Unnamed: 0 int64 0 40.3k | problem stringlengths 10 5.15k | ground_truth stringlengths 1 1.22k | solved_percentage float64 0 100 |
|---|---|---|---|
17,300 | Given that the math test scores, X, follow a normal distribution with a mean of 100 and a standard deviation of δ, and the number of students scoring between 80 and 120 points accounted for about 3/4 of the total 1600 students, calculate the number of students who scored no less than 80 points in this final exam. | 1400 | 61.71875 |
17,301 | There are $2012$ backgammon checkers (stones, pieces) with one side is black and the other side is white.
These checkers are arranged into a line such that no two consequtive checkers are in same color. At each move, we are chosing two checkers. And we are turning upside down of the two checkers and all of the checke... | 1006 | 89.84375 |
17,302 | Compute $63 \times 57$ in your head. | 3591 | 73.4375 |
17,303 | In ancient China, when determining the pentatonic scale of Gong, Shang, Jiao, Zhi, and Yu, a method of calculation involving a three-part loss and gain was initially used. The second note was obtained by subtracting one-third from the first note, and the third note was obtained by adding one-third to the second note, a... | 54 | 69.53125 |
17,304 | Consider a rectangular region of 2x1 unit squares at the center of a large grid of unit squares. Each subsequent ring forms around this rectangle by one unit thickness. Determine the number of unit squares in the $50^{th}$ ring around this central rectangle. | 402 | 35.15625 |
17,305 | Three tenths plus four thousandths. | 0.304 | 92.1875 |
17,306 | Let $A$, $B$, $C$, and $D$ be the vertices of a regular tetrahedron with each edge measuring 1 meter. A bug, starting at vertex $A$, follows a rule: at each vertex, it randomly chooses one of the three edges with equal probability and crawls to the opposite vertex. Define $q = \frac m{2187}$ as the probability that the... | 547 | 34.375 |
17,307 | In the plane rectangular coordinate system $xOy$, the parametric equations of curve $C$ are $\left\{\begin{array}{l}{x=\frac{4}{1+{t}^{2}}},\\{y=\frac{4t}{1+{t}^{2}}}\end{array}\right.$ $(t\in R)$.<br/>$(Ⅰ)$ Find the rectangular coordinate equation of curve $C$;<br/>$(Ⅱ)$ Given that the parametric equations of line $l$... | \frac{\sqrt{15}}{3} | 15.625 |
17,308 | Exhibit a $13$ -digit integer $N$ that is an integer multiple of $2^{13}$ and whose digits consist of only $8$ s and $9$ s. | 8888888888888 | 0.78125 |
17,309 | There are 15 girls in a class of 27 students. The ratio of boys to girls in this class is: | 4:5 | 83.59375 |
17,310 | Two positive integers differ by 8 and their product is 168. What is the larger integer? | 14 | 17.96875 |
17,311 | Given a [rational number](https://artofproblemsolving.com/wiki/index.php/Rational_number), write it as a [fraction](https://artofproblemsolving.com/wiki/index.php/Fraction) in lowest terms and calculate the product of the resulting [numerator](https://artofproblemsolving.com/wiki/index.php/Numerator) and [denominator](... | 128 | 28.125 |
17,312 | How many integers are there between $(11.1)^3$ and $(11.2)^3$? | 37 | 37.5 |
17,313 | The letter T is formed by placing two $3\:\text{inch}\!\times\!5\:\text{inch}$ rectangles to form a T shape. The vertical rectangle is placed in the middle of the horizontal one, overlapping it by $1.5$ inches on both sides. What is the perimeter of the new T, in inches? | 20 | 11.71875 |
17,314 | Find $n$ such that $2^6 \cdot 3^3 \cdot n = 10!$. | 1050 | 0 |
17,315 | Find the phase shift of the graph of \( y = \cos(5x - \frac{\pi}{2}) \). | \frac{\pi}{10} | 100 |
17,316 | In how many ways is it possible to arrange the digits of 11250 to get a five-digit multiple of 5? | 21 | 40.625 |
17,317 | In $\triangle ABC$, $BC=a$, $AC=b$, and $a$, $b$ are the roots of the equation $x^{2}-2 \sqrt{3}x+2=0$, $2\cos (A+B)=1$
$(1)$ Find the degree of angle $C$.
$(2)$ Find the length of $AB$. | \sqrt{10} | 75.78125 |
17,318 | Given an odd function $f(x)$ on $\mathbb{R}$, for any $x \in \mathbb{R}$, $f(x+1) = -f(x)$, and when $x \in (-1, 1)$, $f(x) = x$. Find the value of $f(3) + f(-7.5)$. | 0.5 | 56.25 |
17,319 | Add $518_{12} + 276_{12}$. Express your answer in base 12, using $A$ for $10$ and $B$ for $11$ if necessary. | 792_{12} | 71.09375 |
17,320 | In the interval \([-6, 6]\), an element \(x_0\) is arbitrarily chosen. If the slope of the tangent line to the parabola \(y = x^2\) at \(x = x_0\) has an angle of inclination \(\alpha\), find the probability that \(\alpha \in \left[ \frac{\pi}{4}, \frac{3\pi}{4} \right]\). | \frac{11}{12} | 28.90625 |
17,321 | Given a sequence $\{a_n\}$ satisfying $a_1=1$, $a_n=\log_n(n+1)$ $(n\geqslant 2,n\in \mathbb{N}^*)$, define: $k$ $(k\in \mathbb{N}^*)$ that makes the product $a_1\cdot a_2\cdot \cdots \cdot a_k$ an integer as "simple number". Then, the sum of all "simple numbers" within the interval $[3,2013]$ is. | 2035 | 71.09375 |
17,322 | Given two complex numbers ${z_1}$ and ${z_2}$ that correspond to points in the complex plane that are symmetric about the origin, and ${z_1 = 2 - i}$, determine the value of the complex number $\frac{z_1}{z_2}$. | -1 | 54.6875 |
17,323 | Given $f(x)={x}^{2023}+a{x}^{3}-\frac{b}{x}-8$, $f\left(-2\right)=10$, find $f\left(2\right)=\_\_\_\_\_\_$. | -26 | 87.5 |
17,324 | In the Cartesian coordinate system $xOy$, the ellipse $C: \frac{x^2}{2} + \frac{y^2}{3} = 1$ has a focus on the positive y-axis denoted as $F$. A line $l$ passing through $F$ with a slope angle of $\frac{3\pi}{4}$ intersects $C$ at points $M$ and $N$. The quadrilateral $OMPN$ is a parallelogram.
(1) Determine the posit... | \frac{4}{5}\sqrt{6} | 0 |
17,325 | Given point $A(0,2)$, and $P$ is any point on the ellipse $\frac{x^2}{4}+y^2=1$, then the maximum value of $|PA|$ is ______. | \frac{2\sqrt{21}}{3} | 34.375 |
17,326 | Yao Ming has a free throw shooting percentage of 90% during games. What is the probability that he misses one free throw out of three attempts? | 0.243 | 71.09375 |
17,327 | Given that $\sin \alpha + \cos \alpha = -\frac{\sqrt{5}}{2}$ and $\frac{5\pi}{4} < \alpha < \frac{3\pi}{2}$, find the value of $\cos \alpha - \sin \alpha$. | \frac{\sqrt{3}}{2} | 74.21875 |
17,328 | In the polar coordinate system, the curve $C_1$: $\rho=2\cos\theta$, and the curve $$C_{2}：\rho\sin^{2}\theta=4\cos\theta$$．Establish a Cartesian coordinate system xOy with the pole as the origin and the polar axis as the positive half-axis of x, the parametric equation of curve C is $$\begin{cases} x=2+ \frac {1}{2}t ... | \frac {11}{3} | 1.5625 |
17,329 | Given an arithmetic sequence $\{a_n\}$ where the sum of the first $n$ terms is $S_n = (a+1)n^2 + a$, if the sides of a certain triangle are $a_2$, $a_3$, and $a_4$, then the area of this triangle is ________. | \frac{15}{4} \sqrt{3} | 0 |
17,330 | You want to paint some edges of a regular dodecahedron red so that each face has an even number of painted edges (which can be zero). Determine from How many ways this coloration can be done.
Note: A regular dodecahedron has twelve pentagonal faces and in each vertex concur three edges. The edges of the dodecahedron a... | 2048 | 8.59375 |
17,331 | Determine the longest side of the polygon formed by the system:
$$
\begin{cases}
x + y \leq 4 \\
x + 2y \geq 4 \\
x \geq 0 \\
y \geq 0
\end{cases}
$$ | 2\sqrt{5} | 3.125 |
17,332 | Given $sn(α+ \frac {π}{6})= \frac {1}{3}$, and $\frac {π}{3} < α < \pi$, find $\sin ( \frac {π}{12}-α)$. | - \frac {4+ \sqrt {2}}{6} | 0 |
17,333 | Cagney can frost a cupcake every 15 seconds and Lacey can frost a cupcake every 45 seconds. Working together, calculate the number of cupcakes they can frost in 10 minutes. | 53 | 30.46875 |
17,334 | How many even integers between 3000 and 6000 have four different digits? | 784 | 10.9375 |
17,335 | If \(a\), \(b\), and \(c\) are positive numbers with \(ab = 24\sqrt[3]{3}\), \(ac = 40\sqrt[3]{3}\), and \(bc = 15\sqrt[3]{3}\), find the value of \(abc\). | 120\sqrt{3} | 86.71875 |
17,336 | In $\vartriangle ABC$, the lengths of the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, given that $c=2$, $C=\dfrac{\pi }{3}$.
(1) If the area of $\vartriangle ABC$ is equal to $\sqrt{3}$, find $a$ and $b$;
(2) If $\sin B=2\sin A$, find the area of $\vartriangle ABC$. | \dfrac{2 \sqrt{3}}{3} | 95.3125 |
17,337 | Determine the smallest positive integer $n$ such that $4n$ is a perfect square and $5n$ is a perfect cube. | 25 | 17.96875 |
17,338 | Let the function $f(x)=(x-3)^3 +x-1$. The sequence $\{a_n\}$ is an arithmetic sequence with a non-zero common difference. If $f(a_1)+f(a_2) + \ldots +f(a_7) =14$, then $a_1 +a_2 +\ldots +a_7 =$ ______. | 21 | 85.15625 |
17,339 | The absolute value of -9 is ; the reciprocal of -3 is . | -\frac{1}{3} | 42.1875 |
17,340 | A circle with radius $\frac{\sqrt{2}}{2}$ and a regular hexagon with side length 1 share the same center. Calculate the area inside the circle, but outside the hexagon. | \frac{\pi}{2} - \frac{3\sqrt{3}}{2} | 19.53125 |
17,341 | A point $(x,y)$ is randomly and uniformly chosen inside the rectangle with vertices (0,0), (0,3), (4,3), and (4,0). What is the probability that $x + 2y < 6$? | \dfrac{2}{3} | 55.46875 |
17,342 | A line parallel to the side $AC$ of a triangle $ABC$ with $\angle C = 90$ intersects side $AB$ at $M$ and side $BC$ at $N$ , so that $CN/BN = AC/BC = 2/1$ . The segments $CM$ and $AN$ meet at $O$ . Let $K$ be a point on the segment $ON$ such that $MO+OK = KN$ . The bisector of $\angle ABC$ mee... | 90 | 51.5625 |
17,343 | Given a rectangular pan of brownies measuring 15 inches by 24 inches, cut into triangular pieces with a base of 3 inches and a height of 4 inches, determine the number of triangular pieces that can be cut from the pan. | 60 | 84.375 |
17,344 | Given $a > 1$, $b > 1$, and $$\frac {1}{a-1} + \frac {1}{b-1} = 1$$, find the minimum value of $a + 4b$. | 14 | 77.34375 |
17,345 | Let the sequence $\{a_n\}$ have a sum of the first $n$ terms denoted as $S_n$, and define $T_n=\frac{S_1+S_2+\cdots +S_n}{n}$ as the "ideal number" of the sequence $a_1,a_2,\cdots,a_n$. It is known that the "ideal number" of the sequence $a_1,a_2,\cdots,a_{504}$ is 2020. Determine the "ideal number" of the sequence $2,... | 2018 | 67.1875 |
17,346 | Calculate the nearest integer to $(3+\sqrt{5})^6$. | 20608 | 2.34375 |
17,347 | Find the minimum value of $n (n > 0)$ such that the function \\(f(x)= \begin{vmatrix} \sqrt {3} & \sin x \\\\ 1 & \cos x\\end{vmatrix} \\) when shifted $n$ units to the left becomes an even function. | \frac{5\pi}{6} | 46.875 |
17,348 | Triangle $ABC$ is isosceles with $AB=AC$ . The bisectors of angles $ABC$ and $ACB$ meet at $I$ . If the measure of angle $CIA$ is $130^\circ$ , compute the measure of angle $CAB$ .
*Proposed by Connor Gordon* | 80 | 64.0625 |
17,349 | What is the smallest integer $n$, greater than $1$, such that $n^{-1}\pmod{2310}$ is defined? | 13 | 92.96875 |
17,350 | Given a rectangle divided into a 2x4 grid of equally spaced points, calculate the total number of distinct triangles that can be formed using three of these points as vertices. | 48 | 11.71875 |
17,351 | Let $f(x) = \cos(x + \theta) + \sqrt{2}\sin(x + \phi)$ be an even function, where $\theta$ and $\phi$ are acute angles, and $\cos\theta = \frac{\sqrt{6}}{3}\sin\phi$. Then, evaluate the value of $\theta + \phi$. | \frac{7\pi}{12} | 62.5 |
17,352 | Stock investor Li Jin bought shares of a certain company last Saturday for $27 per share. The table below shows the price changes of the stock within the week.
| Day of the Week | Monday | Tuesday | Wednesday | Thursday | Friday | Saturday |
|-----------------|--------|---------|-----------|----------|--------|-------... | 24.5 | 10.9375 |
17,353 | Simplify the expression $\dfrac{45}{28} \cdot \dfrac{49}{75} \cdot \dfrac{100}{63}$. | \frac{5}{3} | 51.5625 |
17,354 | John has 15 marbles of different colors, including one red, one green, one blue, and three yellow marbles. In how many ways can he choose 5 marbles, if he must choose exactly one marble that is red, green, blue, or yellow? | 756 | 0 |
17,355 | If $α$ and $β$ are acute angles, and they satisfy $\cos α= \frac{4}{5}$ and $\cos (α+β)= \frac{5}{13}$, calculate the value of $\sin β$. | \frac{33}{65} | 32.8125 |
17,356 | There are $13$ positive integers greater than $\sqrt{15}$ and less than $\sqrt[3]{B}$ . What is the smallest integer value of $B$ ? | 4097 | 35.15625 |
17,357 | Find the sum of all positive integers $n$ such that $1.5n - 6.3 < 7.5$. | 45 | 96.09375 |
17,358 | In a similar game setup, there are 30 boxes, each containing one of the following values: \begin{tabular}{|c|c|}\hline\$.01&\$1,000\\\hline\$1&\$5,000\\\hline\$5&\$10,000\\\hline\$10&\$25,000\\\hline\$25&\$50,000\\\hline\$50&\$75,000\\\hline\$75&\$100,000\\\hline\$100&\$200,000\\\hline\$200&\$300,000\\\hline\$300&\$400... | 18 | 39.84375 |
17,359 | City A has 2 attractions, $A$ and $B$, while City B has 3 attractions, $C$, $D$, and $E$. When randomly selecting attractions to visit, find the probability of the following events:
1. Selecting exactly 1 attraction in City A.
2. Selecting exactly 2 attractions in the same city. | \frac{2}{5} | 49.21875 |
17,360 | In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively. It is given that $a\sin 2B=\sqrt{3}b\sin A$.
$(1)$ Find the magnitude of angle $B$;
$(2)$ If $\cos A=\frac{1}{3}$, find the value of $\sin C$. | \frac{2\sqrt{6}+1}{6} | 81.25 |
17,361 | Given the function $f(x) = x + 1 + |3 - x|$, where $x \geq -1$.
1. Find the solution set for the inequality $f(x) \leq 6$.
2. If the minimum value of $f(x)$ is $n$, and the positive numbers $a$ and $b$ satisfy $2nab = a + 2b$, find the minimum value of $2a + b$. | \frac{9}{8} | 46.09375 |
17,362 | Given that an ellipse and a hyperbola $(x^{2}-y^{2}=1)$ share the same foci and the eccentricity is $\frac{\sqrt{2}}{2}$.
(I) Find the standard equation of the ellipse;
(II) A line passing through point $P(0,1)$ intersects the ellipse at points $A$ and $B$. $O$ is the origin. If $\overrightarrow{AP}=2\overrightarrow{PB... | \frac{\sqrt{126}}{8} | 0 |
17,363 | Given the function $f(x) = \sqrt[3]{x}$, calculate the value of $\lim\limits_{\Delta x \to 0} \frac{f(1-\Delta x)-f(1)}{\Delta x}$. | -\frac{1}{3} | 62.5 |
17,364 | What is the product of the numerator and the denominator when $0.\overline{012}$ is expressed as a fraction in lowest terms? | 1332 | 89.84375 |
17,365 | The ratio of the area of a square inscribed in a semicircle to the area of a square inscribed in a full circle is: | 2: 5 | 0 |
17,366 | A high school offers three separate elective classes for the senior two-grade mathematics course. After the selection process, four students request to change their math class. However, each class can accept at most two more students. Determine the number of different ways the students can be redistributed among the cl... | 54 | 28.125 |
17,367 | Given an inverted cone with a base radius of $15 \mathrm{cm}$ and a height of $15 \mathrm{cm}$, and a cylinder with a horizontal base radius of $18 \mathrm{cm}$, determine the height in centimeters of the water in the cylinder after $10\%$ of the water is lost from the cone. | 3.125 | 84.375 |
17,368 | What is the area of the smallest square that can contain a circle of radius 6? | 144 | 99.21875 |
17,369 | Define a function $g$ from the positive integers to the positive integers with the following properties:
(i) $g$ is increasing.
(ii) $g(mn) = g(m)g(n)$ for all positive integers $m$ and $n$.
(iii) If $m \neq n$ and $m^n = n^m$, then $g(m) = n$ or $g(n) = m$.
Compute all possible values of $g(88).$ | 7744 | 18.75 |
17,370 | What is the smallest positive value of $x$ such that $x + 4321$ results in a palindrome? | 13 | 86.71875 |
17,371 | If it costs two cents for each plastic digit used to number each locker and it costs $294.94 to label all lockers up to a certain number, calculate the highest locker number labeled. | 3963 | 10.9375 |
17,372 | Given that the odd function $f(x)$ and the even function $g(x)$ defined on $\mathbb{R}$ satisfy $f(x) + g(x) = a^x - a^{-x} + 2$, and $g(2) = a$, find the value of $f(2)$. | \frac{15}{4} | 93.75 |
17,373 | Find the sum of all real numbers $x$ that are not in the domain of the function $$g(x) = \frac{1}{2 + \frac{1}{2 + \frac{1}{x}}}.$$ | -\frac{9}{10} | 92.96875 |
17,374 | Six students are arranged into two rows, with 3 students per row, and the number of different arrangements must be calculated. | 720 | 86.71875 |
17,375 | Find the equation of the line that passes through the intersection of the lines $2x+3y+5=0$ and $2x+5y+7=0$, and is parallel to the line $x+3y=0$. Also, calculate the distance between these two parallel lines. | \frac{2\sqrt{10}}{5} | 79.6875 |
17,376 | A woman invests in a property for $12,000 with the aim of receiving a $6\%$ return on her investment after covering all expenses including taxes and insurance. She pays $360 annually in taxes and $240 annually for insurance. She also keeps aside $10\%$ of each month's rent for maintenance. Calculate the monthly rent. | 122.22 | 54.6875 |
17,377 | Given $sin(\alpha-\frac{\pi}{6})=\frac{3}{5}$, calculate $cos(\frac{2\pi}{3}-\alpha)$. | \frac{3}{5} | 75 |
17,378 | Given \(\sum^{100}_{i=1} \sum^{100}_{j=1} (i+j)\), find the value of the expression. | 1010000 | 100 |
17,379 | $(1)$ Find the value of $x$: $4\left(x+1\right)^{2}=49$;<br/>$(2)$ Calculate: $\sqrt{9}-{({-1})^{2018}}-\sqrt[3]{{27}}+|{2-\sqrt{5}}|$. | \sqrt{5} - 3 | 38.28125 |
17,380 | Given the hexadecimal system, determine the product of $A$ and $B$. | 6E | 24.21875 |
17,381 | Let the complex number $z = \cos\tfrac{1}{1000} + i \sin\tfrac{1}{1000}.$ Find the smallest positive integer $n$ so that $z^n$ has an imaginary part which exceeds $\tfrac{1}{2}.$ | 524 | 100 |
17,382 | A circle passes through the three vertices of an isosceles triangle that has two sides of length 5 and a base of length 4. What is the area of this circle? Express your answer in terms of $\pi$. | \frac{13125}{1764}\pi | 0 |
17,383 | Cyclic quadrilateral $ABCD$ satisfies $\angle ABD = 70^\circ$ , $\angle ADB=50^\circ$ , and $BC=CD$ . Suppose $AB$ intersects $CD$ at point $P$ , while $AD$ intersects $BC$ at point $Q$ . Compute $\angle APQ-\angle AQP$ . | 20 | 57.8125 |
17,384 | Given P(A) = 0.65, P(B) = 0.2, and P(C) = 0.1, calculate the probability of the event "the drawn product is not a first-class product". | 0.35 | 53.125 |
17,385 | In the diagram, $ABCD$ is a trapezoid with an area of $18.$ $CD$ is three times the length of $AB.$ What is the area of $\triangle ABD?$
[asy]
draw((0,0)--(1,4)--(9,4)--(18,0)--cycle);
draw((9,4)--(0,0));
label("$D$",(0,0),W);
label("$A$",(1,4),NW);
label("$B$",(9,4),NE);
label("$C$",(18,0),E);
[/asy] | 4.5 | 85.15625 |
17,386 | A train took $X$ minutes ($0 < X < 60$) to travel from platform A to platform B. Find $X$ if it's known that at both the moment of departure from A and the moment of arrival at B, the angle between the hour and minute hands of the clock was $X$ degrees. | 48 | 9.375 |
17,387 | Sanitation workers plan to plant 7 trees in a row on one side of a road, choosing only from plane trees and willow trees. What is the number of planting methods where no two adjacent trees are both willows? | 34 | 74.21875 |
17,388 | The expression \(\frac{3}{10}+\frac{3}{100}+\frac{3}{1000}\) can be simplified by converting each fraction to a decimal, and then calculating the sum. | 0.333 | 98.4375 |
17,389 | Let $(a_1, a_2, a_3,\ldots,a_{13})$ be a permutation of $(1,2,3,\ldots,13)$ for which
$$a_1 > a_2 > a_3 > a_4 > a_5 > a_6 > a_7 \mathrm{\ and \ } a_7 < a_8 < a_9 < a_{10} < a_{11} < a_{12} < a_{13}.$$
Find the number of such permutations. | 924 | 18.75 |
17,390 | A larger grid is considered for the next challenge, where each segment must still only be traversed in a rightward or downward direction. Starting from point $A$, located at the top-left of a 3x3 grid, to point $B$ at the bottom-right. How many different routes can be taken? | 20 | 83.59375 |
17,391 | How many four-digit numbers have the property that the second digit is the average of the first and third digits, and the digits are all even? | 50 | 13.28125 |
17,392 | A biased coin with the probability of landing heads as 1/3 is flipped 12 times. What is the probability of getting exactly 9 heads in the 12 flips? | \frac{1760}{531441} | 9.375 |
17,393 | Suppose
\[\frac{1}{x^3 - 2x^2 - 13x + 10} = \frac{A}{x+2} + \frac{B}{x-1} + \frac{C}{(x-1)^2}\]
where $A$, $B$, and $C$ are real constants. What is $A$? | \frac{1}{9} | 82.8125 |
17,394 | Let $T$ be a positive integer whose only digits are 0s and 1s. If $X = T \div 24$ and $X$ is an integer, what is the smallest possible value of $X$? | 4625 | 25 |
17,395 | A pair of standard $6$-sided dice is rolled to determine the side length of a square. What is the probability that the numerical value of the area of the square is less than the numerical value of the perimeter? | \frac{1}{12} | 19.53125 |
17,396 | If four consecutive natural numbers are all composite numbers, find the smallest sum of these four numbers. | 102 | 71.875 |
17,397 | Given that the function $f(x)$ defined on $\mathbb{R}$ is an odd function and satisfies $f(1+x)=f(3+x)$. When $0\leq x\leq 1$, $f(x)=x^{3}-x$. Find $f(\frac{11}{2})+f(6)$. | \frac{3}{8} | 22.65625 |
17,398 | There is a stack of 200 cards, numbered from 1 to 200 from top to bottom. Starting with the top card, the following operations are performed in sequence: remove the top card, then place the next card at the bottom of the stack; remove the new top card, then place the next card at the bottom of the stack… This process i... | 145 | 0 |
17,399 | If $\frac{\sin\theta + \cos\theta}{\sin\theta - \cos\theta} = 2$, calculate the value of $\sin\theta \cdot \cos\theta$. | \frac{3}{10} | 92.1875 |
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