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 \( x \geq 0 \) and \( y \geq 0 \), and \( x^{2} + y^{2} = 4 \), find the minimum value of \( xy - 4(x + y) - 2 \). | -8\sqrt{2} | 55 | 7 |
math | Simplify the expression $(-\frac{1}{16})^{-3/4}$. | -8 | 20 | 2 |
math | Let $\overrightarrow{a}$ and $\overrightarrow{b}$ be two vectors, with $|\overrightarrow{a}|=1$ and $|\overrightarrow{b}|=2$, and $(\overrightarrow{a}+ \overrightarrow{b}) \perp \overrightarrow{a}$. Find the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$. | \frac{2\pi}{3} | 83 | 9 |
math | If $-\frac{\pi}{2} < \alpha < 0$, the point $(\tan \alpha, \cos \alpha)$ is located in the \_\_\_\_\_\_ quadrant. | 2 | 42 | 1 |
math | On a plane, points are colored in the following way:
1. Choose any positive integer \( m \), and let \( K_{1}, K_{2}, \cdots, K_{m} \) be circles with different non-zero radii such that \( K_{i} \subset K_{j} \) or \( K_{j} \subset K_{i} \) for \( i \neq j \).
2. Points chosen inside the circles are colored differently... | 2019 | 145 | 4 |
math | Given the function $f(x)=4\cos \left(x- \frac {\pi}{2}\right)\cdot \sin \left(x- \frac {\pi}{3}\right)-1$.
$(1)$ Find the smallest positive period of the function $y=f(x)$.
$(2)$ Given in $\triangle ABC$, the sides opposite angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and $a$, $b$, $c$ form a geometric sequ... | [-2,-1] | 118 | 5 |
math | The tiling pattern shown uses two types of tiles, regular hexagons and equilateral triangles, with the length of each side of the equilateral triangles equal to half the length of each side of the hexagons. A large number of tiles is used to cover a floor. Which of the following is closest to the fraction of the floor ... | \frac{1}{13} | 127 | 8 |
math | A list of $3042$ positive integers has a unique mode, which occurs exactly $15$ times. Calculate the least number of distinct values that can occur in the list. | 218 | 39 | 3 |
math | A novel begins on page 101 and ends on page 599. Each page number is printed once in the book. How many more times is the digit 5 printed compared to the digit 2 in the page numbers? | 0 | 49 | 1 |
math | A right triangle $DEF$ has an angle of $60^\circ$ at $D$. If $DE = 4$ unit (length along $60^\circ$ angle), determine the area of triangle $DEF$. | \frac{8\sqrt{3}}{3} | 47 | 12 |
math | An equation $x^{-2} = \left(\frac{1}{2}\right)^x$ has how many solutions. | 3 | 26 | 1 |
math | If one of the 13 provinces or territories is chosen at random, what is the probability that it joined Canadian Confederation between 1890 and 1969? | $\frac{4}{13}$ | 38 | 8 |
math | A hunting dog spots a fox 10 meters ahead and immediately starts chasing it. As soon as the fox starts running away, the dog catches up by 10 meters for every meter the fox runs. If the hunting dog and the fox move along the same path, how many meters in total will the hunting dog have traveled when it catches the fox?... | \frac{100}{9} | 81 | 9 |
math | A wire has a length of 6 meters and has 5 nodes that divide the wire into 6 equal parts. If a node is randomly selected to cut the wire, what is the probability that both resulting pieces will have lengths not less than 2 meters? | \frac{3}{5} | 53 | 7 |
math | Given the real coefficient equation \(x^{3} + 2(k-1)x^{2} + 9x + 5(k-1) = 0\) has a complex root with a modulus of \(\sqrt{5}\), find the value of \(k\) and solve the equation. | k = 3 | 62 | 4 |
math | Given the function $f(x) = x\ln x - k(x-1)$, where $k \in \mathbb{R}$.
$(1)$ When $k=1$, find the extreme value of the function $f(x)$.
$(2)$ Is there a positive integer $k$ such that $f(x) + x > 0$ holds for all $x \in (1, +\infty)$? If yes, find the maximum value of $k$; if not, explain the reason. | 3 | 112 | 1 |
math | Athletes A and B are playing a table tennis match with a best-of-five format, where the first player to win $3$ games wins the match. The outcomes of each game are independent of each other. The probability of A winning a game is $\frac{2}{3}$, and the probability of B winning a game is $\frac{1}{3}$.
$(1)$ Find the pr... | \frac{107}{27} | 121 | 10 |
math | An agricultural science institute has a rice experimental field. Last year, the harvest was 800 kilograms. It is expected that this year's harvest will increase by 160 kilograms. What is the percentage increase of this year's harvest compared to last year? | 20\% | 54 | 4 |
math | A "zero cleverly number" is defined as a four-digit number for which the hundreds digit is 0. When removing this 0, the resulting three-digit number, when multiplied by 9, equals the original four-digit number. List all "zero cleverly numbers". | 2025, 4050, 6075 | 56 | 16 |
math | A billiard table is in the shape of a $2 \times 1$ rectangle, with pockets located at the corners and midpoints of the long sides. What is the minimum number of balls that need to be placed inside the rectangle so that each pocket is collinear with some two balls? | 4 | 60 | 1 |
math | The sum of the positive divisors of a positive integer of the form $2^i 3^j 5^k$ is equal to $1200$. What is the value of $i + j + k$? | 7 | 49 | 1 |
math | Given the binomial ${(x+\frac{2}{x})^n}$, if the sum of the binomial coefficients in its expansion is $16$, then $n=$____, and the constant term in the expansion is ____. | 24 | 49 | 2 |
math | What is the greatest integer less than 150 for which the greatest common divisor of that integer and 18 is 6? | 144 | 28 | 3 |
math | Given two vectors $\overrightarrow{a}=(3,-2m)$ and $\overrightarrow{b}=(1,m-2)$ in a plane Cartesian coordinate system, and any vector $\overrightarrow{c}$ in the plane can be uniquely expressed as $\overrightarrow{c}=\lambda \overrightarrow{a}+\mu \overrightarrow{b}$, where $\lambda$ and $\mu$ are real numbers, determ... | (-\infty, \frac{6}{5})\cup(\frac{6}{5},+\infty) | 97 | 25 |
math | Given the complex plane equations $z^{3}-8=0$ and $z^{3}-8z^{2}-8z+64=0$, find the greatest distance between a point of set $A$ and a point of set $B$. | 2\sqrt{21} | 53 | 7 |
math | A finite set $\mathcal{S}$ of distinct real numbers has the following properties: the mean of $\mathcal{S}\cup\{1\}$ is $13$ less than the mean of $\mathcal{S}$, and the mean of $\mathcal{S}\cup\{2001\}$ is $27$ more than the mean of $\mathcal{S}$. Find the mean of $\mathcal{S}$. | 651 | 97 | 3 |
math | A deck of $2n$ cards numbered from $1$ to $2n$ is shuffled and n cards are dealt to $A$ and $B$ . $A$ and $B$ alternately discard a card face up, starting with $A$ . The game when the sum of the discards is first divisible by $2n + 1$ , and the last person to discard wins. What is the probability that $... | 0 | 117 | 1 |
math | Given a triangle $PQR$ with sides $PQ = 7$, $PR = 8$, and $QR = 6$, calculate the value of:
\[\frac{\cos \frac{P - Q}{2}}{\sin \frac{R}{2}} - \frac{\sin \frac{P - Q}{2}}{\cos \frac{R}{2}}.\] | 2 | 82 | 1 |
math | Find the distance from the point \( M_{0} \) to the plane passing through the three points \( M_{1}, M_{2}, M_{3} \).
$$
\begin{aligned}
& M_{1}(3, 10, -1) \\
& M_{2}(-2, 3, -5) \\
& M_{3}(-6, 0, -3) \\
& M_{0}(-6, 7, -10)
\end{aligned}
$$ | 7 | 109 | 1 |
math | There are 10 bags of wheat, each bag weighing 150 kilograms. The excess or shortage of kilograms in each bag is recorded as follows: $-6$, $-3$, $-1$, $-2$, $+7$, $+3$, $+4$, $-3$, $-2$, $+1$. Compare to the standard weight, how many kilograms in total are the 10 bags of wheat over or under? What is the total weight of... | 1498 \text{ kilograms} | 111 | 9 |
math | Given that $S_{n}$ is the sum of the first $n$ terms of an arithmetic sequence $\{a_{n}\}$, $a_{5}=2$, and $a_{n-1}+a_{n+1}=a_{5}a_{n}$ $(n\geqslant 2)$, and $a_{3}$ is the geometric mean between $a_{1}$ and $- \dfrac {8}{5}$.
$(1)$ Find the general formula for the sequence $\{a_{n}\}$.
$(2)$ If $a_{1}$ is an integ... | \dfrac {n}{3n+3} | 186 | 10 |
math | Seven people are sitting at a round table. Let \( w \geq 0 \) be the number of people sitting next to at least one woman, and \( m \geq 0 \) be the number of people sitting next to at least one man. Count the number of possible ordered pairs \( (w, m) \). | 5 | 70 | 1 |
math | A graph has \( 12k \) vertices. Each vertex is connected by \( 3k+6 \) edges. For any two vertices, the number of vertices connected to both of them is the same. Determine the value of \( k \). | 3 | 53 | 1 |
math | In the rectangular coordinate system xOy, the parametric equation of circle C is $$\begin{cases} x=cos\phi \ y=a+sin\phi \end{cases}$$ (a∈R, φ is the parameter). Establish a polar coordinate system with the coordinate origin as the pole and the positive semi-axis of the x-axis as the polar axis. The polar equation of t... | \frac{\sqrt{3}}{4} | 194 | 10 |
math | Given a parabola $x^2=4y$ with focus $F$ and the point $A(-1, 8)$, if $P$ is a point on the parabola, then the minimum value of $|PA| + |PF|$ is \_\_\_\_\_. | 9 | 63 | 1 |
math | Given the set $M=\{x|1\leqslant x\leqslant 10, x\in N\}$, for its non-empty subset $A$, each element $k$ in $A$ is multiplied by $\left(-1\right)^{k}$ and then summed up, determine the total sum of these sums for all non-empty subsets of $M$. | 2560 | 83 | 4 |
math | Given the proposition $p$: For all $x$ in $\mathbb{R}$, $x^2 + 2x + 2 > 0$. The negation of proposition $p$, $\neg p$, is: _______. | \exists x \in \mathbb{R},\ x^2 + 2x + 2 \leq 0 | 50 | 27 |
math | Let $x$, $y$, and $z$ be real numbers such that
\[\tan x + \tan y + \tan z = 0\quad \text{and} \quad \sec x + \sec y + \sec z = 3.\]
Find the sum of all possible values of $\sec 2x + \sec 2y + \sec 2z$. | 3 | 83 | 1 |
math | In $\triangle ABC$, point $D$ is the midpoint of side $BC$. Point $E$ is on $AC$ such that $AE:EC = 2:3$. Point $F$ is on $AD$ such that $AF:FD = 2:1$. If the area of $\triangle DEF$ is 12, determine the area of $\triangle ABC$. | 180 | 80 | 3 |
math | In a "Knowledge Contest" activity, there are four questions labeled A<sub>1</sub>, A<sub>2</sub>, B, and C, among which A<sub>1</sub> and A<sub>2</sub> are easy questions of the same difficulty, B is a medium question, and C is a difficult question. Now, two students, A and B, each need to randomly select one question ... | \frac{3}{8} | 149 | 7 |
math | Given an arithmetic sequence {a_n} with the first term a_1 and common difference d, the sum of its first n terms is denoted by S_n. If the line y = a_1x + m has two intersections with the circle (x-2)^2 + y^2 = 1 that are symmetric with respect to the line x + y + d = 0, find the value of S_n. | -n^2 + 2n | 88 | 7 |
math | The number of values of $x$ satisfying the equation
\[\frac {2x^2 - 10x}{x^2 - 5x} = x - 3\]is: | 0 | 43 | 1 |
math | Given the line $y=kx+4-3k$ and the function $f(x)=\frac{4x-11}{x-3}$, their graphs intersect at points $A$ and $B$. Let $P(x,y)$ be a moving point on the plane such that $|\overrightarrow{PA}+\overrightarrow{PB}|=2$. Determine the range of values for $x^2+y^2$. | [16, 36] | 91 | 8 |
math | Design a set of stamps with the following requirements: The set consists of four stamps of different denominations, with denominations being positive integers. Moreover, for any denomination value among the consecutive integers 1, 2, ..., R, it should be possible to achieve it by appropriately selecting stamps of diffe... | 14 | 82 | 2 |
math | Triangle $PQR$ has side lengths $PQ=6$, $QR=8$, and $PR=9$. Two bugs start simultaneously from $P$ and crawl along the perimeter of the triangle in opposite directions at the same speed. They meet at point $S$. What is $QS$? | 5.5 | 62 | 3 |
math | Find the maximum and minimum values of the function:
1) \( z = x^{2} - y^{2} + 2a^{2} \) in the circle \( x^{2} + y^{2} \leqslant a^{2} \);
2) \( v = 2x^{3} + 4x^{2} + y^{2} - 2xy \) in the closed region bounded by the lines \( y = x^{2} \) and \( y = 4 \). | 0 \text{ and } 32 | 111 | 9 |
math | How many triangles with positive area have all their vertices at points $(i,j)$ in the coordinate plane, where $i$ and $j$ are integers between $1$ and $5$, inclusive?
$\textbf{(A)}\ 2128 \qquad\textbf{(B)}\ 2148 \qquad\textbf{(C)}\ 2160 \qquad\textbf{(D)}\ 2200 \qquad\textbf{(E)}\ 2300$ | 2148 | 114 | 4 |
math | Find the $2023^{\text{th}}$ digit past the decimal point in the decimal expansion of $\dfrac{7}{18}$. | 3 | 34 | 1 |
math | We roll a die \( n \) times. What is the probability that among the rolled numbers, there are two that are the same? | P_n = \frac{6^n - \frac{6!}{(6-n)!}}{6^n} | 28 | 24 |
math | A point $(x, y)$ is randomly selected such that $0 \leq x \leq 3$ and $0 \leq y \leq 3$. What is the probability that $x + 2y \leq 6$? Express your answer as a common fraction. | \frac{1}{4} | 63 | 7 |
math | Find the number of ordered integer pairs \((m, n)\), where \(1 \leqslant m \leqslant 99\), \(1 \leqslant n \leqslant 99\), such that \((m+n)^2 + 3m + n\) is a perfect square. | 98 | 71 | 2 |
math | Determine the range of the function $f(x) = 2x + \sqrt{1 - x}$. | (-\infty, \frac{17}{8}] | 24 | 13 |
math | Simply the expression
\[\frac{(\sqrt{2} - 1)^{1 - \sqrt{3}}}{(\sqrt{2} + 1)^{1 + \sqrt{3}}},\]writing your answer as $a - b \sqrt{c},$ where $a,$ $b,$ and $c$ are positive integers, and $c$ is not divisible by the square of a prime. | 3 - 2 \sqrt{2} | 89 | 9 |
math | Calculate the definite integral:
$$
\int_{0}^{\pi / 4} \frac{4-7 \operatorname{tg} x}{2+3 \operatorname{tg} x} \, dx
$$ | \ln \left( \frac{25}{8} \right) - \frac{\pi}{4} | 49 | 24 |
math | The numbers $-3, 5, 7, 10, 15$ are rearranged according to the following rules:
1. The largest number is not in the last place, but it is within the last three places.
2. The smallest number is not in the first place, but it is within the first three places.
3. The median number is neither in the first nor in the last... | 12 | 98 | 2 |
math | Given \( k \in \mathbb{R} \), find the range of real values of \( x \) that satisfy the equation \( x^{4} - 2kx^{2} + k^{2} + 2k - 3 = 0 \). | [-\sqrt{2}, \sqrt{2}] | 58 | 11 |
math | Let $\triangle ABC$ be an isosceles triangle such that $BC = 40$ and $AB = AC$. The incenter $I$ of $\triangle ABC$ satisfies $IC = 26$. Determine the length of the inradius of the triangle. | 2\sqrt{69} | 57 | 7 |
math | The distance from point P(1, -3, 2) to the xOy plane is $∣2∣$. | 2 | 26 | 1 |
math | The rhombus $ABCD$ is given. Let $E$ be one of the points of intersection of the circles $\Gamma_B$ and $\Gamma_C$ , where $\Gamma_B$ is the circle centered at $B$ and passing through $C$ , and $\Gamma_C$ is the circle centered at $C$ and passing through $B$ . The line $ED$ intersects $\Gamma_B$ at... | 60^\circ | 132 | 4 |
math | Investigate the formula of \\(\cos nα\\) and draw the following conclusions:
\\(2\cos 2α=(2\cos α)^{2}-2\\),
\\(2\cos 3α=(2\cos α)^{3}-3(2\cos α)\\),
\\(2\cos 4α=(2\cos α)^{4}-4(2\cos α)^{2}+2\\),
\\(2\cos 5α=(2\cos α)^{5}-5(2\cos α)^{3}+5(2\cos α)\\),
\\(2\cos 6α=(2\cos α)^{6}-6(2\cos α)^{4}+9(2\cos α)^{2}-2\\),
\\(2\... | 28 | 303 | 2 |
math | In a newly built road in a city, there are 12 street lamps. To save electricity without affecting normal lighting, three of them can be turned off. However, the lamps at both ends cannot be turned off, nor can two adjacent lamps be turned off. How many methods are there to turn off the lamps? | 56 | 65 | 2 |
math | Erika assembles 3 calculators in the same amount of time that Nick assembles 2 calculators. Also, Nick assembles 1 calculator in the same amount of time that Sam assembles 3 calculators. Determine the total number of calculators that can be assembled by Nick, Erika, and Sam in the same amount of time as Erika assembles... | 33 | 81 | 2 |
math | A certain store is having a promotion on two types of chocolates, $A$ and $B$. According to statistics, there were a total of 480 customers who came to buy these two types of chocolates on a certain day, with each customer buying one box of chocolate. The selling prices of types $A$ and $B$ chocolates are $90$ yuan and... | 150 \text{ yuan} | 338 | 8 |
math | Given a function $f(x) = x^2 - 4x + c$ has only one zero, and the function $g(x) = x(f(x) + mx - 5)$ is not monotonic on the interval (2, 3), find the range of the real number $m$. | (-\frac{1}{3}, \frac{5}{4}) | 64 | 15 |
math | Given real numbers $u$, $v$, $x$, $y$ satisfying $u^2+v^2=1$ and
\[
\begin{cases}
& x+y-1\geqslant 0, \\
& x-2y+2\geqslant 0, \\
& x\leqslant 2, \\
\end{cases}
\]
find the maximum value of $z=ux+vy$. | 2\sqrt{2} | 96 | 6 |
math | In a division problem, the dividend is 2016 and the remainder is 7. Find the number of possible divisors. | 4 | 28 | 1 |
math | Alexio now has 120 cards numbered from 1 to 120, inclusive, and places them in a box. He then chooses a card from the box at random. What is the probability that the number on the card he chooses is a multiple of 2, 3, or 5? Express your answer as a common fraction. | \frac{11}{15} | 73 | 9 |
math | A geologist challenges participants in a contest to guess the weight of a meteorite by providing clues. The weight, in grams, must consist of the digits 1, 1, 3, 5, 5, and 8, with the condition that the weight starts with an odd number.
How many valid guesses are possible for the meteorite's weight? | 180 | 75 | 3 |
math | Calculate the following:<br/>$(1)9-5-\left(-4\right)+2$;<br/>$(2)(-\frac{3}{4}+\frac{7}{12}-\frac{5}{9})÷(-\frac{1}{36})$;<br/>$(3)-{2}^{4}-(-5\frac{1}{2})×\frac{4}{11}+(-2)^{3}÷|-3^{2}+1|$;<br/>$(4)99\frac{71}{72}×(-36)$. | -3599\frac{1}{2} | 129 | 12 |
math | There is an unlimited supply of congruent equilateral triangles made of colored paper. Each triangle is a solid color with the same color on both sides of the paper. A large equilateral triangle is constructed from four of these paper triangles as shown. Two large triangles are considered distinguishable if it is not p... | 960 | 115 | 3 |
math | What is the length of the diagonal and the area of a square with side length $30\sqrt{3}$ cm? Express your answer in simplest form. | 2700 \text{ square cm} | 33 | 10 |
math | A square is divided into \( n^{2} \) equal smaller squares. For a certain polyline, it is known that it passes through the centers of all the smaller squares (the polyline may intersect itself). What is the minimum number of segments of this polyline? | 2n - 2 | 53 | 5 |
math | In the London 2012 Olympic torch relay, the transfer route in Greece was divided into 6 segments, carried out by 6 different torchbearers. If the first torchbearer can only be chosen from A, B, or C, and the last torchbearer can only be chosen from A or B, then the total number of different transfer schemes is \_\_\_\_... | 48 + 48 = 96 | 84 | 10 |
math | If $1998$ is written as a product of two positive integers whose difference is as small as possible, then the difference is | 17 | 28 | 2 |
math | One vertex of an equilateral triangle is at the origin, and the other two vertices are on the parabola $x^2 = 2y$. What is the length of the side of this equilateral triangle? | 4\sqrt{3} | 45 | 6 |
math | Given that two of the roots of the cubic equation \[ax^3 + bx^2 + cx + d = 0\] are \(1 + i\) and \(1 - i\), and \(a \neq 0\), compute \(\frac{b + c}{a}\). | 2 | 63 | 1 |
math | There are two red balls and one white ball of the same size in a pocket. Balls are drawn with replacement, and a sequence $\{a\_n\}$ is defined as follows: $a\_n=\begin{cases} -1, \text{if a red ball is drawn on the nth draw}, \\ 1, \text{if a white ball is drawn on the nth draw}. \end{cases}$ If $S\_n$ is the sum of t... | \frac{28}{3^{6}} | 179 | 10 |
math | An equilateral triangle is divided into two 30-60-90 triangles by drawing a line from one vertex to the midpoint of the opposite side. If the length of the median to the longest side (hypotenuse in the 30-60-90 triangle) is $15$ units, what is the length of the shortest side of one of the 30-60-90 triangles, in units? | 15 \text{ units} | 93 | 7 |
math | Given the function $f(x)= e^x - ax - 2$,
1. Find the monotonic intervals of $f(x)$.
2. If $a=1$ and $k$ is an integer, and given that $(x-k)f'(x) + x + 1 > 0$ when $x > 0$, find the maximum value of $k$. | 2 | 80 | 1 |
math | Triangle $ABC$ has $AB=24$, $AC=26$ and $BC=25$. Points $D$ and $E$ are located on $\overline{AB}$ and $\overline{AC}$, respectively, such that $\overline{DE}$ is parallel to $\overline{BC}$ and contains the center of the inscribed circle of triangle $ABC$. Find the length of $DE$ in the form $m/n$, where $m$ and $n$ a... | 53 | 116 | 2 |
math | If $\cos(\alpha - \frac{\pi}{3}) = \frac{2}{3}$ and $\alpha$ is an acute angle, find the value of $\sin(\alpha)$.
A) $\frac{\sqrt{15}}{6}$
B) $\frac{\sqrt{5} - \sqrt{3}}{6}$
C) $\frac{2\sqrt{3} - \sqrt{5}}{6}$
D) $\frac{4 - \sqrt{15}}{6}$ | \frac{2\sqrt{3} - \sqrt{5}}{6} | 110 | 18 |
math | A regular dodecagon $Q_1 Q_2 \dotsb Q_{12}$ is present in the coordinate plane with $Q_1$ at $(1,0)$ and $Q_7$ at $(-1,0)$. If $Q_n$ denotes the point $(x_n,y_n)$, compute the numerical value of the product:
\[(x_1 + y_1 i)(x_2 + y_2 i)(x_3 + y_3 i) \dotsm (x_{12} + y_{12} i).\] | 1 | 123 | 1 |
math | If the graph of $y=x^{2}-2x+m$ passes through three points $A(-1$,$y_{1})$, $B(2$,$y_{2})$, $C(3$,$y_{3})$, then the correct relationship among $y_{1}$, $y_{2}$, $y_{3}$ is ______. | y_{2} < y_{1} = y_{3} | 76 | 14 |
math | In the polar coordinate system, $O$ is the pole, and the center of circle $C$ is $(1, \frac{3\pi}{4})$, with radius $r=1$. Point $P$ moves on circle $C$.
(I) Find the polar coordinate equation of circle $C$;
(II) Establish a rectangular coordinate system with the pole $O$ as the origin and the positive semi-axis of the... | (x - \frac{1}{2} + \frac{\sqrt{2}}{4})^{2} + (y - \frac{\sqrt{2}}{4})^{2} = \frac{1}{4} | 140 | 48 |
math | Given the number of ways to arrange two people is 25 when person A is not in position A, determine the value of $n$. | 6 | 29 | 1 |
math | Find the remainder when $3^{3^{3^3}}$ is divided by 1000.
| 387 | 23 | 3 |
math | Let $\alpha \in (\pi, 2\pi)$, if $\tan\left(\alpha + \frac{\pi}{6}\right) = 2$, then the value of $\cos\left(\frac{\pi}{6} - 2\alpha\right)$ is \_\_\_\_\_. | \frac{4}{5} | 65 | 7 |
math | In $\triangle ABC$, $BC=AB$, $\angle ABC=120^{\circ}$, find the eccentricity of the hyperbola with foci at $A$ and $B$ and passes through point $C$. | \frac{\sqrt{3}+1}{2} | 49 | 12 |
math | A town-planner has built an isolated city whose road network consists of $2N$ roundabouts, each connecting exactly three roads. A series of tunnels and bridges ensure that all roads in the town meet only at roundabouts. All roads are two-way, and each roundabout is oriented clockwise.
Vlad has recently passed his dr... | N | 141 | 2 |
math | Xiao Ming and Xiao Hong learned how to study polynomial multiplication using the area of shapes, and they conducted the following mathematical explorations: cutting a piece of wire into two pieces,
Exploration 1: Xiao Ming cut two pieces of wire of different lengths and used them to form two squares. It is known that ... | \frac{x+y}{2} | 201 | 7 |
math | Find the number of natural numbers \( k \) not exceeding 353500 such that \( k^{2} + k \) is divisible by 505. | 2800 | 38 | 4 |
math | Given the function $f(x)=2\sin (\omega x+\frac{\pi }{3})$ ($\omega > 0$), determine the range of $\omega$ such that the graph of $f(x)$ has exactly two maximum points in the interval $[0,1]$. | [\frac{13\pi }{6},\frac{25\pi }{6}) | 61 | 22 |
math | A line is parameterized by a parameter $t$, such that the vector on the line at $t = -2$ is $\begin{pmatrix} 2 \\ 6 \\ 16 \end{pmatrix}$, and the vector on the line at $t = 1$ is $\begin{pmatrix} -1 \\ -5 \\ -10 \end{pmatrix}.$ Find the vector on the line at $t = 4.$ | \begin{pmatrix} -16 \\ -60 \\ -140 \end{pmatrix} | 97 | 24 |
math | Solve the equations $z^2 = 3 + 3\sqrt{7}i$ and $w^2 = 5 + 5\sqrt{3}i$ in the complex plane. Determine the area of the parallelogram formed by the vertices which are solutions to both equations.
A) $2\sqrt{6} + 4\sqrt{14}$
B) $2\sqrt{6} + 6\sqrt{14}$
C) $3\sqrt{6} + 8\sqrt{14}$
D) $4\sqrt{6} + 5\sqrt{14}$ | 2\sqrt{6} + 6\sqrt{14} | 138 | 15 |
math | Suppose that \( p \) is a prime number and \( 2017_p + 405_p + 114_p + 206_p + 7_p = 253_p + 372_p + 452_p \). Determine how many possible values of \( p \) are there? | 0 | 74 | 1 |
math | The function $g(x) = x^3 - 3ax - a$ is not monotonic in the interval $(0,1)$. The range of $a$ is | (0,1) | 37 | 5 |
math | The function $g$ is defined on the set of integers and satisfies
\[
g(n)= \begin{cases}
n-4 & \mbox{if }n\ge 500 \\
g(g(n+5)) & \mbox{if }n<500
\end{cases}
\]
Find $g(72)$. | 498 | 78 | 3 |
math | Given $A=(a-\frac{{a}^{2}}{a+b})\div \frac{{a}^{2}{b}^{2}}{{a}^{2}-{b}^{2}}$.
$(1)$ Simplify $A$;
$(2)$ If point $P\left(a,b\right)$ is the intersection point of the line $y=x-2$ and the inverse proportion function $y=\frac{1}{x}$, find the value of $A$. | 2 | 103 | 1 |
math | In a class of 150 students, all students took the same test. What is the maximum number of students who can receive a grade above the class average? | 149 | 34 | 3 |
math | In a certain mall, there are two boxes of candies, A and B, priced differently. Box A contains $m$ kg of candies at a price of $a$ yuan/kg, and Box B contains $n$ kg of candies at a price of $b$ yuan/kg. The mall decides to sell the candies from both boxes mixed at a combined price of $\frac{am+bn}{m+n}$ yuan/kg (the m... | 24\, \text{kg} | 270 | 9 |
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