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 | An ant starts at the origin, facing in the positive $x$-direction. Each second, it moves 1 unit forward, then turns counterclockwise by $\sin ^{-1}\left(\frac{3}{5}\right)$ degrees. What is the least upper bound on the distance between the ant and the origin? (The least upper bound is the smallest real number $r$ that ... | \sqrt{10} | 99 | 6 |
math | Find distinct positive integers $n_1<n_2<\dots<n_7$ with the least possible sum, such that their product $n_1 \times n_2 \times \dots \times n_7$ is divisible by $2016$ . | 31 | 62 | 2 |
math | There are 18 identical cars in a train. In some cars, exactly half of the seats are free, in others, exactly one-third of the seats are free, and in the remaining cars, all seats are occupied. At the same time, exactly one-ninth of all seats in the whole train are free. How many cars have all seats occupied? | 13 | 73 | 2 |
math | Positive integers $a$, $b$, and $c$ are randomly and independently selected with replacement from the set $\{1, 2, 3,\dots, 2010\}$. Determine the probability that $abc + ab + a$ is divisible by $3$. | \frac{13}{27} | 59 | 9 |
math | Find the value of $\sin 21^{\circ}\cos 39^{\circ} + \cos 21^{\circ}\sin 39^{\circ}$. The possible answers are:
A) $\frac{\sqrt{2}}{2}$
B) $\frac{1}{2}$
C) $\frac{\sqrt{3}}{2}$
D) $1$ | \frac{\sqrt{3}}{2} | 86 | 10 |
math | A right circular cone is inscribed in a right rectangular prism. The base of the cone is tangent to all four sides of the rectangular base of the prism. If the base of the rectangle has side lengths $a$ and $b$, and the height of both the prism and the cone is $h$, what is the ratio of the volume of the cone to the vol... | \frac{\pi}{12} | 92 | 8 |
math | As a clock has a circular face with a radius of $30$ cm and a smaller circular disk with a radius of $15$ cm rolls externally tangent to the clock face, starting from the $12$ o'clock position, determine the position on the clock face when the disk will be tangent when the arrow next points vertically upwards. | 6 \ \text{o'clock} | 70 | 7 |
math | Determine the product of the real parts of the complex solutions for the equation $x^2 + 2x = -i$. | \frac{1-\sqrt{2}}{2} | 27 | 12 |
math | A school organized students to study at the Jinggangshan Revolution Museum. The travel agency charges $400$ yuan per person. When the number of study participants exceeds $50$, the travel agency offers two discount options:<br/>Option 1: The study group pays $1500$ yuan upfront, and then each person is charged $320$ yu... | \text{Option 2} | 199 | 7 |
math | Calculate the following:<br/>$(1)(-5)+(-2)+(+9)-(-8)$;<br/>$(2)-15+(+3)-(-15)+(+7)-(+2)+(-8)$;<br/>$(3)0.85+(+0.75)-(+2\frac{3}{4})+(-1.85)+(+3)$;<br/>$(4)2-[{(-5\frac{2}{3})-(-\frac{1}{3})}]$;<br/>$(5)1-(-\frac{1}{2})+(-\frac{1}{3})-\frac{3}{4}$;<br/>$(6)|5\frac{1}{11}-3\frac{4}{17}|+4\frac{4}{17}-\frac{1}{11}$. | 6 | 186 | 1 |
math | Given that $x, y > 0$ and $\frac{1}{x} + \frac{1}{y} = 2$, find the minimum value of $x + 2y$. | \frac{3 + 2\sqrt{2}}{2} | 42 | 15 |
math | Given that $S=\{(x,y) : x\in \{0,1,2,3,4,5\}, y\in \{0,1,2,3,4,5\}\}$, let $T$ be the set of all right triangles whose vertices are in $S$. For every right triangle $t=\triangle{ABC}$ with vertices $A$, $B$, and $C$ in counter-clockwise order and where the right angle can be at any vertex, find the product for all $f(t... | 1 | 123 | 1 |
math | Find the integrals:
1) $\int \frac{2 x \, dx}{x^{4}+3}$
2) $\int \frac{\sin x \, dx}{\sqrt{1+2 \cos x}}$
3) $\int \frac{x \, dx}{\sqrt[3]{x^{2}+a}}$
4) $\int \frac{\sqrt{1+\ln x}}{x} \, dx$
5) $\int \frac{d y}{\sqrt{e^{y}+1}}$
6) $\int \frac{d t}{\sqrt{\left(1-t^{2}\right)^{3}}}$ | \frac{t}{\sqrt{1 - t^2}} + C | 142 | 16 |
math | Consider the graph of $y = g(x)$, where $g(x) = \frac{(x-4)(x-2)(x)(x+2)(x+4)}{120} + 2$. The function $g(x)$ is defined only on the domain $-5 \leq x \leq 5$. Determine the sum of all integers $c$ for which the equation $g(x) = c$ has exactly 4 solutions. | 2 | 98 | 1 |
math | In a regular tetrahedron \( P-ABCD \), where each face is an equilateral triangle with side length 1, points \( M \) and \( N \) are the midpoints of edges \( AB \) and \( BC \), respectively. Find the distance between the skew lines \( MN \) and \( PC \). | \frac{\sqrt{2}}{4} | 71 | 10 |
math | A convex quadrilateral \(EFGH\) has vertices \(E, F, G, H\) lying respectively on the sides \(AB, BC, CD,\) and \(DA\) of another quadrilateral \(ABCD\). It satisfies the equation \(\frac{AE}{EB} \cdot \frac{BF}{FC} \cdot \frac{CG}{GD} \cdot \frac{DH}{HA} = 1\). Given that points \(E, F, G,\) and \(H\) lie on the sides... | \lambda | 164 | 2 |
math | Given that "$x > k$" is a sufficient but not necessary condition for "$\frac {3}{x+1} < 1$", determine the range of values for $k$. | [2, +\infty) | 38 | 8 |
math | 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 | 42 | 4 |
math | Let \( f(x) = x^3 - 3x^2 + 5x - 7 \). Find the polynomial \( g(x) \) of the least degree such that
\[ f(2) = g(2), \quad f(2-\sqrt{2}) = g(2-\sqrt{2}), \quad f(2+\sqrt{2}) = g(2+\sqrt{2}) \] | 3x^2 - 5x - 3 | 90 | 11 |
math | If $\log_2 x^3 + \log_{1/3} x = 6$, compute $x$. | 2^{\frac{6 \log_2 3}{3 \log_2 3 - 1}} | 25 | 24 |
math | Given a right triangle \(ABC\) with the right angle at vertex \(B\). A median \(BD\) is drawn from \(B\). Let \(K\) be the point of tangency of side \(AD\) of triangle \(ABD\) with the incircle of this triangle. Find the acute angles of triangle \(ABC\) if \(K\) bisects \(AD\). | 30^\circ, 60^\circ | 78 | 10 |
math | A complex quartic polynomial \( Q \) is quirky if it has four distinct roots, one of which is the sum of the other three. There are four complex values of \( k \) for which the polynomial \( Q(x) = x^4 - k x^3 - x^2 - x - 45 \) is quirky. Compute the product of these four values of \( k \). | 720 | 83 | 3 |
math | Given a $9 \times 9$ chess board, we consider all the rectangles whose edges lie along grid lines (the board consists of 81 unit squares, and the grid lines lie on the borders of the unit squares). For each such rectangle, we put a mark in every one of the unit squares inside it. When this process is completed, how man... | 56 | 84 | 2 |
math | Solve
\[\arccos 2x + \arccos 3x = \frac{\pi}{2}.\] | \pm \frac{1}{\sqrt{13}} | 29 | 13 |
math | Given that $\overrightarrow{a}$ and $\overrightarrow{b}$ are unit vectors, and vector $\overrightarrow{c}$ satisfies $| \overrightarrow{c}-( \overrightarrow{a}+ \overrightarrow{b})|=| \overrightarrow{a}- \overrightarrow{b}|$, determine the maximum value of $| \overrightarrow{c}|$. | 2 \sqrt {2} | 79 | 6 |
math | A square with integer side length is cut into 10 squares, all of which have integer side length and at least 8 of which have area 1. Determine the smallest possible value of the length of the side of the original square. | 4 | 49 | 1 |
math | Given that $f'(x)=3$, determine the value of $\lim\limits_{\Delta x\to0} \frac{f(x+\Delta x)-f(x)}{\Delta x}$. | 3 | 42 | 1 |
math | When a water tank is $40\%$ full, it contains 36 gallons less than when it is $25\%$ empty. How many gallons of water does the tank hold when it is full? | 103\text{ gallons} | 46 | 8 |
math | Given a positive integer $n$ not exceeding $100$ chosen with probabilities $p$ for $n\le 50$ and $3p$ for $n > 50$, calculate the probability that a perfect square is chosen. | 0.08 | 52 | 4 |
math | A dog and a cat simultaneously grab a sausage from different ends with their teeth. If the dog bites off its piece and runs away, the cat will get 300 grams more than the dog. If the cat bites off its piece and runs away, the dog will get 500 grams more than the cat. How much sausage is left if both bite off their piec... | 400 | 81 | 3 |
math | Let $a$, $b$, and $c$ be three positive real numbers whose sum is 2. If no one of these numbers is more than three times any other, find the minimum value of the product $abc$. | \frac{2}{9} | 46 | 7 |
math | Given the regression equation $y=0.75x-68.2$ and a student's height of $x=170$ cm, calculate the student's weight in kg. | 59.3 | 41 | 4 |
math | (1) In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $A=60^{\circ}$, $b=1$, and the area of $\triangle ABC$ is $\sqrt{3}$, then $a=$ .
(2) The sum of the first $n$ terms of the sequence $\{a_n\}$ is $S_n=3n^2-2n$. Then, the general term formula of the sequ... | c\in( \sqrt{5},3) | 227 | 11 |
math | If the equation $ax^2-x-1=0$ has exactly one solution in the interval $(0,1)$, then the range of the real number $a$ is \_\_\_\_\_\_. | a>2 | 44 | 3 |
math | The sum of the first $n$ terms of the sequence $\{a_n\}$ is denoted by $S_n$, where $a_1=1$ and $a_n + a_{n+1} = 3 \times 2^{n-1}$. Find $S_{2017}$. | S_{2017} = 2^{2017} - 1 | 68 | 19 |
math | Find the distance between the intersections of $x=y^3$ and $x+y=2$. Provide the answer in the form $\sqrt{u+v\sqrt{w}}$ and determine the ordered triple $(u, v, w)$. | (u,v,w) = (0,0,1) | 50 | 12 |
math | I have 6 shirts and 6 pairs of pants, each available in 6 colors. There are 12 hats in total; 6 of these hats have patterns and 6 hats are the solid colors that match the shirts and pants. I refuse to wear an outfit where all three items are the same color or where a patterned hat is worn with two solid items of differ... | 246 | 88 | 3 |
math | If the function $y=x^2+2(a-1)x+2$ is monotonically decreasing in the interval $(-\infty, 4]$, then the range of the real number $a$ is | a \leq -3 | 46 | 6 |
math | Let $p(x): ax^2 + 2x + 1 > 0$. If for all $x \in \mathbb{R}$, $p(x)$ is true, then the range of values for $a$ is ______. | a > 1 | 52 | 4 |
math | Given two circles $C_1: (x-a)^2 + (y+2)^2 = 4$ and $C_2: (x+b)^2 + (y+2)^2 = 1$ intersect, find the equation of the line where their common chord lies. | (2a+2b)x + 3 + b^2 - a^2 = 0 | 60 | 21 |
math | There is a unique two-digit positive integer \(s\) for which the last two digits of \(13 \cdot s\) are \(52\). | 04 | 31 | 2 |
math | Joey and his five brothers are ages $3$, $5$, $7$, $9$, $11$, and $13$. One afternoon two of his brothers whose ages sum to $16$ went to the movies, two brothers younger than $10$ went to play baseball, and Joey and the $5$-year-old stayed home. How old is Joey? | 11 | 79 | 2 |
math | Let $f(x)$ be an odd function on $\mathbb{R}$ , such that $f(x)=x^2$ when $x\ge 0$ . Knowing that for all $x\in [a,a+2]$ , the inequality $f(x+a)\ge 2f(x)$ holds, find the range of real number $a$ . | a \in [\sqrt{2}, +\infty) | 86 | 13 |
math | A \(23 \times 23\) square is divided into smaller squares of dimensions \(1 \times 1\), \(2 \times 2\), and \(3 \times 3\). What is the minimum possible number of \(1 \times 1\) squares? | 1 | 59 | 1 |
math | There are no more than 100 workers in the workshop, a third of them are women, and 8% of the workers have a reduced workday. How many workers are there in the workshop? How many of them are women, and how many people have a reduced workday? | 6 | 60 | 1 |
math | Given that $θ∈( \dfrac {π}{2},π)$, and $\dfrac {1}{\sin θ }+ \dfrac {1}{\cos θ }=2 \sqrt {2}$, find the value of $\cos (2θ+ \dfrac {π}{3})$. | \dfrac { \sqrt {3}}{2} | 68 | 11 |
math | There are $2^{10} = 1024$ possible $10$ -letter strings in which each letter is either an A or a B. Find the number of such strings that do not have more than $3$ adjacent letters that are identical. | 548 | 56 | 3 |
math | The line $l: x-2y+5=0$ intersects the circle $C: x^2+y^2=9$ at points A and B. If point D is another point on circle C different from A and B, then the maximum area of $\triangle ABD$ is \_\_\_\_\_. | 6 + 2\sqrt{5} | 67 | 9 |
math | Rani, James, and Fiona start a game with $$3$ each. A bell rings every $10$ seconds, and at each ring, each of the players who has more than $1 dollar will give $$1$ to one of the other two players chosen at random. What is the probability that after the bell has rung $2025$ times, each player will have exactly $$3$?
A... | \frac{2}{27} | 138 | 8 |
math | Let F be the focus of the parabola y^{2}=4x, point A lies on C, and point B(3,0). If |AF|=|BF|, then calculate the length of |AB|. | 2\sqrt{2} | 47 | 6 |
math | Define an ordered triple $(A, B, C)$ of sets to be minimally intersecting if $|A \cap B| = |B \cap C| = |C \cap A| = 1$ and $A \cap B \cap C = \emptyset$. For example, $(\{1,2\},\{2,3\},\{1,3,4\})$ is a minimally intersecting triple. Let $N$ be the number of minimally intersecting ordered triples of sets for which each... | 760 | 154 | 3 |
math | A chord AB passes through point P(2, -2) on the parabola $x^2 = -2y$ with the angles of inclination of PA and PB being complementary. Find the slope of chord AB. | 2 | 46 | 1 |
math | If the function $f(x)=\log_{a}(x^{2}-ax+3)$ $(a > 0$ and $a\neq 1)$, satisfies for any $x_{1}$, $x_{2}$, when $x_{1} < x_{2}\leqslant \frac {a}{2}$, then $f(x_{1})-f(x_{2}) > 0$, the range of the real number $a$ is \_\_\_\_\_\_. | (1,2 \sqrt {3}) | 108 | 9 |
math | Let \( a_{1}, a_{2}, \cdots, a_{105} \) be a permutation of \( 1, 2, \cdots, 105 \), satisfying the condition that for any \( m \in \{3, 5, 7\} \), for all \( n \) such that \( 1 \leqslant n < n+m \leqslant 105 \), we have \( m \mid (a_{n+m}-a_{n}) \). How many such distinct permutations exist? (Provide the answer as a... | 3628800 | 131 | 7 |
math | Find the equation of the line passing through point A (1, 2) which is parallel to the line $2x - 3y + 5 = 0$. | 2x - 3y + 4 = 0 | 36 | 12 |
math | A convex hexagon has interior angles with measures $x+2$, $2x+3$, $3x+4$, $4x+5$, $5x+6$, and $6x+7$ degrees. What is the measure of the largest angle? | 205 | 56 | 3 |
math | A circle centered at $O$ has a radius of 2 and contains the point $A$ on its circumference. Segment $AB$ is tangent to the circle at $A$ and $\angle AOB = \phi$. If point $D$ lies on $\overline{OA}$ such that $\overline{BD}$ bisects $\angle ABO$, express $OD$ in terms of the sine and cosine of $\phi$. | \frac{2}{1 + \sin \phi} | 90 | 12 |
math | In a circle of unit radius, $\triangle ABC$ is inscribed with sides $a$, $b$, and $c$ opposite to angles $A$, $B$, and $C$ respectively. Given the equation $2(\sin^2 A - \sin^2 C) = (\sqrt{2}a - b)\sin B$,
(1) Find the measure of angle $C$;
(2) Find the maximum area of $\triangle ABC$. | \frac{\sqrt{2}}{2} + \frac{1}{2} | 96 | 18 |
math | Compute
\[
\prod_{k = 1}^{8} \prod_{j = 1}^{6} (e^{2 \pi ji/7} - e^{2 \pi ki/9}).
\] | 1 | 49 | 1 |
math | (1) Given that $f(\sqrt{x} + 1) = x + 2\sqrt{x}$, find the analytical expression of $f(x)$;
(2) Given that $f(x)$ is a first-degree function and satisfies $3f(x+1) - 2f(x-1) = 2x + 17$, find the analytical expression of $f(x)$. | f(x) = 2x + 7 | 85 | 10 |
math | Given a sequence $\left\{a_n\right\}$, the sum of its first $n$ terms is $S_n$, where $S_n=2a_n-2$.
$(1)$ Find the general formula for the sequence $\left\{a_n\right\}$.
$(2)$ Let $b_n=\log_2(a_n)$, $c_n= \frac{1}{b_nb_{n+1}}$, and denote the sum of the first $n$ terms of the sequence $\left\{c_n\right\}$ as $T_n$. I... | \left[\frac{1}{9},+\infty\right) | 165 | 15 |
math | Using $[x]$ to denote the greatest integer less than or equal to a real number $x$, find the number of real roots of the equation $\lg^{2} x - [\lg x] - 2 = 0$. | 3 | 48 | 1 |
math | What is the remainder when $2(11085 + 11087 + 11089 + 11091 + 11093 + 11095 + 11097 + 11099)$ is divided by $14$? | 2 | 71 | 1 |
math | In the plane rectangular coordinate system $xOy$, points $A(1,0)$, $B(4,0)$ are given. If there exists a point $P$ on the curve $C: x^{2}+y^{2}-2ax-4ay+5a^{2}-9=0$ such that $|PB|=2|PA|$, then the range of real number $a$ is ____. | [-\sqrt{5}, -\frac{\sqrt{5}}{5}] \cup [\frac{\sqrt{5}}{5}, \sqrt{5}] | 91 | 34 |
math | If $9:x^2 = x:25$, what is the value of $x$? | \sqrt[3]{225} | 21 | 9 |
math | A traffic light runs repeatedly through the following cycle: green for 30 seconds, then yellow for 3 seconds, and then red for 30 seconds. Leah picks a random three-second time interval to watch the light. What is the probability that the color changes while she is watching? | \frac{1}{7} | 61 | 7 |
math | Let $S$ be the set of complex numbers of the form $x + yi$, where $x$ and $y$ are real numbers such that $\frac{1}{2} \le x \le \frac{\sqrt{2}}{2}$ and $y \ge \frac{1}{2}$. Find the smallest positive integer $m$ such that for all positive integers $n \ge m$, there exists a complex number $z \in S$ such that $z^n = 1$. | 24 | 106 | 2 |
math | Let the function $f(x)=2\sin x\cos^2 \frac{\varphi}{2}+\cos x\sin \varphi-\sin x$ $(0 < \varphi < \pi)$ take its minimum value at $x=\pi$.
$(I)$ Find the value of $\varphi$ and simplify $f(x)$;
$(II)$ In $\triangle ABC$, $a$, $b$, and $c$ are the sides opposite angles $A$, $B$, and $C$ respectively, given that $a=1... | \frac{\pi}{12} | 147 | 8 |
math | Define $\mathbf{A} = \begin{pmatrix} 0 & 1 \\ 4 & 0 \end{pmatrix}.$ Find the vector $\mathbf{v}$ such that
\[(\mathbf{A}^7 + \mathbf{A}^5 + \mathbf{A}^3 + \mathbf{A} + \mathbf{I}) \mathbf{v} = \begin{pmatrix} 5 \\ 0 \end{pmatrix}.\] | \begin{pmatrix} -\frac{5}{28899} \\ \frac{1700}{28899} \end{pmatrix} | 112 | 39 |
math | Given the ellipse $\frac{x^2}{4} + \frac{y^2}{3} = 1$, let $(F_1, F_2, P)$ represent the left and right foci and any point on the ellipse, respectively. Determine the range of values for $|PF_1||PF_2|$. | [3, 4] | 70 | 6 |
math | Given a 2x2 grid of unit squares, find the area of the triangle formed by connecting one vertex at the bottom left corner of the bottom left square to the top right corner of the top right square and another vertex at the top left corner of the bottom right square. | 2 | 56 | 1 |
math | For an upcoming holiday, the weather forecast predicts a $30\%$ probability of rain on the first day and a $40\%$ probability on the second day. If it rains on the first day, the probability of rain on the second day increases to $70\%$. What is the probability that it rains on at least one of the two days, assuming th... | 49\% | 89 | 4 |
math | In the Cartesian coordinate system $xOy$, the parametric equation of line $l$ is $$\begin{cases} x= \sqrt {3}- \frac { \sqrt {3}}{2}t \\ y= \frac {1}{2}t \end{cases}$$ (where $t$ is the parameter). In the polar coordinate system with the origin as the pole and the positive x-axis as the polar axis, the polar equation o... | \frac {3+ \sqrt {3}}{2} | 188 | 13 |
math | Given that $\sin\alpha = \frac{1}{2} + \cos\alpha$, and $\alpha \in (0, \frac{\pi}{2})$, find the value of $\frac{\cos 2\alpha}{\sin(\alpha - \frac{\pi}{4})}$. | -\frac{\sqrt{14}}{2} | 63 | 11 |
math | 1. \(\lim _{x \rightarrow 1}(1-x) \operatorname{tg} \frac{\pi x}{2}\)
2. \(\lim _{x \rightarrow \frac{\pi}{4}}\left(\frac{\pi}{4}-x\right) \operatorname{cosec}\left(\frac{3}{4} \pi+x\right)\)
3. \(\lim _{x \rightarrow+\infty} x \operatorname{arcctg} x\)
4. \(\lim _{x \rightarrow-\infty} x\left(\frac{\pi}{2}+\operatorna... | -1 | 147 | 2 |
math | Four positive integers $a$, $b$, $c$, and $d$ sum up to $1105$. If no two of these numbers are the same and each pair of these numbers must have a common divisor greater than 1, what is the largest possible value of $\gcd(a, b, c, d)$? | 221 | 68 | 3 |
math | Given a geometric progression $\{a_n\}$ with the first term $a_1=2$ and the sum of the first $n$ terms as $S_n$, and the equation $S_5 + 4S_3 = 5S_4$ holds, find the maximum term of the sequence $\left\{ \frac{2\log_{2}a_n + 1}{\log_{2}a_n - 6} \right\}$. | 15 | 100 | 2 |
math | How many different ways can six people line up according to the following different requirements?
1. Six people form a circle;
2. All stand in a row, with A and B must be adjacent;
3. All stand in a row, with A and B not adjacent;
4. All stand in a row, with A, B, and C in order from left to right;
5. All stand in a r... | 504 | 116 | 3 |
math | Given $a=\ln 2, b=\ln 3, c=\ln 5$, determine the correct ordering of the fractions $\dfrac{a}{2}, \dfrac{b}{3}, \dfrac{c}{5}$. | \dfrac{c}{5} < \dfrac{a}{2} < \dfrac{b}{3} | 52 | 25 |
math | For a function $f(x)$ defined on $\mathbb{R}$ which satisfies $f(-x) = -f(x)$, $f(x + 1) = f(1 - x)$, and $f(x) = 2^{x} + \frac{6}{5}$ for $x \in (-1, 0)$, find the value of $f(\log_{2}20)$. | -2 | 89 | 2 |
math | Given a large circular clock face with a radius of $30$ cm and a smaller circular disk with a radius of $15$ cm, determine the position on the clock face where the disk is tangent when the arrow is next pointing rightward after rolling from the $3$ o'clock position on the clock face clockwise without slipping. | 9 \text{ o'clock} | 68 | 7 |
math | A merchant buys goods at 25% off the list price. He wants to mark the goods so that he can offer a discount of 25% on the marked price and still achieve a profit of 30% on the selling price. What percentage of the list price must he mark the goods. | 142.86 | 63 | 6 |
math | The maximum value of the function $y=a^x$ ($a>0$, $a\neq1$) in the interval $[1, 2]$ is $\frac{a}{3}$ greater than its minimum value. Find the value of $a$. | a=\frac{2}{3} | 56 | 8 |
math | Given the function $f(x) = x^2 + mx - 1$, find the range of the real number $m$ such that for any $x \in [m, m+1]$, it holds that $f(x) < 0$. | \left(- \dfrac{\sqrt{2}}{2}, 0\right) | 54 | 19 |
math | In triangle \(ABC\), it is known that \(\angle BAC = \alpha\), \(\angle BCA = \gamma\), \(AB = c\). Find the area of triangle \(ABC\). | \frac{c^2 \sin \alpha \sin (\alpha + \gamma)}{2 \sin \gamma} | 45 | 25 |
math | Consider the quadratic equation $x^2 + 4x\sqrt{2} + k = 0$ where $k$ is a parameter. Determine the value of $k$ such that the discriminant is zero and the roots are real and equal. | 8 | 53 | 1 |
math | Given the function $f(x) = \begin{cases} 2x+a, & x<1 \\ -x-2a, & x\geq1 \end{cases}$, if $f(1-a)=f(1+a)$, then the value of $a$ is \_\_\_\_\_\_. | -\frac{3}{4} | 69 | 7 |
math | Non-negative real numbers \( x_{1}, x_{2}, \cdots, x_{2016} \) and real numbers \( y_{1}, y_{2}, \cdots, y_{2016} \) satisfy:
(1) \( x_{k}^{2}+y_{k}^{2}=1 \) for \( k=1,2, \cdots, 2016 \);
(2) \( y_{1}+y_{2}+\cdots+y_{2016} \) is odd.
Find the minimum value of \( x_{1}+x_{2}+\cdots+x_{2016} \). | 1 | 149 | 1 |
math | Given the function $f(x) = \frac{1}{3}x^{3} - ax + 1$.
(1) When $a = 1$, find the equation of the tangent line at $x = 0$.
(2) If the minimum value of $f(x)$ on the interval $[0, 1]$ is $\frac{11}{12}$, determine the value of $a$. | a = \frac{1}{4} | 91 | 9 |
math | Given the parabola $y^2=2px$ intersects with the line $ax+y-4=0$ at points $A$ and $B$, where the coordinates of point $A$ are $(1,2)$. Let the focus of the parabola be $F$. Find the value of $|FA|+|FB|$. | 7 | 74 | 1 |
math | The median of a set of consecutive odd integers is 126. If the greatest integer in the set is 153, what is the smallest integer in the set? | 100 | 37 | 3 |
math | João and Maria each have a large jar with one liter of water. On the first day, João puts $1 \mathrm{ml}$ of water from his jar into Maria's jar. On the second day, Maria puts $2 \mathrm{ml}$ of water from her jar into João's jar. On the third day, João puts $3 \mathrm{ml}$ of water from his jar into Maria's jar, and s... | 900 | 111 | 3 |
math | An employee receives an average of two requests per hour. Assuming a simple flow of requests, what is the probability of receiving four requests in four hours? | 0.0572 | 30 | 6 |
math | Given four one-inch squares are placed in a horizontal line, the third square from the left is lifted, rotated 45 degrees and repositioned in line such that its left diagonal point touches the right edge of the second square. Calculate the greatest vertical distance from the original line to any point on the reposition... | \frac{\sqrt{2}}{2} | 67 | 10 |
math | Twelve points are spaced around a $3 \times 3$ square at intervals of one unit. Two of the 12 points are chosen at random. Find the probability that the two points are one unit apart. | \frac{2}{11} | 45 | 8 |
math | Given a box containing 3 screw base bulbs and 7 bayonet base bulbs, all of which have the same shape and power and are placed with their bases down, calculate the probability that the electrician gets a bayonet base bulb on his third try. | \frac{7}{120} | 52 | 9 |
math | Let $A=\{x|0<x<m\}$, and $B=\{x|0<x<1\}$. Given that $B \subseteq A$, find the range of values for $m$. | m \geq 1 | 44 | 6 |
math | Let $A,$ $B,$ and $C$ be points on a circle of radius $15.$ If $\angle ACB = 90^\circ,$ what is the circumference of the minor arc ${AB}$? Express your answer in terms of $\pi.$ | 7.5\pi | 55 | 5 |
math | Call a set of integers "spacy" if it contains no more than one out of any four consecutive integers. How many subsets of $\{1, 2, 3, \dots, 15\}$, including the empty set, are spacy? | 181 | 55 | 3 |
math | Rosencrantz and Guildenstern are playing a game where they repeatedly flip coins. Rosencrantz wins if 1 head followed by 2009 tails appears. Guildenstern wins if 2010 heads come in a row. They will flip coins until someone wins. What is the probability that Rosencrantz wins? | \frac{2^{2009}-1}{3 \cdot 2^{2008}-1} | 74 | 25 |
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