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 | For a function $f(x)$ defined on the interval $[0,1]$ that satisfies the following two conditions, we call it a G-function:
1. For any $x \in [0,1]$, $f(x) \geq 0$;
2. For $x_1 \geq 0, x_2 \geq 0, x_1 + x_2 \leq 1$, it holds that $f(x_1+x_2) \geq f(x_1) + f(x_2)$. Given the functions $g(x)=x^2$ and $h(x) = 2^x - b$, bo... | \{1\} | 192 | 5 |
math | Given $f(x)={e^x}+\frac{1}{{{e^x}}}$ and $g(x)=\ln[(3-a){e^x}+1]-\ln3a-2x$.
$(1)$ Find the minimum value of the function $f(x)$ on the interval $\left[0,+\infty \right)$.
$(2)$ For any $x_{1}$, $x_{2}\in \left[0,+\infty \right)$, if $g(x_{1})\leqslant f(x_{2})-2$ holds, find the range of values for $a$. | [1,3] | 138 | 5 |
math | Write number 2013 in a sum of m composite numbers. What is the largest value of m? | 502 | 23 | 3 |
math | Given $p$: $\sqrt{2x-1}\leqslant 1$, $q$: $(x-a)[x-(a+1)]\leqslant 0$, determine the range of the real number $a$ such that $q$ is a necessary but not sufficient condition for $p$. | [0, \frac{1}{2}] | 66 | 10 |
math | Given the set $\{a, b, c\} = \{0, 1, 2\}$, and the following three conditions: ① $a \neq 2$; ② $b = 2$; ③ $c \neq 0$ are correct for only one of them, then $10a + 2b + c$ equals $\boxed{?}$. | 21 | 91 | 2 |
math | Calculate the area of $\triangle AMC$ in rectangle $ABCD$ where $AB=10$, $AD=12$, and $AM=9$ meters. | 45 | 35 | 2 |
math | The number of real solutions to the equation $\frac{x}{100} = \sin x$ is to be determined. | 63 | 26 | 2 |
math | A survey of participants was conducted at the Olympiad. $ 90\%$ of the participants liked the first round, $60\%$ of the participants liked the second round, $90\%$ of the participants liked the opening of the Olympiad. Each participant was known to enjoy at least two of these three events. Determine the percenta... | 40\% | 89 | 4 |
math | Monica tosses a fair 8-sided die. If the roll is a prime number less than 5, she wins that amount of dollars. If the roll is a multiple of 4, she loses that amount of dollars. If the roll is an even number less than 4, she wins twice the amount rolled. Otherwise, she wins nothing. What is the expected value of her winn... | -\$0.38 | 97 | 6 |
math | Define a new operation "$\ast$" as follows: when $a \geqslant b$, $a \ast b = ab + b$; when $a \leqslant b$, $a \ast b = ab - a$. If $(2x-1) \ast (x+2) = 0$, then $x=$ ______. | -1, \frac{1}{2} | 76 | 10 |
math | Factorize $a^2-2a-15$ using the cross-multiplication method. | (a+3)(a-5) | 21 | 8 |
math | Given circle $O: x^2+y^2=r^2(r>0)$, $A(x_1, y_1)$, $B(x_2, y_2)$ are two points on circle $O$, satisfying $x_1+y_1=x_2+y_2=3$, $x_1x_2+y_1y_2=-\frac{1}{2}r^2$, calculate the value of $r$. | 3\sqrt{2} | 97 | 6 |
math | The equation $x^2+y^2+x+2my+m=0$ represents a circle. Find the range of values for $m$. | m \neq \frac{1}{2} | 30 | 11 |
math | Given the four propositions, determine the number of incorrect propositions. | 1 | 12 | 1 |
math | Given the function $f(x)=x+\sin \pi x-3$, calculate the value of $f\left( \dfrac {1}{2015}\right)+f\left( \dfrac {2}{2015}\right)+f\left( \dfrac {3}{2015}\right)+\ldots+f\left( \dfrac {4029}{2015}\right)$. | -8058 | 96 | 5 |
math | $\frac{M}{5} = \frac{5}{N}$ and $M=2K$ for some integer $K$, where $M$ and $N$ are positive integers. Find the number of ordered pairs of $(M, N)$. | \textbf{(A)}\ 0 | 53 | 9 |
math | The kindergarten received flashcards for reading: some say "MA", and the others say "NYA". Each child took three cards to form words. It turned out that 20 children can form the word "MAMA" from their cards, 30 children can form the word "NYANYA", and 40 children can form the word "MANYA". How many children have all th... | 10 | 87 | 2 |
math | For all integers $n$ greater than 1, define $a_n = \dfrac{1}{\log_n 1001}$. Let $b = a_2 + a_3 + a_6 + a_7$ and $c = a_{15} + a_{16} + a_{17} + a_{18} + a_{19}$. Find $b - c$. | -1 | 92 | 2 |
math | In triangle $ABC$, $BC = 30 \sqrt{2}$ and $\angle C = 45^\circ$. Let the perpendicular bisector of $BC$ intersect $BC$ and $AC$ at $D$ and $E$, respectively. Find the length of $DE$. | 15 | 61 | 2 |
math | Convert the decimal number 34 to binary. | 100010_2 | 10 | 8 |
math | Given a moving circle $P$ and a fixed circle $C$: $(x+2)^{2}+y^{2}=1$ that are externally tangent to each other, and also tangent to a fixed line $L$: $x=1$, determine the equation of the trajectory of the center $P$ of the moving circle. | y^{2}=-8x | 69 | 7 |
math | Suppose $c$ is inversely proportional to $d$ and $c \cdot d = k$ for a constant $k$. Given $c_1$ and $c_2$ are two values of $c$ such that $\frac{c_1}{c_2} = \frac{4}{5}$, and corresponding values $d_1$ and $d_2$ for $d$, find the ratio $\frac{d_1}{d_2}$. Also, if $c_3 = 2c_1$, find the corresponding value of $d_3$. | \frac{d_1}{2} | 126 | 9 |
math | Given the quadratic function $f(x)=x^{2}+ax+b$ satisfies $f(0)=6$, $f(1)=5$
$(1)$ Find the expression for the function $f(x)$
$(2)$ Find the maximum and minimum values of the function $f(x)$ when $x \in [-2,2]$. | 14 | 72 | 2 |
math | Find the maximum real number \(\lambda\) such that for the real-coefficient polynomial
$$
f(x) = x^3 + ax^2 + bx + c
$$
with all roots being non-negative real numbers, the inequality
$$
f(x) \geqslant \lambda(x - a)^3 \quad \text{for all} \; x \geqslant 0
$$
holds. Also, determine when equality holds in this inequality... | -\frac{1}{27} | 100 | 8 |
math | Given the curve $C_1:y^2=tx$ $(y>0,t>0)$, the tangent line at the point $M\left(\frac{4}{t},2\right)$ is also tangent to the curve $C_2:y=e^{x+l}-1$. Find the value of $t$. | 4e | 68 | 2 |
math | Given two points $A(-3,-4)$ and $B(6,3)$, the distance from these points to the line $l: ax+y+1=0$ are equal. Find the value of the real number $a$. | a = -\frac{7}{9} \text{ or } a = -\frac{1}{3} | 50 | 25 |
math | Given a matrix $M$ with an eigenvalue $\lambda_{1}=8$ and its corresponding eigenvector $a_{1}=[1, 1]$, and the transformation corresponding to the matrix $M$ transforms the point $(-1,2)$ into $(-2,4)$
(I) Find the matrix $M$;
(II) If line $l$ is transformed into line $l′$: $x-2y=4$ under the linear transformation cor... | x+3y+2=0 | 113 | 8 |
math | Find all \( n \in\{1,2, \ldots, 999\} \) such that \( n^{2} \) is equal to the cube of the sum of the digits of \( n \). | 1 \text{ and } 27 | 49 | 9 |
math | Two circles of radius 12 cm overlap such that the distance between their centers is 16 cm. How long, in cm, is the common chord of the two circles? | 8\sqrt{5} \text{ cm} | 37 | 11 |
math | Suppose $x$ is an integer that satisfies the following congruences:
\begin{align*}
2+x &\equiv 3^2 \pmod{2^4}, \\
3+x &\equiv 2^3 \pmod{3^4}, \\
4+x &\equiv 3^3 \pmod{2^3}.
\end{align*}
What is the remainder when $x$ is divided by $24$? | 23 | 97 | 2 |
math | Transform the equation $x(x+2)=5$ into the general form $ax^{2}+bx+c=0$ and identify the values of $a$, $b$, and $c$. | 1, 2, -5 | 41 | 7 |
math | If five boys and three girls are randomly divided into two four-person teams, calculate the probability that all three girls will end up on the same team. | \frac{1}{7} | 30 | 7 |
math | It is known that in 3 out of 250 cases, twins are born, and in one out of three of these cases, the twins are identical (monozygotic) twins. What is the a priori probability that a particular pregnant woman will give birth to twins - a boy and a girl? | \frac{1}{250} | 65 | 9 |
math | Given \( M=\{(x, y) \mid x^{2}+2y^{2}=3\} \) and \( N=\{(x, y) \mid y=mx+b\} \). For all \( m \in \mathbb{R} \),... | b \in \left[-\frac{\sqrt{6}}{2}, \frac{\sqrt{6}}{2}\right] | 60 | 28 |
math | 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 | 51 | 3 |
math | Given $y=\log_a(2-ax)$ is a decreasing function of $x$ in the interval $(0,1)$, find the range of values for $a$. | (1,2] | 37 | 5 |
math | The coefficients of the polynomial
\[a_{10} x^{10} + a_9 x^9 + a_8 x^8 + \dots + a_2 x^2 + a_1 x + a_0 = 0\]are all integers, and its roots $r_1,$ $r_2,$ $\dots,$ $r_{10}$ are all integers. Furthermore, the roots of the polynomial
\[a_0 x^{10} + a_1 x^9 + a_2 x^8 + \dots + a_8 x^2 + a_9 x + a_{10} = 0\]are also $r_1,$... | 11 | 304 | 2 |
math | In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $bcosB$ is the arithmetic mean of $acosC$ and $ccosA$. Then, angle $B$ equals \_\_\_\_\_\_. | \frac{\pi}{3} | 68 | 7 |
math | The sum of the first $n$ terms of the sequence $\{a_{n}\}$ is $S_{n}$, $a_{1}=1$, and the point $(S_{n}, a_{n+1})$ lies on the line $y=2x+1$. <br/>$(1)$ Find the general formula for $\{a_{n}\}$; <br/>$(2)$ For an arithmetic sequence $\{b_{n}\}$ with positive terms, the sum of the first $n$ terms is $T_{n}$, and $T_{3}=... | T_{n}=n^2+2n | 171 | 10 |
math | Find the derivative.
$$
y=e^{a x}\left(\frac{1}{2 a}+\frac{a \cdot \cos 2 b x+2 b \cdot \sin 2 b x}{2\left(a^{2}+4 b^{2}\right)}\right)
$$ | e^{ax} \cos^2(bx) | 64 | 11 |
math | Point $P_{}$ is located inside triangle $ABC$ so that angles $PAB, PBC,$ and $PCA$ are all congruent. The sides of the triangle have lengths $AB=13, BC=14,$ and $CA=15,$ and the tangent of angle $PAB$ is $m/n,$ where $m_{}$ and $n_{}$ are relatively prime positive integers. Find $m+n.$ | 463 | 94 | 3 |
math | In the space rectangular coordinate system Oxyz, the coordinates of the point symmetric to point A (-3, -4, 5) with respect to the plane xOz are given by the formula (x, y, -z), where (x, y) are the coordinates of point A. However, since the x-coordinate of the symmetric point is x, not -x, the correct formula should b... | (-3, 4, 5) | 191 | 9 |
math | Four distinct vertices of a tetrahedron are chosen at random. Find the probability that the plane determined by these three vertices contains points inside the tetrahedron. | 1 | 34 | 1 |
math | When $x \in (0, +\infty)$, the power function $y=(m^2-m-1) \cdot x^{-5m-3}$ is a decreasing function, then the value of the real number $m$ is | 2 | 52 | 1 |
math | Given a cyclic quadrilateral $ABCD$ with $\angle BAD = 80^\circ$ and $\angle ADC = 58^\circ$. The side $AB$ is extended through $B$ to a point $E$. Determine $\angle EBC$.
A) $56^\circ$
B) $58^\circ$
C) $60^\circ$
D) $62^\circ$
E) $64^\circ$ | 58^\circ | 99 | 4 |
math | Given the condition $|a-b|=|a|+|b|$, determine the relationship between the real numbers a and b. | ab\leq0 | 27 | 5 |
math | Given the function $f(x) = x \ln x + (1 - k) x + k$, where $k \in \mathbb{R}$.
1. When $k = 1$, find the intervals where the function $f(x)$ is monotonically increasing or decreasing.
2. When $x > 1$, find the maximum integer value of $k$ such that the inequality $f(x) > 0$ always holds. | k = 3 | 94 | 4 |
math | A class has a group of 7 students, and now select 3 of them to swap seats with each other, while the remaining 4 students keep their seats unchanged. Calculate the number of different ways to adjust their seats. | 70 | 46 | 2 |
math | In the Cartesian coordinate system, the equation of circle $C_{1}$ is $(x-1)^{2}+y^{2}=9$, and the equation of circle $C_{2}$ is $(x+1)^{2}+y^{2}=1$. The moving circle $C$ is internally tangent to circle $C_{1}$ and externally tangent to circle $C_{2}$.
$(1)$ Find the equation of the trajectory $E$ of the center $C$ ... | 6 | 177 | 1 |
math | Given $f(x)$ is an odd function defined on $\mathbb{R}$, when $x > 0$, $(x^2+1)f'(x)+2xf(x)<0$, and $f(-1)=0$. Determine the solution set of the inequality $f(x) > 0$. | (-\infty, -1) \cup (0, 1) | 64 | 16 |
math | Given the imaginary unit $i$, if a complex number $z$ satisfies $z(1+i)^2$, find the imaginary part of the complex number. | 2a | 32 | 2 |
math | Sheila has been invited to a picnic tomorrow, which will happen regardless of the weather. She also has another event to attend. If it rains, there is a $30\%$ probability that Sheila will decide to go to the picnic, but if it is sunny, there is a $70\%$ probability that she will go to the picnic. The forecast for tomo... | 0.5 = 50\% | 125 | 9 |
math | If we divide number $19250$ with one number, we get remainder $11$ . If we divide number $20302$ with the same number, we get the reamainder $3$ . Which number is that? | 53 | 61 | 2 |
math | In the plane rectangular coordinate system $xOy$, the parametric equations of the line $l$ are $\left\{\begin{array}{l}x=1+\frac{1}{2}t\\ y=\frac{\sqrt{3}}{2}t\end{array}\right.$ (where $t$ is a parameter). Taking $O$ as the pole and the positive half-axis of the $x$-axis as the polar axis, the polar coordinate equatio... | \frac{\sqrt{7}}{4} | 213 | 10 |
math | On an $8 \times 8$ board, several dominoes (rectangles of $2 \times 1$) can be placed without overlapping. Let $N$ be the number of ways to place 32 dominoes in this manner, and $T$ be the number of ways to place 24 dominoes in this manner. Which is greater: $-N$ or $M$? Configurations that can be derived from one anot... | T | 107 | 1 |
math | Given: \\
- $p$: $|x+1| \leqslant 3$,
- $q$: $x^2 - 2x + 1 - m^2 \leqslant 0$,
- $m > 0$.
(I) If $m=2$, for the proposition "$p$ or $q$" to be true and "$p$ and $q$" to be false, find the range of real numbers $x$.
(II) If $p$ is a necessary but not sufficient condition for $q$, find the range of real numbers $m$. | (0,1] | 128 | 5 |
math | In $\triangle ABC$, $AB = 10$, $BC = 9$, $CA = 7.5$, and side $BC$ is extended to a point $P$ so that $\triangle PAB$ is similar to $\triangle PCA$. Find the length of $PC$. | 27 | 60 | 2 |
math | Find the number of triples $(x,y,z)$ of real numbers such that
\begin{align*}
x &= 3000 - 3001 \operatorname{sign}(y + z + 3), \\
y &= 3000 - 3001 \operatorname{sign}(x + z + 3), \\
z &= 3000 - 3001 \operatorname{sign}(x + y + 3).
\end{align*}
Note:
\[\operatorname{sign} (a) = \left\{
\begin{array}{cl}
1 & \text{if $a ... | 3 | 183 | 1 |
math | Given 5 people stand in a row, and there is exactly 1 person between person A and person B, determine the total number of possible arrangements. | 36 | 31 | 2 |
math | Given $a > 1$, determine the range of the eccentricity $e$ for the hyperbola $\dfrac{x^2}{a^2} - \dfrac{y^2}{(a+1)^2} = 1$. | (\sqrt{2}, \sqrt{5}) | 53 | 10 |
math | Given the parametric equation of curve $C_{1}$ as $\begin{cases} x=2t-1 \\ y=-4t-2 \end{cases}$ (where $t$ is the parameter), and establishing a polar coordinate system with the origin $O$ as the pole and the positive half-axis of $x$ as the polar axis, the polar equation of curve $C_{2}$ is $\rho= \frac {2}{1-\cos \th... | \frac {3 \sqrt {5}}{10} | 171 | 13 |
math | Consider the simplified form of the expression \[(x+y+z)^{2008}+(x-y-z)^{2008}.\] After expanding and combining like terms, how many terms are there in the simplified expression? | 1{,}010{,}025 | 49 | 13 |
math | Convert the expression $-\sin \alpha + \sqrt{3} \cos \alpha$ into the form $A \sin(\alpha + \varphi)$, where $A > 0$ and $\varphi \in (0, 2\pi)$. | 2\sin(\alpha + \frac{2\pi}{3}) | 56 | 15 |
math | The circumference of a circle $D$ is 72 feet. A chord $\widehat{EF}$ subtends an angle of $45^\circ$ at the center of the circle. Calculate the length of the arc $\widehat{EF}$ and the area of the sector formed by $\widehat{EF}$. | 9 \text{ feet} | 67 | 6 |
math | Given the function $f(x)= \frac{x^{2}}{1+x^{2}}$.
(1) Calculate the values of $f(2)+f( \frac{1}{2})$, $f(3)+f( \frac{1}{3})$, $f(4)+f( \frac{1}{4})$, and conjecture a general conclusion (proof is not required);
(2) Calculate the value of $2f(2)+2f(3)+…+2f(2017)+f( \frac{1}{2})+f( \frac{1}{3})+…f( \frac{1}{2017})+ \frac... | 4032 | 197 | 4 |
math | Two dice are rolled consecutively, and the numbers obtained are denoted as $a$ and $b$.
(Ⅰ) Find the probability that the point $(a, b)$ lies on the graph of the function $y=2^x$.
(Ⅱ) Using the values of $a$, $b$, and $4$ as the lengths of three line segments, find the probability that these three segments can form... | \frac{7}{18} | 96 | 8 |
math | Given that the points $(1, -2)$, $(3, 4)$, and $(6, m/3)$ are on the same straight line, find the value(s) of $m$.
**A)** $27$
**B)** $39$
**C)** $52$
**D)** $15$ | 39 | 71 | 2 |
math | Let $a$, $b$, $c$, $d$, and $e$ be positive integers with $a+b+c+d+e=2020$ and let $M$ be the largest of the sum $a+b$, $b+c$, $c+d$ and $d+e$. What is the smallest possible value of $M$? | 1010 | 74 | 4 |
math | How many 0.1s are there in 1.9? How many 0.01s are there in 0.8? | 80 | 31 | 2 |
math | Worker A and worker B produce the same type of component. Worker A produces 8 more components per day than worker B. It takes the same number of days for worker A to produce 600 components as it does for worker B to produce 400 components. If worker B produces x components per hour, the equation can be written as _____... | \frac{600}{x + 8} = \frac{400}{x} | 73 | 22 |
math | Find the range of $x$ in $(0, 2\pi)$ such that $\sin x > |\cos x|$. | \left( \frac{\pi}{4}, \frac{3\pi}{4} \right) | 27 | 22 |
math | Given $f(x)=ax-\frac{a}{x}-5\ln x, g(x)=x^{2}-mx+4$,
(1) If $x=2$ is the extreme point of function $f(x)$, find the value of $a$;
(2) When $a=2$, if $\exists x_{1}\in(0,1)$, $\forall x_{2}\in[1,2]$, such that $f(x_{1})\geqslant g(x_{2})$ holds, find the range of the real number $m$. | [8-5\ln 2,+\infty) | 125 | 13 |
math | A fifth number, $n$, is added to the set $\{ 3,6,9,10 \}$ to make the mean of the set of five numbers equal to its median. The number of possible values of $n$ is | 3 | 50 | 1 |
math | Betty has two containers. Initially, the first container is $\tfrac{3}{5}$ full of water and the second container is empty. She pours all the water from the first container into the second container, and finds that the second container is now $\tfrac{2}{3}$ full. What is the ratio of the volume of the first container t... | \frac{10}{9} | 139 | 8 |
math | Solve
\[\arctan \frac{1}{x} + \arctan \frac{1}{x^3} = \frac{\pi}{4}.\] | \frac{1 + \sqrt{5}}{2} | 39 | 13 |
math | Read the following material and answer the questions. For any three-digit number $n$, if $n$ satisfies that the digits in each place are all different and not zero, then this number is called a "special number". By swapping any two digits of a "special number", we can obtain three different new three-digit numbers. Let... | -7 | 338 | 2 |
math | A straight line joins the points $(-3,3)$ and $(2,10)$. Its $x$-intercept is:
A) $-\frac{36}{7}$
B) $-\frac{7}{36}$
C) $\frac{5}{8}$
D) $2$
E) $5$ | -\frac{36}{7} | 74 | 8 |
math | Let $n \ge 2$ be a fixed integer. [list=a] [*]Determine the largest positive integer $m$ (in terms of $n$ ) such that there exist complex numbers $r_1$ , $\dots$ , $r_n$ , not all zero, for which \[ \prod_{k=1}^n (r_k+1) = \prod_{k=1}^n (r_k^2+1) = \dots = \prod_{k=1}^n (r_k^m+1) = 1. \] [*]For this value of $... | \{2^n\} | 183 | 6 |
math | 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 | 82 | 7 |
math | In $\triangle ABC$, the lengths of the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $\overrightarrow {BA} \cdot \overrightarrow {AC} = 6$, $b-c=2$, and $\tan A = -\sqrt {15}$, find the length of the altitude drawn from $A$ to side $BC$. | \frac{3\sqrt{15}}{4} | 91 | 13 |
math | Given that $F_1$ and $F_2$ are the left and right foci of the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1(a>b>0)$, and the focal length of the ellipse is $2c(c > 0)$, $A(0,-\sqrt{3}c)$, if the midpoint $M$ of the line segment $AF_1$ lies on the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, determine the eccentric... | \sqrt{3} - 1 | 137 | 8 |
math | Given $$\overrightarrow {a} = (x-1, y)$$, $$\overrightarrow {b} = (x+1, y)$$, and $$|\overrightarrow {a}| + |\overrightarrow {b}| = 4$$
(1) Find the equation of the trajectory C of point M(x, y).
(2) Let P be a moving point on curve C, and F<sub>1</sub>(-1, 0), F<sub>2</sub>(1, 0), find the maximum and minimum valu... | \frac {2 \sqrt {21}}{7} | 200 | 13 |
math | On the faces of a six-sided die are the numbers $6, 7, 8, 9, 10, 11$. The die is rolled twice. The first time, the sum of the numbers on the four "vertical" faces (i.e., excluding the bottom and top faces) was 33, and the second time it was 35. What number could be on the face opposite the face with the number 7? Find ... | 9 \text{ or } 11 | 100 | 9 |
math | A food factory regularly purchases flour. It is known that the factory needs 6 tons of flour per day, the price of each ton of flour is 1800 yuan, and the storage and other costs for flour are an average of 3 yuan per ton per day. Each time flour is purchased, a shipping fee of 900 yuan is required. How often should th... | 10 | 90 | 2 |
math | Given the geometric sequence $\{a_n\}$, $a_1=2$, $a_8=4$. Let $f(x)=x(x-a_1)(x-a_2)...(x-a_8)$. Find $f'(0)$. | 2^{12} | 55 | 5 |
math | In a crown, five diamonds are embedded at five positions, equally spaced around a circle. If there are three different colors of diamonds to choose from, how many different ways are there to embed the diamonds? Assume that the five positions are indistinguishable from each other. | 51 | 54 | 2 |
math | Let $a,$ $b,$ and $c$ be positive real numbers. Find the minimum value of
\[\frac{(a^2 + 4a + 2)(b^2 + 4b + 2)(c^2 + 4c + 2)}{abc}.\] | 216 | 64 | 3 |
math | Given the formula $\cos 3\theta =4\cos ^{3}\theta -3\cos \theta $, $\theta \in R$, with the help of this formula, we can find the range of the function $f\left(x\right)=4x^{3}-3x-2(x\in [0$,$\frac{\sqrt{3}}{2}])$. Then the range of this function is ____. | [-3, -2] | 92 | 6 |
math | In the rectangular parallelepiped shown, $AB = 3$, $BC = 1$, and $CG = 2$. Point $M$ is the midpoint of $\overline{FG}$. What is the volume of the rectangular pyramid with base $BCHE$ and apex $M$? | \frac{4}{3} | 62 | 7 |
math | Two circles with radius 1 intersect at points $X$ and $Y$, with a distance of 1 between them. From point $C$ on one circle, tangents $CA$ and $CB$ are drawn to the other circle. Line $CB$ intersects the first circle again at point $A'$. Find the distance $AA'$. | \sqrt{3} | 72 | 5 |
math | At 7:25, calculate the degree measure of the smaller angle formed by the hands of a clock. | 72.5^\circ | 23 | 6 |
math | Find all positive integers $n$ which satisfy the following tow conditions:
(a) $n$ has at least four different positive divisors;
(b) for any divisors $a$ and $b$ of $n$ satisfying $1<a<b<n$ , the number $b-a$ divides $n$ .
*(4th Middle European Mathematical Olympiad, Individual Competition, Problem 4)* | 6, 8, 12 | 94 | 8 |
math | Let $r$ be the speed in miles per hour at which a wheel, $11$ feet in circumference, travels. If the time for a complete rotation of the wheel is shortened by $\frac{1}{4}$ of a second, the speed $r$ is increased by $5$ miles per hour. Then $r$ is: | 10 | 71 | 2 |
math | Distinct lines \(\ell\) and \(m\) lie in the xy-plane. They intersect at the origin. Point \(Q(3, -2)\) is first reflected about line \(m\), then the resulting point \(Q'\) is reflected about line \(\ell\) to point \(Q''\). The equation of line \(\ell\) is \(2x - 5y = 0\), and the coordinates of \(Q''\) are \((-2, 3)\)... | 5x + 2y = 0 | 111 | 9 |
math | Given that $f(x) = \begin{cases} \frac{a}{x}, & x \geqslant 1 \\ -x + 3a, & x < 1 \end{cases}$ is a monotonic function on $\mathbb{R}$, find the range of values for the real number $a$. | [\frac{1}{2}, +\infty) | 71 | 12 |
math | There are two types of containers: 27 kg and 65 kg. How many containers of the first and second types were there in total, if the load in the containers of the first type exceeds the load of the container of the second type by 34 kg, and the number of 65 kg containers does not exceed 44 units? | 66 | 74 | 2 |
math | A traffic light runs through a repeated cycle: green for 45 seconds, then yellow for 5 seconds, and then red for 40 seconds. Joseph picks a random five-second time interval to watch the light. What is the probability that the color changes while he is watching? | \frac{1}{6} | 58 | 7 |
math | The average of Amy's, Ben's, and Chris's ages is 12. Six years ago, Chris was the same age as Amy is now. In 3 years, Ben's age will be $\frac{3}{4}$ of Amy's age at that time. How old is Chris now? | 17 | 63 | 2 |
math | If a series of functions have the same analytic expression and range but different domains, they are referred to as "twin functions." Therefore, how many twin functions are there whose expression is $y=2x^2+1$ and range is $\{5,19\}$? | 9 | 59 | 1 |
math | Find all integer sequences of the form $ x_i, 1 \le i \le 1997$ , that satisfy $ \sum_{k\equal{}1}^{1997} 2^{k\minus{}1} x_{k}^{1997}\equal{}1996\prod_{k\equal{}1}^{1997}x_k$ . | x_i = 0 | 89 | 6 |
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