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 the function $f(x) = \begin{cases} x+4, & x<a \\ x^{2}-2x, & x\geq a \end{cases}$, if the range of the function $f(x)$ is $\mathbb{R}$, then the range of the real number $a$ is ______. | [-5, 4] | 71 | 6 |
math | Lucy was born on Wednesday, December 1st, 2004. That Wednesday was the first day of her life. Her parents held a party for her on the $1000$th day of her life. On which day of the week was the party? | \text{Monday} | 59 | 5 |
math | Determine the interval in which the function $f(x)={(\dfrac{1}{2})}^{\sqrt{x-{{x}^{2}}}}$ is monotonically decreasing. | \left[0, \dfrac{1}{2}\right] | 41 | 15 |
math | Sergey arranged several (more than two) pairwise distinct real numbers in a circle in such a way that each number is equal to the product of its neighbors. How many numbers could Sergey have arranged? | 6 | 41 | 1 |
math | Given a sequence $\{a\_n\}$ where all terms are positive numbers, it satisfies the equation $\log\_2 a\_n = 1 + \log\_2 a\_{n-1}, n \in \mathbb{N}^*, n \geq 2$, and $a\_1 = 2$.
(I) Find the general formula for the sequence $\{a\_n\}$;
(II) Let $c\_n = (3n - 1) \cdot a\_n$, find the sum of the first $n$ terms of the seq... | m = 2, n = 12 | 243 | 10 |
math | If the function $f(x) = kx^2 + (k - 1)x + 3$ is an even function, find the decreasing interval of $f(x)$. | (-\infty, 0] | 38 | 8 |
math | In the given equations, determine the number of linear equations. | 3 | 12 | 1 |
math | Find the value of \(x\) between 0 and 180 such that
\[
\tan(150^\circ - x^\circ) = \frac{\sin 150^\circ - \sin x^\circ}{\cos 150^\circ - \cos x^\circ}.
\] | 120^\circ | 68 | 5 |
math | An ellipse has its center at the origin, with foci $F\_1$ and $F\_2$ on the $x$-axis. Point $P(2, \sqrt{3} )$ is on the ellipse, and the distances $|PF\_1 |$, $|F\_1 F\_2 |$, $|PF\_2 |$ form an arithmetic sequence. Determine the equation of the ellipse.
A) $\frac{{x}^{2}}{8}+ \frac{{y}^{2}}{6}=1$
B) $\frac{{x}^{2}}{16... | \frac{{x}^{2}}{8} + \frac{{y}^{2}}{6} =1 | 197 | 25 |
math | Given an ellipse $E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ with an eccentricity of $\frac{\sqrt{3}}{2}$ and passing through the point $(1, \frac{\sqrt{3}}{2})$.
(1) Find the equation of $E$;
(2) Determine if there exists a line $l: y = kx + m$ that intersects $E$ at points $P$ and $Q$, and satisfies the ... | y = -x + \sqrt{2} | 201 | 10 |
math | For how many ordered pairs of positive integers \( (a, b) \) with \( a < b \) is the harmonic mean of \( a \) and \( b \) equal to \( 5^{10} \)? | 10 | 48 | 2 |
math | Given the parabola $C: y^{2}=2px (p > 0)$ and its directrix $l$, a line passing through $M(1,0)$ with a slope of $\sqrt{3}$ intersects $l$ at point $A$ and intersects $C$ at one point $B$. If $\overrightarrow{AM}= \overrightarrow{MB}$, then $p=$ ______. | 2 | 87 | 1 |
math | In a pot, there are 6 sesame-filled dumplings, 5 peanut-filled dumplings, and 4 red bean paste-filled dumplings. These three types of dumplings look exactly the same from the outside. If 4 dumplings are randomly selected from the pot, the probability that at least one dumpling of each type is selected is. | \dfrac{48}{91} | 71 | 9 |
math | A bamboo pole of 10 feet in height is broken. If the top part is bent to the ground, the tip of the pole will be 3 feet away from the base. What is the length of the broken part? | 5 \frac{9}{20} | 47 | 9 |
math | Given the function $f(x)= \begin{cases} |2^{x}-1|, & x < 2 \\ \frac {3}{x-1}, & x\geqslant 2 \end{cases}$, find the number of zeros of the function $g(x)=f[f(x)]-2$. | 4 | 69 | 1 |
math | Given $$f(x)= \frac {a-x}{x-a-1}$$, the center of symmetry of the graph is (3, -1). Find the value of the real number $a$. | 2 | 42 | 1 |
math | Explore the rules and complete the related questions. Define the "$*$" operation: $(+2)*\left(+4\right)=+(2^{2}+4^{2})$; $(-4)*\left(-7\right)=[\left(-4\right)^{2}+\left(-7\right)^{2}]$; $(-2)*\left(+4\right)=-[\left(-2\right)^{2}+\left(+4\right)^{2}]$; $(+5)*\left(-7\right)=-[\left(+5\right)^{2}+\left(-7\right)^{2}]$;... | 5 | 365 | 1 |
math | The negation of the proposition "For all $x$ in $\mathbb{R}$, $|x-2| < 3$" is __________. | \exists x \in \mathbb{R}, |x-2| \geq 3 | 34 | 21 |
math | A sphere with center $O$ has radius $6$. A triangle with sides of length $15, 15,$ and $24$ is situated in space so that each of its sides is tangent to the sphere. What is the distance between $O$ and the plane determined by the triangle? | $2\sqrt{5}$ | 63 | 7 |
math | Let \( m \) and \( n \) be positive integers, and let \( p \) be a prime number. Find all triplets \((m, n, p)\) that satisfy the equation \(\mathrm{C}_{m}^{3} - 4 = p^{n}\). | (7, 1, 31), (6, 4, 2) | 62 | 19 |
math | In triangle $XYZ \sim$ triangle $PQR$, with $YZ = 35$ cm, $XY = 20$ cm, where $\triangle XYZ$ is a right triangle with $X$ being the right angle. If $QR = 14$ cm, find the length of $PR$. Express your answer to the nearest tenth. | 8.0 \text{ cm} | 74 | 8 |
math | Calculate the distance between the points $(3, 7)$ and $(-5, 3)$, and find the coordinates of their midpoint. | (-1, 5) | 29 | 6 |
math | In the planar quadrilateral ABCD, we have AB = CD and AD = DC = CB = 1. The areas of triangles ABD and BCD are respectively denoted as S and T. Find the maximum value of $S^2 + T^2$. | \frac{1}{2} | 56 | 7 |
math | Given the point $(\cos \theta, \sin \theta)$, the distance from this point to the line $x \sin \theta + y \cos \theta - 1 = 0$ is $\frac{1}{2}$ $(0 \leqslant \theta \leqslant \frac{\pi}{2})$. Find the value of $\theta$ | \frac{\pi}{12} \text{ or } \frac{5\pi}{12} | 79 | 23 |
math | Given the set \( A=\{x \mid x^{2}+2x-8>0\} \) and \( B=\{x \mid x^{2}-2ax+4 \leq 0\} \), if \( a>0 \) and there is exactly one integer in the intersection of \( A \) and \( B \), then the range of \( a \) is \(\qquad\). | \left[\frac{13}{6}, \frac{5}{2}\right) | 91 | 19 |
math | Two externally tangent circles with centers at points $A$ and $B$ have radii of lengths $7$ and $2$, respectively. A line externally tangent to both circles intersects the extension of ray $AB$ at point $C$ past $B$. Find the length of $BC$. | 3.6 | 60 | 3 |
math | Given the set $A=\{-1,-\frac{1}{2},-\frac{1}{3},0,\frac{2}{3},2,3\}$, $B=\{y=f\left(x\right)|f\left(x\right)=x^{k}$, $k\in A$ and the graph of $y=f\left(x\right)$ is symmetric about the $y$-axis$\}$, then the number of elements in set $B$ is ______. | 3 | 105 | 1 |
math | Given $f(x)=\log_{2}(4^x+1)-kx$ $(k\in \mathbb{R})$.
$(1)$ If $f(x)$ is an even function, find the value of the real number $k$;
$(2)$ Let $g(x)=f(x)-a+1$, $k=2$, if the function $g(x)$ has a zero, find the range of values for $a$; | (1,+\infty) | 96 | 7 |
math | The minimum positive period and the maximum value of the function $f(x)=3\sin\left(\frac{x}{4}+\frac{π}{6}\right)-1$ are, respectively, what are the minimum positive period and the maximum value? | 2 | 52 | 1 |
math | Three circles of radius $s$ are drawn in the first quadrant of the $xy$-plane. The first circle is tangent to both axes, the second is tangent to the first circle and the $x$-axis, and the third is tangent to the first circle and the $y$-axis. A circle of radius $r>s$ is tangent to both axes and to the second and third... | 9 | 257 | 1 |
math | Find the smallest natural number $n$ that the following statement holds :
Let $A$ be a finite subset of $\mathbb R^{2}$ . For each $n$ points in $A$ there are two lines including these $n$ points. All of the points lie on two lines. | n = 6 | 72 | 4 |
math | Given two propositions P: For any real number x, ax^2 + ax + 1 > 0 always holds; Q: The equation x^2 - x + a = 0 has real roots with respect to x;
(1) What kind of condition is "a=0" for P?
(2) If only one of P and Q is true, find the range of real number a. | (- \infty, 0) \cup (\frac{1}{4}, 4) | 84 | 20 |
math | (1) Evaluate $ \int_{\minus{}\sqrt{3}}^{\sqrt{3}}( x^2\minus{}1)dx,\ \int_{\minus{}\sqrt{3}}^{\sqrt{3}} (x\minus{}1)^2dx,\ \int_{\minus{}\sqrt{3}}^{\sqrt{3}} (x\plus{}1)^2dx$ .
(2) If a linear function $ f(x)$ satifies $ \int_{\minus{}\sqrt{3}}^{\sqrt{3}} (x\minus{}1)f(x)dx\equal{}5\sqrt{3},\ \int_{\minus{}\sqrt... | f(x) = 2x - \frac{1}{2} | 230 | 15 |
math | Given that triangle ABC is a triangle with altitude AD and median BE, and the lengths of BC, AD, and AC are known, determine the length of EC. | \frac{1}{2}AC | 33 | 8 |
math | Given $\triangle ABC$ where $\overline{CA} = \overline{CB}$, and square $BCDE$ is constructed on $CB$ away from the triangle, if $x$ is the number of degrees in $\angle DAB$, express $x$ in terms of the angles of $\triangle ABC$. | 45^\circ | 66 | 4 |
math | A ball is dropped from 45 feet high and bounces back up to a third of the distance it just fell each time. After how many bounces will the ball first reach a maximum height less than 2 feet? | 4 | 46 | 1 |
math | What is the coefficient of $x^3$ when $7x^4-3x^3 -3x^2-8x + 1$ is multiplied by $8x^4+2x^3 - 7x^2 + 3x + 4$ and the like terms are combined? | 37 | 67 | 2 |
math | In a basket, there are 41 apples: 10 green, 13 yellow, and 18 red. Alyona is sequentially taking out one apple at a time from the basket. If at any point the number of green apples she has taken out is less than the number of yellow apples, and the number of yellow apples is less than the number of red apples, then she... | 39 | 126 | 2 |
math | Given that a coin is tossed 9 times and lands heads up each time, calculate the probability that it lands heads up on the 10th toss. | \frac{1}{2} | 32 | 7 |
math | Given that 20% of the participants scored 60 points, 25% scored 75 points, 15% scored 85 points, 30% scored 90 points, and the rest scored 95 points, calculate the difference between the mean and median score of the participants' scores on this competition. | 5 | 72 | 1 |
math | The base of a triangle is 20; the medians drawn to the lateral sides are 18 and 24. Find the area of the triangle. | 288 | 34 | 3 |
math | When the function $f(x)$ is an even function defined on $\mathbb{R}$, $f(x+1)$ is an odd function, and $f(0)=0$, and given that $f(x)= \log_{2}x$ when $x\in(0,1]$, find the real number $x$ in the interval $(8,10)$ that satisfies the equation $f(x)+1=f(1)$. | \dfrac{17}{2} | 94 | 8 |
math | Given the functions $y=\sin x+\cos x$ and $y=2 \sqrt {2}\sin x\cos x$, determine the correct conclusion(s) from the following options:
$①$ Both functions' graphs are centrally symmetric about the point $(-\frac{π}{4},0)$;
$②$ Both functions' graphs are axially symmetric about the line $x=-\frac{π}{4}$;
$③$ Both func... | ③⑤ | 148 | 4 |
math | Given the line $4x-3y+a=0$ intersects with the circle $x^2+y^2+4x=0$ at points $A$ and $B$, and $\angle AOB=120^\circ$, find the value of the real number $a$. | 3 | 61 | 1 |
math | Inside triangle $ABC$, there are 1997 points. Using the vertices $A, B, C$ and these 1997 points, into how many smaller triangles can the original triangle be divided? | 3995 | 45 | 4 |
math | In the Cartesian coordinate system, it is known that the point $P(3m-6, m+1)$. Find the coordinates of point $P$ according to the following conditions:<br/>$(1)$ If point $P$ lies on the $x$-axis, find the coordinates of point $P$;<br/>$(2)$ Given that the coordinates of point $A$ are $(-1,2)$, and $AP$ is parallel to ... | (-1, \frac{8}{3}) | 111 | 10 |
math | Let $a, b \in \mathbb{R}^{+}$, and $a + 2b = 3$. Find the minimum value of $\dfrac{1}{a} + \dfrac{1}{b}$. | 1 + \dfrac{2\sqrt{2}}{3} | 51 | 15 |
math | A digit is appended to the left of the number $864$. If the resulting four-digit number is divisible by $5$, how many possibilities are there for the digit that was appended? | 0 | 39 | 1 |
math | Given the function $f(x) = x^2 - bx + 3$, and $f(0) = f(4)$.
(1) Find the zeros of the function $y = f(x)$, and write the set of $x$ values for which $f(x) < 0$;
(2) Find the maximum and minimum values of the function $y = f(x)$ in the interval $[0, 3]$. | 3 | 96 | 1 |
math | Find the range of the expression $\frac{a \cos x - b \sin x + 2c}{\sqrt{a^2 + b^2 + c^2}}$ where $x, a, b,$ and $c$ are arbitrary numbers such that $a^2 + b^2 + c^2 \neq 0$. | [-\sqrt{5}, \sqrt{5}] | 74 | 11 |
math | In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are denoted as $a$, $b$, $c$ respectively, and it is given that $a^2-(b-c)^2=(2-\sqrt{3})bc$, $\sin A\sin B=\cos^2\frac{C}{2}$.
$(1)$ Find the magnitude of angle $B$;
$(2)$ If the arithmetic sequence $\{a_n\}$ has a non-zero common difference, and $a_1\co... | S_n=\frac{n}{n+1} | 181 | 10 |
math | In triangle $ABC$, $AB = 12$, $AC = 10$, and $BC = 16$. The centroid $G$ of triangle $ABC$ divides each median in the ratio $2:1$. Calculate the length $GP$, where $P$ is the foot of the perpendicular from point $G$ to side $BC$. | \frac{\sqrt{3591}}{24} | 74 | 14 |
math | As shown in Figure 1, a line segment of length 1 is divided into two parts, $x$ and $y$. Then, the segment of length $x$ is bent into a semicircular arc $ACB$, and the segment of length $y$ is folded into a rectangle $ABDE$ along three sides $(BD, DE, EA)$, forming a closed "curved shape" $ACBDEA$. What is the maximum ... | \frac{1}{2(\pi+4)} | 100 | 11 |
math | Triangle $ABC$ has side-lengths $AB = 12, BC = 24,$ and $AC = 18.$ The line through the incenter of $\triangle ABC$ parallel to $\overline{BC}$ intersects $\overline{AB}$ at $M$ and $\overline{AC}$ at $N.$ What is the perimeter of $\triangle AMN?$
$\textbf{(A)}\ 27 \qquad \textbf{(B)}\ 30 \qquad \textbf{(C)}\ 33 \qqua... | 30 | 145 | 2 |
math | Let \( g : \mathbb{R} \to \mathbb{R} \) be a function such that
\[ g(x) g(y) - g(xy) = x - y \]
for all real numbers \( x \) and \( y \).
Let \( n \) be the number of possible values of \( g(3) \), and let \( s \) be the sum of all possible values of \( g(3) \). Find \( n \times s \). | -2 | 104 | 2 |
math | Given the function $f(x) = x^3 + 3x^2 + 6x + 4$, and given that $f(a) = 14$ and $f(b) = -14$, calculate the value of $a + b$. | -2 | 56 | 2 |
math | Let ellipse $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ have its left and right foci as $F_1$ and $F_2$, respectively. Point $P$ is on $C$, and $PF_2 \perpendicular F_1F_2$, $\angle PF_1F_2 = 30^{\circ}$. Calculate the eccentricity of $C$. | \frac{\sqrt{3}}{3} | 108 | 10 |
math | Find the slope and the angle of inclination \\(\theta\\) of the line passing through the following two points: \\(C(m,n)\\) and \\(D(m,-n)\\) where \\(n \neq 0\\). | 90^\circ | 52 | 4 |
math | Determine the value of $a$ that satisfies the equation $375_{a} + 592_{a} = 9C7_{a}$, where $C_{a}=12_{10}$. | 13 | 49 | 2 |
math | If $(x+2)^2(x-3)=40$, find the sum of the possible values of $x$. | -1 | 25 | 2 |
math | Express the limit of the sum $$\lim_{n\rightarrow \infty} \frac {1^{p}+2^{p}+3^{p}+\ldots+n^{p}}{n^{p+1}}$$ (where $p>0$) as a definite integral. | \int_{0}^{1} x^{p} \, dx | 62 | 15 |
math | A \(101 \times 101\) grid is given, where all cells are initially colored white. You are allowed to choose several rows and paint all the cells in those rows black. Then, choose exactly the same number of columns and invert the color of all cells in those columns (i.e., change white cells to black and black cells to wh... | 5100 | 92 | 4 |
math | Let \( L_{1} \) be the line with equation \( 6x - y + 6 = 0 \). Let \( P \) and \( Q \) be the points of intersection of \( L_{1} \) with the \( x \)-axis and \( y \)-axis, respectively. A line \( L_{2} \) that passes through the point \( (1,0) \) intersects the \( y \)-axis and \( L_{1} \) at \( R \) and \( S \), resp... | y = -3x + 3, \quad y = -10x + 10 | 175 | 21 |
math | In triangle $PQR,$ points $M$ and $N$ are on $\overline{PQ}$ and $\overline{PR},$ respectively, such that $PM = 2,$ $MQ = 6,$ $PN = 3,$ and $NR = 9.$ The angle bisector $\overline{PS}$ intersects $\overline{MN}$ at $T.$ Compute $\frac{PT}{PS}.$ | \frac{5}{20} = \frac{1}{4} | 90 | 16 |
math | Given the function $f(x) = x \ln x + (1 - k)x + k$, where $k \in \mathbb{R}$.
$(1)$ When $k = 1$, determine the intervals on which the function $f(x)$ is monotonic.
$(2)$ When $x > 1$, find the greatest integer value of $k$ such that the inequality $f(x) > 0$ holds true for all $x$. | 3 | 96 | 1 |
math | Given that all terms in the sequence $\{a_n\}$ are positive numbers, and the sum of the first $n$ terms, denoted as $S_n$, satisfies the equation $2S_n = (a_n + 3)(a_n - 2)$ for all $n \in \mathbb{N}^*$.
(1) Find the general term formula for the sequence $\{a_n\}$.
(2) Find the sum of the first $n$ terms, denoted as ... | \frac{n}{6n+9} | 138 | 9 |
math | Let $T$ be a subset of $\{1, 2, 3, \ldots, 75\}$ such that no pair of distinct elements in $T$ has a sum divisible by 5. What is the maximum number of elements in $T$? | 30 | 58 | 2 |
math | We know the following about a one-story apartment: a) there is at most 1 door between any two rooms; b) from any room, at most 1 door opens to the outside of the apartment; c) there are 12 doors in the apartment. What is the minimum number of rooms in the apartment? | 5 | 66 | 1 |
math | Consider the set $S$ of $100$ numbers: $1; \frac{1}{2}; \frac{1}{3}; ... ; \frac{1}{100}$ .
Any two numbers, $a$ and $b$ , are eliminated in $S$ , and the number $a+b+ab$ is added. Now, there are $99$ numbers on $S$ .
After doing this operation $99$ times, there's only $1$ number on $S$ . What val... | 100 | 137 | 3 |
math | If \( x \) and \( y \) are real numbers with \( (x+y)^{4}=x-y \), what is the maximum possible value of \( y \)? | \frac{3\sqrt[3]{2}}{16} | 37 | 15 |
math | In the Cartesian coordinate system, given vector $\overrightarrow{a} = (-1, 2)$, and points A$(8, 0)$, B$(n, t)$, C$(k\sin\theta, t)$ $(0 \leq \theta \leq \frac{\pi}{2})$.
(1) If $\overrightarrow{AB} \perp \overrightarrow{a}$ and $|\overrightarrow{AB}| = \sqrt{5}|\overrightarrow{OA}|$ (where $O$ is the origin), find th... | \overrightarrow{OA} \cdot \overrightarrow{OC} = 32 | 189 | 18 |
math | Determine how many of the first 30 rows of Pascal's triangle, excluding row 0 and row 1, consist only of even numbers (with the exception of the 1's at either end). Recall that each number in Pascal's triangle is the sum of the two numbers above it. | 4 | 61 | 1 |
math | Suppose the solution set of the inequality $x(1 + \ln x) + 2k > kx$ with respect to $x$ is $A$, and $(2, +\infty) \subseteq A$. What is the maximum value of $k$? | 4 | 58 | 1 |
math | Given that $F\_1$ and $F\_2$ are the left and right foci of the hyperbola $C: \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 (a > 0, b > 0)$, and point $P$ is on hyperbola $C$. If $PF\_1 \perp F\_1 F\_2$ and $PF\_1 = F\_1 F\_2$, find the eccentricity of hyperbola $C$. | \sqrt{2} + 1 | 117 | 8 |
math | In triangle $ABC$, $AB = AC$, and point $D$ is the midpoint of both $\overline{BC}$ and $\overline{AE}$. $\overline{CE}$ is 15 units long. Calculate the length of $\overline{BD}$. Additionally, if the height from point $A$ perpendicular to $BC$ is 12 units, find the area of triangle $ABC$. Express $\overline{BD}$ as a ... | 90 | 111 | 2 |
math | Four different positive integers are placed on a grid and then covered up. The product of the integers in each row and column is shown in the diagram.
What is the sum of the four integers?
A 10
B 12
C 13
D 14
E 15 | 13 | 63 | 2 |
math | A small bookstore received a new shipment of a popular book series "Adventurers Unleashed". The manager decided to display 30% of the books while storing the remaining 45 books in the backroom. How many books were included in the shipment? | 64 | 54 | 2 |
math | On Children's Day, Xiao Ming and his parents went to the Wenzhou Zoo to participate in the Animal Festival and had a great time. However, when they returned home, they found their clothes were very dirty. Therefore, the family initiated a "Love Cleanliness, Emphasize Hygiene" campaign, changing and washing all the dirt... | 20 | 232 | 2 |
math | Mr. $X$ owns a home worth $$20,000$. He sells it to Mr. $Y$ at a $15\%$ profit. Mr. $Y$ then sells the house back to Mr. $X$ at a $15\%$ loss. How much did Mr. $X$ gain or lose in the end?
A) Mr. X gains $3500$
B) Mr. X gains $3450$
C) Mr. X loses $3450$
D) Mr. X gains $3200$ | 3450 | 125 | 4 |
math | The graph of the function $f(x)=\sqrt{3}\cos 2x-\sin 2x$ can be obtained by translating the graph of the function $f(x)=2\sin 2x$ by an unspecified distance. | \dfrac{\pi}{6} | 50 | 7 |
math | In a new setup, three identical square crates are packed with cylindrical pipes, each with a diameter of $12 \text{ cm}$. Pipes are stacked in a triangular formation in each crate but with heights of the stacks varying. In Crate C, exactly three pipes are stacked such that they touch each other, as shown in the side vi... | H = 12 + 12\sqrt{3} \text{ cm} | 125 | 19 |
math | Seventy percent of a train's passengers are men, and fifteen percent of those men are in the business class. What is the number of men in the business class if the train is carrying 300 passengers? | 32 | 45 | 2 |
math | $\left(\frac{1+i}{1-i}\right)^{4}$ | 1 | 16 | 1 |
math | If $2\tan \alpha=3\tan \frac{\pi}{8}$, then $\tan \left( \alpha - \frac{\pi}{8} \right)=$\_\_\_\_$.$ | \frac{1+5 \sqrt{2}}{49} | 45 | 15 |
math | Write out the digits from 0 to 9 in such a way that in the new sequence of sums of two adjacent digits, each term, starting from the second, is 1 more than the previous one. | 5, 0, 6, 1, 7, 2, 8, 3, 9, 4 | 43 | 28 |
math | Given that the angle $θ$ between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is $120^{\circ}$, and $|\overrightarrow{a}|=4$, $|\overrightarrow{b}|=2$, compute the following:
$(1) \overrightarrow{a} \cdot \overrightarrow{b};$
$(2) (\overrightarrow{a} + \overrightarrow{b}) \cdot (\overrightarrow{a} - 2\overrigh... | 2\sqrt{3} | 126 | 6 |
math | In the cartesian coordinate system $(xOy)$, the parametric equations of line $l$ are $\begin{cases} x=-2-3t, \\ y=2-4t. \end{cases} (t \text{ is the parameter})$, and it intersects with curve $C$: $(y-2)^{2}-x^{2}=1$ at points $A$ and $B$.
(1) Find the length of $|AB|$.
(2) In the polar coordinate system established ... | \frac {30}{7} | 174 | 8 |
math | The ratio of boys to girls in Mr. Smith's class is 3:4, and there are 42 students in total. What percent of the students are boys. | 42.86\% | 36 | 7 |
math | Given the vector $v = (a_{n+1} - a_n, a_{n+1} - a_n)$, and $v$ is the direction vector of the line $y = x$. If $a_1 = 5$, calculate the sum of the first 10 terms of the sequence $\{a_n\}$. | 50 | 73 | 2 |
math | A point $ M$ is taken on the perpendicular bisector of the side $ AC$ of an acute-angled triangle $ ABC$ so that $ M$ and $ B$ are on the same side of $ AC$ . If $ \angle BAC\equal{}\angle MCB$ and $ \angle ABC\plus{}\angle MBC\equal{}180^{\circ}$ , find $ \angle BAC.$ | 30^\circ | 103 | 4 |
math | Quadrilateral $ABCD$ is inscribed in a circle with segment $AC$ a diameter of the circle. If $m\angle DAC = 15^\circ$ and $m\angle BAC = 45^\circ$, determine the ratio of the area of $ABCD$ to the area of the circle and express it as a common fraction in simplest radical form in terms of $\pi$ as $\frac{a+\sqrt{b}}{c\p... | 5 | 128 | 1 |
math | Given an ellipse $C$ with its center at the origin and the major axis along the $y$-axis, passing through points $M(4,2)$ and $N(\sqrt{6},3)$.<br/>$(1)$ Find the equation of the ellipse $C$;<br/>$(2)$ Let $R(x_{0},y_{0})$ be any point on the ellipse $C$. Two tangents are drawn from the origin $O$ to the circle $R:(x-x_... | 36 | 169 | 2 |
math | Given the fraction $\frac{1}{3}$, determine by how much it is greater than or less than the decimal $0.333333333$. | \frac{1}{3\cdot 10^9} | 37 | 14 |
math | Given Miki has 15 apples and 10 bananas, She uses her juicer to extract 9 ounces of apple juice from 3 apples and 10 ounces of banana juice from 2 bananas. She makes a banana-apple juice blend using 5 apples and 4 bananas. Calculate the percent of the blend that is apple juice. | 42.86\% | 72 | 7 |
math | Compare the number \(a\) with one, if
$$
a = 0.99999^{1.00001} \cdot 1.00001^{0.99999}.
$$ | a < 1 | 52 | 4 |
math | In the expansion of $(x + y)^n$, if the coefficient of the seventh term is the largest, then the possible value(s) for $n$ are $\_\_\_\_\_\_\_\_$. | n = 11, 12, 13 | 43 | 13 |
math | Find the derivative of the following functions:
(1) $y=(x+1)^2(x-1)$;
(2) $y=x^2\sin x$;
(3) $y=\frac{e^x+1}{e^x-1}$;
(4) $f(x)=\frac{e^x}{x-2}$. | \frac{e^x(x-3)}{(x-2)^2} | 81 | 17 |
math | Calculate the sum $(-1)^{-11} + (-1)^{-10} + (-1)^{-9} + \cdots + (-1)^9 + (-1)^{10} + (-1)^{11}$. | 0 | 52 | 1 |
math | For the partial solutions of the binary linear equation in $x$ and $y$, $ax+by=c$, as shown in the table:<br/>
| $x$ | $\ldots $ | $-2$ | $-1$ | $0$ | $1$ | $2$ | $\ldots $ |
|-----|-----------|------|------|----|----|----|----------|
| $y$ | $\ldots $ | $-1$ | $0$ | $1$| $2$| $3$| $\ldots $ |
$(1)$ The binary linea... | m \geqslant -2 | 287 | 8 |
math | The following $100$ numbers are written on the board: $$ 2^1 - 1, 2^2 - 1, 2^3 - 1, \dots, 2^{100} - 1. $$ Alice chooses two numbers $a,b,$ erases them and writes the number $\dfrac{ab - 1}{a+b+2}$ on the board. She keeps doing this until a single number remains on the board.
If the sum of all possible number... | 100 | 162 | 3 |
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