | 64 |
Compute the sum $\frac{4}{3} + \frac{8}{9} + \frac{18}{27} + \frac{40}{81} + \frac{88}{243} - 5$.
**A** $-\frac{1220}{972}$
**B** $-\frac{610}{486}$
**C** $-\frac{305}{243}$
**D** $-\frac{152.5}{121.5}$
**E** $-\frac{76.25}{60.75}$ | -\frac{305}{243} | math | 142 |
Given $\log 216$, evaluate the expression in terms of logarithms of prime factors. | 3 \log 6 | math | 20 |
Determine the range of the function $f(x) = \log_{3}(8^{x}+1)$. | (0,\infty) | math | 25 |
Given $\sin(x + \frac{\pi}{4}) = -\frac{3}{5}$, find the value of $\sin 2x$. | \frac{-7}{25} | math | 32 |
Given the vectors $\overrightarrow{a}=(\cos x, \sin x)$, $\overrightarrow{b}=(\cos x + 2\sqrt{3}, \sin x)$, and $\overrightarrow{c}=(0, 1)$, where $x \in \mathbb{R}$:
1. If $\overrightarrow{a} \perp \overrightarrow{c}$, find the value of $\cos 2x$.
2. If the function $f(x) = \overrightarrow{a} \cdot (\overrightarrow{b}... | 5 | math | 149 |
If $1998$ is written as a product of two positive integers whose difference is as small as possible, then the difference is | 17 | math | 28 |
Given a line $l$ passes through point $P(2,3)$, and is intersected by two parallel lines $l_{1}: 3x+4y-7=0$ and $l_{2}: 3x+4y+8=0$ to form a line segment of length $d$.
$(1)$ Find the minimum value of $d$;
$(2)$ Calculate the value of $d$ when line $l$ is parallel to the x-axis. | 5 | math | 103 |
What is the area of the region enclosed by the graph of the equation $x^2 - 20x + 5y + 100 = 25 + 15y - y^2$ that lies above the line $y = x - 6$? | 25\pi | math | 60 |
Given that $49$ of the first $50$ counted were red and $7$ out of every $8$ counted thereafter were red, find the maximum value of $n$, given that $90$% or more of the balls counted were red. | 210 | math | 55 |
Given the arithmetic sequence $\{a_{n}\}$ with a common difference $d < 0$, let $S_{n}$ represent the sum of its first $n$ terms. It is known that $3 \sqrt {5}$ is the geometric mean between $-a_{2}$ and $a_{9}$, and $S_{10}=20$. Find the value of $d$. | d = -2 | math | 84 |
Define the sequence $(x_{n})$ : $x_{1}=\frac{1}{3}$ and $x_{n+1}=x_{n}^{2}+x_{n}$ . Find $\left[\frac{1}{x_{1}+1}+\frac{1}{x_{2}+1}+\dots+\frac{1}{x_{2007}+1}\right]$ , wehere $[$ $]$ denotes the integer part. | 2 | math | 109 |
A group of 12 friends decides to form a committee of 5. Calculate the number of different committees that can be formed. Additionally, if there are 4 friends who refuse to work together, how many committees can be formed without any of these 4 friends? | 56 | math | 55 |
A point $P$ lies in the same plane as a given square of side $2$. Let the vertices of the square, taken counterclockwise, be $A, B, C$ and $D$. Also, let the distances from $P$ to $B, C$ and $D$, respectively, be $v, w$ and $t$. What is the greatest distance that $P$ can be from $A$ if $v^2 + w^2 = t^2$?
A) $\sqrt{8}$
... | \sqrt{10} | math | 139 |
The volume of the parallelepiped $A B C D A_1 B_1 C_1 D_1$ is $V$. Find the volume of the pyramid $A B C C_1$. | \frac{1}{6} V | math | 43 |
What is the hundreds digit of $(25! - 20!)$? | 0 | math | 17 |
How many positive cubes divide $3!\cdot 5!\cdot 7!\,$? | 6 | math | 20 |
If $f(x) = x^2 + 2\int_{0}^{1}f(x)dx,$ then $\int_{0}^{1}f(x)dx=$ | -\frac{1}{3} | math | 39 |
Find all positive integers $n$ such that $1! + 2! + \ldots + n!$ is a perfect square. | 1 \text{ and } 3 | math | 29 |
The "Tuning Day Method" is a procedural algorithm for seeking precise fractional representations of numbers. Suppose the insufficient approximation and the excessive approximation of a real number $x$ are $\dfrac{b}{a}$ and $\dfrac{d}{c}$ ($a,b,c,d \in \mathbb{N}^*$) respectively, then $\dfrac{b+d}{a+c}$ is a more accu... | \dfrac{22}{7} | math | 162 |
Find the area of the region in the \(xy\)-plane satisfying \(x^{6} - x^{2} + y^{2} \leq 0\). | \frac{\pi}{2} | math | 36 |
Bill’s age is one third larger than Tracy’s age. In $30$ years Bill’s age will be one eighth larger than Tracy’s age. How many years old is Bill? | 24 | math | 40 |
Given a complex number $Z$ that satisfies $Z(i - 1) = 2i$ (where $i$ is the imaginary unit), determine the conjugate of $Z$. | 1 + i | math | 39 |
The distance between the two intersections of $x=y^4$ and $x+y^2=1$ is $\sqrt{u+v\sqrt5}$. Find the ordered pair, $(u,v)$. | (-2,2) | math | 44 |
Find the domain of the following functions (the result should be expressed as a set or interval):
(1) $$y= \frac { \sqrt {x-4}}{|x|-5}$$
(2) $y=\log_{a}(2-x)$ ($a>0$ and $a\neq 1$)
(3) $$y= \sqrt {1-\left( \frac {1}{2}\right)^{x}}$$． | [0, +\infty) | math | 99 |
A thousand points form the vertices of a convex polygon with 1000 sides. Inside this polygon, there are another 500 points placed such that no three of these 500 points are collinear. The polygon is triangulated in such a way that all of these 1500 points are vertices of the triangles, and none of the triangles have an... | 1998 | math | 90 |
The original price of "Fun Math Stories" is 25 yuan, and the current price is 20 yuan. The current price is % of the original price. | 80\% | math | 38 |
Compute
$$\sum_{k=1}^{1000} k(\lceil \log_{\sqrt{2}}{k}\rceil- \lfloor\log_{\sqrt{2}}{k} \rfloor).$$ | 499477 | math | 53 |
In triangle $PQR$, the angle bisectors are $PL$, $QM$, and $RN$, which intersect at the incenter $I$. If $\angle PRQ = 24^\circ$, find the measure of $\angle PIM$, in degrees. | 78^\circ | math | 54 |
A pedestrian departed from point \( A \) to point \( B \). After walking 8 km, a second pedestrian left point \( A \) following the first pedestrian. When the second pedestrian had walked 15 km, the first pedestrian was halfway to point \( B \), and both pedestrians arrived at point \( B \) simultaneously. What is the ... | 40 | math | 84 |
The sum of the roots of the equation $x^2+3ax+3a+1=0$ in terms of their tangents is equal to $\tan(\alpha+\beta)$. | \frac{\pi}{4} | math | 40 |
The numbers $203$ and $298$ divided with the positive integer $x$ give both remainder $13$ . Which are the possible values of $x$ ? | 19 \text{ or } 95 | math | 48 |
A straight line is tangent to the graphs of the functions $y=\ln x$ and $y=e^{x}$ at points $P(x_{1}, y_{1})$ and $Q(x_{2}, y_{2})$ respectively. The value of $(1-e^{y_1})(1+x_2)$ is ______. | 2 | math | 70 |
If line $l$ passes through the point $( \sqrt {3}, -3)$ and has an inclination angle of $30°$, determine the equation of line $l$. | y+3= \frac {\sqrt{3}}{3}(x- \sqrt{3}) | math | 37 |
Let $S_n$ be the sum of the first $n$ terms of the sequence $\{a_n\}$, given that $a_1=3$, $a_{n+1}=2S_n+3$.
$(1)$ Find the general formula for the sequence $\{a_n\}$.
$(2)$ Let $b_n=(2n-1)a_n$, find the sum of the first $n$ terms of the sequence $\{b_n\}$, denoted as $T_n$. | T_n=3+(n-1)\cdot3^{n+1} | math | 108 |
How many lattice points are exactly twice as close to $(0,0)$ as they are to $(15,0)$ ? (A lattice point is a point $(a,b)$ such that both $a$ and $b$ are integers.) | 12 | math | 59 |
Given a regular tetrahedron with an edge length of $1$, calculate its volume. | \frac{\sqrt{2}}{12} | math | 19 |
Given a geometric sequence $\{a_n\}$ with the first term $a_1$ and common ratio $q$, its general term $a_n$ is \_\_\_\_\_\_. | a_{1}q^{n-1} | math | 40 |
Given the equation \( m\left(x^{2}+y^{2}+2 y+1\right) = (x - 2y + 3)^{2} \), determine the range of values for \( m \) such that the equation represents an ellipse. | (5,+\infty) | math | 58 |
A rectangular yard has two flower beds in the form of congruent isosceles right triangles. The rest of the yard is a trapezoid. The parallel sides of the trapezoid measure $18$ and $30$ meters. Determine the fraction of the yard occupied by the flower beds. | \frac{1}{5} | math | 65 |
Given proposition p: The real number $m$ satisfies $m^2 - 7ma + 12a^2 < 0$ ($a > 0$), and proposition q: The equation $\frac{x^2}{m-1} + \frac{y^2}{2-m} = 1$ represents an ellipse with foci on the y-axis. If $\neg p$ is a necessary but not sufficient condition for $\neg q$, find the range of values for the real number ... | \left[\frac{1}{3}, \frac{3}{8}\right] | math | 108 |
The bottoms of two vertical poles are 20 feet apart on a flat ground. One pole is 8 feet tall and the other is 18 feet tall. Simultaneously, the ground between the poles is sloped, with the base of the taller pole being 2 feet higher than the base of the shorter pole due to the slope. Calculate the length in feet of a ... | \sqrt{544} | math | 95 |
How many different rectangles with sides parallel to the grid can be formed by connecting four of the dots in a $5\times 5$ square array of dots? (Two rectangles are different if they do not share all four vertices.) | 100 | math | 47 |
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} | math | 125 |
Tyrone had $97$ marbles and Eric had $11$ marbles. Tyrone then gave some of his marbles to Eric so that Tyrone ended with twice as many marbles as Eric. Calculate the number of marbles Tyrone gave to Eric. | 25 | math | 57 |
Calculate the value of $-1+2-3+4-5+6+\ldots -2021+2022-2023$. | -1012 | math | 36 |
Given the circle O: $x^2+y^2=1$ and a fixed point A(2, 1), a point P outside circle O draws a tangent PQ to circle O, satisfying $|PQ|=|PA|$. If the circle with center P and radius r has a common point with circle O, calculate the minimum value of r. | \frac{3\sqrt{5}}{5} - 1 | math | 74 |
In the Cartesian coordinate system $xOy$, with $O$ as the pole and the positive half-axis of the $x$-axis as the polar axis, establish a polar coordinate system. Given that the parametric equations of line $l$ are $\begin{cases}x=-1+t \\ y=1+t\end{cases}$ (with $t$ as the parameter), and the general equation of curve $... | S_{\triangle PAB} = \frac{1}{2} \times 3\sqrt{2} \times 2\sqrt{2} = 6 | math | 204 |
In the Mathematical Competition of HMS (Hellenic Mathematical Society) take part boys and girls who are divided into two groups : *Juniors* and *seniors.*The number of the boys taking part of this year competition is 55% of the number of all participants. The ratio of the number of juniors ... | \frac{11}{9} | math | 150 |
Given a geometric sequence $\{a_n\}$ satisfies $a_2a_6=64$ and $a_3a_4=32$, find the common ratio $q=\boxed{\text{answer}}$; and the sum $a_1^2+a_2^2+\ldots+a_n^2=\boxed{\text{answer}}$. | \frac{4^n-1}{3} | math | 78 |
Given $2^{202} +202$ is divided by $2^{101}+2^{51}+1$, calculate the remainder. | 201 | math | 36 |
If the monotonically decreasing interval of the function $f(x)=a(x^{3}-x)$ is $\left(- \frac{ \sqrt{3}}{3}, \frac{ \sqrt{3}}{3}\right)$, then the range of values for $a$ is $\_\_\_\_\_\_\_\_\_\_.$ | a > 0 | math | 72 |
The solution interval for the equation $\log_{3}x + x = 3$ is $(2,3)$. | (2,3) | math | 25 |
If \( A \) and \( B \) are relatively prime numbers, then the greatest common divisor of \( A^{3} + B^{3} \) and \( A^{2} + B^{2} \) is either 1 or 2. | 1 \text{ or } 2 | math | 54 |
Let $\square B_1B_2B_3B_4$ be a square, and define $B_{n+4}$ to be the midpoint of the line segment $B_nB_{n+1}$ for all positive integers $n$. Find the measure of $\measuredangle B_{101}B_{102}B_{100}$.
A) $45^\circ$
B) $60^\circ$
C) $90^\circ$
D) $120^\circ$
E) $135^\circ$ | 90^\circ | math | 122 |
Ice-cream-o-rama has expanded its range of basic flavors to include mint, along with the traditional chocolate, vanilla, and strawberry. Using these four basic flavors, they now make "new" flavors by blending together five scoops of ice cream. Different proportions of the basic flavors create different new flavors.
Ho... | 56 | math | 80 |
Given the hyperbola $C: \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 (a > 0, b > 0)$ with an eccentricity of $\sqrt{5}$, and the distance from point $P(\sqrt{a^{2} + b^{2}}, 0)$ to its asymptotes is $8$, determine the length of the real axis of $C$. | 8 | math | 98 |
Find the greatest integer $A$ for which in any permutation of the numbers $1, 2, \ldots , 100$ there exist ten consecutive numbers whose sum is at least $A$ . | 505 | math | 49 |
Calculate $\frac{(1+i)^{2}}{(1-i)^{3}}$. | -\frac{1}{2}-\frac{1}{2}i | math | 18 |
Given triangle $\text{ABC}$ with internal angles $\text{A}$, $\text{B}$, and $\text{C}$, and their respective opposite sides $a$, $b$, and $c$. Let $a=4$, $c=\sqrt{13}$, and $\sin\text{A}=4\sin\text{B}$.
(1) Find the length of side $b$.
(2) Find the measure of angle $\text{C}$. | \text{C}=60^\circ | math | 102 |
In the first year of high school, Hua Hua found that for a finite set $A$, it is generally possible to find two non-empty sets $A_{1}$ and $A_{2}$, satisfying $A_{1}$ ⋂ $A_{2}$ = $\varnothing$, and $A_{1} \cup A_{2}$ = $A$. The set $\{A_{1}$, $A_{2}\}$ is called a "partition set" of $A$. If there are $n$ ($n \geq 2$, $... | 2^{n-1} - 1 | math | 159 |
The time right now is exactly midnight. What time will it be in 2405 minutes? | 4\!:\!05 \text{ p.m.} | math | 21 |
John's bookshelf holds seven books with the following widths in centimeters: $7$, $\frac{3}{4}$, $1.25$, $3$, $8$, $2.5$, and $12$. What is the average book width, in centimeters?
A) $\frac{241}{49}$ cm
B) $5$ cm
C) $4$ cm
D) $3$ cm
E) $3.5$ cm | \frac{241}{49} | math | 102 |
A regular triangle \( \triangle ABC \) with side length 3 is divided into three equal parts on each side. Parallel lines to each side are drawn through these division points, creating a total of 10 intersection points within and on the edges of the triangle, known as grid points. Determine the minimum number \( n \) of... | 5 | math | 103 |
Soda is now sold in packs of 8, 15, and 30 cans. Find the minimum number of packs needed to buy exactly 130 cans of soda. | 6 | math | 39 |
Given $$\frac{\cos(\pi - 2\alpha)}{\sin(\alpha - \frac{\pi}{4})} = -\frac{\sqrt{2}}{2},$$ find the value of $\sin 2\alpha$. | -\frac{3}{4} | math | 51 |
The number of integers between 208 and 2008 ending with 1 is: | 180 | math | 21 |
Observe the following equations about converting repeating decimals to fractions: (Note: the numbers with dots on top) $0.\dot{3} = \frac{3}{9} = \frac{1}{3}$, $1.\dot{1}\dot{8} = \frac{18}{99} = \frac{2}{11}$, $0.\dot{3}\dot{5}\dot{2} = \frac{352}{999}$, $0.000\dot{5}\dot{9} = \frac{1}{1000} \times \frac{59}{99} = \fr... | \frac{7}{30} | math | 178 |
In the diagram, triangles $ABC$ and $CBD$ are isosceles. Triangle $ABC$ has $\angle BAC = 2 \times \angle ABC$. The perimeter of $\triangle CBD$ is $21$, the perimeter of $\triangle ABC$ is $26$, and the length of $BD$ is $9$. What is the length of $AB$? | 10 | math | 80 |
Let $\mathbf{p}$ be the projection of vector $\mathbf{v}$ onto vector $\mathbf{u},$ and let $\mathbf{q}$ be the projection of $\mathbf{p}$ onto $\mathbf{u}.$ If $\frac{\|\mathbf{p}\|}{\|\mathbf{v}\|} = \frac{3}{4},$ then find $\frac{\|\mathbf{q}\|}{\|\mathbf{u}\|}.$ | \frac{9}{16} | math | 106 |
The minimum value of the function $y=\sin x+ \sqrt{3}\cos x$ is to be found. | -2 | math | 25 |
Given that $|\overset{→}{a}| = 1$, $|\overset{→}{b}| = 2$, the angle between $\overset{→}{a}$ and $\overset{→}{b}$ is $120^{\circ}$, and $\overset{→}{a} + \overset{→}{b} + \overset{→}{c} = \overset{→}{0}$, find the angle between $\overset{→}{a}$ and $\overset{→}{c}$. | 90^{\circ} | math | 117 |
The point $O$ is the center of the circle circumscribed about $\triangle ABC$, with $\angle BOC = 150^{\circ}$ and $\angle AOB = 130^{\circ}$, as shown. What is the degree measure of $\angle ABC$? | 40^\circ | math | 62 |
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 | math | 101 |
In the arithmetic sequence $\{a\_n\}$, $a\_1 = -60$ and $a\_17 = -12$.
(1) Find the nth term, $a\_n$;
(2) Find the sum of the absolute values of the first 30 terms in this sequence. | 765 | math | 68 |
Find all values of the parameter $m$ such that the equations $x^2 = 2^{|x|} + |x| - y - m = 1 - y^2$ have only one root. | m = 0 | math | 50 |
A smooth sphere with a radius of 1 cm was dipped in blue paint and placed between two absolutely smooth concentric spheres with radii of 4 cm and 6 cm, respectively (the sphere was outside the smaller sphere but inside the larger one). When in contact with both spheres, the sphere leaves a blue trace. While moving, the... | 38.25 | math | 128 |
Let $x,$ $y,$ and $z$ be angles such that
\begin{align*}
\sin x &= \cot y, \\
\sin y &= \cot z, \\
\sin z &= \cot x.
\end{align*}
Find the minimum possible value of $\cos x.$ | \sqrt{\frac{3 - \sqrt{5}}{2}} | math | 64 |
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} | math | 41 |
Find the minimum value of
\[(13 - x)(11 - x)(13 + x)(11 + x) + 1000.\] | 424 | math | 36 |
Given an arithmetic sequence $\{a_n\}$ with the first term $a_1=1$ and a common difference $d>0$, and a geometric sequence $\{b_n\}$, satisfying $b_2=a_2$, $b_3=a_5$, $b_4=a_{14}$.
(1) Find the general term of the sequences $\{a_n\}$ and $\{b_n\}$.
(2) Let the sequence $\{c_n\}$ satisfy $c_n=2a_n-18$, find the mini... | -40 | math | 157 |
Given a finite sequence $a_1, a_2, a_3, \ldots, a_n$ (where $n$ is a positive integer) that satisfies the conditions $a_1 = a_n$, $a_2 = a_n-1$, $\ldots$, $a_n = a_1$, i.e., $a_k = a_{n-k+1}$ (for $k=1, 2, \ldots, n$), we call it a "symmetric sequence". Let $\{b_n\}$ be a symmetric sequence with 7 terms, where $b_1, b_... | 2, 5, 8, 11, 8, 5, 2 | math | 182 |
When $x \in [1,2]$, the inequality $2^{x}-\log_{\frac{1}{2}}x+m\leqslant 0$ always holds, then the range of the real number $m$ is \_\_\_\_\_\_. | (-\infty,-5] | math | 59 |
Given the power function $f(x)=({{m^2}-m-1}){x^{\frac{1}{m}}}$ is monotonically increasing on $(0,+\infty)$, the value of the real number $m$ is ____. | 2 | math | 55 |
Given that the perimeter of a sector of a circle is $16$, find the radius and central angle of the sector when its area is at its maximum. Also, find the maximum area. | 16 | math | 39 |
\(8.469 \sin^{4} x + 2 \cos^{3} x + 2 \sin^{2} x - \cos x + 1 = 0\). | x = \pi(2k + 1) | math | 42 |
Find the number of odd digits in the base-4 representation of $317_{10}$. | 4 | math | 22 |
A larger deck of cards contains 104 cards divided into 4 suits, each of which has 26 cards. Two of the suits ($\heartsuit$ and $\diamondsuit$) are red, and the other two ($\spadesuit$ and $\clubsuit$) are black. The cards in this deck are shuffled randomly. What is the probability that the first three cards drawn from ... | \frac{425}{3502} | math | 90 |
A point $P(x, y)$ is randomly selected from the set $M={(x,y)∣(|x|−1)^2+(|y|−1)^2 < 4$, $x$, $y∈Z}$. If the probability that $xy≥k (k > 0)$ is $\frac{6}{25}$, what is the maximum value of $k$? | 2 | math | 83 |
In $\triangle ABC$, let $A$, $B$, and $C$ be the three interior angles, and $a$, $b$, and $c$ be the sides opposite these angles, respectively. Given that:
\[2 \sqrt{2}\left(\sin ^{2} A-\sin ^{2} C\right) = (a-b) \sin B,\]
and the circumradius of $\triangle ABC$ is $\sqrt{2}$.
(1) Find angle $C$.
(2) Find the maximum... | \frac{3\sqrt{3}}{2} | math | 125 |
Let $x$ be chosen at random from the interval $(0,1)$. What is the probability that
$\lfloor\log_{10}5x\rfloor - \lfloor\log_{10}x\rfloor = 0$?
Here $\lfloor x\rfloor$ denotes the greatest integer that is less than or equal to $x$.
A) $\frac{1}{7}$
B) $\frac{1}{9}$
C) $\frac{1}{10}$
D) $\frac{1}{8}$
E) $\frac{2}{9}$ | \frac{1}{9} | math | 130 |
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