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 | Bob, a spherical person, is floating around peacefully when Dave the giant orange fish launches him straight up 23 m/s with his tail. If Bob has density 100 $\text{kg/m}^3$ , let $f(r)$ denote how far underwater his centre of mass plunges underwater once he lands, assuming his centre of mass was at water level when ... | 3 | 150 | 1 |
math | Given the function $f(x)=\left\{{\begin{array}{l}{{x^2}-m({2x-1})+{m^2},x≤2,}\\{{2^{x+1}},x＞2,}\end{array}}\right.$ find the range of $m$ such that $f(x)$ reaches its minimum value when $x=2$. | [2,4] | 83 | 5 |
math | In the diagram below, lines $m$ and $n$ are parallel. Find the measure of angle $y$ in degrees. [asy]
size(200);
import markers;
pair A = dir(-22)*(0,0);
pair B = dir(-22)*(4,0);
pair C = dir(-22)*(4,2);
pair D = dir(-22)*(0,2);
pair F = dir(-22)*(0,1.3);
pair G = dir(-22)*(4,1.3);
pair H = dir(-22)*(2,1);
pair I = dir... | 135^\circ | 362 | 5 |
math | A section of a rectangular parallelepiped $A B C D A_{1} B_{1} C_{1} D_{1}$ by a plane passing through its vertex $D$ intersects the lateral edges $A A_{1}$, $B B_{1}$, and $C C_{1}$ at points $K$, $M$, and $N$, respectively. Find the ratio of the volume of the pyramid with vertex at point $P$ and base $D K M N$ to the... | \frac{m}{3(m+n)} | 147 | 9 |
math | There are $2024$ cities in a country, every two of which are bidirectionally connected by exactly one of three modes of transportation - rail, air, or road. A tourist has arrived in this country and has the entire transportation scheme. He chooses a travel ticket for one of the modes of transportation and the city fr... | 1012 | 150 | 4 |
math | Several points, including points \( A \) and \( B \), are marked on a line. We consider all possible segments with endpoints at the marked points. Vasya counted that point \( A \) is inside 40 of these segments, and point \( B \) is inside 42 segments. How many points were marked? (The endpoints of a segment are not co... | 14 | 83 | 2 |
math | Given that there are \( C \) integers that satisfy the equation \( |x-2| + |x+1| = B \), find the value of \( C \). | 4 | 37 | 1 |
math | When $x < \frac{5}{4}$, the range of the function $f(x)=8x+\frac{1}{4x-5}$ is . | (-\infty,10-2 \sqrt{2}] | 41 | 14 |
math | Let \( b \) be a real number randomly selected from the interval \( [1, 25] \). Then, \( m \) and \( n \) are two relatively prime positive integers such that \( \frac{m}{n} \) is the probability that the equation \( x^4 + 36b^2 = (9b^2 - 15b)x^2 \) has \textit{at least} two distinct real solutions. Find the value of \... | 2 | 108 | 1 |
math | The Bank of Springfield has introduced a new plan called the "Ultra Savings Account" which compounds annually at a rate of 2%. If Lisa decides to invest $1500 in this new account, how much interest will she earn after 10 years? | 328.49 | 53 | 6 |
math | Let \( n \) be a positive integer greater than two. What are the largest value \( h \) and the smallest value \( H \) such that
$$
h < \frac{a_{1}}{a_{1}+a_{2}}+\frac{a_{2}}{a_{2}+a_{3}}+\ldots+\frac{a_{n}}{a_{n}+a_{1}} < H
$$
for any positive numbers \( a_{1}, a_{2}, \ldots, a_{n} \)? | h = 1 \text{ and } H = n-1 | 119 | 14 |
math | $n$ is a natural number. Given $3n \cdot 3n$ table, the unit cells are colored white and black such that starting from the left up corner diagonals are colored in pure white or black in ratio of 2:1 respectively. ( See the picture below). In one step any chosen $2 \cdot 2$ square's white cells are colored orang... | n | 136 | 2 |
math | To calculate $41^2$, Tom mentally computes $40^2$ and adds a number. To find $39^2$, Tom subtracts a number from $40^2$. What number does he add to calculate $41^2$ and subtract to calculate $39^2$? | 79 | 66 | 2 |
math | Several schoolchildren went mushroom picking. The schoolchild who gathered the most mushrooms collected \( \frac{1}{5} \) of the total amount of mushrooms, while the one who gathered the least collected \( \frac{1}{7} \) of the total amount. How many schoolchildren were there? | 6 | 63 | 1 |
math | Given the function $f(x) = 2\sin(\omega x + \varphi)$ ($\omega > 0$, $|\varphi| < \frac{\pi}{2}$) passes through point B(0, -1), and is monotonic in the interval $\left(\frac{\pi}{18}, \frac{\pi}{3}\right)$, the graph of $f(x)$ coincides with its original graph after being shifted left by $\pi$ units. When $x_1, x_2 \i... | -1 | 179 | 2 |
math | If each face of a tetrahedron is not an isosceles triangle, calculate the minimum number of edges of different lengths. | 3 | 28 | 1 |
math | A point inside the circle $x^2 + y^2 = 10x$, identified as $(5,3)$, has $k$ chords whose lengths form an arithmetic sequence. The shortest chord length is the first term $a_1$ of the sequence, and the longest chord length is the last term $a_k$. If the common difference $d$ is within the interval $\left[\frac{1}{3}, \f... | 5 | 106 | 1 |
math | There exists a scalar \(d\) so that
\[\mathbf{i} \times (\mathbf{w} \times \mathbf{j}) + \mathbf{j} \times (\mathbf{w} \times \mathbf{k}) + \mathbf{k} \times (\mathbf{w} \times \mathbf{i}) = d \mathbf{w}\]for all vectors \(\mathbf{w}.\) Find \(d.\) | 0 | 99 | 1 |
math | Given that the graph of the function $f(x) = 3\sin\left(\frac{x}{2}\right) - 4\cos\left(\frac{x}{2}\right)$ is symmetric with respect to the line $x=\theta$, find $\sin\theta$. | \sin\theta = -\frac{24}{25} | 59 | 15 |
math | Let \( n \in \mathbf{Z}_{+} \). Does there exist a positive real number \(\varepsilon=\varepsilon(n)\) such that for any \( x_{1}, x_{2}, \cdots, x_{n} > 0 \), the following inequality holds?
$$
\begin{array}{l}
\left(\prod_{i=1}^{n} x_{i}\right)^{\frac{1}{n}} \\
\leqslant(1-\varepsilon) \frac{1}{n} \sum_{i=1}^{n} x_{i... | \varepsilon = \frac{1}{n} | 171 | 11 |
math | The equation \(x^2 + 16x = 100\) has two solutions. The positive solution has the form \(\sqrt{c} - d\) for positive natural numbers \(c\) and \(d\). What is \(c+d\)? | 172 | 55 | 3 |
math | Find the distance from the center of the circle $(x+1)^{2}+y^{2}=2$ to the line $y=2x+3$. | \frac{\sqrt{5}}{5} | 35 | 10 |
math | What is the average of all the integer values of $N$ such that $\frac{N}{84}$ is strictly between $\frac{4}{9}$ and $\frac{2}{7}$? | 31 | 42 | 2 |
math | Given the equation \((a-1)(\sin 2x+\cos x)+(a-1)(\sin x-\cos 2x)=0\), where \( a < 0 \), find the number of solutions for \( x \) in the interval \( (-\pi, \pi) \). | 4 | 65 | 1 |
math | Given that
\[
2^{-\frac{5}{3} + \sin 2\theta} + 2 = 2^{\frac{1}{3} + \sin \theta},
\]
compute \(\cos 2\theta.\) | -1 | 54 | 2 |
math | Given that the eccentricity of the ellipse $C:\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 (a > b > 0)$ is $\frac{1}{2}$, and point $M(\sqrt{3},\frac{\sqrt{3}}{2})$ is on ellipse $C$.
(I) Find the equation of the ellipse $C$;
(II) If line $l$, which does not pass through the origin $O$, intersects ellipse $C$ at points ... | \sqrt{3} | 156 | 5 |
math | Given that the cost per unit of a certain product is $6$ yuan, and the selling price per unit is $x$ yuan $(x > 6)$, with an annual sales volume of $u$ ten thousand units. If $21-u$ is directly proportional to $x^2-6x$, and when the selling price per unit is $10$ yuan, the annual sales volume is $\frac{23}{3}$ ten thou... | 36 | 161 | 2 |
math | 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 | 57 | 7 |
math | Find all triples of natural numbers $(a, b, c)$ for which the number $$ 2^a + 2^b + 2^c + 3 $$ is the square of an integer. | (1, 1, 1) | 47 | 9 |
math | The longer leg of a right triangle is $3$ feet shorter than three times the length of the shorter leg. The area of the triangle is $72$ square feet. What is the length of the hypotenuse, in feet? | \sqrt{505} | 49 | 7 |
math | If the parabola \( C_{m}: y = x^{2} - m x + m + 1 \) intersects line segment \( AB \) with endpoints \( A(0,4) \) and \( B(4,0) \) at exactly two points, what is the range of \( m \)? | \left[3, \frac{17}{3}\right] | 68 | 15 |
math | Given that a positive number $x$ has the property that $x\%$ of $x^2$ is $16$, find the value of $x$. | 12 | 35 | 2 |
math | The probability of A not losing is $\dfrac{1}{3} + \dfrac{1}{2}$. | \dfrac{1}{6} | 25 | 7 |
math | Let $ a,\ b$ are postive constant numbers.
(1) Differentiate $ \ln (x\plus{}\sqrt{x^2\plus{}a})\ (x>0).$
(2) For $ a\equal{}\frac{4b^2}{(e\minus{}e^{\minus{}1})^2}$ , evaluate $ \int_0^b \frac{1}{\sqrt{x^2\plus{}a}}\ dx.$ | \ln \left( \frac{(e - e^{-1}) (\sqrt{\frac{4}{(e - e^{-1})^2} + 1} + 1)}{2} \right) | 105 | 45 |
math | Given that the equation in terms of $x$, $\frac{2x-m}{x-3}-1=\frac{x}{3-x}$ has a positive solution, the range of values for $m$ is ____. | m > 3 \text{ and } m \neq 9 | 45 | 15 |
math | Given $\overrightarrow{a}=( \sqrt {3}\sin x,-1)$ and $\overrightarrow{b}=(1,\cos x)$,
(1) If $\overrightarrow{a} \perp \overrightarrow{b}$, find the value of $\tan x$.
(2) If $f(x)= \overrightarrow{a} \cdot \overrightarrow{b}$, find the smallest positive period and the maximum value of the function $f(x)$. | 2 | 101 | 1 |
math | Given that in triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, if $4\sin A - b\sin B = c\sin \left(A-B\right)$. Find the value of $a$.
$(1)$ Find the value of $a$;
$(2)$ If the area of $\triangle ABC$ is $\frac{\sqrt{3}(b^{2}+c^{2}-a^{2})}{4}$, find the maximu... | \text{Maximum Perimeter} = 12 | 123 | 11 |
math | Three people want to travel from city $A$ to city $B$, which is located 45 kilometers away from city $A$. They have two bicycles. The speed of a cyclist is 15 km/h, and the speed of a pedestrian is 5 km/h. What is the minimum time it will take for them to reach city $B$, given that the bicycles cannot be left unattende... | 3 \text{ hours} | 86 | 6 |
math | A machine can operate at various speeds, and some of the products it produces may have defects. The number of defective products produced per hour varies with the operating speed of the machine. Let $x$ denote the speed (in revolutions per second) and $y$ denote the number of defective products per hour. The following ... | 14 | 374 | 2 |
math | Consider the 100th, 101st, and 102nd rows of Pascal's triangle, denoted as sequences $(p_i)$, $(q_i)$, and $(r_i)$ respectively. Calculate:
\[
\sum_{i = 0}^{100} \frac{q_i}{r_i} - \sum_{i = 0}^{99} \frac{p_i}{q_i}.
\] | \frac{1}{2} | 97 | 7 |
math | Given three positive numbers $a$, $b$, and $c$, where $c = \frac{a + b}{2}$, determine the relationship between the arithmetic mean and geometric mean of these three numbers. | \frac{a+b+c}{3} \geq \sqrt[3]{abc} | 43 | 19 |
math | Given a function $f(x)$ defined on $R$ such that $f(x+2)=f(x)$, and when $x\in [-1,1]$, $f(x)=e^{1-|x|}-2$. Four statements are made about the function $f(x)$: ① $f(x)$ is an even function; ② The graph of $f(x)$ is symmetric about the line $x=2$; ③ The equation $f(x)=1-|x|$ has two distinct real roots; ④ $f(\frac{1}{2}... | ①② | 148 | 4 |
math | If three factors of $3x^4 - mx^2 + nx + p$ are $x-3$, $x+1$, and $x-2$, find the value of $|3m - 2n|$. | 25 | 49 | 2 |
math | 6 people stand in a row, with person A not at either end. Find the number of ways to arrange them so that person A and person B are not adjacent. | 288 | 34 | 3 |
math | Consider a set of 16 integers from -8 to 7, inclusive, arranged to form a 4-by-4 square. Find the value of the common sum of the numbers in each row, column, and main diagonal. | -2 | 48 | 2 |
math | A ray of light is emitted from point A (-3, 5) to the x-axis, and after reflection, it passes through point B (2, 10). Find the distance from A to B. | 5 \sqrt {10} | 44 | 7 |
math | Let $x, y$ be integers that satisfy the equation $y^2 + 3x^2y^2 = 30x^2 + 517$. Then, $3x^2y^2 =$ ? | 588 | 50 | 3 |
math | Determine for which \( n \in \mathbf{N} \), the product \( \pi_{n} \) of the first \( n \) terms of the geometric sequence with first term \( a_1 = 1536 \) and common ratio \( q = -\frac{1}{2} \) is maximized. | 12 | 73 | 2 |
math | There is a closed sack with apples in it. Three friends tried to lift the sack and guess how many fruits were inside. The first guessed there were 18 apples, the second guessed 21, and the third guessed 22. When they opened the sack, they found out that one of them was off by 2, another by 3, and another by 6. How many... | 24 | 93 | 2 |
math | In the Cartesian coordinate system $xOy$, the equation of curve $C$ is $x^{2}-2x+y^{2}=0$. Taking the origin as the pole and the positive $x$-axis as the polar axis, a polar coordinate system is established. The polar equation of line $l$ is $\theta= \dfrac {\pi}{4}(\rho\in\mathbb{R})$.
(Ⅰ) Write the polar equation o... | 1 | 170 | 1 |
math | The product of two positive integers plus their sum is 187. The two integers are relatively prime, and each is less than 30. What is the sum of these two integers? | 49 | 40 | 2 |
math | Find the minimum value of the function \(f(x) = 7x^2 - 28x + 2003\) and evaluate \(f\) at \(x = 3\). | 1982 | 42 | 4 |
math | Given the function $f(x)=ae^{x}\cdot\cos x-x\sin x$, and the tangent line to the curve $y=f(x)$ at $(0,f(0))$ is parallel to $x-y=0$.
$(1)$ Find the value of $a$;
$(2)$ When $x\in\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$, investigate the number of zeros of the function $y=f(x)$, and explain the reason. | 2 | 109 | 1 |
math | The equation of the ellipse that has the same foci as the hyperbola $3x^{2}-y^{2}=3$ and an eccentricity reciprocal to it is $ \dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}=1$, where a and b are positive numbers. | \dfrac{x^{2}}{16}+\dfrac{y^{2}}{12}=1 | 74 | 23 |
math | Find the largest positive integer \( n \) such that the system of equations
$$
(x+1)^{2}+y_{1}^{2}=(x+2)^{2}+y_{2}^{2}=\cdots=(x+k)^{2}+y_{k}^{2}=\cdots=(x+n)^{2}+y_{n}^{2}
$$
has integer solutions \(\left(x, y_{1}, y_{2}, \cdots, y_{n}\right)\). | 3 | 114 | 1 |
math | Let $f(x)$ be an increasing function defined on $(0, +\infty)$, satisfying $f\left(\frac{x}{y}\right) = f(x) - f(y)$.
(1) Find the value of $f(1)$;
(2) Solve the inequality: $f(x-1) < 0$;
(3) If $f(2) = 1$, solve the inequality $f(x+3) - f\left(\frac{1}{x}\right) < 2$. | x \in (0, 1) | 112 | 9 |
math | A road of 1500 meters is being repaired. In the first week, $\frac{5}{17}$ of the total work was completed, and in the second week, $\frac{4}{17}$ was completed. What fraction of the total work was completed in these two weeks? And what fraction remains to complete the entire task? | \frac{8}{17} | 73 | 8 |
math | Given the function $f(x)=3\sin(2x-\frac{\pi}{3})$ with its graph denoted as $C$, determine the correct conclusion(s) from the following options (write out the numbers of all correct conclusions).
$①$ The graph $C$ is symmetrical about the line $x=\frac{11\pi}{12}$;
$②$ The graph $C$ is symmetrical about the point $(\fr... | ①② | 197 | 4 |
math | Find the derivative of the function $f(x)={\log_2}\frac{1}{x}$. | -\frac{1}{{x\ln2}} | 23 | 11 |
math | Decomposing the positive integer $12$ into the product of two positive integers can be done in three ways: $1 \times 12$, $2 \times 6$, and $3 \times 4$. Among these, $3 \times 4$ has the smallest absolute difference between the two numbers, and we call $3 \times 4$ the optimal decomposition of $12$. When $p \times q$ ... | 3^{50} - 1 | 225 | 8 |
math | Given the function $f(x)=ax+\ln x-1$, where $a$ is a constant,
(1) When $a\in(-\infty,-\frac{1}{e})$, if the maximum value of $f(x)$ in the interval $(0,e)$ is $-3$, find the value of $a$;
(2) When $a=-\frac{1}{e}$, if the function $g(x)=|f(x)|-\frac{\ln x}{x}-\frac{b}{2}$ has zero points, find the range of values for ... | [2-\frac{2}{e},+\infty) | 129 | 13 |
math | Let $A_1A_2 \dots A_{4000}$ be a regular $4000$ -gon. Let $X$ be the foot of the altitude from $A_{1986}$ onto diagonal $A_{1000}A_{3000}$ , and let $Y$ be the foot of the altitude from $A_{2014}$ onto $A_{2000}A_{4000}$ . If $XY = 1$ , what is the area of square $A_{500}A_{1500}A_{2500}A_{3500}$ ?
*Pro... | 2 | 168 | 1 |
math | Find $(-1)^{-11} + (-1)^{-10} + (-1)^{-9} + \cdots + (-1)^9 + (-1)^{10} + (-1)^{11}$. | 0 | 50 | 1 |
math | Let \( S = \{ 1, 2, \cdots, 2005 \} \). Find the smallest number \( n \) such that in any subset of \( n \) pairwise coprime numbers from \( S \), there is at least one prime number. | 16 | 61 | 2 |
math | One of the asymptotes of a hyperbola has equation \( y = 2x \). The foci of the hyperbola have the same \( x \)-coordinate, which is \( 4 \). Find the equation of the other asymptote of the hyperbola, giving your answer in the form \( y = mx + b \). | y = -2x + 16 | 72 | 9 |
math | In rectangle $ABCD$, $\overline{AB}=20$ and $\overline{BC}=10$. Let $E$ be a point on $\overline{CD}$ such that $\angle CBE=15^\circ$. What is $\overline{AE}$? | 20 | 60 | 2 |
math | Fill in the blanks with appropriate numbers.
80 grams = ___ kilograms
165 centimeters = ___ meters
4 jiao 9 fen = ___ yuan
13 yuan 7 fen = ___ yuan
5 tons 26 kilograms = ___ tons. | 5.026 | 55 | 5 |
math | If the equation $mx^2+3x-4=3x^2$ is a quadratic equation in one variable with respect to $x$, then determine the set of values for $m$. | m \neq 3 | 41 | 6 |
math | Same as question 4, but now we want one of the rectangle's sides to be along the hypotenuse. | 3 | 24 | 1 |
math | Given the sequence $\{a_n\}$, $a_1=1$, and $a_{n+1} = \frac{2a_n}{a_n+2}$ ($n\in\mathbb{N}^*$), determine the $m$ term of this sequence for which $\frac{2}{101}$ is the $m$ term. | 100 | 78 | 3 |
math | Given the set \( S = \{1, 2, 3, \ldots, 100\} \), the set \( S_1 \) is a non-empty subset of \( S \), and the difference between the largest and smallest elements in \( S_1 \) is called the "diameter" of \( S_1 \). Determine the total sum of the diameters of all subsets of \( S \) whose diameter is 91. | 9 \times 91 \times 2^{90} | 99 | 14 |
math | Solve in $\mathbb{C}^{3}$ the system
$$
\left\{\begin{array}{l}
a^{2}+a b+c=0 \\
b^{2}+b c+a=0 \\
c^{2}+c a+b=0
\end{array}\right.
$$ | (0, 0, 0) \ \text{and} \ \left( -\frac{1}{2}, -\frac{1}{2}, -\frac{1}{2} \right) | 68 | 46 |
math | The probability it will snow on Friday is $40\%$, and the probability it will snow on Saturday is $30\%$. If the probability of snow on a given day is independent of the weather on any other day, what is the probability it will snow on both days, expressed as a percent? | 12\% | 64 | 4 |
math | Given a geometric sequence with positive terms $\{a_n\}$, where $a_4=9$ and $a_6=27$, let $b_n=\log_{\sqrt{3}}(3a_n)$. Find the sum of the first $2017$ terms of the sequence $\{b_n\}$. | 2017 \times 1011 | 72 | 11 |
math | If \( x \) and \( y \) are real numbers such that \( x + y = 4 \) and \( xy = -2 \), evaluate the value of \( x + \frac{x^3}{y^2} + \frac{y^3}{x^2} + y \). | 440 | 65 | 3 |
math | Given $F(x) = \int_{0}^{x} (t^2 + 2t - 8)dt$,
(1) Find the monotonic intervals of $F(x)$;
(2) Find the maximum and minimum values of $F(x)$ on $[1, 3]$. | F(2) = -\frac{28}{3} | 66 | 14 |
math | Given a point $P$ inside an equilateral triangle with a side length of $1$, the distances from this point to the three sides of the triangle are $a$, $b$, and $c$ ($a, b, c > 0$), respectively. Find the range of possible values for $ab + bc + ca$. | (0, \frac{1}{4}] | 69 | 10 |
math | In the complex plane, three vertices of a square correspond to the complex numbers $\frac {3+i}{1-i}$, $-2+i$, and $0$, respectively. The complex number corresponding to the fourth vertex is \_\_\_\_\_\_. | -1+3i | 52 | 5 |
math | There are four rectangles, each 7 cm long and 5 cm wide. These rectangles can be arranged to form a larger rectangle (without overlapping). How many different ways can this be done (illustrate with sketches and list all solutions)? What is the maximum and minimum possible perimeter of the larger rectangle in centimeter... | 66 \text{ cm} \text{ (maximum)}, 48 \text{ cm} \text{ (minimum)} | 64 | 27 |
math | Let $d(n)$ denote the number of positive divisors of the positive integer $n$. What is the smallest positive real value of $c$ such that $d(n) \leq c \cdot \sqrt{n}$ holds for all positive integers $n$? | \sqrt{3} | 55 | 5 |
math | Given 1 gram, 2 grams, 3 grams, and 5 grams weights, each available in one piece, you can measure weights from 1 gram to 11 grams. Some weights can be measured in more than one way; for example, 3 grams can be measured using the 3 grams weight or using both 1 gram and 2 grams weights. What is the minimum weight that re... | 9 | 94 | 1 |
math | Given a set of data $1$, $x$, $5$, $7$ with a unique mode and a median of $6$, find the average value of the set. | 5 | 36 | 1 |
math | Cory has $4$ apples, $2$ oranges, and $2$ bananas. If Cory decides to eat one piece of his fruit per day for eight days, and he must eat all apples before finishing all the oranges and bananas, in how many orders can Cory eat the fruit? | 6 | 59 | 1 |
math | Let the system of equations be satisfied for positive numbers \( x, y, z \):
\[
\left\{
\begin{array}{l}
x^{2} + x y + y^{2} = 108 \\
y^{2} + y z + z^{2} = 49 \\
z^{2} + x z + x^{2} = 157
\end{array}
\right.
\]
Find the value of the expression \( x y + y z + x z \). | 84 | 111 | 2 |
math | The public security traffic police departments at all levels in Yunfu City remind citizens that they must strictly abide by the regulations of "one helmet per person" when riding a bike. A certain shopping mall in Yunan County simultaneously purchased two types of helmets, $A$ and $B$. It is known that purchasing $3$ t... | 2048 | 277 | 4 |
math | Many residents of a town gathered to spend a day in nature for a holiday. They hired all available vans, ensuring that each van carried the same number of people. Halfway through the journey, 10 vans broke down, so each of the remaining vans had to take one extra person. When it was time to return home, unfortunately, ... | 900 | 103 | 3 |
math | Given a moving point P, the lengths of the tangents from P to the two circles $x^2+y^2-1=0$ and $x^2+y^2-8x-8y+31=0$ are equal. Find the equation of the trajectory of point P. | x+y-4=0 | 63 | 6 |
math | Between 1000 and 9999, find the number of four-digit integers with distinct digits where the absolute difference between the first and last digit is 2. | 840 | 37 | 3 |
math | Given a set of 10 scores: 10, 10, 10, 9, 10, 8, 8, 10, 10, 8, calculate the standard deviation of the scores. | 0.9 | 53 | 3 |
math | $16.$ If the graph of the function $f(x) = (1-x^2)(x^2 + ax + b)$ is symmetric about the line $x = -2$, then the maximum value of $f(x)$ is $\_\_\_\_$. | f(-2 - \sqrt{5}) = f(-2 + \sqrt{5}) = 16 | 56 | 23 |
math | Given the sequence $\{a_n\}$ with the sum of its first $n$ terms $S_n = 3n^2 - 2n$,
(1) Find the general formula for the sequence $\{a_n\}$;
(2) Determine whether the sequence $\{a_n\}$ is an arithmetic sequence. If it is, find the first term and the common difference; if not, explain why. | 6 | 89 | 1 |
math | Given the equation
\[ x_{1}^{2} + x_{2}^{2} = (x_{1} + x_{2})^{2} - 2 x_{1} x_{2} \]
and Vieta's formulas:
\[
\left\{
\begin{array}{l}
x_{1} + x_{2} = -(m+1) \\
x_{1} x_{2} = 2 m - 2
\end{array}
\right.
\]
where \( D = (m+3)^{2} \geq 0 \), find the value of \( x_{1}^{2} + x_{2}^{2} \). Then, determine \( y = (m-1)... | 4 \text{ at } m=1 | 178 | 9 |
math | Calculate the limit of the numerical sequence:
$$\lim _{n \rightarrow \infty} \frac{\sqrt{n^{3}+1}-\sqrt{n-1}}{\sqrt[3]{n^{3}+1}-\sqrt{n-1}}$$ | \infty | 56 | 3 |
math | Let $f(x) = x^2 - 3x$. How many distinct real numbers $c$ satisfy $f(f(f(f(c)))) = 6$? | 16 | 35 | 2 |
math | Line segment $\overline{AB}$ is extended past $B$ to point $P$ such that the ratio $AP:PB$ is $7:2$. If
\[\overrightarrow{P} = t \overrightarrow{A} + u \overrightarrow{B},\]
determine the constants $t$ and $u$ to represent the position vector $\overrightarrow{P}$ in terms of $\overrightarrow{A}$ and $\overrightarrow{B}... | \left(\frac{7}{9}, \frac{2}{9}\right) | 100 | 18 |
math | Given that Ace runs at a constant speed and Flash accelerates from rest at a rate of \(a\) yards/s\(^2\), where \(a > 0\), determine the time \(t\) in seconds that it takes Flash to catch up to Ace, given that Ace's speed is \(v\) yards per second and Flash has a head start of \(y\) yards. | \frac{v + \sqrt{v^2 + 2ay}}{a} | 78 | 19 |
math | The square of $x$ and the cube root of $y$ vary inversely. If $x = 3$ when $y = 64$, find $y$ when $xy = 90$. | 15 \cdot \sqrt[5]{15} | 45 | 12 |
math | Let \(z\) and \(w\) be complex numbers such that \(|3z - w| = 15\), \(|z + 3w| = 9\), and \(|z + w| = 5\). Find \(|z|\). | 4 | 57 | 1 |
math | If the graph of the function $f(x) = \sin \omega x \cos \omega x + \sqrt{3} \sin^2 \omega x - \frac{\sqrt{3}}{2}$ ($\omega > 0$) is tangent to the line $y = m$ ($m$ is a constant), and the abscissas of the tangent points form an arithmetic sequence with a common difference of $\pi$.
(Ⅰ) Find the values of $\omega$ and... | \frac{11\pi}{3} | 145 | 10 |
math | A turtle crawled out of its home and crawled in a straight line at a constant speed of 5 m/hour. After an hour, it turned $90^{\circ}$ (either right or left) and continued crawling, then crawled for another hour, then turned again $90^{\circ}$ (either right or left), and so on. It crawled for a total of 11 hours, turni... | 5 \text{ m} | 111 | 6 |
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