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 quadratic equation \( ax^2 + bx + c \) and the table of values \( 6300, 6481, 6664, 6851, 7040, 7231, 7424, 7619, 7816 \) for a sequence of equally spaced increasing values of \( x \), determine the function value that does not belong to the table. | 6851 | 100 | 4 |
math | Given a pyramid with an equilateral triangular base (each side has length 3) and isosceles triangular lateral faces, the apex of the pyramid is directly above the centroid of the base at a height of 9 units from the base. Find the edge length of the cube positioned within the pyramid such that its one vertex is at the ... | 3 | 92 | 1 |
math | Find the result of the following operation: $999,999,999,999 - 888,888,888,888 + 111,111,111,111$.
A) $111,111,111,111$
B) $222,222,222,222$
C) $333,333,333,333$
D) $444,444,444,444$
E) $555,555,555,555$ | 222,222,222,222 | 155 | 15 |
math | The square of an integer is 224 greater than the integer itself. What is the sum of all integers for which this is true? | 2 | 29 | 1 |
math | Given that square and square root operations are inverse operations, for example, $(a^{2}±2ab+b^{2})=(a±b)^{2}$, then $\sqrt {a^{2}±2ab+b^{2}}=|a±b|$. How to simplify the double square root expression $\sqrt {a±2 \sqrt {b}}$ $(a > 0,b > 0,a±2 \sqrt {b} > 0)$? If we can find two numbers $m$ and $n$ $(m > 0,n > 0)$ such ... | \frac { \sqrt {10}+ \sqrt {6}}{2} | 387 | 18 |
math | Given the function $f(x)=\ln (1+x)- \frac {x(1+λx)}{1+x}$, if $f(x)\leqslant 0$ when $x\geqslant 0$, calculate the minimum value of $λ$. | \frac {1}{2} | 59 | 7 |
math | Given that the complex number $z$ satisfies $\overline{z}=\frac{-2i}{z}+2$, find the value of $|z|$. | \sqrt{2} | 35 | 5 |
math | Find the minimum value of $(x^2y-1)^2 + (x+y-1)^2$ for real numbers $x$ and $y$. | 1 | 33 | 1 |
math | Let \(a, b, c, d, e, f, g, h\) be real numbers such that \(abcd = 8\) and \(efgh = 16.\) Find the minimum value of
\[(ae)^2 + (bf)^2 + (cg)^2 + (dh)^2 + (ab)^2 + (cd)^2 + (ef)^2 + (gh)^2.\] | 64 | 87 | 2 |
math | In triangle $PQR$, $\angle PQR$ is $60^\circ$, and $\overline{PS}$ bisects $\angle QPR$, while $\overline{SR}$ bisects $\angle QRP$. Point $T$ is on $\overline{QR}$ such that $\overline{ST}$ extended bisects $\angle QSR$. The measure of angle $PST$ is
A) $45^\circ$
B) $60^\circ$
C) $75^\circ$
D) $90^\circ$
E) $105^\cir... | 60^\circ | 128 | 4 |
math | Let the function $f(x) = \sin(2x + \varphi)$ where $-\pi < \varphi < 0$, and the graph of $y=f(x)$ has a line of symmetry which is the line $x= \frac{\pi}{8}$.
(1) Find $\varphi$;
(2) Find the intervals where $y=f(x)$ is decreasing. | \left[\frac{5\pi}{8} + k\pi, \frac{9\pi}{8} + k\pi\right] | 85 | 32 |
math | Given the function $f(x)=\sin 2x$, its graph intersects with the line $2kx-2y-kπ=0 (k > 0)$ at exactly three common points. The x-coordinates of these points in ascending order are $x_{1}$, $x_{2}$, $x_{3}$. Calculate the value of $(x_{1}-x_{3})\tan (x_{2}-2x_{3})$. | -1 | 97 | 2 |
math | Given the function $f(x) = 2\sin x \cos x - 2\sin^2 x + 1 (x \in \mathbb{R})$.
1. Find the smallest positive period and the monotonically increasing interval of the function $f(x)$.
2. In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, with $a = \sqrt{3}$, $A$ being ... | \frac{3(\sqrt{3} + \sqrt{2})}{4} | 153 | 18 |
math | Fido's leash is tied to a stake at the center of his yard, which is in the shape of an equilateral triangle. His leash is exactly long enough to reach the midpoint of each side of the triangle. If the fraction of the area of Fido's yard that he is able to reach while on his leash is expressed in simplest radical form a... | 27 | 97 | 2 |
math | A painting measures 20 inches by 30 inches, with the longer side placed vertically. The frame width at the top and bottom is three times the width of the wood on the sides. If the total area of the frame is the same as the area of the painting itself, find the ratio of the shorter to the longer dimension of the framed ... | 1:2 \text{ (B)} | 107 | 9 |
math | In triangle $\triangle ABC$, the lengths of the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, $c$, $c=2$, the area is $S$, and $bcos\frac{A}{2}=S$. Find:<br/>
$(1)$ The measure of angle $A$.<br/>
$(2)$ The range of values for $\frac{{b+c}}{a}$. | (1, 2] | 92 | 6 |
math | How many ways are there to put 6 balls in 2 boxes if the balls are distinguishable but the boxes are not? | 42 | 26 | 2 |
math | Given vectors $m=(\sin \frac{x}{4}, \cos \frac{x}{4})$ and $n=(\sqrt{3}\cos \frac{x}{4}, \cos \frac{x}{4})$, let $f(x)=m\cdot n$;
(1) If $f(x)=1$, find the value of $\cos(x+ \frac{\pi}{3})$;
(2) In $\triangle ABC$, where the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and it satisfie... | (1, \frac{3}{2}) | 151 | 10 |
math | Square $EFGH$ has sides of length 4. Segments $EP$ and $EQ$ divide the square's area into three equal parts. Determine the length of segment $EP$. | \sqrt{\frac{208}{9}} | 40 | 11 |
math | Fill in the blanks with appropriate numbers.
4 liters 25 milliliters = ___ milliliters
6.09 cubic decimeters = ___ cubic centimeters
4.9 cubic decimeters = ___ liters ___ milliliters
2.03 cubic meters = ___ cubic meters ___ cubic decimeters. | 30 | 65 | 2 |
math | Two cylindrical poles, with diameters of $10$ inches and $30$ inches respectively, are placed side by side and bound together with a wire. Calculate the length of the shortest wire that will go around both poles.
**A)** $20\sqrt{3} + 24\pi$
**B)** $20\sqrt{3} + \frac{70\pi}{3}$
**C)** $30\sqrt{3} + 22\pi$
**D)** $16\sq... | 20\sqrt{3} + \frac{70\pi}{3} | 146 | 18 |
math | Convert the point $(1, -2\sqrt{3}, 4)$ in rectangular coordinates to spherical coordinates. Enter your answer in the form $(\rho,\theta,\phi),$ where $\rho > 0,$ $0 \le \theta < 2\pi,$ and $0 \le \phi \le \pi.$ | \left( \sqrt{29}, 2\pi - \tan^{-1}(2\sqrt{3}), \cos^{-1}\left(\frac{4}{\sqrt{29}}\right) \right) | 68 | 48 |
math | Given that the function $f(x+1)$ is an odd function, and the function $f(x-1)$ is an even function, and $f(0)=2$, then $f(4)=$ ? | -2 | 45 | 2 |
math | For which natural numbers \(a\) is it true that the greatest common divisor of \(\left(a^{4}+3a^{2}+1, a^{3}+2a\right)\) is 1? | 1 | 47 | 1 |
math | If there are exactly $3$ integer solutions for the system of one-variable linear inequalities about $x$: $\left\{\begin{array}{l}x+1>0\\ 3x-a\leq 0\end{array}\right.$, then the range of values for $a$ is ____. | 6 \leq a < 9 | 66 | 8 |
math | Melinda will roll two standard six-sided dice and make a two-digit number with the two numbers she rolls. For example, if she rolls a 6 and a 3, she can either form 36 or 63. What is the probability that she will be able to make an integer between 21 and 30, inclusive? Express your answer as a common fraction. | \frac{5}{18} | 80 | 8 |
math | Bill buys a stock that decreases by $15\%$ on the first day, and then on the second day the stock increases by $25\%$ of its value at the end of the first day. What was the overall percent increase or decrease in Bill's stock over the two days? | 6.25\% | 62 | 6 |
math | Given that \(3^8 \cdot 5^2 \cdot 2 = a^b\), where both \(a\) and \(b\) are positive integers, find the smallest possible value for \(a + b\). | 812 | 47 | 3 |
math | Square $IJKL$ has one vertex on each side of square $WXYZ$. Point $I$ is on $WZ$ such that $WI = 9 \cdot IZ$. Determine the ratio of the area of square $IJKL$ to the area of square $WXYZ$.
A) $\frac{1}{20}$
B) $\frac{1}{50}$
C) $\frac{1}{40}$
D) $\frac{1}{64}$
E) $\frac{1}{80}$ | \frac{1}{50} | 115 | 8 |
math | Given the ellipse C: $$\frac{{x}^{2}}{{a}^{2}}+ \frac{{y}^{2}}{{b}^{2}}=1$$ (where $a>b>0$) passes through the point $(0, \sqrt{2})$, and satisfies $a+b=3\sqrt{2}$.
1. Find the equation of ellipse C.
2. If a line with a slope of $\frac{1}{2}$ intersects ellipse C at two distinct points A and B, and the point M has coor... | 0 | 159 | 1 |
math | Given a rectangular box with dimensions $2 \times b \times c$, where $b$ and $c$ are integers with $b \leq c$, find the number of ordered pairs $(b, c)$ such that the sum of the volume and the surface area of the box equals 120. | 0 | 64 | 1 |
math | Given a quadrilateral $ABCD$ inscribed in a circle with side $AB$ extended beyond $B$ to point $E$ and side $CD$ extended beyond $D$ to point $F$, if $\measuredangle BAD=85^\circ$ and $\measuredangle ADC=75^\circ$, find $\measuredangle EBC$ and $\measuredangle EFC$.
A) $65^\circ \quad$ B) $75^\circ \quad$ C) $85^\circ ... | 75^\circ | 133 | 4 |
math | Show Nesbitt's inequality: for all real numbers \( a, b, c > 0 \), we have
$$
\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b} \geqslant \frac{3}{2}
$$
Also, find the case of equality. | \frac{a}{b+c} + \frac{b}{c+a} + \frac{c}{a+b} = \frac{3}{2} | 73 | 34 |
math | Given that the interior angles \(A, B, C\) of triangle \(\triangle ABC\) are opposite to the sides \(a, b, c\) respectively, and that \(A - C = \frac{\pi}{2}\), and \(a, b, c\) form an arithmetic sequence, find the value of \(\cos B\). | \frac{3}{4} | 71 | 7 |
math | As Emily is riding her bicycle on a long straight road, she spots Emerson skating in the same direction $1/2$ mile in front of her. After she passes him, she can see him in her rear mirror until he is $1/2$ mile behind her. Emily rides at a constant rate of $12$ miles per hour, and Emerson skates at a constant rate of ... | 15 | 96 | 2 |
math | The graph of the function $f(x)$ is shown. How many values of $x$ satisfy $f(f(x)) = 5$? [asy]
import graph; size(7.4cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-4.4,xmax=5.66,ymin=-1.05,ymax=6.16;
for(int i = -4; i <= 5; ++i) {
draw((i,-1)--(i,6), dashed+mediumgre... | 3 | 437 | 1 |
math | Given $f(x)=x\ln x+(1-a)x+a$, if there exists $n > 0$ such that $f(m)=n$ for any $m\in\left(1,+\infty\right)$, then the maximum value of the integer $a$ is $\_\_\_\_\_\_\_$. | 3 | 70 | 1 |
math | The function $f(x)=x^{2}+ax+3$ is decreasing on the interval $(-\infty,2]$. Determine the range of values for the real number $a$. | a\leq-4 | 41 | 6 |
math | A natural number greater than 0 is called a perfect number if the sum of all its factors equals twice the number itself. For example, the factors of 6 are \(1, 2, 3, 6\), and \(1 + 2 + 3 + 6 = 12\). Therefore, 6 is the smallest perfect number. The question of whether there are infinitely many perfect numbers remains an... | 16256 | 130 | 5 |
math | Let $a$ be a positive real number, $n$ a positive integer, and define the *power tower* $a\uparrow n$ recursively with $a\uparrow 1=a$ , and $a\uparrow(i+1)=a^{a\uparrow i}$ for $i=1,2,3,\ldots$ . For example, we have $4\uparrow 3=4^{(4^4)}=4^{256}$ , a number which has $155$ digits. For each positive in... | x_{42} | 196 | 6 |
math | John recorded times of 120, 125, 112, 140, and 130 seconds in five attempts for swimming 100 meters. After two more attempts, he managed to decrease his median time to 125 seconds. If one of the new times recorded was 122 seconds, what was his time in the other new attempt? | 124 | 84 | 3 |
math | Given that $n$ is a natural number such that the leftmost digits in the decimal representations of $2^n$ and $3^n$ are the same, find all possible values of the leftmost digit. | \{1, 2, 3, 4, 5, 6, 7, 8, 9\} | 49 | 29 |
math | Let $N$ be the number of consecutive $0$s at the right end of the decimal representation of the product $1!2!3! \cdots 49!50!$. Find the remainder when $N$ is divided by $100$. | 12 | 56 | 2 |
math | In triangle $\triangle ABC$, the lengths of the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, where $a=5$ and $b=6$. <br/>$(1)$ If $\cos B=-\frac{4}{5}$, find the values of $A$ and the radius $R$ of the circumcircle of $\triangle ABC$; <br/>$(2)$ If the area of the triangle $S_{\triangle}=... | c = 4 \text{ or } c = \sqrt{106} | 126 | 18 |
math | Solve the following equation:
$$
2x^{4} + 2y^{4} - 4x^{3}y + 6x^{2}y^{2} - 4xy^{3} + 7y^{2} + 7z^{2} - 14yz - 70y + 70z + 175 = 0.
$$ | (x, y, z) = (0, 0, -5) | 85 | 16 |
math | Given that $F\_1$ and $F\_2$ are the left and right foci of the hyperbola $C\_1$: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 (a > 0, b > 0)$, and a circle $C\_2$ with center $F\_1$ and radius $|F\_1F\_2|$ intersects the right branch of the hyperbola at points $P$ and $Q$. If the area of $\triangle PF\_1F\_2$ is $4$ and $\a... | (x + 2)^2 + y^2 = 16 | 153 | 14 |
math | Senya has three straight sticks, each 24 centimeters long. Senya broke one of them into two pieces in such a way that, with the two broken pieces and the two whole sticks, he was able to form the outline of a right triangle. How many square centimeters is the area of this triangle? | 216 | 65 | 3 |
math | Given the function $f(x)=\sin x+\sqrt{3}\cos x (x\in \mathbb{R})$, first compress the x-coordinates of all points on the graph of $y=f(x)$ to half of their original length, then move all points of the obtained graph to the right by $\theta (\theta > 0)$ units. If the resulting graph is symmetric about the line $x=\frac... | \frac{\pi}{6} | 103 | 7 |
math | Given $x = \dfrac{1+i\sqrt{3}}{2}$, where $i = \sqrt{-1}$, calculate $\dfrac{1}{x^2 + x}$. | \frac{-i\sqrt{3}}{3} | 43 | 12 |
math | Given the function $f(x)$ is an odd function defined on $\mathbb{R}$, and its derivative is denoted by $f'(x)$. If for any real number $x$, we have $x^2f'(x) > 2xf(-x)$, find the solution set for the inequality $x^2f(x) < (3x-1)^2f(1-3x)$. | (-\infty, \frac{1}{4}) | 89 | 12 |
math | Find all the values of \( m \) for which the zeros of \( 2x^2 - mx - 8 \) differ by \( m - 1 \). | 6, -\frac{10}{3} | 36 | 11 |
math | If $\sqrt{x-1}$ is meaningful, then the value of $x$ can be _____. (Write down one value) | 2 | 27 | 1 |
math | The graph of the function $y=\sin (2x+\varphi) (0 < \varphi < \pi)$ is shifted to the right by $\frac{\pi}{8}$ and then is symmetric about the $y$-axis. Determine the possible value(s) of $\varphi$. | \frac{3\pi}{4} | 62 | 9 |
math | In △ABC, it is known that 2B = A + C and c = 2a.
1. Find the measures of angles A, B, and C.
2. Let the sequence {a\_n} satisfy a\_n = n|cos(nC)|, and the sum of the first n terms is S\_n. If S\_n = 20, find the value of n. | 9 | 87 | 1 |
math | Let $A B C D$ be a convex quadrilateral such that $\angle A B D=\angle B C D=90^{\circ}$, and let $M$ be the midpoint of segment $B D$. Suppose that $C M=2$ and $A M=3$. Compute $A D$. | \sqrt{21} | 66 | 6 |
math | Let $AB$ be a diameter of a circle centered at $O$. Let $E$ be a point on the circle, and let the tangent at $B$ intersect the tangent at $A$ and $AE$ at $C$ and $D$, respectively. If $\angle BAE = 53^\circ$, find $\angle CED$, in degrees. | 37^\circ | 76 | 4 |
math | Let $h(x) = x^6 - x^5 + x^4 - x^3 + x^2 - x + 1$. What is the remainder when the polynomial $h(x^{10})$ is divided by the polynomial $h(x)$? | 7 | 56 | 1 |
math | Given that $a\_n$ is the coefficient of the $x^{2}$ term in the expansion of $(2+x)^{n}$ $(n∈N^{},n≥slant 2,x∈R)$, find the limit: $\lim\_{n→∞}( \frac {2^{2}}{a_{2}}+ \frac {2^{3}}{a_{3}}+…+ \frac {2^{n}}{a_{n}})=$ _____ . | 8 | 102 | 1 |
math | For certain real values of $a, b, c,$ and $d,$ the equation $x^4+ax^3+bx^2+cx+d=0$ has four non-real roots. The product of two of these roots is $13+i$ and the sum of the other two roots is $3+4i,$ where $i=\sqrt{-1}.$ Find $b.$ | 26 | 98 | 2 |
math | Given the function $f(x)=\sqrt{1+x}+\sqrt{1-x}$.
$(1)$ Find the domain and range of $f(x)$;
$(2)$ Let $h(x)=\frac{{f(x)}}{{2\sqrt{1-{x^2}}+6}}$. If the inequality $h(x)≤\frac{3}{4}{m^2}-\frac{1}{2}am$ holds for any $x\in \left[-1,1\right]$ and any $a\in \left[-1,1\right]$, find the range of the real number $m$. | (-\infty, -1]\cup[1, +\infty) | 135 | 17 |
math | Given the expansion $(1+x)^6(1-2x)^5 = a_0 + a_1x + a_2x^2 + \ldots + a_{11}x^{11}$, calculate:
(1) $a_1 + a_2 + a_3 + \ldots + a_{11}$;
(2) $a_0 + a_2 + a_4 + \ldots + a_{10}$. | a_0 + a_2 + a_4 + \ldots + a_{10} = -32 | 101 | 25 |
math | A four-digit number is divisible by 9, and the three-digit number obtained by removing the last digit is exactly a multiple of 4. What is the last digit of the largest such four-digit number? | 3 | 42 | 1 |
math | Find the shortest distance from a point on the curve $y=\ln x$ to the line $2x-y+3=0$. | \frac{4 + \ln 2}{\sqrt{5}} | 28 | 15 |
math | Daniel works at an electronics store and suggests that the popularity of a toaster (measured in number of sales) is inversely proportional to its cost, but with a promotional twist: if a toaster's cost decreases by 25%, sales increase by an additional 10% beyond the usual increase due to the price drop. If 15 customers... | 22 | 104 | 2 |
math | We consider a chessboard of size $8 \times 8$ where the squares are alternately colored white and black. An infernal rook is a piece that can attack the squares of its color located on its row, as well as the squares of the other color located in its column. What is the maximum number of infernal rooks that can be plac... | 16 | 92 | 2 |
math | Given \( y = \frac{1+i\sqrt{3}}{2} \), calculate \( \dfrac{1}{y^3-y} \). | -\frac{1}{2} + \frac{i\sqrt{3}}{6} | 34 | 19 |
math | A sequence $b_1, b_2, b_3, \dots,$ is defined recursively by $b_1 = 3,$ $b_2 = 2,$ and for $k \ge 3,$
\[ b_k = \frac{1}{4} b_{k-1} + \frac{2}{5} b_{k-2}. \]
Evaluate $b_1 + b_2 + b_3 + \dotsb.$ | \frac{85}{7} | 99 | 8 |
math | Given an arithmetic sequence $\{a_n\}$, the sum of the first $n$ terms is $S_n$, and $a_1 = -20$. If the minimum value of $S_n$ is only $S_6$, then the range of the common difference $d$ is \_\_\_\_\_\_. | \left(\frac{10}{3}, 4\right) | 69 | 15 |
math | Find all integers $\,a,b,c\,$ with $\,1<a<b<c\,$ such that \[ (a-1)(b-1)(c-1) \] is a divisor of $abc-1.$ | (2, 4, 8) | 53 | 9 |
math | A torus (donut) having inner radius $3$ and outer radius $5$ rests on a flat table. Determine the radius of the largest spherical ball that can be placed on top of the center of the torus such that the ball still touches the horizontal plane. (Assume the torus is generated by revolving a circle in the $xz$-plane center... | 4 | 122 | 1 |
math | In triangle $\triangle ABC$, $\tan B = \frac{4}{3}$. Calculate the length of side $AC$ if side $AB$ is 3 units. | 5 | 36 | 1 |
math | Given the hyperbola $C: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ ($a>b>0$) with semi-focal distance $c$. A line passing through the focus and with a slope of $1$ intersects both branches of the hyperbola $C$ at distinct points. If the chord of the hyperbola $C$ cut by the directrix of the parabola $y^2=4cx$ has a length ... | e = \sqrt{3} | 152 | 7 |
math | Given the plane vectors $a, b,$ and $c$ such that:
$$
\begin{array}{l}
|\boldsymbol{a}|=|\boldsymbol{b}|=|\boldsymbol{c}|=2, \boldsymbol{a}+\boldsymbol{b}+\boldsymbol{c}=\mathbf{0}.
\end{array}
$$
If $0 \leqslant x \leqslant \frac{1}{2} \leqslant y \leqslant 1$, then find the minimum value of
$$
|x(\boldsymbol{a}-\bo... | \frac{1}{2} | 158 | 7 |
math | Marisa has a collection of $2^8-1=255$ distinct nonempty subsets of $\{1, 2, 3, 4, 5, 6, 7, 8\}$ . For each step she takes two subsets chosen uniformly at random from the collection, and replaces them with either their union or their intersection, chosen randomly with equal probability. (The collection is allowed to... | \frac{1024}{255} | 132 | 12 |
math | Find all positive integers \( m \), \( n \) such that \( \frac{1}{m} + \frac{1}{n} - \frac{1}{mn} = \frac{2}{5} \). | (3, 10), (10, 3), (4, 5), (5, 4) | 48 | 26 |
math | A large sphere has a volume of $450\pi$ cubic units. A smaller sphere has a volume which is $27.04\%$ of the volume of the large sphere. What is the ratio of the radius of the smaller sphere to the radius of the larger sphere? Express your answer as a decimal. | 0.646 | 68 | 5 |
math | Let $1 \leq n \leq 2021$ be a positive integer. Jack has $2021$ coins arranged in a line where each coin has an $H$ on one side and a $T$ on the other. At the beginning, all coins show $H$ except the nth coin. Jack can repeatedly perform the following operation: he chooses a coin showing $T$ , and turns over... | n = 1011 | 143 | 7 |
math | Given $m, n \in \mathbb{R}$, determine the condition for the equation $\frac{x^{2}}{m} + \frac{y^{2}}{n} = 1$ to represent a hyperbola. | mn < 0 | 51 | 4 |
math | Given a regular tetrahedral prism $P-ABCD$ where the base edge length is equal to the height, point $G$ is the centroid of the face $\triangle PBC$. Find the sine of the angle between the line $AG$ and the base $ABCD$. | \frac{\sqrt{38}}{19} | 58 | 12 |
math | An uncrossed belt, without any slack, is fitted around two circular pulleys, with one having a radius of $10$ inches and the other $6$ inches. The distance between the points where the belt touches the pulleys is $26$ inches. What is the distance between the centers of the two pulleys in inches?
A) $2\sqrt{178}$
B) $2\... | 2\sqrt{173} | 118 | 8 |
math | Find all triples of real numbers \((a, b, c)\) such that \(a^{2} + 2b^{2} - 2bc = 16\) and \(2ab - c^{2} = 16\). | (4, 4, 4), (-4, -4, -4) | 53 | 18 |
math | A lake ecologist needs to estimate the number of turtles in a pond. On June 1, he tags 80 turtles and releases them back into the pond. On October 1, he captures a random sample of 50 turtles and finds that 2 of them are tagged. To estimate the number of turtles in the pond on June 1, he assumes that 30% of these turtl... | 1000 | 154 | 4 |
math | Each triangle is a 30-60-90 triangle, and the hypotenuse of one triangle is the longer leg of an adjacent triangle. There are now five 30-60-90 triangles in sequence, and the hypotenuse of the largest triangle is 16 centimeters. What is the length of the longer leg of the smallest triangle? | 4.5\sqrt{3} \text{ cm} | 78 | 13 |
math | Given the function $f(x) = \sin(\omega x + \varphi)$ (where $\omega > 0$, $|\varphi| < \frac{\pi}{2}$), if the graph of the function is shifted to the left by $\frac{\pi}{12}$ units and the resulting graph is symmetric about the y-axis, and if the graph of the function is shifted to the right by $\frac{\pi}{6}$ units a... | 4 | 116 | 1 |
math | How many orderings $(a_{1}, \ldots, a_{8})$ of $(1,2, \ldots, 8)$ exist such that $a_{1}-a_{2}+a_{3}-a_{4}+a_{5}-a_{6}+a_{7}-a_{8}=0$ ? | 4608 | 73 | 4 |
math | In the Cartesian coordinate system $xOy$, points $M(-2 \sqrt {2}, 0)$ and $N(2 \sqrt {2}, 0)$ are given. If line $m$ is perpendicular to $MN$ at point $D$, and point $C$ is a moving point on line $m$. Let $H$ be the midpoint of segment $CD$, and $\overrightarrow{NH} \cdot \overrightarrow{MC} = 8$. The locus of point $H... | 2 | 231 | 1 |
math | A running track is formed by two concentric circles. If the difference in the circumferences of the two circles is $24\pi$ feet, determine both the width of the track and the area enclosed between these two circles. | 12 | 47 | 2 |
math | Given the inequality $-\frac{1}{4}{x^2}≤ax+b≤{e^x}$ holds for any $x\in R$, then the maximum value of $a+b$ is ______. | 2 | 45 | 1 |
math | Given the function $f(x) = A\sin(\omega x + \varphi)$, where $x \in \mathbb{R}$, $A > 0$, $\omega > 0$, $0 < \varphi < \frac{\pi}{2}$, the graph intersects with the x-axis, and the distance between two adjacent intersection points is $\frac{\pi}{2}$. Also, there is a lowest point on the graph $M\left( \frac{2\pi}{3}, -... | [-1, 2] | 200 | 6 |
math | If the inequality \( x^{2} - \log_{a} x < 0 \) is always satisfied on the interval \( \left(0, \frac{1}{2}\right] \), what is the range of values for the real number \( a \)? | \left(\frac{1}{16}, 1\right) | 57 | 15 |
math | Given $m \in \mathbb{R}$, the complex number $z= \frac{m(m+2)}{m-1}+(m^2+2m-3)i$, for what value(s) of $m$ does:
(1) $z$ belong to $\mathbb{R}$ (i.e., $z$ is a real number)?
(2) $z$ belong to the set of imaginary numbers?
(3) $z$ is a pure imaginary number?
(4) The point corresponding to $z$ lie in the second q... | m = 0 \text{ or } m = -2 | 153 | 13 |
math | Three different numbers are randomly selected from the set $T = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13\}$. The probability that their sum is 15 would be greater if the number $m$ had first been removed from set $T$. What is the value of $m$? | 5 | 94 | 1 |
math | Let $D$, $E$, $F$ be the midpoints of the sides $BC$, $CA$, $AB$ of $\triangle ABC$, respectively. Calculate the sum of the vectors $\overrightarrow{DA}$, $\overrightarrow{EB}$, and $\overrightarrow{FC}$. | \overrightarrow{0} | 61 | 6 |
math | If the function $f(x) = x^2 + 2(a-1)x + 2$ is decreasing on $(-\infty, 4)$, then the range of the real number $a$ is. | a \leq -3 | 47 | 6 |
math | Determine the number of solutions to the equation:
\[3\sin^3 x - 10 \sin^2 x + 3 \sin x = 0\]
in the range $0 \leq x \leq 2\pi$. | 7 | 54 | 1 |
math | Given the complex number $z_1 = m^2 + 2m - 3 + (m - 1)i$ is a pure imaginary number $(m \in \mathbb{R})$.
$(1)$ Find the value of $m$;
$(2)$ If the complex number $z_2 = \frac{4 - 2i}{(1 + \frac{1}{4}z_1) \cdot i}$, find $|z_2|$. | \sqrt{10} | 103 | 6 |
math | In triangle $ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. Given that $b= \sqrt {2}$, $c=3$, $B+C=3A$.
(1) Find the length of side $a$;
(2) Find the value of $\sin (B+ \frac {3π}{4})$. | \frac{\sqrt{10}}{10} | 88 | 12 |
math | Given 2tanθ - tan(θ + $\frac{π}{4}$) = 7, solve for tanθ. | 2 | 27 | 1 |
math | In the Cartesian coordinate system $xoy$, the curve $C_{1}$ is defined by $\begin{cases} x=t\cos\alpha \\ y=t\sin\alpha \end{cases}$ (where $t$ is a parameter, $t{\neq }0$), and $0{\leqslant }\alpha{ < }\pi$. In the polar coordinate system with $O$ as the pole and the positive $x$-axis as the polar axis, the curve $C_{... | 4 | 215 | 1 |
math | 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) | 58 | 7 |
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