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 | Find the equation of the line passing through the point $(-2,1)$ and parallel to the line $2x-3y+5=0$. | 2x-3y+7=0 | 32 | 9 |
math | The standard equation of a circle with the center at $(2, -1)$ and passing through the point $(-1, 3)$ is $(x-2)^2+(y+1)^2=25$. | (x-2)^2+(y+1)^2=25 | 45 | 14 |
math | A $10 \times 10$ square is divided into unit squares. How many triangles are formed after drawing one diagonal? | 110 | 27 | 3 |
math | Given a circle $M$ passes through the point $P(1, -\sqrt{3})$ and is symmetric to the circle $C$: $(x+2)^{2}+y^{2}=r^{2} (r > 0)$ with respect to the $y$-axis.
$(I)$ Find the equation of circle $M$;
$(II)$ If there are two mutually perpendicular lines $l_{1}$ and $l_{2}$, both passing through the point $A(-1,0)$, an... | 28 | 171 | 2 |
math | Given the sequences $\{a_n\}$ and $\{b_n\}$ that satisfy $(a_1a_2...a_n=(\sqrt{2})^{b_n}(n∈N^∗)).$ If $\{a_n\}$ is a geometric sequence, with $a_1=2, b_3=6+b_2,$ let $(C_n=\frac{1}{a_n}-\frac{1}{b_n}(n∈N^∗)),$ and denote the sum of the first $n$ terms of the sequence $\{C_n\}$ as $(S_n).$ If $k$ is a positive integer s... | 4 | 164 | 1 |
math | Let \( A, B, C \) be the angles of a triangle where \( A = 45^\circ \) and \( A + B + C = 180^\circ \). Compute:
\[
\begin{vmatrix}
\tan A & 1 & 1 \\
1 & \tan B & 1 \\
1 & 1 & \tan C
\end{vmatrix}.
\] | 2 | 91 | 1 |
math | There are $5 \cdot 338$ singers participating in an arts festival, and you need to schedule $m$ performances, with 4 singers performing in each. Ensure that any two of the 8 singers perform together the same number of times. Design a schedule that minimizes the number of performances $m$. | 14 | 66 | 2 |
math | Let $P$ be a cubic monic polynomial with roots $a$ , $b$ , and $c$ . If $P(1)=91$ and $P(-1)=-121$ , compute the maximum possible value of \[\dfrac{ab+bc+ca}{abc+a+b+c}.\]
*Proposed by David Altizio* | 7 | 87 | 1 |
math | Three people each have a known amount of écus. The first person gives to the other two as much as each of them has. After that, the second person gives to the other two as much as each of them has. Finally, the third person gives to the other two as much as each of them has. After this, each person has 8 écus. How much... | (13, 7, 4) | 84 | 10 |
math | Given the function $f(x)=-x^{3}+ax^{2}+bx+c$, the equation of the tangent line at point $P(1,m)$ is $y=-3x+1$.
1. If the function $f(x)$ has an extreme value at $x=-2$, find the expression of $f(x)$.
2. If the function $f(x)$ is monotonically increasing in the interval $[-2,0]$, find the range of values for the real n... | [4,+\infty) | 109 | 7 |
math | Given the function $f(x)=e^{x}-x+ \frac {1}{2}x^{2}$ (where $e$ is the base of the natural logarithm) and the function $g(x)= \frac {1}{2}x^{2}+ax+b$ (where $a\in\mathbb{R}$ and $b\in\mathbb{R}$).
1. Find the extreme values of $f(x)$.
2. If $f(x)\geqslant g(x)$, find the maximum value of $b(a+1)$. | \frac {e}{2} | 123 | 7 |
math | Simplify $\cot 20 + \tan 10.$ | \csc 20 | 14 | 6 |
math | The negation of the proposition "$\exists x \in \mathbb{R}, f(x) < 0$" is $\forall x \in \mathbb{R}, f(x) \geq 0$. | \forall x \in \mathbb{R}, f(x) \geq 0 | 46 | 19 |
math | Three fair eight-sided dice are rolled. What is the probability that the values shown on two of the dice sum to the value shown on the remaining die?
A) $\frac{267}{512}$
B) $\frac{89}{512}$
C) $\frac{1}{3}$
D) $\frac{89}{216}$
E) $\frac{5}{24}$ | \frac{267}{512} | 91 | 11 |
math | Given the fractional equation about $x$: $\frac{a+2}{x+1}=1$ has a non-positive solution, then the range of $a$ is ____. | a \leqslant -1 \text{ and } a \neq -2 | 37 | 19 |
math | Consider the graph of the quadratic function \(y = g(x)\) defined by \(g(x) = \frac{x^2}{2} - x - 2\). The distance between grid lines is 1 unit.
Find the sum of all distinct numbers \(x\) such that \(g(g(g(x))) = -2\). | 4 | 69 | 1 |
math | Yura placed in a row 2001 coins with denominations of 1, 2, and 3 kopecks. It turned out that between any two 1-kopeck coins there is at least one coin, between any two 2-kopeck coins there are at least two coins, and between any two 3-kopeck coins there are at least three coins. How many 3-kopeck coins could Yura have... | 500 | 96 | 3 |
math | Given the function $f(x) = x^3 - ax - 1$.
(1) If $a = 3$, find the intervals of monotonicity for $f(x)$;
(2) If $f(x)$ is monotonically increasing on the set of real numbers $\mathbb{R}$, find the range of the real number $a$;
(3) Is there a real number $a$ such that $f(x)$ is monotonically decreasing on the interva... | a \geq 3 | 128 | 6 |
math | Let
\[f(x) = \left\{
\begin{array}{cl}
x^2 + 3 & \text{if $x < 15$}, \\
3x - 2 & \text{if $x \ge 15$}.
\end{array}
\right.\]
Find $f^{-1}(10) + f^{-1}(49).$ | \sqrt{7} + 17 | 84 | 9 |
math | On Ming's way to the swimming pool, there are 200 trees. On his round trip, Ming marked some trees with red ribbons. On his way to the swimming pool, he marked the 1st tree, the 6th tree, the 11th tree, and so on, marking every 4th tree. On his way back, he marked the 1st tree he encountered, the 9th tree, the 17th tre... | 140 | 121 | 3 |
math | Let \( z = \frac{1+\mathrm{i}}{\sqrt{2}} \). Then calculate the value of \( \left(\sum_{k=1}^{12} z^{k^{2}}\right)\left(\sum_{k=1}^{12} \frac{1}{z^{k^{2}}}\right) \). | 36 | 75 | 2 |
math | For what value of $m$ does the quadratic equation in $x$, $(2m+1)x^{2}+4mx+2m-3=0$, have:
$(1)$ two distinct real roots;
$(2)$ two equal real roots;
$(3)$ no real roots. | m \in (-\infty, -\frac{3}{4}) | 60 | 16 |
math | Given that $a$ is a positive integer constant, let set $A=\{x||x-a| < a+ \frac{1}{2},x∈\mathbb{Z}\}$, and set $B=\{x||x| < 2a,x∈\mathbb{Z}\}$, find the sum of all elements in set $A∪B$. | 2a | 79 | 2 |
math | If $\mathbf{B} = \begin{pmatrix} 2p & 2q \\ 2r & 2s \end{pmatrix},$ then its transpose is given by
\[\mathbf{B}^T = \begin{pmatrix} 2p & 2r \\ 2q & 2s \end{pmatrix}.\] Given that $\mathbf{B}^T = 4\mathbf{B}^{-1},$ find $p^2 + q^2 + r^2 + s^2.$ | 2 | 123 | 1 |
math | In the Cartesian coordinate system $xOy$, the parameter equation of circle $C$ is $\begin{cases} x=2\cos \alpha+ \sqrt {3} \\ y=2\sin \alpha+1 \end{cases} (\alpha \text{ is the parameter})$. Establish a polar coordinate system with the origin $O$ as the pole and the positive half-axis of $x$ as the polar axis.
$(1)$ ... | \sqrt {3} | 167 | 5 |
math | When the real number \( a \) is in the specified range, there does not exist a real number \( x \) such that \( |x + a + 1| + |x + a^{2} - 2| < 3 \). | (-\infty, -2] \cup [0, 1] \cup [3, \infty) | 53 | 25 |
math | Calculate the arc length of the curve described by the equation in the rectangular coordinate system.
$$
y = \ln \left(1-x^{2}\right), \quad 0 \leq x \leq \frac{1}{4}
$$ | \frac{1}{2} \ln \left( \frac{5}{3} \right) + \frac{1}{4} | 52 | 30 |
math | The 4th World Internet Conference was held in Wuzhen, Zhejiang Province from December 3rd to 5th, 2017. During the conference, a company's unmanned supermarket provided a brand new experience of new retailing brought by the internet. Little Zhang bought 15 items in total, including keychains and plush toys. After leavi... | 36 \text{ yuan} | 140 | 7 |
math | Given an ellipse $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ passing through point $A(-2,0)$, with its right focus at $F(1,0)$.
$(1)$ Find the equation of the ellipse $C$;
$(2)$ Let $P$ be a moving point on the ellipse $C$ (not on the $x$-axis), $M$ be the midpoint of $AP$, a line parallel to $AP$ passing through ... | t = 4 | 181 | 4 |
math | Given the equation $\tan(2x)=\sin(x)$, determine the number of solutions on the interval $[0,2\pi]$. | 4 | 31 | 1 |
math | The distance between the two intersections of $x=y^4$ and $x+y^2=1$ is $\sqrt{u+v\sqrt5}$. Find the ordered pair, $(u,v)$. | (-2,2) | 44 | 5 |
math | The domain of the function $f(x)=\frac{(2x-1)^{0}}{\sqrt{2-x}}$ is $(-\infty,\frac{1}{2})\cup(\frac{1}{2},2]$. | (-\infty, \frac{1}{2}) \cup (\frac{1}{2}, 2) | 52 | 24 |
math | In the diagram, $AB$ is a line segment, and point $C$ is outside the line segment creating two angles $\angle ACB = x^\circ$ and $\angle BCA = 30^\circ$ with point $D$ on segment $AB$ such that $\angle ACD = 90^\circ$ and $\angle DCB = 60^\circ$. Find the value of $x$.
[asy]
draw((0,0)--(10,0),black+linewidth(1));
dra... | 90^\circ | 282 | 4 |
math | Given the ellipse $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1$ (a > b > 0) and the hyperbola $\frac{x^{2}}{m^{2}}- \frac{y^{2}}{n^{2}}=1$ (m > 0, n > 0) share common foci $F_1$, $F_2$, and intersect at point P in the first quadrant. The eccentricities of the ellipse and hyperbola are $e_1$ and $e_2$, respectively. If $... | \frac{\sqrt{3}}{2} | 209 | 10 |
math |
What is the minimum number of points in which 5 different non-parallel lines, not passing through a single point, can intersect? | 10 | 28 | 2 |
math | Given the universal set U = R, set A = {x | 1 < x ≤ 4}, and set B = {x | 6 - a < x < 2a - 1}:
1. If a = 4, find A ∪ B and B ∩ (U - A).
2. If A ⊆ B, find the range of values for a. | a \geq 5 | 82 | 6 |
math | A triangle has sides of length $1$, $\sqrt {5}$, and $2 \sqrt {2}$. Calculate the sum of its largest and smallest angles. | 135^{\circ} | 34 | 7 |
math | Sixty bricks, each measuring $2''\times6''\times15''$, are to be laid one on top of the other to construct a column. Each brick can be oriented so it contributes $2''$, $6''$, or $15''$ to the total height of the column. Determine how many distinct total heights can be reached using all sixty bricks. | 781 | 78 | 3 |
math | Given $m$, $x$, $y$ satisfy:
1. $(x-5)^2$ and $|m-1|$ are opposites of each other,
2. $-2ab^{y+1}$ and $4ab^5$ are like terms,
First simplify $(2x^2 - 3xy - 4y^2) - m(3x^2 - xy + 9y^2)$, then find the value of this expression. | -273 | 100 | 4 |
math | If two moving points A and B on the circle $x^{2}+y^{2}=5$ such that $|AB|=\sqrt{15}$, and point M moves on the line $2x+y-5=0$, calculate the minimum value of $|\overrightarrow{MA}+\overrightarrow{MB}|$. | \sqrt{5} | 70 | 5 |
math | Given a hyperbola $\frac{x^2}{4} - y^2 = 1$ and a straight line $l$ passing through the right focus of the hyperbola intersecting it at points A and B, if there exist exactly three such lines for which the length $|AB|=a$, then find the range of values for the real number $a$. | 4 | 77 | 1 |
math | Given that the solution set of the inequality $|x| > ax + 1$ is a subset of $\{x | x \leq 0\}$, find the range of values of $a$. | a \geq 1 | 44 | 6 |
math | Given the set of scores 90, 89, 90, 95, 93, 94, 93, find the median and the mean of this set of data. | 93, 92 | 45 | 6 |
math | From a barrel, 4 liters of wine are drawn, and this is replaced with 4 liters of water. From the resulting mixture, 4 liters are drawn again and replaced with 4 liters of water. This operation is repeated a total of three times, and the final result is that there are 2.5 liters more water than wine. How many liters of ... | 16 | 82 | 2 |
math | Let $\mathcal{P}$ be the parabola defined by $y = (x-1)^2 - 3$, with its vertex $V_1$ and focus $F_1$. Points $A$ and $B$ lie on $\mathcal{P}$ such that the angle $\angle AV_1B = 90^\circ$. Let $\mathcal{Q}$ be the locus of the midpoint of line segment $\overline{AB}$. Denote the vertex and focus of $\mathcal{Q}$ as $V... | \frac{2}{2} = 1 | 146 | 10 |
math | From the 20 numbers 11, 12, 13, 14, ... 30, how many numbers at a minimum must be taken to ensure that among the selected numbers, there will definitely be a pair whose sum is a multiple of ten? | 11 | 58 | 2 |
math | A coin is flipped multiple times until we get an odd number of heads followed by a tail. Given $n \in \mathbb{N}^{*}$, find the number of sequences of $n$ flips. | F_{n-1} | 45 | 6 |
math | Find the area of the region bounded by the graph of $r = 2\sec\theta,$ the graph of $r = 2\csc\theta,$ the $x$-axis, and the $y$-axis. | 4 | 50 | 1 |
math | Let $b_n$ be the integer obtained by writing all the integers from 1 to $n$ consecutively from left to right. For example, $b_3 = 123$ and $b_{11} = 1234567891011$. Compute the remainder when $b_{55}$ is divided by $55$. | 0 | 82 | 1 |
math | You are given the numbers $0$, $2$, $3$, $4$, $6$. Use these numbers to form different combinations and calculate the following:
$(1)$ How many unique three-digit numbers can be formed?
$(2)$ How many unique three-digit numbers that can be divided by $3$ can be formed? (Note: Write the result of each part in data... | 20 | 81 | 2 |
math | Xiaohu uses 6 equilateral triangles with side lengths of 1 to form shapes on a table without overlapping. Each triangle must share at least one side fully with another triangle. As shown in the image, there are two possible shapes formed. Among all the possible shapes, what is the smallest perimeter? | 6 | 63 | 1 |
math | Given two linear functions \( f(x) \) and \( g(x) \) such that the graphs \( y = f(x) \) and \( y = g(x) \) are parallel lines, and not parallel to the coordinate axes. Find the minimum value of the function \( (g(x))^2 + 8 f(x) \), if the minimum value of the function \( (f(x))^2 + 8 g(x) \) is -29. | -3 | 97 | 2 |
math | The left and right foci of the ellipse $\dfrac {x^{2}}{a^{2}}+ \dfrac {y^{2}}{b^{2}}=1(a > b > 0)$ are $F_{1}$ and $F_{2}$, respectively, and its eccentricity is $\dfrac {1}{2}$. Point $P$ is a moving point on the ellipse, and the maximum area of $\triangle F_{1}PF_{2}$ is $\sqrt {3}$.
$(1)$ Find the equation of the ... | 0 | 241 | 1 |
math | A certain sports brand store is preparing to purchase two types of sports suits, A and B. The quantity of type A sports suits purchased for $8100 is the same as the quantity of type B sports suits purchased for $9000. If purchasing one set of each type of sports suit costs a total of $380, find the cost price of each s... | 200 | 87 | 3 |
math | Given Professor Chang has nine different language books lined up on a bookshelf: two Arabic, three German, and four Spanish, calculate the number of ways to arrange the books on the shelf keeping the Arabic books together and the Spanish books together. | 5760 | 48 | 4 |
math | Find the coordinates of the intersection points between the line $$\begin{cases} \left.\begin{matrix}x=2+t \\ y=-1-t\end{matrix}\right.\end{cases}$$ (where $t$ is a parameter) and the curve $$\begin{cases} \left.\begin{matrix}x=3\cos\alpha \\ y=3\sin\alpha\end{matrix}\right.\end{cases}$$ (where $\alpha$ is a parameter)... | \left( \frac {1- \sqrt {17}}{2}, \frac {1+ \sqrt {17}}{2}\right) | 105 | 33 |
math | Given $p$: $0 < x < 2$, $q$: $x < a$, if $p$ is a sufficient but not necessary condition for $q$, then the range of the real number $a$ is. | [2, +\infty) | 47 | 8 |
math | Convert $110_{(5)}$ to binary. | 11110_{(2)} | 13 | 9 |
math | Given the ratio of the surface areas of two spheres is 1:3, find the ratio of their volumes. | 1:3\sqrt{3} | 23 | 8 |
math |
An investor has an open brokerage account with an investment company. In 2021, the investor received the following income from securities:
- Dividends from shares of the company PAO “Winning” amounted to 50,000 rubles.
- Coupon income from government bonds OFZ amounted to 40,000 rubles.
- Coupon income from corporate... | 11050 | 176 | 5 |
math | Given the piecewise function \\(f(x)=\begin{cases} & \lg x, & x > 0 \\ & x+\int_{0}^{a}{3t^{2}dt}, & x\leqslant 0 \end{cases}\\), if \\(f[f(1)]=1\\), determine the constant term in the expansion of \\((4^{x}-2^{-x})^{a+5}\\). | 15 | 94 | 2 |
math | In the polar coordinate system, the polar equation of curve \\(C\\) is given by \\(\rho = 6\sin \theta\\). The polar coordinates of point \\(P\\) are \\((\sqrt{2}, \frac{\pi}{4})\\). Taking the pole as the origin and the positive half-axis of the \\(x\\)-axis as the polar axis, a Cartesian coordinate system is establis... | 3\sqrt{2} | 181 | 6 |
math | Given positive real numbers \( x \) and \( y \) that satisfy
\[
\left(2x + \sqrt{4x^2 + 1}\right)\left(\sqrt{y^2 + 4} - 2\right) \geq y,
\]
find the minimum value of \( x + y \). | 2 | 72 | 1 |
math | Given the line $l: y = \frac{m}{n}x - \frac{1}{n}$, determine the condition for its graph to pass through the first, second, and fourth quadrants simultaneously. | m > 0 \text{ and } n < 0 | 45 | 13 |
math | Given the function $f(x)=|x+a|+|x-2|$, and the solution set of $f(x)\leqslant |x-4|$ contains $[1,2]$, the range of values for $a$ is _______. | [-3,0] | 55 | 5 |
math | Alice can finish a task in 4 hours, and Bob can complete the same task in 6 hours. Alice and Bob work together on the task and also take two short breaks of 15 minutes each (half-hour total break time). Let $t$ be the total time in hours required for them to finish the task, including their breaks. Determine the equati... | \left(\frac{5}{12}\right)(t-\frac{1}{2}) = 1 | 180 | 23 |
math | If line $l_1: y-2=(k-1)x$ and line $l_2$ are symmetric about the line $y=x+1$, find the fixed point through which line $l_2$ always passes. | (1,1) | 49 | 5 |
math | Given the function $f(x)=\sin 2x+2\cos ^{2}x-1$.
$(1)$ Find the smallest positive period of $f(x)$;
$(2)$ When $x∈[0,\frac{π}{2}]$, find the minimum value of $f(x)$ and the corresponding value of the independent variable $x$. | \frac{\pi}{2} | 77 | 7 |
math | Let \( S = \{1, 2, \ldots, 98\} \). Find the smallest natural number \( n \) such that in any \( n \)-element subset of \( S \), it is always possible to select 10 numbers, and no matter how these 10 numbers are divided into two groups of five, there will always be a number in one group that is coprime with the other f... | 50 | 120 | 2 |
math | For some integers $m$ and $n$, the expression $(x+m)(x+n)$ is equal to a quadratic expression in $x$ with a constant term of -12. Which of the following cannot be a value of $m$? | 5 | 51 | 1 |
math | Find the simplified form of the expression $\frac{2^{4y-1}}{7^{-1} + 2^{-1}}$.
A) $2^{4y-1} \cdot \frac{9}{14}$
B) $2^{4y-1}$
C) $2^{4y-1} \cdot \frac{14}{9}$
D) $2^{4y-2} \cdot \frac{14}{9}$
E) $\frac{2^{4y-1}}{9}$ | 2^{4y-1} \cdot \frac{14}{9} | 119 | 17 |
math | Find the value of $\sin 30^{\circ}\cos 15^{\circ}+\cos 30^{\circ}\sin 15^{\circ}$. | \frac{\sqrt{2}}{2} | 39 | 10 |
math | Given the function $f\left(x\right)=x^{3}+x+1$, if $f\left(1-x\right)+f\left(2x\right) > 2$, determine the range of $x$. | (-1,+\infty) | 51 | 7 |
math | Jenny places a total of 30 red Easter eggs in several green baskets and a total of 45 orange Easter eggs in some blue baskets. Each basket must contain at least 5 eggs. Determine the number of eggs Jenny placed in each basket. | 15 | 52 | 2 |
math | On a drawing with a scale of 50:1, the distance between points A and B on the drawing is 2 cm. What is the actual distance between A and B? | 0.04 | 38 | 4 |
math | Let $S$ be the set of ordered triples $(x,y,z)$ of real numbers for which
\begin{align*} \label{eq:1} \log_{10}(x+y) = z \\
\log_{10}(x^{2}+y^{2}) = z+1 \end{align*} There are real numbers $a$ and $b$ such that for all ordered triples $(x,y.z)$ in $S$ we have $x^{3}+y^{3}=a \cdot 10^{3z} + b \cdot 10^{2z}.$ What is the... | \frac{29}{2} | 140 | 8 |
math | The inclination angle of the line $x+\sqrt{3}y+2=0$ is what. | 150^{\circ} | 22 | 7 |
math | How many numbers in the set $\{7, 17, 27, 37, \ldots\}$ can be written as the difference of two primes? | 1 | 37 | 1 |
math | C is a circle and L a line not meeting it. M and N are variable points on L such that the circle diameter MN touches C but does not contain it. Show that there is a fixed point P such that the ∠MPN is constant. | P | 52 | 1 |
math | Given the inequality system about $x$: $\left\{\begin{array}{l}{x>-1}\\{x\leq 1-k}\end{array}\right.$
$(1)$ When $k=-2$, find the solution set of the inequality system.
$(2)$ If the solution set of the inequality system is $-1 < x \leq 4$, find the value of $k$.
$(3)$ If the inequality system has three integer so... | -2 < k \leq -1 | 110 | 9 |
math | In an equilateral triangle ABC with side length 2, let points D, F be on AB and points E, G on AC such that DE and FG are parallel to BC. Triangle ADE and trapezoids DFGE and FBCG have the same perimeter. Find DE+FG. | 2 | 61 | 1 |
math | Let \( M \) be a set consisting of a finite number of positive integers, and suppose \( M = A_{1} \cup A_{2} \cup \cdots \cup A_{20} = B_{1} \cup B_{2} \cup \cdots \cup B_{20} \), where \( A_{i} \neq \varnothing, B_{i} \neq \varnothing \) for \( i = 1, 2, \ldots, 20 \). For any \( 1 \leq i < j \leq 20 \), we have \( A_... | 180 | 279 | 3 |
math | How many positive integers less than 900 can be written as a product of two or more consecutive prime numbers? | 14 | 24 | 2 |
math | Given a three-digit number $M$ in base $5$ expressed as $\overline{def}_5$ and a three-digit number in base $8$ expressed as $\overline{fed}_8$, find the last digit of $M$ when expressed in base $5$. | 0 | 58 | 1 |
math | The Happy Valley Kennel has 4 chickens, 3 dogs, and 5 cats. In how many ways can the 12 animals be placed in a row of 12 cages, such that all of the animals of each type are in adjacent cages, and all groups of chickens must appear before any group of cats? (Two animals of the same species are distinguishable.) | 34,\!560 | 78 | 7 |
math | A cylinder has a radius of 4 cm and a height of 9 cm. What is the longest segment, in centimeters, that would fit inside this cylinder? | \sqrt{145}\, \text{cm} | 34 | 13 |
math | Let $e_1 = b^3 + 3^b + b \cdot 3^{(b+1)/2}$ and $e_2 = b^3 + 3^b - b \cdot 3^{(b+1)/2}$. If $1 \le b \le 500$, how many integral values of $b$ ensure that $e_1 \cdot e_2$ is a multiple of $7$? | 71 | 98 | 2 |
math | In recent years, a company has been spending approximately 240,000 yuan on electricity annually. To save energy and reduce emissions, the company decided to install a solar power generation device that can be used for 15 years and connected to the company's power grid. The installation cost (in 10,000 yuan) of this dev... | 57.5 | 286 | 4 |
math | A and B each hold 7 cards, with the numbers 1, 2, 3, 4, 5, 6, and 7. If each person draws one card, what is the probability that the sum of the numbers on the two cards is 8? | \frac{1}{7} | 59 | 7 |
math | Two circles with equal radii intersect in such a way that the area of the shaded region is equal to the sum of the areas of the two unshaded regions. If the area of the shaded region is $216 \pi$, find the circumference of each circle. | 36\pi | 56 | 4 |
math | Let $A$ be a set of natural numbers, for which for $\forall n\in \mathbb{N}$ exactly one of the numbers $n$ , $2n$ , and $3n$ is an element of $A$ . If $2\in A$ , show whether $13824\in A$ . | 13824 \notin A | 84 | 10 |
math | Given that $\{a_n\}$ is a geometric sequence with first term $32$, $S_n$ is the sum of its first $n$ terms, and $\frac{S_6}{S_3} = \frac{65}{64}$, find the sum of the first $10$ terms of the sequence $\{|\log_2 a_n|\}$. | 58 | 82 | 2 |
math | A paper equilateral triangle of side length $2$ on a table has vertices labeled $A,B,C.$ Let $M$ be the point on the sheet of paper halfway between $A$ and $C.$ Over time, point $M$ is lifted upwards, folding the triangle along segment $BM,$ while $A,B,$ and $C$ on the table. This continues until $A$... | 1 | 131 | 1 |
math | On the island of Friends and Foes, every citizen is either a Friend (who always tells the truth) or a Foe (who always lies). Seven citizens are sitting in a circle. Each declares "I am sitting between two Foes". How many Friends are there in the circle? | 3 | 59 | 1 |
math | 1. Find the imaginary part of the complex number $2+\frac{1}{3i}$.
2. Given that $\cos{\frac{π}{3}}=\frac{1}{2}$, $\cos{\frac{π}{5}}\cos{\frac{2π}{5}}=\frac{1}{4}$, $\cos{\frac{π}{7}}\cos{\frac{2π}{7}}\cos{\frac{3π}{7}}=\frac{1}{8}$, ..., find a general expression for the $n^{th}$ equation based on the given equations.... | (-1,0)\cup(1,+\infty) | 214 | 13 |
math | Given the function $f(x)=ax^{2}-(a+2)x+\ln x$, where $a\in \mathbb{R}$.
(I) When $a=1$, find the equation of the tangent line at the point $(1,f(1))$ on the curve $y=f(x)$.
(II) When $a > 0$, if the minimum value of $f(x)$ in the interval $[1,e]$ is $-2$, find the range of values for $a$. | [1,+\infty) | 109 | 7 |
math | Solve for $n$: $0.07n + 0.12(30 + n) + 0.04n = 20.4$. | n = 73.0434782609 | 38 | 16 |
math | Find all values of the parameter \(a\) for which the equation \(x^{2} + 2x + 2|x + 1| = a\) has exactly two roots. | a > -1 | 38 | 4 |
math | Using the six digits 0, 1, 2, 3, 4, 5,
(1) How many distinct three-digit numbers can be formed?
(2) How many distinct three-digit odd numbers can be formed? | 48 | 49 | 2 |
math | Compute $\sin 45^\circ$ and $\cos 45^\circ$. | \frac{1}{\sqrt{2}} | 18 | 10 |
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