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 | Consider the following four propositions:
1. The negation of the statement "For all $x$ in $\mathbb{R}$, $\cos(x) > 0$" is "There exists an $x$ in $\mathbb{R}$ such that $\cos(x) \leq 0$";
2. If $\log_{10}(a) + \log_{10}(b) = \log_{10}(a+b)$, then the maximum value of $a+b$ is 4;
3. A defined odd function $f(x)$ on $\m... | 1, 3, 4 | 215 | 7 |
math | Determine the real values of $x$ such that the triangle with sides $5$ , $8$ , and $x$ is obtuse. | (3, \sqrt{39}) \cup (\sqrt{89}, 13) | 37 | 21 |
math | What is the smallest positive integer $n$ such that all the roots of $z^4 + z^3 + 1 = 0$ are $n^{\text{th}}$ roots of unity? | 5 | 44 | 1 |
math | Determine the perimeter, in cm, of quadrilateral \( ABCD \) if \( \overline{AB} \perp \overline{BC} \), \( \overline{DC} \perp \overline{BC} \), \( AB = 15 \) cm, \( DC = 7 \) cm, and \( BC = 10 \) cm. | 32 + \sqrt{164} \text{ cm} | 83 | 15 |
math | Determine the values of $p$ and $q$ if $18^3 = \frac{27^2}{3} \cdot 2^{9p} \cdot 3^{3q}$. | \frac{1}{3} | 46 | 7 |
math | An interesting math problem: A snail is in a well, 1.1 meters below the opening. Each day, the snail climbs up 40 cm during the daytime and slips down 20 cm at night. Eventually, the snail climbs out of the well. It took the snail ____ days to climb out. | 4 \text{ days} | 69 | 6 |
math | In the parallelepiped $ABCD-A_{1}B_{1}C_{1}D_{1}$, if $\overrightarrow{{A_1}M}=2\overrightarrow{MC}$ and $\overrightarrow{AM}=x\overrightarrow{AB}+y\overrightarrow{AD}+z\overrightarrow{A{A_1}}$, determine the values of the real numbers $x$, $y$, and $z$. | \frac{2}{3}, \frac{2}{3}, \frac{1}{3} | 95 | 21 |
math | In the complex plane, the vectors corresponding to the complex numbers $-3-i$ and $5+i$ are $\overrightarrow{OA}$ and $\overrightarrow{OB}$, respectively, where $O$ is the origin. Find the complex number corresponding to the vector $\overrightarrow{OA}+\overrightarrow{OB}$ and $\overrightarrow{BA}$, and the distance be... | 2\sqrt{17} | 87 | 7 |
math | Given functions $f(x)=-2x$ for $x<0$ and $g(x)=\frac{x}{\ln x}+x-2$. If $f(x_{1})=g(x_{2})$, find the minimum value of $x_{2}-2x_{1}$. | 4\sqrt{e}-2 | 64 | 7 |
math | Two identical polygons were cut out of cardboard, overlaid, and pinned together at a certain point. When one polygon is rotated around this "axis" by $25^{\circ} 30^{\prime}$, it coincides again with the second polygon. What is the minimum possible number of sides of such polygons? | 240 | 67 | 3 |
math | Find the number of real solutions to the equation
\[
\frac{x}{50} = \cos x.
\] | 31 | 26 | 2 |
math | Given that \( \cos A + \cos B + \cos C = \sin A + \sin B + \sin C = 0 \), find the value of \( \cos^4 A + \cos^4 B + \cos^4 C \). | \frac{9}{8} | 55 | 7 |
math | Let the function $f(x)=\ln (x+1)+a(x^{2}-x)$, where $a\in \mathbb{R}$,
(I) Discuss the number of extreme points of the function $f(x)$ and explain the reason;
(II) If $\forall x > 0$, $f(x)\geqslant 0$ holds, find the range of values for $a$. | [0,1] | 89 | 5 |
math | Given the proposition "There exists $x \in [1, 2]$ such that $x^2 + 2x + a \geq 0$" is true, find the range of values for $a$. | [-8, +\infty) | 46 | 8 |
math | Triangle \(P Q R\) has been divided into twenty-five congruent right-angled triangles, as shown. The length of \(R P\) is \(2.4 \, \text{cm}\). What is the length of \(P Q\)?
A) \(3 \, \text{cm}\)
B) \(3.2 \, \text{cm}\)
C) \(3.6 \, \text{cm}\)
D) \(4 \, \text{cm}\)
E) \(4.8 \, \text{cm}\) | 3 \, \text{cm} | 122 | 8 |
math | Given $\tan\alpha = -\frac{3}{2}$, where $\alpha$ is an angle in the second quadrant
$(1)$ Find the value of $\frac{\sin(-\alpha - \frac{\pi}{2})\cos(\frac{3}{2}\pi + \alpha)\tan(\pi - \alpha)}{\tan(-\alpha - \pi)\sin(-\pi - \alpha)}$;
$(2)$ Find the value of $\frac{1}{\cos\alpha\sqrt{1 + \tan^2\alpha}} + \sqrt{\fr... | 2 | 159 | 1 |
math | The original price of a bicycle is 760 yuan. After a 75% discount, the price of the bicycle is \_\_\_\_\_\_ yuan. | 570 | 36 | 3 |
math | Class 4(1) students participate in interest groups. More than half of the class joined the basketball training, the remaining half plus 2 joined the table tennis training, the remaining half plus 3 joined the chess group training, and the last 2 people joined the broadcasting group. Please calculate how many students a... | 50 | 72 | 2 |
math | The range of the function \( y = \arcsin[\sin x] + \arccos[\cos x] \) (for \( x \in [0, 2\pi) \) and where \([x]\) denotes the greatest integer less than or equal to \( x \)) is ____. | \left\{0, \frac{\pi}{2}, \pi\right\} | 66 | 19 |
math | A quadratic polynomial with real coefficients, $p(x)$, is termed as *mysterious* if $p(p(x))=0$ has exactly four real roots, including multiplicities. Find the polynomial $p(x)$ such that its leading coefficient is $1$, the sum of its roots is minimal, and calculate $p(0)$.
**A)** $2$
**B)** $3$
**C)** $4$
**D)** $5$ | 4 | 94 | 1 |
math | In a certain country, the airline system is arranged so that each city is connected by airlines to no more than three other cities, and from any city, it's possible to reach any other city with no more than one transfer. What is the maximum number of cities that can exist in this country? | 10 | 60 | 2 |
math | In the trapezoid \(ABCD\), if \(AB = 8\), \(DC = 10\), the area of \(\triangle AMD\) is 10, and the area of \(\triangle BCM\) is 15, then the area of trapezoid \(ABCD\) is \(\quad\). | 45 | 72 | 2 |
math | A club has between 150 and 250 members. Every month, all the members meet up for a group activity that requires the members to be divided into seven distinct groups. If one member is unable to attend, the remaining members can still be evenly divided into the seven groups. Calculate the sum of all possible numbers of m... | 2807 | 73 | 4 |
math | Nails are affixed to a wooden board such that they are 1 unit apart in both horizontal and vertical directions. A rubber band is looped around four nails as shown to form a quadrilateral. Calculate the area of this quadrilateral (in square units). | 6 | 53 | 1 |
math | Given a line $l$ with an inclination angle of $\frac{\pi}{6}$ passing through the focus $F$ of the parabola $C: y^{2}=2px (p > 0)$, determine the x-coordinate of a point $P$ on the parabola $C$ that is symmetric to a point $Q(5,0)$ on the $x$-axis with respect to the line $l$. | 3 | 92 | 1 |
math | Translate the function $f(x) = 3\sin(2x + \varphi)$, where $\varphi \in (0, \pi)$, by moving its graph to the right along the x-axis by $\frac{\pi}{6}$ units to obtain the graph of function $g(x)$. If the function $g(x)$ satisfies $g(|x|) = g(x)$, then determine the value of $\varphi$. | \frac{5\pi}{6} | 93 | 9 |
math | A certain water plant's reservoir has 400 tons of water at the beginning of each day at midnight. While supplying water to residents, it also pumps water into the reservoir at a rate of 60 tons per hour. The total amount of water supplied to residents within $t$ hours is $120\sqrt{6t}$ $(0 \leqslant t \leqslant 24)$.
... | 8 | 146 | 1 |
math | A function \( f \) is defined by \( f(z) = -i\overline{z} \), where \( i^2 = -1 \) and \( \overline{z} \) is the complex conjugate of \( z \). How many values of \( z \) satisfy both \( |z| = 3 \) and \( f(z) = z \)? | 2 | 82 | 1 |
math | A group of 8 boys and 8 girls was paired up randomly. Find the probability that there is at least one pair with two girls. Round your answer to the nearest hundredth. | 0.98 | 38 | 4 |
math | Find the least positive integer $k$ so that $k + 25973$ is a palindrome (a number which reads the same forward and backwards). | 89 | 38 | 2 |
math | If $-\frac{π}{2}＜α＜\frac{π}{2}$, $-\frac{π}{2}＜β＜\frac{π}{2}$, and $\tan \alpha $, $\tan \beta $ are two roots of the equation ${x^2}+3\sqrt{3}x+4=0$, then $\alpha +\beta =$____. | -\frac{2π}{3} | 84 | 8 |
math | In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $c\sin A - 2b\sin C = 0$ and $a^2 - b^2 - c^2 = \frac{\sqrt{5}}{5}ac$.
1. Find the value of $\cos A$.
2. If $b = \sqrt{5}$, find the area of $\triangle ABC$. | 3 | 109 | 1 |
math | In a school there are $1200$ students. Each student is part of exactly $k$ clubs. For any $23$ students, they are part of a common club. Finally, there is no club to which all students belong. Find the smallest possible value of $k$ . | 23 | 69 | 2 |
math | Let $f(x) = (x+1)\ln(x+1)$.
(1) Find the minimum value of $f(x)$.
(2) If for any $x \geq 0$, $f(x) \geq ax$ holds, find the range of the real number $a$. | (-\infty, 1] | 67 | 8 |
math | Given that the sequence $\{ \frac {1}{a_{n}}\}$ is an arithmetic sequence, and $a_{3}= \frac {1}{8}$, $a_{2}=4a_{7}$.
(Ⅰ) Find the general term formula for $\{a_{n}\}$;
(Ⅱ) If $b_{n}=a_{n}a_{n+1}(n∈N^{*})$, find the sum of the first n terms $S_{n}$ of the sequence $\{b_{n}\}$. | S_{n}=\frac {n}{6n+4} | 116 | 14 |
math | Determine the order of magnitude for the three numbers: $(0.3)$, $2^{0.3}$, and $\log _{2}0.3$. | \log _{2}0.3 < (0.3)^{2} < 2^{0.3} | 36 | 26 |
math | Given the complex number $z=(2+i)(a+2i^3)$ corresponds to a point in the fourth quadrant on the complex plane, determine the range of the real number $a$. | (-1,4) | 40 | 5 |
math | Given that a calculator displays the number 00016, find the fewest number of times you must depress the reciprocal key to display the number 00016. | 2 | 38 | 1 |
math | Let \( F \) be the set consisting of all functions \( f \) such that \( f: P(S) \rightarrow \mathbb{R} \) and for all \( X, Y \subseteq P(S) \), \( f(X \cap Y) = \min (f(X), f(Y)) \), where \( S \) is a finite set and \( P(S) \) is the power set of \( S \). Find \( \max _{f \in F} \mid \operatorname{Im}(f) \mid \), whe... | n+1 | 138 | 3 |
math | Suppose a parabola has its vertex at $\left(-\frac{1}{3}, -\frac{1}{9}\right)$ and follows the equation $y = ax^2 + bx + c$, with $a > 0$ and $2a + b + 3c$ being an integer. Find the smallest possible value of $a$. | \frac{1}{3} | 76 | 7 |
math | (The full score of this question is 14 points)
When making an investment plan, one should consider not only the potential profits but also the possible losses. An investor plans to invest in two projects, A and B. According to predictions, the maximum possible profits for projects A and B are 100% and 50%, respective... | 40,000 \text{ yuan and } 60,000 \text{ yuan} | 147 | 24 |
math | A regular triangle \( \triangle ABC \) with side length 3 is divided into three equal parts on each side. Parallel lines to each side are drawn through these division points, creating a total of 10 intersection points within and on the edges of the triangle, known as grid points. Determine the minimum number \( n \) of... | 5 | 103 | 1 |
math | What is the greatest common divisor of $122^2 + 234^2 + 346^2 + 458^2$ and $121^2 + 233^2 + 345^2 + 457^2$? | 1 | 65 | 1 |
math | Given vectors $\overrightarrow {a}$ = (4, -7) and $\overrightarrow {b}$ = (3, -4), find the projection of $\overrightarrow {a}$ - $2\overrightarrow {b}$ in the direction of $\overrightarrow {b}$. | -2 | 59 | 2 |
math | Let \( V \) be a rectangular prism with integer side lengths. The largest face has an area of 240, the smallest face has an area of 48, and a third face has an area \( x \), where \( x \) is not equal to 48 or 240. What is the sum of all possible values of \( x \)? | 260 | 79 | 3 |
math | Find all solutions $x$ (real and otherwise) to the equation
\[x^6 + 64 = 0.\] | \sqrt{3} + i, 0 + 2i, -\sqrt{3} + i, -\sqrt{3} - i, 0 - 2i, \sqrt{3} - i | 28 | 47 |
math | In the Wuyang Middle School Math Competition, the full score is 120 points. It is stipulated that those who score no less than 100 points will receive a gold medal, and those who score between 80 and 99 points will receive a silver medal. It was found that the number of gold medals is 8 less than the number of silver m... | 125 | 191 | 3 |
math | Solve the inequality:
$$
x \log _{1 / 10}\left(x^{2}+x+1\right)>0
$$ | (-\infty, -1) | 33 | 8 |
math | Given the parametric equations
\begin{align*}
x &= \cos t + \frac{t}{3}, \\
y &= \sin 2t,
\end{align*}
determine how many times the graph intersects itself for $x$ values between 1 and 50. | 48 | 62 | 2 |
math | The equation of the curve $y=\ln |x|$ passes through the origin with two tangent lines given by ____, ____. | x + ey = 0 | 26 | 6 |
math | Find all integers \( n \) for which the number \( \left|n^{2} - 6n - 27\right| \) is prime. | -4, -2, 8, 10 | 35 | 12 |
math | Given that the even function $f(x)$ defined on $\mathbb{R}$ is decreasing on $[0,+\infty)$, and the inequality $f(-ax+x^{3}+1)+f(ax-x^{3}-1)\geqslant 2f(1)$ holds for $x\in(0, \sqrt {2}]$, determine the range of values for the real number $a$. | [2, 3] | 88 | 6 |
math | For the sequence $\{a_n\}$, the sum of the first $n$ terms $S_n = 3n - 2n^2$ ($n \in \mathbb{N}^*$), then $a_n =$; at this time, the relationship between $S_n$ and $n a_n$ is. | -4n + 5 | 71 | 6 |
math | The prime factorization of 1386 is $2 \times 3 \times 3 \times 7 \times 11$. How many ordered pairs of positive integers $(x,y)$ satisfy the equation $xy = 1386$, and both $x$ and $y$ are even? | 12 | 66 | 2 |
math | A pair of natural numbers \( (a, p) \) is called "good" if the number \( a^{3} + p^{3} \) is divisible by \( a^{2} - p^{2} \), with \( a > p \).
(a) Indicate any possible value of \( a \) for which the pair \( (a, 13) \) is good.
(b) Find the number of good pairs for which \( p \) is a prime number less than 20. | 8 | 108 | 1 |
math | Four complex numbers are vertices of a square in the complex plane. The first three vertices are $1+i, -1+3i$, and $-3-i$. Determine the fourth vertex.
A) $3+i$
B) $-3+i$
C) $-1-i$
D) $3-i$ | -1-i | 65 | 3 |
math | A math competition consists of three problems, each of which receives an integer score from 0 to 7. For any two competitors, it is known that there is at most one problem in which they received the same score. Find the maximum number of competitors in this competition. | 64 | 55 | 2 |
math | Given quadrilateral ABCD, there are four conditions: AB∥CD, AB=CD, BC∥AD, BC=AD. Determine the total number of ways to select two conditions from these that can make quadrilateral ABCD a parallelogram. | 4 | 51 | 1 |
math | Given matrix $A= \begin{bmatrix} 1 & 2 \\ -1 & 4 \end{bmatrix}$, $B= \begin{bmatrix} 5 \\ 3 \end{bmatrix}$.
(1) Find the eigenvalues $\lambda_1$, $\lambda_2$ and the corresponding eigenvectors $\alpha_1$, $\alpha_2$ of $A$;
(2) Calculate $A^4B$. | \begin{bmatrix} 145 \\ 113 \end{bmatrix} | 100 | 21 |
math | Two standard dice are rolled. What is the expected number of 6's obtained, given that rolling a 4 resets any accumulated count of 6’s back to zero? Express your answer as a common fraction. | \frac{5}{18} | 43 | 8 |
math | Given that $\theta$ is a real number, if the complex number $z = \sin(2\theta) - 1 + i(\sqrt{2}\cos\theta - 1)$ is a pure imaginary number, determine the imaginary part of $z$. | -2 | 55 | 2 |
math | What is the sum of all integer solutions to \( |n| < |n+4| < 15 \)? | 54 | 25 | 2 |
math | Square $ABCD$ has side length $5$ and arc $BD$ with center $A$ . $E$ is the midpoint of $AB$ and $CE$ intersects arc $BD$ at $F$ . $G$ is placed onto $BC$ such that $FG$ is perpendicular to $BC$ . What is the length of $FG$ ? | 2 | 100 | 1 |
math | Let \( a \) and \( b \) be positive real numbers. Find the minimum value of
\[
a^2 + 2b^2 + \frac{2}{(a + 2b)^2}.
\] | 2 | 49 | 1 |
math | Given in the geometric sequence $\{a_n\}$, it is always true that $a_{n+1} > a_n$ for $n\in\mathbb{N}^*$, and $a_1a_4=8$, $a_2+a_3=6$.
(I) Find the general formula for the sequence $\{a_n\}$.
(II) If the sequence $\{b_n\}$ satisfies $\dfrac{a_1}{b_1} + \dfrac{3a_2}{b_2} + \ldots + \dfrac{(2n-1)a_n}{b_n} = n$ ($n\in\m... | S_n=(2n-3)\cdot2^n+3 | 178 | 13 |
math | Samantha departs from Town A at 7:45 AM heading towards Town B at a steady speed of 15 miles per hour, and Adam leaves Town B at 8:15 AM heading towards Town A on the same 75-mile route at a constant speed of 20 miles per hour. Determine the time they meet and the distance traveled by Samantha at this time. | 10:11 \text{ AM, } 36 \text{ miles} | 82 | 19 |
math | Find all positive integer solutions \((x, y, z, t)\) for the equation \(2^{y} + 2^{z} \times 5^{t} - 5^{x} = 1\). | (2,4,1,1) | 48 | 9 |
math | Find a positive integer that is divisible by 14 and has a square root between 26 and 26.5. | 700 | 27 | 3 |
math | Solve the equation $\sqrt{2x + 16} - \frac{8}{\sqrt{2x + 16}} = 4$.
A) $x = 4\sqrt{3}$
B) $x = 3\sqrt{4}$
C) $x = -4\sqrt{3}$
D) $x = 2\sqrt{12}$ | x = 4\sqrt{3} | 87 | 9 |
math | Given $a \gt 1$, $b \gt 0$, and $a+b=2$, calculate the minimum value of $\frac{1}{{a-1}}+\frac{1}{{2b}}$. | \frac{3}{2} + \sqrt{2} | 47 | 13 |
math | Find the remainder when $7 \times 17 \times 27 \times 37 \times \ldots \times 297 \times 307$ is divided by $6$. | 1 | 45 | 1 |
math | During a flood control emergency, a large gasoline canister drifting downstream is to be detonated using a shooting method. It is known that there are only 5 bullets available, and the first successful shot can only cause gasoline to leak, while a second successful hit is required for detonation. The probability of hit... | \frac{232}{243} | 96 | 11 |
math | Find the equation of the circle passing through the three points $A(4,1)$, $B(-6,3)$, $C(3,0)$, and then find the radius and the coordinates of the center of this circle. | (-\frac{1}{2}, \frac{9}{2}) | 50 | 15 |
math | Let $ S$ be the set of points in the Cartesian plane that satisfy
\[ \Big|\big|{|x| \minus{} 2}\big| \minus{} 1\Big| \plus{} \Big|\big|{|y| \minus{} 2}\big| \minus{} 1\Big| \equal{} 1.
\]
If a model of $ S$ were built from wire of negligible thickness, then the total length of wire required would be $ a\sqrt {b... | 10 | 157 | 2 |
math | A point has rectangular coordinates $(2,-1,-2)$ and spherical coordinates $(\rho, \theta, \phi).$ Find the rectangular coordinates of the point with spherical coordinates $(\rho, \theta, 2 \phi).$ | \left( -\frac{8}{3}, \frac{4}{3}, -\frac{1}{3} \right) | 50 | 29 |
math | If
\[\tan x = \frac{a^2 + b^2}{2ab},\]where $a > b > 0$ and $x$ is such that $45^\circ < x < 90^\circ$, then find $\cos x$ in terms of $a$ and $b$. | \frac{2ab}{\sqrt{a^4 + 6a^2b^2 + b^4}} | 69 | 26 |
math | To celebrate 2019, Faraz gets four sandwiches shaped in the digits 2, 0, 1, and 9 at lunch. The four digits get reordered (but not flipped or rotated) on his plate, and he notices that they form a 4-digit multiple of 7. What is the greatest possible number that could have been formed? | 1092 | 75 | 4 |
math | The coefficients of the polynomial
\[a_{12} x^{12} + a_{11} x^{11} + \dots + a_1 x + a_0 = 0\]
are all integers, and its roots $r_1, r_2, \dots, r_{12}$ are all integers. Furthermore, the roots of the polynomial
\[a_0 x^{12} + a_1 x^{11} + \dots + a_{11} x + a_{12} = 0\]
are also $r_1, r_2, \dots, r_{12}$. Find the num... | 13 | 170 | 2 |
math | As shown in the figure, in rectangle \(ABCD\), \(AB=4\), \(BC=6\), points \(E, F, G, H\) lie on \(AB, BC, CD, DA\) respectively, and the ratios \(AE:EB=3:1\), \(BF:FC=2:1\), \(DG:GC=1:3\), \(AH:HD=1:2\). Point \(P\) lies on \(HF\), and the area of quadrilateral \(AEPH\) is 5. Find the area of quadrilateral \(PFCG\). | 8 | 128 | 1 |
math | Let the real numbers \( a_{1}, a_{2}, \ldots, a_{2016} \) satisfy \( 9a_{i} > 11a_{i+1}^2 \) for \( i = 1, 2, \ldots, 2015 \). Find the maximum value of \[ \left(a_{1}-a_{2}^{2}\right) \cdot \left(a_{2}-a_{3}^{2}\right) \cdots \cdot \left(a_{2015}-a_{2016}^{2}\right) \cdot \left(a_{2016}-a_{1}^{2}\right) \]. | \left(\frac{1}{4}\right)^{2016} | 156 | 17 |
math | In the diagram, $\triangle ABE$, $\triangle BCE$ and $\triangle CDE$ are right-angled at $E$, with $\angle AEB = \angle BEC = \angle CED = 60^\circ$, and $AE=36$. Find the perimeter of the quadrilateral $ABCD$. | 31.5\sqrt{3} + 40.5 | 67 | 15 |
math | In triangle $\triangle ABC$, the corresponding sides of angles $A$, $B$, and $C$ are $a$, $b$, $c$, and $\cos C = -\frac{1}{4}$. If $2\sin A + \sin B = \frac{{\sqrt{15}}}{2}$, calculate $\sin B$. | \frac{\sqrt{15}}{8} | 74 | 11 |
math | Let us call a number \( \mathrm{X} \) "50-podpyirayushchim" if for any 50 real numbers \( a_{1}, \ldots, a_{50} \) whose sum is an integer, there exists at least one \( a_i \) such that \( \left|a_{i}-\frac{1}{2}\right| \geq X \).
Find the greatest 50-podpyirayushchee \( X \), rounded to the nearest hundredth accordin... | 0.01 | 120 | 4 |
math | Spinners $C$ and $D$ are spun. Spinner $C$ is divided into four sections numbered 1, 3, 5, and 7. Spinner $D$ is divided into three sections numbered 2, 4, and 6. What is the probability that the sum of the two spinners' numbers is divisible by 3? | \frac{1}{4} | 75 | 7 |
math | Given functions $f\left(x\right)=x^{2}-ax-2\left(a\in R\right)$ and $g\left(x\right)=-x^{2}+x+a$.
$(1)$ If $x=-1$ is a real root of the equation $f\left(x\right)=0$, find the range of the function $f\left(x\right)$.
$(2)$ If for any $x_1∈[\frac{1}{4},1]$, there exists $x_{2}\in \left[1,2\right]$ such that $g(x_{1})... | (1, +\infty) | 152 | 8 |
math | In 1900, a reader asked the following question in 1930: He knew a person who died at an age that was $\frac{1}{29}$ of the year of his birth. How old was this person in 1900? | 44 | 58 | 2 |
math | Given $\overrightarrow{a}=(1,λ,2)$ and $\overrightarrow{b}=(2,-1,1)$, with an angle of $60^{\circ}$ between $\overrightarrow{a}$ and $\overrightarrow{b}$, find the value of $λ$. | λ = -17 \text{ or } λ = 1 | 62 | 14 |
math | A number $x$ is randomly chosen from the interval $[0,π]$. What is the probability that the event $\sin x + \sqrt{3} \cos x \leqslant 1$ occurs? | p = \frac{1}{2} | 47 | 9 |
math | We cut a circular sheet of paper into $n$ congruent sectors, and then each of those sectors is transformed into a conical surface. For which $n$ will the combined volume of the cones determined by these conical surfaces be maximized? | 2 | 51 | 1 |
math | Given the exponential function $y=g(x)$ satisfies $g(3)=27$, and the function $f(x)=\frac{\mathbf{n{-}g(x)}}{\mathbf{m{+}3}\mathbf{g(x)}}$ with domain $R$ is an odd function.
(1) Determine the analytical expressions for $y=g(x)$ and $y=f(x)$;
(2) If for any $t∈(1,4)$, the inequality $f(2t-3)+f(t-k) > 0$ always holds,... | [9,+\infty) | 131 | 7 |
math | In a local debate club, there are 5 teams with 8 members each. Each of the 5 teams rotates to organize monthly debates. During each debate, each team selects three members to be on the debate organizing committee, except the organizing team, which selects four members. How many possible 16-member debate organizing comm... | 3,\!442,\!073,\!600 | 70 | 16 |
math | Let the three-digit number \( n = \overline{abc} \). If \( a \), \( b \), and \( c \) can form an isosceles (including equilateral) triangle, how many such three-digit numbers are there? | 165 | 53 | 3 |
math | How many non-empty subsets $T$ of $\{1,2,3,\ldots,17\}$ have the following two properties?
$(1)$ No two consecutive integers belong to $T$.
$(2)$ If $T$ contains $k$ elements, then $T$ contains no number less than $k+1$. | 594 | 70 | 3 |
math | Given $F_{1}$ and $F_{2}$ are the left and right foci of the hyperbola $C:\frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1(a＞0,b＞0)$, $P$ is a point on the hyperbola in the first quadrant, and $∠{F}_{1}P{F}_{2}=\frac{π}{3}$, $|{F}_{1}{F}_{2}|=2\sqrt{3}$, the symmetric point $Q$ of $F_{1}$ with respect to the bisector ... | \sqrt{3} | 165 | 5 |
math | In triangle \(ABC\), it is known that \(AB = c\), \(AC = b\), and the angle bisector extending from angle \(A\) has length \(l\). Find the third side of the triangle. | BC = (b+c) \sqrt{\frac{bc - l^2}{bc}} | 47 | 19 |
math | A girl has the following ten coins in her pocket: $3$ pennies, $3$ nickels, $3$ dimes, $1$ quarter, and $1$ half-dollar. She takes out two coins at a time, records the sum of their values, and then puts them back. She repeats this process for different pairs of coins. What is the maximum number of different sums she ca... | 15 | 85 | 2 |
math | The doctor has told Cal O'Ree that during his ten weeks of working out at the gym, he can expect each week's weight loss to be $1\%$ of his weight at the end of the previous week. His weight at the beginning of the workouts is $244$ pounds. How many pounds does he expect to weigh at the end of the ten weeks? Express yo... | 221 | 87 | 3 |
math | Given that the graph of the linear function $y=(5-a)x+a+1$ passes through the first, second, and third quadrants, and the fractional equation in terms of $x$ $\frac{10}{2-x}=2-\frac{ax}{x-2}$ has integer solutions, the sum of all integers $a$ that satisfy the conditions is ______. | 7 | 78 | 1 |
math | Given the expression \[2 - (-3) - 4 - (-5) - 6 - (-7) \times 2,\] calculate its value. | -14 | 34 | 3 |
math | For which integer $c$ does $x^2 + x + c$ divide $x^{13} - x + 106$? | 2 | 32 | 1 |
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