problem stringlengths 14 4.09k | solution stringlengths 1 802k | task_type stringclasses 3
values | problem_tokens int64 5 895 |
|---|---|---|---|
Given that $\overrightarrow{e\_1}$ and $\overrightarrow{e\_2}$ are unit vectors with an angle of $60^{\circ}$ between them, $\overrightarrow{a} = 2\overrightarrow{e\_1} + \overrightarrow{e\_2}$, and $\overrightarrow{b} = -3\overrightarrow{e\_1} + 2\overrightarrow{e\_2}$, find the angle between $\overrightarrow{a}$ and ... | 120 | math | 111 |
Find the number of positive integers \(n \le 2000\) that can be expressed in the form
\[
\lfloor x \rfloor + \lfloor 2x \rfloor + \lfloor 4x \rfloor = n
\]
for some real number \(x\). | 1140 | math | 66 |
Point \( B \) lies on segment \( AC \), with \( AB = 14 \) and \( BC = 28 \). Semicircles are constructed on segments \( AB \), \( BC \), and \( AC \) as diameters in the same half-plane. Find the radius of the circle that is tangent to all three semicircles. | 7 | math | 75 |
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} | math | 66 |
Given that Carl has 4 cubes each having side length 3, and Kate has 6 cubes each having side length 4, calculate the total volume of these cubes. | 492 | math | 35 |
Let $ABCD$ be a cyclic quadrilateral with circumradius $100\sqrt{3}$ and $AC=300$ . If $\angle DBC = 15^{\circ}$ , then find $AD^2$ .
*Proposed by Anand Iyer* | 60000 | math | 69 |
Given that the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{4} = 1$ ($a > 0$) has an eccentricity of $\frac{\sqrt{5}}{2}$, and the points $F_1$ and $F_2$ are its foci on the left and right side, respectively. Point $P$ has coordinates $(5, y_0)$ and point $Q$ is the point on the hyperbola symmetrical to $P$ with respe... | 6\sqrt{5} | math | 137 |
Given a number $a$ is randomly selected from the interval $[-2, 4]$, find the probability that $1$ belongs to the set ${x|2x^2+ax-a^2>0}$. | \frac{1}{2} | math | 47 |
Given that a real number $x$ is chosen at random from the interval $[-2,3]$, calculate the probability that this number is less than $1$. | \frac{3}{5} | math | 34 |
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 | math | 74 |
Given points A(3, -4) and B(5, 2) are equidistant from line L, and line L passes through the intersection of two lines L<sub>1</sub>: $3x - y - 1 = 0$ and L<sub>2</sub>: $x + y - 3 = 0$, find the equation of line L. | y = -x + 3 | math | 82 |
Katya thought of a natural number. Her friends asked the following questions:
Alyona: Is it divisible by 7?
Lena: Is it divisible by 5?
Rita: Is it less than 9?
Sveta: Is it divisible by 35?
Katya answered affirmatively to only two out of the four questions. What numbers could Katya have thought of? List all possi... | 5 \; \text{and} \; 7 | math | 86 |
Solve for $y$: $\left(\frac{1}{8}\right)^{3y+9} = (32)^{2y+7}$. | -\frac{62}{19} | math | 35 |
In our school's November exam for senior year students majoring in science, about 1000 students took the math test. The scores $\xi$ follow a normal distribution $N(100, a^2)$ ($a>0$, full score 150). It was found that approximately 60% of the students scored between 80 and 120. How many students scored at least 120? | 200 | math | 93 |
Given the function $f(x)= \begin{cases} (5-a)x-3,x < 1 \\ \log _{a}x,x\geqslant 1 \end{cases}$, find the range of values for the real number $a$ that makes this function an increasing function on $\mathbb{R}$. | [2,5) | math | 71 |
Find the value of $\cos\left(\alpha + \frac{2\pi}{3}\right)$ given that $\sin\left(\alpha + \frac{\pi}{3}\right) + \cos\left(\alpha - \frac{\pi}{2}\right) = -\frac{4\sqrt{3}}{5}$ and $-\frac{\pi}{2} < \alpha < 0$. | \frac{4}{5} | math | 87 |
A ray of light is emitted from point A $(-\frac{1}{2}, 0)$ and reflects off point B $(0, 1)$ on the y-axis. The equation of the line containing the reflected ray is \_\_\_\_\_\_. | 2x+y-1=0 | math | 54 |
What is the sum of the prime numbers less than $20$? | 77 | math | 15 |
The graph of $r = \sin \theta$ represents a circle. Find the smallest value of $t$ such that when $r = \sin \theta$ is plotted for $0 \le \theta \le t,$ the resulting graph covers the entire circle. | 2\pi | math | 55 |
The circles C1: x^2 + y^2 + 2x + y - 2 = 0 and C2: x^2 + y^2 - 4x - 2y + 4 = 0 have a common tangent. Determine the number of common tangent lines. | 4 | math | 62 |
In the tetrahedron $P-ABC$, the side edges $PA=2$, $PB=PC= \sqrt {6}$. Find the surface area of the sphere passing through points $P$, $A$, $B$, $C$ when the sum of the areas of the three side faces of the tetrahedron $P-ABC$ is maximized. | 16\pi | math | 78 |
Calculate $\sec \frac{2 \pi}{9}+\sec \frac{4 \pi}{9}+\sec \frac{6 \pi}{9}+\sec \frac{8 \pi}{9}$. | 4 | math | 46 |
What is the ratio of the area of an equilateral triangle inscribed in a semicircle with radius \(r\) to the area of an equilateral triangle inscribed in a circle with radius \(r\)? Express your answer as a common fraction. | \frac{4}{9} | math | 51 |
Non-zero vectors \(\vec{a}\) and \(\vec{b}\) satisfy \(|\vec{a}| = |\vec{b}| = |\vec{a} - \vec{b}|\). The angle between \(\vec{a}\) and \(\vec{a} + \vec{b}\) is equal to? | 30^\circ | math | 73 |
Given that $\boldsymbol{a}$ and $\boldsymbol{b}$ are two perpendicular unit vectors, and that $|\boldsymbol{c}|=13$, $\boldsymbol{c} \cdot \boldsymbol{a}=3$, and $\boldsymbol{c} \cdot \boldsymbol{b}=4$, find the minimum value of $\left|\boldsymbol{c}-t_{1} \boldsymbol{a}-t_{2} \boldsymbol{b}\right|$ for any real number... | 12 | math | 118 |
On the base \(AC\) of an isosceles triangle \(ABC (AB = BC)\), point \(M\) is marked. It is known that \(AM = 7\), \(MB = 3\), \(\angle BMC = 60^\circ\). Find the length of segment \(AC\). | 17 | math | 67 |
Calculate: \( 3752 \div (39 \times 2) + 5030 \div (39 \times 10) = \) | 61 | math | 38 |
I have 17 distinguishable socks in my drawer: 4 white, 4 brown, 5 blue, and 4 black. In how many ways can I choose a pair of socks, provided that I get two socks of different colors? | 108 | math | 51 |
Given the line $l$: $\begin{cases} x=t \\ y=t+1 \end{cases} (t\text{ is a parameter})$ and the circle $C$: $\rho=2\cos\theta$, calculate the distance from the center of circle $C$ to the line $l$. | \sqrt{2} | math | 65 |
A certain aquatic product dealer purchases a batch of a certain species of freshwater fish at a price of $30$ yuan per kilogram. Based on sales experience, the daily sales volume $y$ (in kilograms) of this species of freshwater fish is known to have a linear relationship with the selling price $x$ (in yuan per kilogram... | 2250 | math | 231 |
In ∆ABC, if a = 1, c = 2, and B = 60°, calculate the area of ∆ABC. | \dfrac{\sqrt{3}}{2} | math | 32 |
Calculate $2013_5 \div 23_5$ in base $5$. | 34_5 | math | 21 |
Given that {a<sub>n</sub>} is an arithmetic sequence, if a<sub>1</sub>+a<sub>5</sub>+a<sub>9</sub>=8π, find the sum of the first 9 terms S<sub>9</sub> and the value of cos(a<sub>3</sub>+a<sub>7</sub>). | - \frac {1}{2} | math | 81 |
Calculate the probability that athlete A cannot run the first leg and athlete B cannot run the last leg in a 4x100 meter relay race selection from 6 short-distance runners, including athletes A and B, to form a team of 4 runners. | \frac{7}{10} | math | 53 |
Convert the binary number $1010_2$ to a decimal number. | 10 | math | 17 |
Each triangle is a 30-60-90 triangle, and the hypotenuse of one triangle is the longer leg of an adjacent triangle. There are now five 30-60-90 triangles in sequence, and the hypotenuse of the largest triangle is 16 centimeters. What is the length of the longer leg of the smallest triangle? | 4.5\sqrt{3} \text{ cm} | math | 78 |
Given that the circumradius of triangle \( \triangle ABC \) is \( R \), and
\[ 2R\left(\sin^2 A - \sin^2 C\right) = (\sqrt{2} a - b) \sin B, \]
where \( a \) and \( b \) are the sides opposite to \( \angle A \) and \( \angle B \) respectively. Determine the measure of \( \angle C \). | 45^\circ | math | 96 |
Given the function $f(x)=-\frac{1}{\sqrt{b}}e^{\sqrt{ax}} (a > 0, b > 0)$, the tangent line to its graph at $x=0$ is tangent to the circle $x^2 + y^2 = 1$. Find the minimum value of $2^a + 2^b$. | 2\sqrt{2} | math | 81 |
Evaluate the expression \( 3000(3000^{2999})^2 \). | 3000^{5999} | math | 24 |
Find all numbers \(a, b, c, d\) such that:
(i) The function \(f(x)=4x^{3} - dx\) satisfies the inequality \( |f(x)| \leq 1 \) for \(x \in [-1,1]\);
(ii) The function \(g(x)=4x^{3} + ax^{2} + bx + c\) satisfies the inequality \( |g(x)| \leq 1 \) for \(x \in [-1,1]\). | d=3, b=-3, a=0, c=0 | math | 106 |
Calculate:<br/>$(1)(-\frac{3}{4}-\frac{5}{8}+\frac{9}{12})×(-24)$;<br/>$(2)-1^{6}+|\left(-2\right)^{3}-10|-\left(-3\right)\div \left(-1\right)^{2023}$. | 14 | math | 80 |
Given the sets $M = \{-1, 1, -2, 2\}$ and $N = \{1, 4\}$, determine the intersection of $M$ and $N$. | 1 | math | 44 |
Read the calculation process of the following student and complete the task.
Simplify first, then evaluate: $[\left(2x+y\right)\left(2x-y\right)-\left(2x-3y\right)^{2}]\div \left(-2y\right)$, where $x=1$, $y=-2$.
Solution: Original expression $=(4x^{2}-y^{2}-4x^{2}-12xy+9y^{2})\div \left(-2y\right)$, Step 1
$=(-1... | -16 | math | 230 |
The ratio of the surface area of a cube to the surface area of its inscribed sphere. | \frac{6}{\pi} | math | 19 |
Team A and Team B each has 7 players who will compete in the predetermined order of a Go competition. The first players from each team compete first, and the loser is eliminated while the winner continues to compete against the next player from the losing team, and so on, until all members of one team are eliminated. H... | 3432 | math | 79 |
A circular disc with a diameter of $2D$ is placed on a $6 \times 6$ checkerboard, aiming for the centers to coincide. Calculate the number of checkerboard squares which are completely covered by the disc. | 16 | math | 47 |
Given \(2^{2000} - 2^{1999} - 3 \times 2^{1998} + 2^{1997} = l \cdot 2^{1997}\), determine the value of \(l\). | -1 | math | 61 |
A circle is defined by the equation $x^{2}+y^{2}+4x-2y-1=0$. There exist two points on the circle that are symmetric with respect to the line $ax-2by+2=0$ ($a > 0, b > 0$). Find the minimum value of $\frac{1}{a}+\frac{4}{b}$. | 9 | math | 85 |
Point $E$ is one-fourth the way along side $\overline{CD}$ of rectangle $ABCD$, and $\overline{BE}$ meets diagonal $\overline{AC}$ at $F$. The area of quadrilateral $AFED$ is $36$. Calculate the area of $ABCD$. | 144 | math | 65 |
Given that each digit in the addition problem has been replaced by a letter, and different letters represent different digits, calculate the value of C. | 1 | math | 28 |
Given that $y=f(x)$ is an odd function defined on $R$, and $f(x)=x-2$ when $x > 0$, find the solution set of the inequality $f(x) < \frac{1}{2}$. | (-\infty, -\frac{3}{2}) \cup [0, \frac{5}{2}) | math | 52 |
Let $x_1$, $x_2$, $x_3$, $x_4$, $x_5$ be positive integers, and $x_1 + x_2 + x_3 + x_4 + x_5 \leq x_1x_2x_3x_4x_5$. Find the maximum value of $x_5$. | 5 | math | 80 |
Numbers $1, 2, 3, \ldots, 16$ are written in a $4 \times 4$ array, one number in each square, such that if two numbers are consecutive, then they occupy squares that share an edge. The numbers in the four corners add up to $34$. What is the number in the center of the matrix? | 10 | math | 78 |
The function $f(x) = x^3 + ax$ ($x \in \mathbb{R}$) has an extremum at $x = l$. Then, the equation of the tangent line to the curve $y = f(x)$ at the origin is ____. | 3x + y = 0 | math | 57 |
Find the number of integers that belong to the range of the function:
$$
f(x) = 2 \cos 2x + 2 \cos x - 2019
$$ | 7 | math | 41 |
If the function $f(x) = kx^2 + (k - 1)x + 3$ is an even function, find the decreasing interval of $f(x)$. | (-\infty, 0] | math | 38 |
John is 24 years younger than his dad. The sum of their ages is 68 years. How many years old is John? | 22 | math | 29 |
Let $x$ and $y$ be real numbers such that $3(x^2 + y^2) = x + y$. Find the maximum value of $x + 2y$. | \sqrt{\frac{5}{18}} + \frac{1}{2} | math | 40 |
Given the triangular pyramid $P-ABC$, point $M$ on segment $PC$ satisfies $PM=\frac{1}{3}PC$, and point $N$ on segment $PB$ satisfies $PN=\frac{2}{3}PB$. Calculate the ratio of the volumes of the triangular pyramids $P-AMN$ and $P-ABC$. | \frac{2}{9} | math | 75 |
A certain organization has 840 staff members. Now, 42 individuals are chosen using systematic sampling for a questionnaire survey. If all 840 individuals are randomly assigned numbers from 1 to 840, determine the number of people among the 42 sampled whose numbers fall within the interval $[61, 120]$. | 3 | math | 75 |
The positive five-digit integers that use each of the five digits $1, 2, 3, 4, 5$ exactly once are ordered from least to greatest. What is the $44^{\text{th}}$ integer in the list? | 25143 | math | 54 |
Ashley has six state quarters. If the collector offers to buy them for 1500% of their face value, calculate the total amount of money she will receive. | 15 \times 1.50 = 22.50 \text{ dollars} | math | 36 |
Find all the numbers from 1 to 100 that have an odd number of positive divisors. | 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 | math | 22 |
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 | math | 106 |
(1) Given $|a|=1$, $|b|=2$, $|c|=3$, and $a > b > c$, then $a+b-c=$ ____.
(2) Given $a$, $b$, $c$, $d$ are rational numbers, $|a-b| \leq 9$, $|c-d| \leq 16$, and $|a-b-c+d|=25$, then $|b-a|-|d-c|=$ ____. | -7 | math | 106 |
How many different ways can six people stand in a row if the following conditions are met?
(1) Person A and Person B must be adjacent;
(2) Person A and Person B are not adjacent;
(3) There are exactly two people between Person A and Person B;
(4) Person A is not at the left end, and Person B is not at the right end. | 504 | math | 79 |
Given an ellipse $C:\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1(a>b>0)$ with eccentricity $\frac{\sqrt{2}}{2}$, point $A\left(2,1\right)$ lies on the ellipse $C$.
$(1)$ Find the standard equation of the ellipse $C$;
$(2)$ A line $l$ passing through point $M\left(0,1\right)$ intersects the ellipse $C$ at point... | 2 | math | 142 |
Evaluate $$\sin \left(1998^{\circ}+237^{\circ}\right) \sin \left(1998^{\circ}-1653^{\circ}\right)$$ | -\frac{1}{4} | math | 49 |
A two-digit integer $AB$ equals $\frac{1}{8}$ of the three-digit integer $AAB$, where $A$ and $B$ represent distinct digits from 1 to 9. What is the smallest possible value of the three-digit integer $AAB$? | 773 | math | 58 |
The center coordinates of the circle $x^2 + y^2 - 4x + 6y = 0$ can be found by rewriting the equation in standard form $(x-h)^2 + (y-k)^2 = r^2$, where $(h,k)$ represents the center of the circle. | (2,-3) | math | 64 |
How many numbers should there be in a lottery for the probability of getting an ambo to be $\frac{5}{473}$, when drawing five numbers? | 44 | math | 34 |
Participation in the local soccer league this year is $10\%$ higher than last year. The number of males increased by $5\%$ and the number of females increased by $20\%$ . What fraction of the soccer league is now female? | \frac{4}{11} | math | 61 |
Give the value of \(0 - 1 + 2 - 3 + 4 - 5 + \ldots - 49 + 50\). Only a numerical answer is expected. | 25 | math | 42 |
What is one-half times two-thirds times three-fourths? | \frac{1}{4} | math | 12 |
The degree of any term of the polynomial is less than or equal to 6. | 6 | math | 17 |
Carlinhos likes writing numbers in his notebook. One day he wrote the numbers from 1 to 999, one after the other, to form the giant number:
$$
123456789101112 \ldots 997998999
$$
Based on this number, the following questions are asked:
(a) How many digits were written?
(b) How many times does the digit 1 appear?
... | 8 | math | 143 |
There are five "equations" of the form \( x^{2} + \ldots x + \ldots = 0 \) written on the board. Two players take turns filling in the blanks with natural numbers from 1 to 10, with each number being used exactly once. The game ends when all numbers are filled in. The player who makes the first move aims to maximize th... | 3 | math | 126 |
Given a hyperbola centered at the origin with asymptotic equations $y = \pm \sqrt{3}x$, and the hyperbola passes through the point $(\sqrt{2}, \sqrt{3})$,
(1) Find the equation of the hyperbola;
(2) Find the distance from the foci of the hyperbola to the asymptotes. | \sqrt{3} | math | 80 |
Q. For integers $m,n\geq 1$ , Let $A_{m,n}$ , $B_{m,n}$ and $C_{m,n}$ denote the following sets: $A_{m,n}=\{(\alpha _1,\alpha _2,\ldots,\alpha _m) \colon 1\leq \alpha _1\leq \alpha_2 \leq \ldots \leq \alpha_m\leq n\}$ given that $\alpha _i \in \mathbb{Z}$ for all $i$ $B_{m,n}=\{(\alpha _1,\alpha _2,\ldots ,... | |A_{m,n}| = \binom{m+n-1}{n-1} | math | 383 |
Let $n$ be a positive integer. A child builds a wall along a line with $n$ identical cubes. He lays the first cube on the line and at each subsequent step, he lays the next cube either on the ground or on the top of another cube, so that it has a common face with the previous one. How many such distinct walls exist... | 2^{n-1} | math | 78 |
Given that the larger root of the equation $2002^2x^2 - 2003 \cdot 2001x - 1 = 0$ is $r$, and the smaller root of the equation $2001x^2 - 2002x + 1 = 0$ is $s$, find the value of $r - s$. | \frac{2000}{2001} | math | 85 |
Given a hyperbola with its center at the origin, foci $F_1$ and $F_2$ on the coordinate axes, eccentricity $\sqrt{2}$, and passing through the point $(4, -\sqrt{10})$. Point $M(3, m)$ is on the hyperbola.
$(1)$ Find the equation of the hyperbola;
$(2)$ Find the area of $\triangle F_1MF_2$. | 6 | math | 99 |
$\int_{0}^{\frac{\pi}{2}}\cos x dx$ | 1 | math | 18 |
In a mathematics competition, 1000 students are numbered as follows: 0001, 0002, 0003, ..., 1000. It is planned to draw a sample of size 50 by dividing into 50 parts using systematic sampling. If the first part includes the numbers 0001, 0002, ..., 0020, and a number 0015 is randomly selected from it, then the 40th num... | 0795 | math | 120 |
Given point P(3, -4), find the distance from point P to the chord AB formed by the two tangent lines drawn from point P to the circle C: $x^2+y^2=9$. | \frac{16}{5} | math | 44 |
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 | math | 98 |
Given the line $2ax-by+2=0$ ($a > 0$, $b > 0$) is intersected by the circle $x^{2}+y^{2}+2x-4y+1=0$ to form a chord of length $4$, find the minimum value of $\dfrac {4}{a}+ \dfrac {1}{b}$. | 9 | math | 84 |
Find three numbers where the first number is 80% of the second number, the ratio of the second number to the third number is 0.5:9/20, and the sum of the first and third numbers is 70 more than the second number. | 80, 100, 90 | math | 57 |
Given any cubic function $f(x)=ax^2+bx^2+cx+d (a\neq 0)$, it has a symmetry center $M(x_0, f(x_0))$. Let the derivative of the function $f(x)$ be $f′(x)$, and the derivative of $f′(x)$ be $f″(x)$. Then, $f″(x_0)=0$. If the function $f(x) = x^3 - 3x^2$, then find the value of $f(\frac{1}{2016}) + f(\frac{2}{2016}) + f(\... | -8062 | math | 177 |
A sphere intersects the xy-plane in a circle centered at $(1,3,0)$ with a radius of 2. The sphere also intersects the yz-plane in a circle centered at $(0,3,-8),$ with radius $r.$ Find $r$ and verify the radius of the sphere. | 2\sqrt{17} | math | 62 |
(The full score for this question is 8 points) Arrange 3 male students and 2 female students in a row,
(1) The number of all different arrangements;
(2) The number of arrangements where exactly two male students are adjacent;
(3) The number of arrangements where male students are of different heights and are ar... | 20 | math | 89 |
For any real numbers $a, b$ ($a < b$), a random variable $X$ is said to follow a normal distribution if $P(a < X \leq b) = \int_{a}^{b} \phi_{\mu\sigma}(x) \,dx$. It is denoted as $X \sim \text{N}(\mu, \sigma^2)$. If $X \sim \text{N}(0,1)$, then $\int_{-1}^{1} \phi_{\mu\sigma}(x) \,dx = \boxed{?}$. | 0.6826 | math | 130 |
A store received shipments of physics and mathematics textbooks. After selling $50\%$ of the mathematics textbooks and $20\%$ of the physics textbooks, totaling 390 books, the remaining number of mathematics textbooks was three times the number of remaining physics textbooks. How many mathematics and physics textbooks ... | 720\text{ and } 150 | math | 69 |
If $z=\frac{2+i}{i}$, find the coordinates of the point corresponding to $z$ in the complex plane. | (1, -2) | math | 28 |
In the expansion of \((x+y+z)^{8}\), find the sum of the coefficients for all terms of the form \(x^{2} y^{a} z^{b}\) (where \(a, b \in \mathbf{N}\)). | 1792 | math | 55 |
Given the function $f(x) = e^x \cos(x) + x^5$, determine the equation of the tangent line to the curve $y = f(x)$ at the point $(0, f(0))$. | y = x + 1 | math | 47 |
In $\triangle ABC$, $\angle ACB$ is an obtuse angle, $AC=BC=1$, and $\overrightarrow {CO}=x \overrightarrow {CA}+y \overrightarrow {CB}$ with $x+y=1$. The minimum value of the function $f(m)=| \overrightarrow {CA}-m \overrightarrow {CB}|$ is $\frac { \sqrt {3}}{2}$. Find the minimum value of $| \overrightarrow {CO}|$. | \frac {1}{2} | math | 104 |
If $f(x) = \frac{1 + x}{1 - 3x}, f_1(x) = f(f(x)), f_2(x) = f(f_1(x)),$ and in general $f_n(x) = f(f_{n-1}(x)),$ then $f_{1993}(3)=$ | \frac{1}{5} | math | 73 |
The largest of 20 consecutive odd integers whose sum is 8000. | 419 | math | 18 |
Randall proposes a new temperature system called Felsius temperature with the following conversion between Felsius \(^{\circ} \mathrm{E}\), Celsius \(^{\circ} \mathrm{C}\), and Fahrenheit \(^{\circ} \mathrm{F}\): \(^{\circ} E=\frac{7 \times{ }^{\circ} \mathrm{C}}{5}+16=\frac{7 \times{ }^{\circ} \mathrm{F}-80}{9}\). For... | -120 | math | 207 |
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