Unnamed: 0
int64 0
40.3k
| problem
stringlengths 10
5.15k
| ground_truth
stringlengths 1
1.22k
| solved_percentage
float64 0
100
|
|---|---|---|---|
18,400 |
Given that the solution set of the inequality $x^{2}-2x+1-m^{2} \lt 0$ is $A$;
$(1)$ Find $A$;
$(2)$ If $0 \lt m \lt 1$, and $A=\{x\left|\right.a \lt x \lt b\}$, find the minimum value of $\frac{1}{{8a+2b}}-\frac{1}{{3a-3b}}$.
|
\frac{2}{5}
| 51.5625 |
18,401 |
Given that the sum of the first $n$ terms of the sequence $\{a_{n}\}$ is $S_{n}$, and $a_{1}=4$, $a_{n}+a_{n+1}=4n+2$ for $n\in \mathbb{N}^{*}$, calculate the maximum value of $n$ that satisfies $S_{n} \lt 2023$.
|
44
| 35.9375 |
18,402 |
Xiao Kang exercises every day by doing several sets of push-ups, 8 per set, and several sets of pull-ups, 5 per set. On the first day, he did a total of 41 reps (combining both exercises), and he increases the number by 1 each day until the 100th day. If the number of sets of push-ups and pull-ups he does each day are both positive integers, then over these 100 days, Xiao Kang did at least $\boxed{100}$ sets of push-ups and at least $\boxed{106}$ sets of pull-ups.
|
106
| 86.71875 |
18,403 |
In the Cartesian coordinate system, establish a polar coordinate system with the coordinate origin as the pole and the non-negative semi-axis of the $x$-axis as the polar axis. Given that point $A$ has polar coordinates $(\sqrt{2}, \frac{\pi}{4})$, and the parametric equations of line $l$ are $\begin{cases} x = \frac{3}{2} - \frac{\sqrt{2}}{2}t \\ y = \frac{1}{2} + \frac{\sqrt{2}}{2}t \end{cases}$ (where $t$ is the parameter), and point $A$ lies on line $l$.
(I) Find the parameter $t$ corresponding to point $A$;
(II) If the parametric equations of curve $C$ are $\begin{cases} x = 2\cos \theta \\ y = \sin \theta \end{cases}$ (where $\theta$ is the parameter), and line $l$ intersects curve $C$ at points $M$ and $N$, find $|MN|$.
|
\frac{4\sqrt{2}}{5}
| 14.0625 |
18,404 |
The bases \(AB\) and \(CD\) of the trapezoid \(ABCD\) are 367 and 6 respectively, and its diagonals are mutually perpendicular. Find the dot product of the vectors \(\overrightarrow{AD}\) and \(\overrightarrow{BC}\).
|
2202
| 48.4375 |
18,405 |
Given that Crystal runs due north for 2 miles, then northwest for 1 mile, and southwest for 1 mile, find the distance of the last portion of her run that returns her directly to her starting point.
|
\sqrt{6}
| 82.03125 |
18,406 |
Find the maximum value of the real constant $C$ such that $x^{2}+y^{2}+1\geq C(x+y)$ , and $ x^{2}+y^{2}+xy+1\geq C(x+y)$ for all reals $x,y$ .
|
\sqrt{2}
| 78.90625 |
18,407 |
(1) Find the domain of the function $f(x) = \log(2\sin 2x - 1)$.
(2) Calculate: $$\log_{2}\cos \frac{\pi}{9} + \log_{2}\cos \frac{2\pi}{9} + \log_{2}\cos \frac{4\pi}{9}.$$
|
-3
| 96.875 |
18,408 |
Given four distinct points P, A, B, C on a plane satisfying $\overrightarrow{PA} + \overrightarrow{PB} + \overrightarrow{PC} = \overrightarrow{0}$ and $\overrightarrow{AB} + \overrightarrow{AC} + m\overrightarrow{AP} = \overrightarrow{0}$, determine the value of the real number $m$.
|
-3
| 86.71875 |
18,409 |
Five friends earned $18, $23, $28, $35, and $45. If they split their earnings equally among themselves, how much will the friend who earned $45 need to give to the others?
|
15.2
| 17.1875 |
18,410 |
Given $tan({θ+\frac{π}{4}})=2tanθ-7$, determine the value of $\sin 2\theta$.
|
\frac{4}{5}
| 87.5 |
18,411 |
In the regular hexagon \( A B C D E F \), two of the diagonals, \( F C \) and \( B D \), intersect at \( G \). The ratio of the area of quadrilateral FEDG to the area of \( \triangle B C G \) is:
|
5: 1
| 0 |
18,412 |
A high school basketball team has 12 players, including a set of twins, John and James. In how many ways can we choose a starting lineup of 5 players if exactly one of the twins must be in the lineup?
|
660
| 0.78125 |
18,413 |
Lucy started with a bag of 180 oranges. She sold $30\%$ of them to Max. From the remaining, she then sold $20\%$ to Maya. Of the oranges left, she donated 10 to a local charity. Find the number of oranges Lucy had left.
|
91
| 65.625 |
18,414 |
Given that the sequence $\{a\_n\}$ is an arithmetic progression, and the sequence $\{b\_n\}$ satisfies $b\_n=a\_n a_{n+1} \cdot a_{n+2} (n \in \mathbb{N}^*)$, let $S\_n$ be the sum of the first $n$ terms of $\{b\_n\}$. If $a_{12}=\frac{3}{8} a_{5} > 0$, find the value of $n$ when $S\_n$ reaches its maximum.
|
16
| 25 |
18,415 |
Let $S$ be the set of 81 points $(x, y)$ such that $x$ and $y$ are integers from $-4$ through $4$ . Let $A$ , $B$ , and $C$ be random points chosen independently from $S$ , with each of the 81 points being equally likely. (The points $A$ , $B$ , and $C$ do not have to be different.) Let $K$ be the area of the (possibly degenerate) triangle $ABC$ . What is the expected value (average value) of $K^2$ ?
|
\frac{200}{3}
| 31.25 |
18,416 |
A right circular cone is sliced into five equal-height sections by planes parallel to its base. What is the ratio of the volume of the second-largest piece to the volume of the largest piece?
|
\frac{37}{61}
| 28.125 |
18,417 |
Given that the focus $F$ of the parabola $x=4y^{2}$ intersects the parabola at points $M$ and $N$, and $|MF|= \frac{1}{8}$, find the value of $|MN|$.
|
\frac{1}{4}
| 78.125 |
18,418 |
If "For all $x \in \mathbb{R}, (a-2)x+1>0$" is a true statement, then the set of values for the real number $a$ is.
|
\{2\}
| 58.59375 |
18,419 |
Given the parabola $C: y^2 = 2px \ (0 < p < 4)$ with focus $F$, and a moving point $P$ on $C$. Let $A(4, 0)$ and $B(p, \sqrt{2}p)$ be such that the minimum value of $|PA|$ is $\sqrt{15}$. Find the value of $|BF|$.
|
\frac{9}{2}
| 60.9375 |
18,420 |
Given $f(x)$ is an even function defined on $\mathbb{R}$ and satisfies $f(x+2)=-\frac{1}{f(x)}$. If $f(x)=x$ for $2\leq x \leq 3$, find the value of $f\left(-\frac{11}{2}\right)$.
|
\frac{5}{2}
| 6.25 |
18,421 |
The number of trailing zeros in 2006! is to be calculated.
|
500
| 100 |
18,422 |
If $\sin\theta + \cos\theta = \frac{2\sqrt{2}-1}{3}$ ($0 < \theta < \pi$), then $\tan\theta = \_\_\_\_\_\_$.
|
-2\sqrt{2}
| 21.09375 |
18,423 |
Let \( f(x) \) be a function from \( \mathbf{R} \) to \( \mathbf{R} \), and for any real numbers, it holds that
$$
f(x^{2}+x) + 2 f(x^{2}-3x+2) = 9x^{2} - 15x,
$$
then the value of \( f(50) \) is ( ).
|
146
| 81.25 |
18,424 |
Of all positive integral solutions $(x,y,z)$ to the equation \[x^3+y^3+z^3-3xyz=607,\] compute the minimum possible value of $x+2y+3z.$ *Individual #7*
|
1215
| 7.8125 |
18,425 |
The circle centered at $(3, -2)$ and with radius $5$ intersects the circle centered at $(3, 4)$ and with radius $\sqrt{17}$. Find $(AB)^2$, where $A$ and $B$ are the points of intersection.
|
\frac{416}{9}
| 65.625 |
18,426 |
What is the average of all the integer values of $M$ such that $\frac{M}{42}$ is strictly between $\frac{5}{14}$ and $\frac{1}{6}$?
|
11
| 74.21875 |
18,427 |
For some complex number $z$ with $|z| = 3,$ there is some real $\lambda > 1$ such that $z,$ $z^2,$ and $\lambda z$ form an equilateral triangle in the complex plane. Find $\lambda.$
|
\frac{1 + \sqrt{33}}{2}
| 7.03125 |
18,428 |
Given that $2x + 5y = 20$ and $5x + 2y = 26$, find $20x^2 + 60xy + 50y^2$.
|
\frac{59600}{49}
| 5.46875 |
18,429 |
A spinner is divided into three equal sections labeled 0, 1, and 5. If you spin the spinner three times, use the outcomes to form a three-digit number (hundreds, tens, units), what is the probability that the resulting number is divisible by 5?
|
\frac{2}{3}
| 98.4375 |
18,430 |
What three-digit positive integer is one more than a multiple of 3, 4, 5, 6, and 7?
|
421
| 87.5 |
18,431 |
Given that the area of $\triangle ABC$ is $S$, and $\overrightarrow{AB} \cdot \overrightarrow{AC} = S$.
(1) Find the values of $\sin A$, $\cos A$, and $\tan 2A$.
(2) If $B = \frac{\pi}{4}, \; |\overrightarrow{CA} - \overrightarrow{CB}| = 6$, find the area $S$ of $\triangle ABC$.
|
12
| 47.65625 |
18,432 |
A line is parameterized by
\[\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 2 \\ 1 \end{pmatrix} + s \begin{pmatrix} 4 \\ -1 \end{pmatrix}.\]
A second line is parameterized by
\[\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 3 \\ 7 \end{pmatrix} + v \begin{pmatrix} -2 \\ 5 \end{pmatrix}.\]
If $\theta$ is the angle formed by the two lines, then find $\cos \theta.$ Also, verify if the point \((5, 0)\) lies on the first line.
|
\frac{-13}{\sqrt{493}}
| 43.75 |
18,433 |
Compute $\arccos(\sin 3)$, where all functions are in radians.
|
3 - \frac{\pi}{2}
| 36.71875 |
18,434 |
Determine how many positive integer multiples of $2002$ can be represented in the form $10^{j} - 10^{i}$, where $i$ and $j$ are integers and $0 \leq i < j \leq 150$.
|
1825
| 2.34375 |
18,435 |
Simplify and then evaluate the expression:
$$( \frac {x}{x-1}- \frac {x}{x^{2}-1})÷ \frac {x^{2}-x}{x^{2}-2x+1}$$
where $$x= \sqrt {2}-1$$
|
1- \frac { \sqrt {2}}{2}
| 0 |
18,436 |
Suppose $a$ is a real number such that $\sin(\pi \cdot \cos a) = \cos(\pi \cdot \sin a)$ . Evaluate $35 \sin^2(2a) + 84 \cos^2(4a)$ .
|
21
| 74.21875 |
18,437 |
An infinite geometric series has a first term of $15$ and a second term of $5$. A second infinite geometric series has the same first term of $15$, a second term of $5+n$, and a sum of three times that of the first series. Find the value of $n$.
|
\frac{20}{3}
| 73.4375 |
18,438 |
Calculate the sum: $\dfrac{2}{100} + \dfrac{5}{1000} + \dfrac{8}{10000} + \dfrac{6}{100000}$.
|
0.02586
| 93.75 |
18,439 |
Given circle $C$: $(x-2)^2 + y^2 = 4$, and line $l$: $x - \sqrt{3}y = 0$, the probability that the distance from point $A$ on circle $C$ to line $l$ is not greater than $1$ is $\_\_\_\_\_\_\_.$
|
\frac{1}{2}
| 19.53125 |
18,440 |
In a sealed box, there are three red chips and two green chips. Chips are randomly drawn from the box without replacement until either all three red chips or both green chips are drawn. What is the probability of drawing all three red chips?
|
$\frac{2}{5}$
| 0 |
18,441 |
What common fraction (that is, a fraction reduced to its lowest terms) is equivalent to $0.1\overline{35}$?
|
\frac{67}{495}
| 67.96875 |
18,442 |
A new window design consists of a rectangle topped with a semi-circle at both ends. The ratio of the length AD of the rectangle to its width AB is 4:3. If AB is 36 inches, calculate the ratio of the area of the rectangle to the combined area of the semicircles.
|
\frac{16}{3\pi}
| 78.125 |
18,443 |
Consider a unit square $WXYZ$ with midpoints $M_1$, $M_2$, $M_3$, and $M_4$ on sides $WZ$, $XY$, $YZ$, and $XW$ respectively. Let $R_1$ be a point on side $WZ$ such that $WR_1 = \frac{1}{4}$. A light ray starts from $R_1$ and reflects off at point $S_1$ (which is the intersection of the ray $R_1M_2$ and diagonal $WY$). The ray reflects again at point $T_1$ where it hits side $YZ$, now heading towards $M_4$. Denote $R_2$ the next point where the ray hits side $WZ$. Calculate the sum $\sum_{i=1}^{\infty} \text{Area of } \triangle WSR_i$ where $R_i$ is the sequence of points on $WZ$ created by continued reflection.
A) $\frac{1}{28}$
B) $\frac{1}{24}$
C) $\frac{1}{18}$
D) $\frac{1}{12}$
|
\frac{1}{24}
| 31.25 |
18,444 |
Given the ellipse $\Gamma: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, its right focus is $F(3,0)$, and its top and bottom vertices are $A$ and $B$ respectively. The line $AF$ intersects $\Gamma$ at another point $M$. If the line $BM$ intersects the $x$-axis at point $N(12,0)$, then the eccentricity of $\Gamma$ is _______.
|
\frac{1}{2}
| 57.8125 |
18,445 |
From 3 male students and 2 female students, calculate the number of different election results in which at least one female student is elected.
|
14
| 10.15625 |
18,446 |
A standard deck of 52 cards is arranged randomly. What is the probability that the top three cards alternate in color, starting with a red card, then a black card, followed by another red card?
|
\frac{13}{102}
| 60.9375 |
18,447 |
Find \(n\) such that \(2^6 \cdot 3^3 \cdot n = 10!\).
|
n = 2100
| 10.15625 |
18,448 |
Given a sequence $\{a_n\}$ where all terms are positive integers, let $S_n$ denote the sum of the first $n$ terms. If $a_{n+1}=\begin{cases} \frac{a_n}{2},a_n \text{ is even} \\\\ 3a_n+1,a_n \text{ is odd} \end{cases}$ and $a_1=5$, calculate $S_{2015}$.
|
4725
| 0 |
18,449 |
A room is 24 feet long and 14 feet wide. Find the ratio of the length to its perimeter and the ratio of the width to its perimeter. Express each ratio in the form $a:b$.
|
7:38
| 31.25 |
18,450 |
Given the system of equations in terms of $x$ and $y$: $\left\{\begin{array}{l}{3x+5y=m+2}\\{2x+3y=m}\end{array}\right.$.<br/>$(1)$ Find the relationship between $x$ and $y$ (express $y$ as an algebraic expression only containing $x$);<br/>$(2)$ If the solution to this system satisfies $x+y=-10$, find the value of $m$.
|
-8
| 89.0625 |
18,451 |
Let $m$ and $n$ satisfy $mn = 6$ and $m+n = 7$. Additionally, suppose $m^2 - n^2 = 13$. Find the value of $|m-n|$.
|
\frac{13}{7}
| 68.75 |
18,452 |
Given that 216 sprinters enter a 100-meter dash competition, and the track has 6 lanes, determine the minimum number of races needed to find the champion sprinter.
|
43
| 70.3125 |
18,453 |
Given that $F\_1(-4,0)$ and $F\_2(4,0)$ are the two foci of the ellipse $\frac{x^2}{25} + \frac{y^2}{9} = 1$, and $P$ is a point on the ellipse such that the area of $\triangle PF\_1F\_2$ is $3\sqrt{3}$, find the value of $\cos\angle{F\_1PF\_2}$.
|
\frac{1}{2}
| 84.375 |
18,454 |
Given that $F\_1$ and $F\_2$ are the left and right foci of the hyperbola $E$: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$, and point $M$ is on $E$, with $MF\_1$ perpendicular to the $x$-axis and $\sin \angle MF\_2F\_1 = \frac{1}{3}$. Find the eccentricity of $E$.
|
\sqrt{2}
| 90.625 |
18,455 |
The scientific notation for 0.000048 is $4.8\times 10^{-5}$.
|
4.8 \times 10^{-5}
| 91.40625 |
18,456 |
Find $(1_8 + 2_8 + 3_8 + \cdots + 30_8) \times 3_{10}$, expressed in base $8$.
|
1604_8
| 66.40625 |
18,457 |
Carefully observe the following three rows of related numbers:<br/>First row: $-2$, $4$, $-8$, $16$, $-32$, $\ldots$;<br/>Second row: $0$, $6$, $-6$, $18$, $-30$, $\ldots$;<br/>Third row: $-1$, $2$, $-4$, $8$, $-16$, $\ldots$;<br/>Answer the following questions:<br/>$(1)$ The $6$th number in the first row is ______;<br/>$(2)$ What is the relationship between the numbers in the second row, the third row, and the first row?<br/>$(3)$ Take a number $a$ from the first row and the other two numbers corresponding to it from the second and third rows, such that the sum of these three numbers is $642$. Find the value of $a$ and state which number in the first row $a$ corresponds to.
|
256
| 67.96875 |
18,458 |
In a new diagram, $A$ is the center of a circle with radii $AB=AC=8$. The sector $BOC$ is shaded except for a triangle $ABC$ within it, where $B$ and $C$ lie on the circle. If the central angle of $BOC$ is $240^\circ$, what is the perimeter of the shaded region?
|
16 + \frac{32}{3}\pi
| 0 |
18,459 |
In a certain entertainment unit, each member can sing or dance at least one of the two. It is known that there are 4 people who can sing and 5 people who can dance. Now, 2 people are selected from them to participate in a social charity performance. Let $\xi$ be the number of people selected who can both sing and dance, and $P(\xi≥1)=\frac{11}{21}$.
$(Ⅰ)$ Find the total number of members in this entertainment unit.
$(Ⅱ)$ Find the probability distribution and the expected value $E(\xi)$ of the random variable $\xi$.
|
\frac{4}{7}
| 27.34375 |
18,460 |
The probability of A not losing is $\dfrac{1}{3} + \dfrac{1}{2}$.
|
\dfrac{1}{6}
| 1.5625 |
18,461 |
The cost of two pencils and three pens is $4.10, and the cost of three pencils and one pen is $2.95. What is the cost of one pencil and four pens?
|
4.34
| 50 |
18,462 |
Determine the period of the function $y = \tan(2x) + \cot(2x)$.
|
\frac{\pi}{2}
| 82.03125 |
18,463 |
A *palindromic table* is a $3 \times 3$ array of letters such that the words in each row and column read the same forwards and backwards. An example of such a table is shown below.
\[ \begin{array}[h]{ccc}
O & M & O
N & M & N
O & M & O
\end{array} \]
How many palindromic tables are there that use only the letters $O$ and $M$ ? (The table may contain only a single letter.)
*Proposed by Evan Chen*
|
16
| 84.375 |
18,464 |
Add $49.213$ to $27.569$, and round your answer to the nearest whole number.
|
77
| 97.65625 |
18,465 |
Randomly select $3$ out of $6$ small balls with the numbers $1$, $2$, $3$, $4$, $5$, and $6$, which are of the same size and material. The probability that exactly $2$ of the selected balls have consecutive numbers is ____.
|
\frac{3}{5}
| 14.0625 |
18,466 |
Calculate:<br/>$(1)-7+13-6+20$;<br/>$(2)-2^{3}+\left(2-3\right)-2\times \left(-1\right)^{2023}$.
|
-7
| 78.125 |
18,467 |
If $f^{-1}(g(x))=x^4-1$ and $g$ has an inverse, find $g^{-1}(f(10))$.
|
\sqrt[4]{11}
| 84.375 |
18,468 |
Given an ellipse $C$: $\frac{{x}^{2}}{3}+{y}^{2}=1$ with left focus and right focus as $F_{1}$ and $F_{2}$ respectively, the line $y=x+m$ intersects $C$ at points $A$ and $B$. Determine the value of $m$ such that the area of $\triangle F_{1}AB$ is twice the area of $\triangle F_{2}AB$.
|
-\frac{\sqrt{2}}{3}
| 32.8125 |
18,469 |
A plane passes through the midpoints of edges $AB$ and $CD$ of pyramid $ABCD$ and divides edge $BD$ in the ratio $1:3$. In what ratio does this plane divide edge $AC$?
|
1:3
| 38.28125 |
18,470 |
In the diagram, square $PQRS$ has side length 2. Points $M$ and $N$ are the midpoints of $SR$ and $RQ$, respectively. The value of $\cos (\angle MPN)$ is
|
$\frac{4}{5}$
| 0 |
18,471 |
From a standard deck of 52 cards, what is the probability that the top four cards include only cards from the black suits ($\spadesuit$ and $\clubsuit$)?
|
\frac{276}{4998}
| 0 |
18,472 |
Given that $P$ is a moving point on the curve $y= \frac {1}{4}x^{2}- \frac {1}{2}\ln x$, and $Q$ is a moving point on the line $y= \frac {3}{4}x-1$, then the minimum value of $PQ$ is \_\_\_\_\_\_.
|
\frac {2-2\ln 2}{5}
| 0 |
18,473 |
Calculate the sum of $0.\overline{6}$ and $0.\overline{7}$ as a common fraction.
|
\frac{13}{9}
| 99.21875 |
18,474 |
In triangle \( \triangle ABC \), \( BC=a \), \( AC=b \), \( AB=c \), and \( \angle C = 90^{\circ} \). \( CD \) and \( BE \) are two medians of \( \triangle ABC \), and \( CD \perp BE \). Express the ratio \( a:b:c \) in simplest form.
|
1 : \sqrt{2} : \sqrt{3}
| 25.78125 |
18,475 |
Determine the number of ways a student can schedule four mathematics courses — algebra, geometry, number theory, and statistics — on an 8-period day, given that no two mathematics courses can be scheduled in consecutive periods.
|
120
| 87.5 |
18,476 |
Given that the domains of functions f(x) and g(x) are both R, and f(x) + g(2-x) = 5, g(x) - f(x-4) = 7. If the graph of y = g(x) is symmetric about the line x = 2, g(2) = 4, determine the value of \sum _{k=1}^{22}f(k).
|
-24
| 0 |
18,477 |
Given that each side of a triangle is an integer and none exceeds 4, determine the number of such distinct triangles.
|
13
| 0 |
18,478 |
Given that $a$, $b$, $c \in \mathbb{R}$, $a + b + c = 3$, $a \geqslant b \geqslant c$, the equation $ax^{2}+bx+c=0$ has real roots. Find the minimum value of $a$.
|
\frac{4}{3}
| 14.84375 |
18,479 |
If a computer executes the following program:
1. Initial values: \( x = 3 \), \( S = 0 \).
2. \( x = x + 2 \).
3. \( S = S + x \).
4. If \( S \geq 10000 \), go to step 5; otherwise, go back to step 2.
5. Print \( x \).
6. Stop.
Then the value printed at step 5 is:
|
201
| 98.4375 |
18,480 |
What is the sum of the digits of the base $7$ representation of $2019_{10}$?
|
15
| 25 |
18,481 |
Given $x > 0, y > 0$, and $2x + 8y - xy = 0$, find the minimum value of $xy$.
|
64
| 98.4375 |
18,482 |
Consider license plates consisting of a sequence of four digits followed by two letters. Assume each arrangement is equally likely for these plates. What is the probability that such a license plate contains at least one palindrome sequence (either the four-digit sequence or the two-letter sequence)? Express your result as a simplified fraction.
|
\frac{5}{104}
| 24.21875 |
18,483 |
What is the remainder when $4x^8 - x^7 + 5x^6 - 7x^4 + 3x^3 - 9$ is divided by $3x - 6$?
|
1119
| 4.6875 |
18,484 |
Given the set $S=\{A, A_1, A_2, A_3, A_4\}$, define the operation $\oplus$ on $S$ as: $A_i \oplus A_j = A_k$, where $k=|i-j|$, and $i, j = 0, 1, 2, 3, 4$. Calculate the total number of ordered pairs $(i, j)$ that satisfy the condition $(A_i \oplus A_j) \oplus A_2 = A_1$ (where $A_i, A_j \in S$).
|
12
| 45.3125 |
18,485 |
How can you cut a 5 × 5 square with straight lines so that the resulting pieces can be assembled into 50 equal squares? It is not allowed to leave unused pieces or to overlap them.
|
50
| 46.875 |
18,486 |
Let \( R \) be a semicircle with diameter \( XY \). A trapezoid \( ABCD \) in which \( AB \) is parallel to \( CD \) is circumscribed about \( R \) such that \( AB \) contains \( XY \). If \( AD = 4 \), \( CD = 5 \), and \( BC = 6 \), determine \( AB \).
|
10
| 42.96875 |
18,487 |
A rectangular piece of paper $ABCD$ has sides of lengths $AB = 1$ , $BC = 2$ . The rectangle is folded in half such that $AD$ coincides with $BC$ and $EF$ is the folding line. Then fold the paper along a line $BM$ such that the corner $A$ falls on line $EF$ . How large, in degrees, is $\angle ABM$ ?
[asy]
size(180); pathpen = rgb(0,0,0.6)+linewidth(1); pointpen = black+linewidth(3); pointfontpen = fontsize(10); pen dd = rgb(0,0,0.6) + linewidth(0.7) + linetype("4 4"), dr = rgb(0.8,0,0), dg = rgb(0,0.6,0), db = rgb(0,0,0.6)+linewidth(1);
pair A=(0,1), B=(0,0), C=(2,0), D=(2,1), E=A/2, F=(2,.5), M=(1/3^.5,1), N=reflect(B,M)*A;
D(B--M--D("N",N,NE)--B--D("C",C,SE)--D("D",D,NE)--M); D(D("M",M,plain.N)--D("A",A,NW)--D("B",B,SW),dd); D(D("E",E,W)--D("F",F,plain.E),dd);
[/asy]
|
30
| 78.125 |
18,488 |
How many perfect squares are between 100 and 400?
|
11
| 97.65625 |
18,489 |
In the plane rectangular coordinate system $xOy$, the parametric equations of curve $C$ are $\left\{{\begin{array}{l}{x=2+3\cos\alpha}\\{y=3\sin\alpha}\end{array}}\right.$ ($\alpha$ is the parameter). Taking the coordinate origin $O$ as the pole and the non-negative half-axis of the $x$-axis as the polar axis, the polar coordinate equation of the line $l$ is $2\rho \cos \theta -\rho \sin \theta -1=0$.
$(1)$ Find the general equation of curve $C$ and the rectangular coordinate equation of line $l$;
$(2)$ If line $l$ intersects curve $C$ at points $A$ and $B$, and point $P(0,-1)$, find the value of $\frac{1}{{|{PA}|}}+\frac{1}{{|{PB}|}}$.
|
\frac{{3\sqrt{5}}}{5}
| 0 |
18,490 |
It is known that the complex number \( z \) satisfies \( |z| = 1 \). Find the maximum value of \( u = \left| z^3 - 3z + 2 \right| \).
|
3\sqrt{3}
| 84.375 |
18,491 |
In triangle \( \triangle ABC \), it is known that \( AC=3 \) and the three interior angles \( A \), \( B \), and \( C \) form an arithmetic sequence.
(1) If \( \cos C= \frac {\sqrt{6}}{3} \), find \( AB \);
(2) Find the maximum value of the area of \( \triangle ABC \).
|
\frac {9 \sqrt {3}}{4}
| 0 |
18,492 |
In how many ways can the digits of $47,\!770$ be arranged to form a 5-digit number, ensuring the number does not begin with 0?
|
16
| 97.65625 |
18,493 |
The sequence $(x_n)$ is determined by the conditions: $x_0=1992,x_n=-\frac{1992}{n} \cdot \sum_{k=0}^{n-1} x_k$ for $n \geq 1$ .
Find $\sum_{n=0}^{1992} 2^nx_n$ .
|
1992
| 40.625 |
18,494 |
What is the absolute value of the difference between the squares of 103 and 97?
|
1200
| 93.75 |
18,495 |
Given the function $f(x)=\frac{1}{2}\sin 2x\sin φ+\cos^2x\cos φ+\frac{1}{2}\sin (\frac{3π}{2}-φ)(0 < φ < π)$, whose graph passes through the point $(\frac{π}{6},\frac{1}{2})$.
(I) Find the interval(s) where the function $f(x)$ is monotonically decreasing on $[0,π]$;
(II) If ${x}_{0}∈(\frac{π}{2},π)$, $\sin {x}_{0}= \frac{3}{5}$, find the value of $f({x}_{0})$.
|
\frac{7-24\sqrt{3}}{100}
| 28.90625 |
18,496 |
Connecting the centers of adjacent faces of a cube forms a regular octahedron. What is the volume ratio of this octahedron to the cube?
|
$\frac{1}{6}$
| 0 |
18,497 |
A student passes through three intersections with traffic lights, labeled \\(A\\), \\(B\\), and \\(C\\), on the way to school. It is known that the probabilities of encountering a red light at intersections \\(A\\), \\(B\\), and \\(C\\) are \\( \dfrac {1}{3} \\), \\( \dfrac {1}{4} \\), and \\( \dfrac {3}{4} \\) respectively. The time spent waiting at a red light at these intersections is \\(40\\) seconds, \\(20\\) seconds, and \\(80\\) seconds, respectively. Additionally, whether or not the student encounters a red light at each intersection is independent of the other intersections.
\\((1)\\) Calculate the probability that the student encounters a red light for the first time at the third intersection on the way to school;
\\((2)\\) Calculate the total time the student spends waiting at red lights on the way to school.
|
\dfrac {235}{3}
| 85.9375 |
18,498 |
If $a$, $b$, $c$, and $d$ are four positive numbers whose product is 1, calculate the minimum value of the algebraic expression $a^2+b^2+c^2+d^2+ab+ac+ad+bc+bd+cd$.
|
10
| 88.28125 |
18,499 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $b\sin A=a\sin C$ and $c=1$, find the value of $b$ and the maximum area of $\triangle ABC$.
|
\frac{1}{2}
| 43.75 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.