problem stringlengths 14 4.09k | solution stringlengths 1 802k | task_type stringclasses 3
values | problem_tokens int64 5 895 |
|---|---|---|---|
The length of the shortest trip from $A$ to $C$ along the edges of a cube shown is the length of 4 edges. How many different 4-edge trips are there from $A$ to $C$?
[asy]
size(4cm,4cm);
pair a1, b1, c1, d1;
a1=(1,1);
b1=(0,1);
c1=(1.6,1.4);
d1=(1,0);
pair e1, f1, g1, h1;
e1=(0,0);
f1=c1-(a1-d1);
g1=b1+(c1-a1);
h1=e1+(... | 12 | math | 240 |
In rectangle $ABCD$, the diagonals intersect at point $O$. If $AB = 12$ and $BC = 16$, what is $\sin \angle BAO$? | 0.6 | math | 41 |
Given the function $f(x)=2|x|+|2x-m|$ where $m>0$, and the graph of the function is symmetric about the line $x=1$.
$(Ⅰ)$ Find the minimum value of $f(x)$.
$(Ⅱ)$ Let $a$ and $b$ be positive numbers such that $a+b=m$. Find the minimum value of $\frac{1}{a}+\frac{4}{b}$. | \frac{9}{4} | math | 96 |
Given the function f(x) = e^x + ae^-x, where 'a' is a constant. If f(x) is an odd function, find the value of 'a'. If f(x) is an increasing function on R, find the range of 'a'. | a \in (-\infty, 0] | math | 57 |
The center of a circle is on the positive half of the $x$-axis, and its radius is the length of the imaginary semi-axis of the hyperbola $\dfrac{x^{2}}{16} - \dfrac{y^{2}}{9} = 1$, and it is tangent to the asymptotes of this hyperbola. The equation of the circle is __________. | (x-5)^{2} + y^{2} = 9 | math | 84 |
Given that the elevation angle of point B in the same direction as observed from point A is 60^{\circ}, and the depression angle of point C is 70^{\circ}, find the measure of angle ∠BAC. | 130^{\circ} | math | 50 |
Given \(\frac{1+\sin x}{\cos x}=\frac{22}{7}\), and \(\frac{1+\cos x}{\sin x}=\frac{m}{n}\) (where \(\frac{m}{n}\) is in simplest form). Find \(m+n\). | 44 | math | 67 |
A geometric sequence contains $5$ terms, each of which is a positive integer less than $100$, and the sum of these $5$ terms is $121$. Find the sum of the terms in odd positions of the sequence. | 91 | math | 51 |
Given two sets $A=\{x \mid 1 < x - 1 < 3\}$ and $B=\{x \mid (x - 3)(x - a) < 0\}$,
1. When $a=5$, find $A \cap B$ and $A \cup B$.
2. If $A \cap B = B$, find the range of the real number $a$. | [2, 4] | math | 90 |
If the power function $f(x) = x^{k}$ is a decreasing function on $(0, +\infty)$, determine the value of $k$. | -1 | math | 34 |
For the complex number $z=\frac{3+i}{1-2i}$ (where $i$ is the imaginary unit), find $|\overline{z}|$. | \sqrt{2} | math | 36 |
$2019$ students are voting on the distribution of $N$ items. For each item, each student submits a vote on who should receive that item, and the person with the most votes receives the item (in case of a tie, no one gets the item). Suppose that no student votes for the same person twice. Compute the maximum possibl... | 1009 | math | 100 |
What is the minimum value of the expression $x^2 + y^2 - 8x + 6y + 20$ for real $x$ and $y$? | -5 | math | 39 |
Given that $x$ is the arithmetic mean between 4 and 16, find the value of $x$. | 10 | math | 24 |
Consider the parabola $C$: $y^{2}=4x$ with focus $F$. The line $l$ passing through $F$ intersects $C$ at points $A$ and $B$. Given point $M(-1,2)$, if $\overrightarrow{MA} \cdot \overrightarrow{MB}=0$, then the slope of line $l$ is $k=$\_\_\_\_\_\_. | k=1 | math | 90 |
In triangle \(ABC\) with a \(120^\circ\) angle at vertex \(A\), the bisectors \(AA_1\), \(BB_1\), and \(CC_1\) are drawn. Find the angle \(C_1 A_1 B_1\). | 90^\circ | math | 60 |
In a box, there are 6 ballpoint pens: 3 are black, 2 are blue, and 1 is red. Three pens are randomly selected from the box.
(1) How many basic events are there in this experiment? If the 3 black ballpoint pens are labeled as A, B, C, the 2 blue ballpoint pens as d, e, and the 1 red ballpoint pen as x, and a basic eve... | \frac{4}{5} | math | 146 |
Triangle $ABC$ has sides tangent to a circle with center $O$. Given that $\angle ABC = 72^\circ$ and $\angle BAC = 67^\circ$, find $\angle BOC$, where the circle is the excircle opposite vertex $C$. | 110.5^\circ | math | 57 |
The graph of the function $y=\sin x$ is translated according to the vector $\overrightarrow{a}=(-\frac{\pi}{2}, 2)$ and then coincides with the graph of the function $g(x)$. Determine the function $g(x)$. | \cos x+2 | math | 57 |
1. Given $\tan \frac{\alpha}{2} = \frac{1}{2}$, find the value of $\sin\left(\alpha + \frac{\pi}{6}\right)$.
2. Given $\alpha \in \left(\pi, \frac{3\pi}{2}\right)$ and $\cos\alpha = -\frac{5}{13}$, $\tan \frac{\beta}{2} = \frac{1}{3}$, find the value of $\cos\left(\frac{\alpha}{2} + \beta\right)$. | -\frac{17\sqrt{13}}{65} | math | 121 |
In a positive geometric sequence $\{a_n\}$, it is known that $a_1a_2a_3=4$, $a_4a_5a_6=12$, and $a_{n-1}a_na_{n+1}=324$. Determine the value of $n$. | 14 | math | 69 |
Given the function $y=\cos \left(2x-\frac{\pi }{6}\right)$, determine the horizontal shift required to obtain the graph of the function $y=\sin \left(2x+\frac{\pi }{6}\right)$. | \frac{\pi }{12} | math | 55 |
If ${x^2}+x=5+\sqrt{5}$, then the value of $x$ is ____. | \sqrt{5} \text{ or } -\sqrt{5} - 1 | math | 26 |
Find all positive integers $k$ , so that there exists a polynomial $f(x)$ with rational coefficients, such that for all sufficiently large $n$ , $$ f(n)=\text{lcm}(n+1, n+2, \ldots, n+k). $$ | k = 1 | math | 62 |
Let $x,$ $y,$ and $z$ be positive real numbers. Find the minimum value of
\[\frac{4z}{2x + y} + \frac{4x}{y + 2z} + \frac{y}{x + z}.\] | 3 | math | 60 |
Given \(\vec{O}P = (2,1)\), \(\vec{O}A = (1,7)\), and \(\vec{O}B = (5,1)\), let \(X\) be a point on the line \(OP\) (with \(O\) as the coordinate origin). Find the measure of the angle \(\angle AXB\) when the dot product \(\vec{X}A \cdot \vec{X}B\) is minimized. | \arccos\left(-\frac{4 \sqrt{17}}{17}\right) | math | 102 |
In the Cartesian coordinate system \( xOy \), the function \( f(x) = a \sin(ax) + \cos(ax) \) (where \( a > 0 \)) is defined. Find the area of the closed region formed by the graph of \( f(x) \) over its smallest positive period and the graph of \( g(x) = \sqrt{a^2 + 1} \). | \frac{2 \pi}{a} \sqrt{a^{2}+1} | math | 86 |
A point $P$ is chosen uniformly at random inside a square of side length 2. If $P_{1}, P_{2}, P_{3}$, and $P_{4}$ are the reflections of $P$ over each of the four sides of the square, find the expected value of the area of quadrilateral $P_{1} P_{2} P_{3} P_{4}$. | 8 | math | 85 |
Among the rational numbers $\left(-1\right)^{2}$, $\left(-1\right)^{3}$, $-1^{2}$, $|-1|$, $-\left(-1\right)$, $\frac{1}{{-1}}$, determine the number of them that equals 1. | 3 | math | 67 |
Given the function $f(x)=\ln (x+1)- \frac{ax}{x+1}-x$, where $a\in R$.
(I) Find the monotonic intervals of the function $f(x)$ when $a > 0$.
(II) If there exists $x > 0$ such that $f(x)+x+1 < - \frac{x}{x+1}$ ($a\in Z$) holds, find the minimum value of $a$. | 5 | math | 103 |
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 | math | 100 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively.
$(1)$ If $\cos \left(A+ \frac {\pi}{6}\right)=\sin A$, find the value of $A$;
$(2)$ If $\cos A= \frac {1}{4}$ and $4b=c$, find the value of $\sin B$. | \frac {1}{4} | math | 98 |
What is the smallest positive integer $n$ such that $23n \equiv 5678 \pmod{11}$? | 2 | math | 30 |
In the arithmetic sequence $\{a_n\}$, if $a_1 + a_4 + a_7 = 39$ and $a_2 + a_5 + a_8 = 33$, calculate the value of $a_3 + a_6 + a_9$. | 27 | math | 64 |
What is the minimum value of the expression $2x^2+3y^2+8x-24y+62$ for real $x$ and $y$? | 6 | math | 39 |
Given the function $f(x) = ax - \ln x$, if $f(x) > 1$ always holds true in the interval $(1, +\infty)$, determine the range of the real number $a$. | [1, +\infty) | math | 48 |
A traffic light cycles as follows: green for 45 seconds, then yellow for 5 seconds, and then red for 50 seconds. Mark chooses a random five-second interval to observe the light. What is the probability that the color changes during his observation? | \frac{3}{20} | math | 54 |
From a square with a side length of 5, four corner unit squares are cut out (see figure). What is the area of the largest square that can be cut out from the remaining part? | 9 | math | 40 |
Given that a line passes through points $A(1,3)$ and $B(2,5)$, calculate the slope of this line. | 2 | math | 30 |
A store increased the original price of a product by $20\%$ and then decreased the new price by $x\%$. The final price turned out to be $88\%$ of the original price. Find the value of $x$. | 26.67 | math | 53 |
Given the non-negative real numbers $x$ and $y$ that satisfy the constraints:
$$
\begin{cases}
x+y-3 \leq 0 \\
2x+y-4 \geq 0
\end{cases}
$$
find the maximum value of $z=2x+3y$. | 8 | math | 67 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, with $C= \frac{\pi}{3}$, $b=8$, and the area of $\triangle ABC$ is $10\sqrt{3}$.
$(1).$ Find the value of $c$; $(2).$ Find the value of $\cos(B-C)$. | \frac{13}{14} | math | 97 |
Given that the radius of the circumcircle of acute triangle ABC is $\frac{\sqrt{3}}{3}BC$, and $AB = 3$, $AC = 4$, find the length of $BC$. | \sqrt{13} | math | 45 |
A rectangular park is one-third as wide as it is long, and it is completely enclosed by 90 meters of fencing. What is the number of square meters in the area of the park? | 379.6875 | math | 40 |
Given a sequence $\{a_{n}\}$ where $a_{1}=1$, and $\left(n+1\right)a_{n+1}-na_{n}=2n+1$, if $b_{n}=[\lg a_{n}]$, and the sum of the first $n$ terms of the sequence $\{b_{n}\}$ is $T_{n}$, calculate $T_{2022}$. | 4959 | math | 92 |
Given $\sin(\alpha - \beta) = \frac{1}{3}$ and $\cos \alpha \sin \beta = \frac{1}{6}$, find $\cos(2\alpha + 2\beta)$. | \frac{1}{9} | math | 49 |
Let $a,$ $b,$ $c$ be positive real numbers such that $a + b + c = 1.$ Find the minimum value of $a^2 + 2b^2 + c^2.$ | \frac{2}{5} | math | 47 |
Let \[f(x) = \left\{
\begin{array}{cl}
x^2 - 4 &\text{ if }x>6, \\
3x + 2 &\text{ if } -6 \le x \le 6, \\
5 &\text{ if } x < -6.
\end{array}
\right.\] If \( x \) is a multiple of 3, add 5 to \( f(x) \). Find \( f(-8) + f(0) + f(9) \). | 94 | math | 118 |
Let's consider a modified version of Lucas numbers defined by the recursion $M_0 = 3, M_1 = 2$, and subsequently $M_n = M_{n-1} + M_{n-2}$. Compute the units digit of $M_{M_{12}}$. | 1 | math | 62 |
The arithmetic sequence $\{a_{n}\}$ has a first term $a_{1}=1$, a common difference $d \neq 0$, and satisfies the condition $a_{3} \cdot a_{4} = a_{12}$.
(1) Find the general term formula for the sequence $\{a_{n}\}$;
(2) Let $b_{n} = a_{n} \cdot 2^{n}$, find the sum of the first $n$ terms, $T_{n}$, for the sequence $\... | T_{n} = (n-1) \cdot 2^{n+1} + 2 | math | 123 |
In Weather Town, the forecast predicts a 75% chance of rain each day during the upcoming five-day festival. On days it doesn’t rain, the weather will be sunny. Jasmine and Lee hope for exactly two sunny days during this time, as they plan indoor activities otherwise. What is the probability they get the weather they wa... | \frac{135}{512} | math | 75 |
Determine the range of the function $f(x) = \left(\frac{1}{4}\right)^x - 3\left(\frac{1}{2}\right)^x + 2$ where $x \in [-2, 2]$. | [-\frac{1}{4}, 6] | math | 55 |
Complex numbers $a$, $b$, $c$ form an equilateral triangle with side length 24 in the complex plane. If $|a + b + c| = 48$, find $|ab + ac + bc|$. | 768 | math | 51 |
Jonas sets his watch correctly at 8:00 AM and notices that his watch reads 9:48 AM at the actual time of 10:00 AM. Assuming his watch loses time at a constant rate, calculate the actual time when his watch will first read 5:00 PM. | 6:00 PM | math | 65 |
Given the function $f(x)=\log_{2}(1-x)-\log_{2}(1+x)$.
(1) Find the domain of the function $f(x)$;
(2) Determine the parity (odd or even) of $f(x)$;
(3) Does the equation $f(x)=x+1$ have any roots? If it does have a root $x_{0}$, find an interval $(a,b)$ with a length of $\frac{1}{4}$ such that $x_{0}\in(a,b)$; otherwi... | \left(-\frac{1}{2},-\frac{1}{4}\right) | math | 139 |
Calculate the arc length of the curve described by the equation in the rectangular coordinate system.
$$
y = \ln \left(1-x^{2}\right), \quad 0 \leq x \leq \frac{1}{4}
$$ | \frac{1}{2} \ln \left( \frac{5}{3} \right) + \frac{1}{4} | math | 52 |
Given that $C_{n+1}^{7} - C_{n}^{7} = C_{n}^{8}$, calculate the value of $n$. | 14 | math | 36 |
Given the function $f(x) = x^2 + 1$, calculate the value of $f(a + 1)$. | a^{2}+2a+2 | math | 27 |
Solve the following system of equations:
$$
\begin{array}{r}
x^{2} + y \sqrt{x y} = 105 \\
y^{2} + x \sqrt{y x} = 70
\end{array}
$$ | (x, y) = (9, 4) | math | 57 |
During the holiday, a school organizes a trip for 360 teachers and students. A bus rental company offers two types of buses for hire: Type A buses have 40 seats each and a rental fee of 400 yuan; Type B buses have 50 seats each and a rental fee of 480 yuan. The minimum rental fee required to hire buses from this compan... | 3520 | math | 86 |
Let $b_1, b_2, \ldots$ be a sequence determined by the rule $b_n= \frac{b_{n-1}}{3}$ if $b_{n-1}$ is divisible by 3, and $b_n = 2b_{n-1} + 2$ if $b_{n-1}$ is not divisible by 3. Determine how many positive integers $b_1 \le 3000$ are such that $b_1$ is less than each of $b_2$, $b_3$, and $b_4$. | 2000 | math | 129 |
Shea and Ara were the same height originally. Shea has grown by 25%, and Ara has grown by one-third the number of inches Shea has grown. If Shea is now 70 inches tall, calculate Ara's current height in inches. | 60.67 | math | 51 |
In the polar coordinate system, given point A (2, $\frac{\pi}{2}$), and point B lying on the line $l: \rho \cos \theta + \rho \sin \theta = 0$ (where $0 \leq \theta \leq 2\pi$), find the polar coordinates of point B when the length of segment AB is the shortest. | (\sqrt{2}, \frac{3\pi}{4}) | math | 82 |
**A park is designed with two circular walking tracks. The smaller track has a diameter of 15 meters. If the diameter of the larger track is 20 meters, what percent increase in area results from the larger track compared to the smaller one?** | 77.78\% | math | 53 |
Let \( x \) and \( y \) be positive real numbers. Find the minimum value of
\[
\frac{\sqrt{(x^2 + y^2)(4x^2 + y^2)}}{xy}.
\] | 3 | math | 50 |
In a certain sequence, the first term is $a_1 = 2010$ and the second term is $a_2 = 2011$. Furthermore, the values of the remaining terms are chosen so that $a_n + a_{n+1} + a_{n+2} = 2n$ for all $n \geq 1$. Determine $a_{1000}$. | 2676 | math | 90 |
Determine the value of \( n \) such that \( 2^7 \cdot 3^4 \cdot n = 10! \). | 350 | math | 32 |
Interior numbers begin in the third row of Pascal's Triangle. The sum of the interior numbers in the fourth row is 6. The sum of the interior numbers of the fifth row is 14. What is the sum of the interior numbers of the seventh row? | 62 | math | 54 |
Given that the probability that each ball falls into bin k is 3^(-k) for k = 1,2,3,..., find the probability that the blue ball falls into a lower-numbered bin than the yellow ball. | \frac{7}{16} | math | 48 |
Given that \(x\) is real and \(x^3 + \frac{1}{x^3} = 116\), find the value of \(x + \frac{1}{x}\). | 4 | math | 44 |
Hexagon $ABCDEF$ has its center at $G$. Each vertex and the center are to be assigned one of the digits $1$ through $7$, with each digit used exactly once, such that the sums of the numbers on the lines $AGC$, $BGD$, and $CGE$ are all equal. How many ways can this be done? | 144 | math | 75 |
Place water in air at a temperature of $\theta$℃ to cool it. If the original temperature of the water is $\theta_1$℃ ($\theta < \theta_1$), the temperature $\theta$℃ of the object after $t$ minutes can be obtained by the formula $\theta = \theta + (\theta_1 - \theta)e^{-kt}$, where $k$ is a positive constant determined... | 22.8\,^\circ\text{C} | math | 239 |
A bookshelf has $6$ different English books and $2$ different math books. If you randomly pick $1$ book, calculate the number of different ways to pick the book. | 6 + 2 = 8 | math | 38 |
On the sides $AB$, $BC$, and $AC$ of triangle $ABC$, whose area is 75, points $M$, $N$, and $K$ are respectively located. It is known that $M$ is the midpoint of $AB$, the area of triangle $BMN$ is 15, and the area of triangle $AMK$ is 25. Find the area of triangle $CNK$. | 15 | math | 91 |
If $B$ is an angle such that $\tan B - \sec B = -1,$ determine all possible values of $\cos B,$ separated by commas. | 1 | math | 33 |
Solve the system of equations:
$$
\begin{aligned}
& \frac{1}{x} = y + z \\
& \frac{1}{y} = z + x \\
& \frac{1}{z} = x + y
\end{aligned}
$$ | \left( \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right), \left( -\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2} \right) | math | 59 |
Given that the sum of the first $n$ terms of a geometric sequence $\{a_n\}$ is $S_n=(a-2)\cdot3^{n+1}+2$, find the constant $a$. | \dfrac {4}{3} | math | 46 |
The opposite of the number $2023$ is $-2023$. | -2023 | math | 19 |
Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that $f(1) = 2$ and
\[f(x^2 + y^2) = (x + y) (f(x) - f(y))\]for all real numbers $x$ and $y$.
Let $n$ be the number of possible values of $f(3),$ and let $s$ be the sum of all possible values of $f(3).$ Find $n \times s.$ | 6 | math | 114 |
Find the value of $s$ for which the vector
\[\bold{u} = \begin{pmatrix} 1 \\ -2 \\ -4 \end{pmatrix} + s \begin{pmatrix} 5 \\ 3 \\ -2 \end{pmatrix}\] is closest to
\[\bold{b} = \begin{pmatrix} 3 \\ 3 \\ 4 \end{pmatrix}.\] | \frac{9}{38} | math | 95 |
Taxi driver Li Shifu's operation on the first afternoon of National Day was conducted on a north-south road. If heading south is denoted as "$-$" and heading north is denoted as "$+$", his driving situation that afternoon was as follows: (unit: kilometers, each trip with passengers $)-2,-3,-6,+8,-9,-7,-5,+13.$
$(1)$ W... | 106.08 \text{ yuan} | math | 303 |
Find \( n > 1 \) such that using stamp denominations of \( n \) and \( n+2 \), it is possible to obtain any value \( \geq 2n + 2 \). | 3 | math | 45 |
An equilateral triangle is inscribed in a circle, then a circle is inscribed in this triangle, a square is then inscribed in this inner circle, and finally, a circle is inscribed in that square. What is the ratio of the area of the smallest circle to the area of the largest circle? | \frac{1}{8} | math | 63 |
Given that the monotonically decreasing function $f(x)$ defined on $(-\infty, 3]$ satisfies $f(1+\sin ^{2}x)\leqslant f(a-2\cos x)$ for all real numbers $x$, determine the range of values for $a$. | (-\infty,-1] | math | 64 |
Given a cuboid with length, width, and height as $2a$, $a$, and $a$ respectively, and all its vertices are on a sphere, calculate the surface area of the sphere. | 6\pi a^2 | math | 42 |
Let point $C(0,p)$ lie on the $y$-axis and another point $D(x,0)$ lie on the $x$-axis, where $x$ and $p$ are positive integers with $D$ located between $O(0,0)$ and $B(12,0)$. Find an expression for the area of $\triangle COD$ in terms of $x$ and $p$. | \frac{xp}{2} | math | 89 |
Using three-digit powers of $3$ and $7$ in a cross-number puzzle, determine the unique possible digit for the intersecting square. | 3 | math | 29 |
First, find the derivative of the following functions and calculate the derivative at \\(x=\pi\\).
\\((1) f(x)=(1+\sin x)(1-4x)\\) \\((2) f(x)=\ln (x+1)-\dfrac{x}{x+1}\\). | \dfrac{\pi}{(\pi+1)^{2}} | math | 66 |
For the inequality $kx^{2}-kx+4\geqslant 0$ to hold for any $x\in R$, the range of values for $k$ is ____. | [0,16] | math | 42 |
Factor the following expressions:
\\((1) 5x^{2}+6xy-8y^{2}\\)
\\((2) x^{2}+2x-15-ax-5a\\) | (x+5)(x-3-a) | math | 46 |
The common chord length of Circle \\(O_{1}\\): \\(x^{2}+y^{2}-2x=0\\) and Circle \\(O_{2}\\): \\(x^{2}+y^{2}-4y=0\\) is to be calculated. | \dfrac {4 \sqrt {5}}{5} | math | 61 |
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 | math | 51 |
In the rectangular coordinate system $(xOy)$, the parametric equations of curve $C_1$ are given by $\begin{cases} x=2+2\cos{\varphi}, \\ y=2\sin{\varphi} \end{cases}$ where $\varphi$ is a parameter. Curve $C_2$ has a polar coordinate equation of $\rho =4\sin{\theta}$ with the origin $O$ as the pole and the positive hal... | \alpha = \frac{3\pi}{4} | math | 259 |
Given a point M(2, 1), draw a line l that intersects the ellipse $\frac{x^2}{16} + \frac{y^2}{4} = 1$ at points A and B. If point M is the midpoint of segment AB, find the equation of line l. | x + 2y - 4 = 0 | math | 64 |
For vectors \(\mathbf{v} = \begin{pmatrix} 2 \\ 3 \end{pmatrix}\) and \(\mathbf{w} = \begin{pmatrix} -4 \\ 1 \end{pmatrix}\), compute \(\text{proj}_{\mathbf{w}} \mathbf{v}\). | \begin{pmatrix} \frac{20}{17} \\ -\frac{5}{17} \end{pmatrix} | math | 75 |
Given the numbers \( x_{1}, \ldots, x_{n} \in \left(0, \frac{\pi}{2}\right) \), find the maximum value of the expression
\[
A = \frac{\sin x_{1} + \ldots + \sin x_{n}}{\sqrt{\operatorname{tg}^{2} x_{1} + \ldots + \operatorname{tg}^{2} x_{n} + n}}
\] | \frac{\sqrt{n}}{2} | math | 104 |
In a plane, \( n \) lines are drawn such that each pair of lines intersects, but no four lines pass through a single point. There are a total of 16 intersection points, and 6 of these points are passed through by three lines. Find \( n \). | 8 | math | 58 |
Find the length of the common chord of circle $C_{1}$: $x^{2}+y^{2}-9=0$ and circle $C_{2}$: $x^{2}+y^{2}-6x+8y+9=0$. | \frac{24}{5} | math | 57 |
An ellipse has its foci at $(1, 1)$ and $(1, 3)$. Given that it passes through the point $(6, 2)$, its equation can be written in the form \[\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\] where $a, b, h, k$ are constants, and $a$ and $b$ are positive. Find $a+k.$ | 7 | math | 103 |
Express the production of new energy vehicles in China in 2022 (7,003,000) in scientific notation. | 7.003 \times 10^{6} | math | 30 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.