task_type stringclasses 4
values | problem stringlengths 14 5.23k | solution stringlengths 1 8.29k | problem_tokens int64 9 1.02k | solution_tokens int64 1 1.98k |
|---|---|---|---|---|
math | Given $\lfloor x \rfloor$ be the greatest integer less than or equal to $x$, determine the number of real solutions to $4x^2-40\lfloor x \rfloor +51=0$. | 4 | 49 | 1 |
math | If $x$ satisfies $\left(x-2\right)^{x+1}=1$, then the value of the integer $x$ is ______. | -1, 3, 1 | 32 | 8 |
math | Palindromic primes are two-digit prime numbers such that the number formed when the digits are reversed is also prime. What is the sum of all palindromic primes less than 50? | 109 | 41 | 3 |
math | Given the function $f(x)= \frac {2x^{2}}{e}+ \frac {e^{2}}{x}$, $g(x)=3e\ln x$, where $e$ is the base of the natural logarithm.
- (I) Discuss the monotonicity of the function $f(x)$.
- (II) Determine whether the curve $y=f(x)$ and $y=g(x)$ have a common point and a common tangent line at that point. If such a point exi... | y=3x | 126 | 4 |
math | In a fuel tank, there are 40 liters of oil. The oil flows out of the tank uniformly at a rate of 0.2 liters per minute. Determine the function relationship between the remaining oil quantity Q (in liters) in the tank and the time t (in minutes) of outflow. | Q=40-0.2t | 63 | 9 |
math | There are lindens and birches planted around the house, with their total quantity being more than 14. If the number of lindens is doubled and the number of birches is increased by 18, there will be more birches than lindens. If the number of birches is doubled without changing the number of lindens, there will now be m... | 11 \text{ limes and } 5 \text{ birch trees} | 99 | 18 |
math | Given an acute triangle $ABC$ with sides $a$, $b$, $c$ opposite to angles $A$, $B$, $C$ respectively. Choose any one of the three conditions below to solve the following problems (if multiple conditions are used, follow the first solution):
$(1)$ Find angle $B$;
$(2)$ If $b=2$ and the area of triangle $ABC$ is $\sqrt{3... | 2 | 98 | 1 |
math | In rectangle $ABCD$, $AB = 10$ cm, $BC = 5$ cm, and $DG = DH$. The area of triangle $DGH$ is one-fifth the area of rectangle $ABCD$. Determine the length in centimeters of segment $GH$. Express your answer in simplest radical form.
[asy]
draw((0,0)--(0,50)--(100,50)--(100,0)--cycle);
draw((40,50)--(100,20));
label("$A... | GH = 2\sqrt{10} \text{ cm} | 196 | 15 |
math | Find \( n \) if \( n\cdot (n+1)! + (n+1)! = 5040 \), where \( n! = n\cdot (n-1)\cdot (n-2)\cdots 2\cdot 1 \). | 5 | 58 | 1 |
math | Given that Sarah and Jill start a swimming race from opposite ends of a 50-meter pool, and they cross paths two minutes after they start, determine the time it takes for them to cross paths for the second time. | 6 | 45 | 1 |
math | Quadrilateral $ABCD$ has $AB = BC = CD$, $m\angle ABC = 70^\circ$ and $m\angle BCD = 170^\circ$. What is the degree measure of $\angle BAD$? | 85 | 53 | 2 |
math | There is a cube of size \(10 \times 10 \times 10\) made up of small unit cubes. A grasshopper is sitting at the center \(O\) of one of the corner cubes. It can jump to the center of a cube that shares a face with the one in which the grasshopper is currently located, provided that the distance to point \(O\) increases.... | \frac{27!}{(9!)^3} | 101 | 13 |
math | If $\cos(α + \frac{π}{3}) = -\frac{\sqrt{3}}{3}$, find the value of $\sin α$. | \frac{\sqrt{6} + 3}{6} | 34 | 13 |
math | Find the coefficient of the $x^2$ term in the expansion of $(2x^3 + 5x^2 - 3x + 1)(3x^2 - 9x - 5)$. | 5 | 47 | 1 |
math | Suppose the line $l$: $\begin{cases} x=1+\frac{1}{2}t \\ y=\frac{\sqrt{3}}{2}t \end{cases} (t \text{ is the parameter})$, and the curve $C_{1}$: $\begin{cases} x=\cos \theta \\ y=\sin \theta \end{cases} (\theta \text{ is the parameter})$.
(1) Let $l$ intersect $C_{1}$ at two points $A$ and $B$, find $|AB|$.
(2) If th... | \frac{\sqrt{6}}{4}(\sqrt{2}-1) | 214 | 17 |
math | Given the function $f(x)=a^2x^2-2ax+1$, if the proposition "$\forall x \in (0,1)$, $f(x) \neq 0$" is a false proposition, then the range of the real number $a$ is __________. | a > 1 | 63 | 4 |
math | The tangent line to the curve $y=x^3+x-2$ at point P is parallel to the line $y=4x-1$. Find the equation of this tangent line. | y=4x-4 \text{ or } y=4x | 39 | 15 |
math | A batch of parts is divided into three grades, with 24 first-grade parts and 36 second-grade parts. Using stratified sampling to draw a sample of size 20, if exactly 10 third-grade parts are drawn, then the number of third-grade parts in this batch is $\_\_\_\_\_\_\_$, and the number of second-grade parts drawn in the ... | 6 | 95 | 1 |
math | Find all real solutions to the system of equations
$$
\begin{aligned}
& x^{2}+y^{2}+z^{2}=1, \\
& x^{3}+y^{3}+z^{3}=1 .
\end{aligned}
$$ | (1, 0, 0), (0, 1, 0), (0, 0, 1) | 58 | 27 |
math | Given the complex number $Z=\frac{-2+i}{i^{2018}}$, find the imaginary part of the conjugate of the complex number $Z$, denoted as $\overline{Z}$. | 1 | 45 | 1 |
math | Given the fraction $\frac{7}{125}$, convert it to a decimal. | 0.056 | 19 | 5 |
math | A chord of the parabola $y^2=4x$ intersects the parabola at points A($x_1$, $y_1$) and B($x_2$, $y_2$). If the length of the chord AB is 7, find the distance from the midpoint M of the chord to the parabola's directrix. | \frac{7}{2} | 77 | 7 |
math | What is the angle opposite side \( c \) in a triangle with sides \( a, b, c \), perimeter \( 2s \), and area \( T \), if
$$
T+\frac{a b}{2}=s(s-c) ?
$$ | 45^\circ | 55 | 4 |
math | Compute $\arccos (\sin 3).$ All functions are in radians. | 3 - \frac{\pi}{2} | 17 | 9 |
math | In the sequence $\{a_{n}\}$, $a_{1}=18$, $a_{2}=24$, $a_{n+2}-a_{n}=-6$.
$(1)$ Find the general formula for $\{a_{n}\}$;
$(2)$ Let the sum of the first $n$ terms of the sequence $\{a_{n}\}$ be $S_{n}$, find the maximum value of $S_{n}$. | 96 | 100 | 2 |
math | A bag contains 4 blue, 3 green, and 5 red chips. If the 12 chips are randomly drawn from the bag, one at a time and without replacement, what is the probability that the chips are drawn in such a way that the 4 blue chips are drawn consecutively, the 3 green chips are drawn consecutively, and the 5 red chips are drawn ... | \frac{1}{4620} | 104 | 10 |
math | Rotate the parabola $y=3x^2-6x+5$ around its vertex by 180°, then translate it along the axis of symmetry to obtain a new parabola that intersects the line $y=-x-2$ at the point $(2, m)$. The equation of the new parabola is ____. | y=-3x^2+6x-4 | 74 | 11 |
math | The quadrilateral $ABCD$ has the following equality $\angle ABC=\angle BCD=150^{\circ}$. Moreover, $AB=18$ and $BC=24$, the equilateral triangles $\triangle APB,\triangle BQC,\triangle CRD$ are drawn outside the quadrilateral. If $P(X)$ is the perimeter of the polygon $X$, then the following equality is true $P(APQRD)=... | 10 | 109 | 2 |
math | A right frustum oil tank can hold 190 liters of kerosene. Given that the lengths of its top and bottom edges are 60cm and 40cm respectively, its depth is \_\_\_\_\_\_\_\_. | 75\text{cm} | 52 | 7 |
math | Determine the angle of inclination, in terms of inverse trigonometric functions, for the line given by the equation $2x+y-1=0$. | \alpha = \pi - \arctan(2) | 32 | 13 |
math | Given a sequence $\{a_n\}$ whose sum of the first $n$ terms is $S_n$, $a_1=\frac{1}{2}$, and $2a_{n+1}=S_n+1$.
(Ⅰ) Find the values of $a_2$ and $a_3$;
(Ⅱ) Let $b_n=2a_n-2n-1$, find the sum of the first $n$ terms of the sequence $\{b_n\}$, denoted as $T_n$. | 2\cdot\left(\frac{3}{2}\right)^{n}-n^2-2n-2 | 115 | 25 |
math | Evaluate \[\frac{3}{\log_3{1000^4}} + \frac{4}{\log_7{1000^4}}\], giving your answer as a fraction in lowest terms. | \log_{10} (3^{1/4} \cdot 7^{1/3}) | 49 | 22 |
math | Given an arithmetic sequence $\{a_n\}$ with a common difference $d > 0$, and $a_2$, $a_5-1$, $a_{10}$ form a geometric sequence. The first term of the sequence is $a_1=5$, and $S_n$ is the sum of the first $n$ terms of the sequence. Find the minimum value of $\frac{2S_n+n+32}{a_n+1}$. | \frac{20}{3} | 99 | 8 |
math | In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $c= \sqrt {3}$, and $(\sqrt {3}+a)(sinC-sinA)=(a+b)sinB$, find the maximum value of the perimeter of $\triangle ABC$. | 2+ \sqrt {3} | 78 | 7 |
math | Given that $m \angle A= 60^\circ$, $BC=12$ units, $\overline{BD} \perp \overline{AC}$, $\overline{CE} \perp \overline{AB}$ and $m \angle DBC = 3m \angle ECB$, the length of segment $EC$ can be expressed in the form $a(\sqrt{b}+\sqrt{c})$ units where $b$ and $c$ have no perfect-square factors. What is the value of $a+b... | 11 | 244 | 2 |
math | Given that the ceiling is 3.0 meters above the ground, Bob is 1.8 meters tall and can reach 50 centimeters above the top of his head, and the light fixture is 15 centimeters below the ceiling, calculate the height of the box in centimeters. | 55 | 61 | 2 |
math | Given $f(x)=\frac{[\sin(\frac{\pi}{2}-x)\tan(\pi-x)]^{2}-1}{4\sin(\frac{3\pi}{2}+x)+\cos(\pi-x)+\cos(2\pi-x)}$,
(1) Find $f(-1860^{\circ})$;
(2) If the equation $f^{2}(x)+(1+\frac{1}{2}a)\sin x+2a=0$ has two roots in $x\in[\frac{\pi}{6},\frac{3\pi}{4}]$, find the range of the real number $a$;
(3) Find the maximum value... | y_\text{max}=2 | 175 | 7 |
math | Given a function $f(x)$ defined on the domain $\mathbb{R}$ that satisfies $f(-x) + f(x) = 0$. If $f(x)$ is a decreasing function on $(0, +\infty)$ and $f(-2) = 0$, find the range of values for $x$ such that $(x-1)f(x) > 0$. | (-2,0)∪(1,2) | 82 | 11 |
math | Given the function $f(x+2) - f(x) = 2f(1)$, and the graph of $y=f(x-1)$ is symmetric about $x=1$, with $f(0)=2$, calculate $f(2015) + f(2016)$. | 2 | 66 | 1 |
math | A community plans to invest $500$ yuan to purchase three types of books, A, B, and C, with type A costing $30$ yuan per book, type B costing $25$ yuan per book, and type C costing $20$ yuan per book. Given that the community must buy at least $5$ books of type A and at most $6$ books of type A, determine the number of ... | 6 | 97 | 1 |
math | We place labeled points on a circle as follows. At step 1, take two points at opposite ends of a diameter and label them both 1. At step \( n > 1 \), place a point at the midpoint of each arc created at step \( n-1 \) and label it with the sum of the labels at the two adjacent points. What is the total sum of the label... | 2 \times 3^{n-1} | 87 | 10 |
math | Let $x,$ $y,$ and $z$ be angles such that
\begin{align*}
\sin x &= \cot y, \\
\sin y &= \cot z, \\
\sin z &= \cot x.
\end{align*}
Find the minimum possible value of $\cos x.$ | \sqrt{\frac{3 - \sqrt{5}}{2}} | 64 | 15 |
math | Given that the function $f(x)$ defined on $[0,+\infty)$ satisfies $f(x)=2f(x+2)$, and when $x\in[0,2)$, $f(x)=-2x^{2}+4x$. Let the maximum value of $f(x)$ on $[2n-2,2n)$ be $a_{n}(n\in\mathbb{N}^{*})$, calculate the sum of the first $n$ terms of $\{a_{n}\}$. | 4- \frac {1}{2^{n-2}} | 114 | 13 |
math | Let $(x,y,z)$ be an ordered triplet of real numbers that satisfies the following system of equations: \begin{align*}x+y^2+z^4&=0,y+z^2+x^4&=0,z+x^2+y^4&=0.\end{align*} If $m$ is the minimum possible value of $\lfloor x^3+y^3+z^3\rfloor$ , find the modulo $2007$ residue of $m$ . | 2004 | 113 | 4 |
math | The ratio $AC:CB$ is $3:4$, in $\triangle ABC$. The external angle bisector of $\angle C$ intersects the extension of $BA$ at $P$, where $A$ is between $P$ and $B$. Find the ratio $PA:AB$. | 3:1 | 60 | 3 |
math | Given a triangle $\Delta ABC$ where $CD$ is the altitude of length 1 centimeter and splits $\Delta ABC$ into two 45-45-90 triangles, find the area of $\Delta ABC$. | \sqrt{2} | 47 | 5 |
math | For the set $A=\{a_1, a_2, \ldots, a_n\}$ ($n \in \mathbb{N}^*$, $n \geq 3$), define the set $S=\{x | x=a_i+a_j, 1 \leq i < j \leq n\}$, and denote the number of elements in set $S$ as $S(A)$.
(1) If set $A=\{1, 2, 3, 4\}$, then $S(A)=$ \_\_\_\_\_\_ .
(2) If $a_1, a_2, \ldots, a_n$ form an arithmetic sequence with ... | 5, 2n-3 | 185 | 7 |
math | Given the 20 vertices of a regular 20-sided polygon inscribed in the unit circle on the complex plane, denoted as \( z_{1}, z_{2}, \cdots, z_{20} \), determine the number of distinct points corresponding to the complex numbers \( z_{1}^{2010.5}, z_{2}^{2005}, \cdots, z_{20}^{2005} \). | 4 | 99 | 1 |
math | For some positive integer \( n \), the number \( 210n^3 \) has \( 210 \) positive integer divisors, including \( 1 \) and the number \( 210n^3 \). How many positive integer divisors does the number \( 289n^5 \) have? | 108 | 73 | 3 |
math | Given the function $f(x)=x+\dfrac{a}{x}+a^{2}-2$ ($a\in \mathbb{R}$).
$(1)$ If $f(x)$ is an odd function, and it is increasing on the interval $(0,+\infty)$, find the value of $a$;
$(2)$ If the equation $|\log_{8}(x+1)|-a^{2}+f(1)=0$ has two different real roots $m$, $n$ in the interval $(-1,1)$, find the range of $a... | -1 | 148 | 2 |
math | If the algebraic expression $\frac{(x-1)^{0}}{\sqrt{x+2}}$ is meaningful, then the range of real number $x$ is ____. | x > -2 \quad \text{and} \quad x \neq 1 | 37 | 19 |
math | A rectangle \( P Q R S \) has side-lengths \( a \) and \( b \), with \( a < b \). The rectangle \( P T U V \) has side-lengths \( c \) and \( d \), with \( c < d \). Also, \( a < d \) and \( c < b \), as shown. The sides \( R S \) and \( T U \) cross at \( X \). Which of these conditions guarantees that \( Q, X \), and... | \frac{a}{d} + \frac{c}{b} = 1 | 217 | 18 |
math | The equation of circle $O$ is ${x}^{2}+{y}^{2}=9$, and $P$ is a moving point on circle $O$. If the perpendicular bisector of segment $OP$ is always covered by the plane region $|x|+|y|\geqslant m$, then the range of values for the real number $m$ is. | (-\infty, \dfrac{3}{2}] | 81 | 13 |
math | Let \(a\) and \(b\) be the roots of the equation \(x^2 - mx + 3 = 0.\) Suppose that \(a^2 + (1/b)\) and \(b^2 + (1/a)\) are the roots of the equation \(x^2 - px + r = 0.\) Determine the value of \(r.\) | \frac{46}{3} | 78 | 8 |
math | In \\(\triangle ABC\\), the sides opposite to angles \\(A\\), \\(B\\), and \\(C\\) are \\(a\\), \\(b\\), and \\(c\\) respectively, and it is given that \\(b\cos C = 3a\cos B - c\cos B\\).
\\((1)\\) Find the value of \\(\cos B\\);
\\((2)\\) If \\(\overrightarrow{BA} \cdot \overrightarrow{BC} = 2\\) and \\(b = 2\sqrt{2... | a = c = \sqrt{6} | 143 | 9 |
math | Let $S$ be a square one of whose sides has a length of $\sqrt{2}$. A point $v=(x,y)$ is chosen uniformly at random over all pairs of real numbers $x$ and $y$ such that $0 \le x \le 5000$ and $0 \le y \le 5000$. Let $T(v)$ be a translated copy of $S$ centered at $v$. What is the probability that the square region determ... | \frac{4}{625} | 178 | 9 |
math | Given two linear functions \( f(x) \) and \( g(x) \) such that the graphs \( y=f(x) \) and \( y=g(x) \) are parallel lines, but not parallel to the coordinate axes. Find the minimum value of the function \( (g(x))^2 - 3f(x) \), if the minimum value of the function \( (f(x))^2 - 3g(x) \) is \( \frac{11}{2} \). | -10 | 102 | 3 |
math | Given the circle $(x+1)^2+y^2=2$, determine the center and radius of the circle. | (-1,0), \sqrt{2} | 24 | 10 |
math | Automobile license plates for a region consist of four letters followed by a dash and three single digits. How many different license plate combinations are possible if exactly two letters are repeated once each, and digits increase consecutively? | 15,600 | 44 | 6 |
math | Given that line $l$ passes through point $P(2, 1)$ with an inclination angle of $135^\circ$, establish a polar coordinate system with the origin $O$ as the pole and the positive semi-axis of $x$ as the polar axis (the unit length is consistent with that of the rectangular coordinate system $xoy$). The equation of circl... | \sqrt{14} | 148 | 6 |
math | Given that the lines $y=k_{1}x$ and $y=k_{2}x(k_{1} > k_{2})$ are two tangent lines of the curve $y=ax+2\ln |x|$, calculate $k_{1}-k_{2}$. | \frac{4}{e} | 59 | 7 |
math | Let $p$, $q$, and $r$ be the roots of $x^3 - 2x^2 - x + 3 = 0$. Find $\frac{1}{p-2} + \frac{1}{q-2} + \frac{1}{r-2}$. | -3 | 65 | 2 |
math | Given the three vertices of triangle $\triangle ABC$ are $A(1,4)$, $B(-2,3)$, and $C(4,-5)$, find the equation of the circumscribed circle, the coordinates of the circumcenter, and the radius of the circumscribed circle. | 5 | 63 | 1 |
math | If each of the variables represents a different digit, what is the value of $a+b+c+d$?
[asy]
label("$a$",(1,0),E);
label("$b$",(2,0),E);
label("$c$",(3,0),E);
label("$d$",(1,-1),E);
label("$c$",(2,-1),E);
label("$a$",(3,-1),E);
label("+",(-2,-1),E);
draw((-2.1,-1.4)--(4.1,-1.4),linewidth(0.5));
label("1",(0,-2),E);
fo... | 18 | 166 | 2 |
math | Given sets $A=\{x|1\leqslant x \lt 5\}$, $B=\{x|-a \lt x\leqslant a+3\}$. If $B\subseteq \left(A\cap B\right)$, determine the range of the real number $a$. | (-\infty, -1] | 68 | 8 |
math | Given $f(x) = x^2 - 2017x + 8052 + |x^2 - 2017x + 8052|$, then find the value of $f(1) + f(2) + f(3) + \ldots + f(2013)$. | 0 | 75 | 1 |
math | Given that $x > 2$, $y > 0$ and they satisfy $2^{x \cdot 2^{y}} = 16$, determine the minimum value of $\frac {2}{x-2} + \frac {2}{y}$. | 4 | 56 | 1 |
math | Find the expected value and the variance of a random variable \(X\), which is uniformly distributed in the interval \([a ; b]\). | \mathrm{Var}(X) = \frac{(b - a)^2}{12} | 29 | 20 |
math | Given that the line $l_{1}$: $x-3y+2=0$ is symmetric to the line $l_{2}$: $mx-y+b=0$ with respect to the $x$-axis, calculate the value of $m+b$. | -1 | 56 | 2 |
math | Find the matrix \(\mathbf{N}\) with real entries such that:
\[
\mathbf{N}^2 - 3\mathbf{N} + 2\mathbf{N} = \begin{pmatrix} 6 & 12 \\ 3 & 6 \end{pmatrix}.
\] | \begin{pmatrix} 2 & 4 \\ 1 & 2 \end{pmatrix} | 72 | 23 |
math | Each of the eight letters in "GEOMETRY" is written on its own square tile and placed in a bag. What is the probability that a tile randomly selected from the bag will have a letter on it that is in the word "ANGLE"? Express your answer as a common fraction. | \frac{1}{4} | 57 | 7 |
math | The inscribed circle of triangle $ABC$ is tangent to $\overline{AB}$ at $P,$ and its radius is $21$. Given that $AP=23$ and $PB=27,$ find the perimeter of the triangle. | 345 | 52 | 3 |
math | A line with a slope of $-1$ is drawn through the right vertex $A$ of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ ($a > 0, b > 0$). This line intersects the two asymptotes of the hyperbola at points $B$ and $C$. If $2\overrightarrow{AB} = \overrightarrow{BC}$, then the eccentricity of the hyperbola is \_\_\_\_... | \sqrt{5} | 117 | 5 |
math | If two numbers are randomly chosen without replacement from $\{2, 4, 6, 8\}$, what is the probability that their product will be a multiple of 8? Express your answer as a common fraction. | \frac{2}{3} | 47 | 7 |
math | Randomly select 10 students from classes A and B of a certain school and measure their heights (unit: cm). The height data obtained are as follows:
Class A: $170\ \ 179\ \ 162\ \ 168\ \ 158\ \ 182\ \ 179\ \ 168\ \ 163\ \ 171$
Class B: $159\ \ 173\ \ 179\ \ 178\ \ 162\ \ 181\ \ 176\ \ 168\ \ 170\ \ 165$
$(1)$ Calc... | 165\ \text{cm} | 191 | 9 |
math | 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 | 69 | 12 |
math | Given a number is called flippy if its digits alternate between two distinct digits from the set {4, 6}, calculate the number of four-digit flippy numbers that are divisible by 4. | 1 | 40 | 1 |
math | Given in $\triangle ABC$, $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ respectively, satisfying $(5a-3b)\cos C=3c\cdot\cos B$.
(Ⅰ) Find $\cos C$;
(Ⅱ) If $c=4$, when the area of triangle $ABC$ is maximized, find the values of $a$ and $b$. | 2\sqrt{5} | 95 | 6 |
math | Diane has one 1-cent stamp, two identical 2-cent stamps, three identical 3-cent stamps, four identical 4-cent stamps, and five identical 5-cent stamps. How many different arrangements can Diane paste exactly 15 cents worth of postage in a row across the top of an envelope, given that she must use at least one of each s... | 120 | 82 | 3 |
math | In the tetrahedron \(ABCD\), the ratios of the lengths are:
\[ BD : CD : AB : AC : AD : BC = \sqrt{3} : \sqrt{2} : 1 : 1 : 1 : 1 \]
Find the angle between \(AD\) and \(BC\). | 60^\circ | 68 | 4 |
math | A rectangle has a perimeter of 40 units, and one side is at least 3 units longer than the other side. Both dimensions are whole numbers. What is the maximum possible area of this rectangle in square units? | 91 | 45 | 2 |
math | If \((2 x+4)^{2 n}=a_{0}+a_{1} x+a_{2} x^{2}+\cdots+a_{2 n} x^{2 n}\) for \(n \in \mathbf{N}^{+}\), then the remainder when \(a_{2}+a_{4}+\cdots+a_{2 n}\) is divided by 3 is . | 1 | 89 | 1 |
math | Given the parabola $y^{2}=8x$ whose directrix intersects with the hyperbola $\dfrac {x^{2}}{a^{2}}- \dfrac {y^{2}}{b^{2}}=1$ ($a > 0,b > 0$) at points $A$ and $B$, and one of the asymptotes of the hyperbola is $y= \dfrac {4 \sqrt {3}}{3}x$, find the standard equation of the hyperbola. | \dfrac {x^{2}}{3}- \dfrac {y^{2}}{16}=1 | 112 | 23 |
math | Given that the value of $\pi = 3.1415926\ldots$, the largest integer not greater than $\pi^3$ is ____. The smallest integer not less than $\pi^3$ is ____. | 32 | 50 | 2 |
math | A single bench section at a school event can hold either $7$ adults or $11$ children. When $N$ bench sections are connected end to end, an equal number of adults and children seated together will occupy all the bench space. What is the least possible positive integer value of $N?$ | 77 | 62 | 2 |
math | A right circular cylinder with its diameter equal to its height is inscribed in a right circular cone. The cone has a diameter of 18 and an altitude of 20. The axes of the cylinder and the cone coincide. Find the radius of the cylinder. Express your answer as a common fraction. | \frac{90}{19} | 62 | 9 |
math | Let's say you deposit $X$ in a special savings account that initially increases your deposit by $r\%$ but then charges a management fee of $s\%$ on the new total. Find the condition on $r$ and $s$ such that the value of the deposit after these operations is still greater than $X$, given that $s < 20$.
A) $r < \frac{100... | r > \frac{100s}{100 - s} | 157 | 16 |
math | To the eight-digit number 20192020, prepend and append one digit each so that the resulting ten-digit number is divisible by 72. Indicate all possible solutions. | 2201920200 \text{ and } 3201920208 | 41 | 26 |
math | Let $A$ denote the set of all integers $n$ such that $1 \le n \le 10000$ , and moreover the sum of the decimal digits of $n$ is $2$ . Find the sum of the squares of the elements of $A$ . | 7294927 | 70 | 7 |
math | From this infinite list of numbers, how many are integers? $$\sqrt{6561},\sqrt[3]{6561},\sqrt[4]{6561},\sqrt[5]{6561},\sqrt[6]{6561},\ldots$$ | 3 | 65 | 1 |
math | Given that $f(x)$ is an odd function defined on $\mathbb{R}$, and $f(x+2)+f(x)=0$. When $x \in [0,1]$, $f(x)=2^x-1$. Find the value of $f(\log_{\frac{1}{8}}125)$. | \frac{1}{4} | 73 | 7 |
math | In the process of selecting test points using the 0.618 method, if the experimental interval is $[1000,2000]$, and the first three test points are $x_1, x_2, x_3$ (with $x_2 < x_1$); and if the result at $x_2$ is better than that at $x_1$, then $x_3$ equals? | 1236 | 95 | 4 |
math | A cylindrical soup can with a radius of 5 cm and a height of 10 cm is cut into a quarter wedge along its height by a planar cut through its axis. Calculate the volume of this quarter wedge. | 196.25 | 45 | 6 |
math | The negation of the proposition "For all $x$ in $[1, 2]$, $x^2 < 4$" is. | \exists x \in [1, 2], x^2 \geq 4 | 31 | 19 |
math | How many tetrahedrons can be formed by choosing four vertices from the six vertices of a triangular prism? | 12 | 22 | 2 |
math | The sides $a$, $b$, and $c$ of a triangle $\triangle ABC$ are opposite to angles $A$, $B$, and $C$, respectively. Given that $a+c=4$, and $\sin A (1+\cos B) = (2-\cos A)\sin B$, find the maximum area of $\triangle ABC$. | \sqrt{3} | 73 | 5 |
math | Let $Q$ be the set of rational numbers, and $Z$ be the set of integers. In the coordinate plane, for a positive integer $m$, define the set of points
$$
A_{m}=\left\{(x, y) \mid x, y \in \mathbf{Q}, x y \neq 0, \frac{x y}{m} \in \mathbf{Z}\right\}.
$$
For a line segment $MN$, define $f_{m}(MN)$ as the number of points... | \lambda = 2 | 215 | 5 |
math | Given that $P(A)=0.7, P(B)=0.2, P(C)=0.1$, calculate the probability of the event "the drawn product is not a first-class product". | 0.3 | 41 | 3 |
math | Given that points $F_1$ and $F_2$ are the left and right foci of the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ $(a > b > 0)$, respectively, a line passing through $F_1$ perpendicular to the $x$-axis intersects the ellipse at points $A$ and $B$. If $\triangle ABF_2$ is an acute triangle, determine the range of the... | (\sqrt{2}-1, 1) | 117 | 10 |
math | Selected Exercise $4-4$: Coordinate Systems and Parametric Equations
In a rectangular coordinate system, a polar coordinate system is established with the origin as the pole and the positive half of the $x$-axis as the polar axis. The polar equation of curve $C$ is given by $\rho \sin^{2}\theta = 2a\cos\theta, (a > 0)... | a=1 | 226 | 3 |
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