instruction_seed stringlengths 2 10.7k | solution stringlengths 0 10.8k ⌀ | answer stringlengths 0 575 ⌀ | problem_type stringclasses 9
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13. In $\triangle A B C$, $\angle B=\frac{\pi}{4}, \angle C=\frac{5 \pi}{12}, A C$ $=2 \sqrt{6}, A C$'s midpoint is $D$. If a line segment $P Q$ of length 3 (point $P$ to the left of point $Q$) slides on line $B C$, then the minimum value of $A P+D Q$ is $\qquad$. | 13. $\frac{\sqrt{30}+3 \sqrt{10}}{2}$.
Given that $B C=6$.
Draw a line $D E / / B C$ through point $D$, intersecting $A B$ at point $E$.
Then $D E=\frac{1}{2} B C=3$.
Thus, quadrilateral $P Q D E$ is a parallelogram, i.e., $D Q=E P$.
Therefore, the problem is reduced to: finding a point on line $B C$ such that $A P+E... | \frac{\sqrt{30}+3\sqrt{10}}{2} | Geometry | math-word-problem | Yes | Yes | olympiads | false |
29. If $n^{2}+202 n$ is a perfect square, then the positive integer $n=$ | \begin{tabular}{c}
$\mathbf{2 9}$ \\
\hline 5000
\end{tabular} | 5000 | Algebra | math-word-problem | Yes | Problem not solved | olympiads | false |
## Task 1 - 070621
The lines $g_{1}, g_{2}, g_{3}$ and $g_{4}$ intersect each other in the manner shown in the figure. Of the magnitudes $\alpha, \beta, \gamma$ and $\delta$ of the angles thus formed, $\alpha=50^{\circ}, \beta=130^{\circ}$ and $\gamma=70^{\circ}$.
Determine $\delta$!
$ that satisfy the equation $x^{4}+y^{4}=2 z^{2}$, and the numbers $x$ and $y$ are coprime. | Solution (Prof. Lucian Petrescu, Tulcea):
We observe that the numbers $x$ and $y$ are odd.
We deduce that $z$ is also odd and is coprime with the number $x y$.
The equation in the statement transforms successively into $x^{8}+2 x^{4} y^{4}+y^{8}=4 z^{4}$, equivalent to $\left(x^{4}-y^{4}\right)^{2}=4 z^{4}-4 x^{4} y... | (1,1,1) | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
What is the smallest positive angle that has the same terminal side as $-2015^\circ$? | First, we express $-2015^\circ$ as a sum of full rotations (each rotation is $360^\circ$) and a residual angle. It is known that one full rotation does not change the terminal side of an angle. Thus, we find the equivalent angle between $0^\circ$ and $360^\circ$ by adding multiples of $360^\circ$ to $-2015^\circ$.
\be... | 145^\circ | Geometry | math-word-problem | Yes | Yes | cn_k12 | false |
7. If the real number $x$ satisfies $\log _{2} x=3+2 \cos \theta$, then $|x-2|+|x-33|$ equals ( ).
(A) $35-2 x$
(B) 31
(C) $2 x-35$
(D) $2 x-35$ or $35-2 x$ | 7. B.
Since $-1 \leqslant \cos \theta \leqslant 1$, we have $1 \leqslant \log _{2} x \leqslant 5$, thus $2 \leqslant x \leqslant 32$. Therefore, $|x-2|+|x-33|=x-2+33-x=31$. | B | Algebra | MCQ | Yes | Yes | cn_contest | false |
Calculate the area of the pentagon with sides of length 17, 22, 30, 26, and 22 units. Assume the pentagon can be divided into a right triangle and a trapezoid for easier computation. | To solve this, let's assume a feasible geometric arrangement and divide the pentagon into a right triangle and a trapezoid. The side lengths are altered as follows for the hypothetical division:
- Right triangle sides: 17 (height) and 22 (base).
- Trapezoid with parallel sides of 26 and 30, and height of 22 (matching t... | 803 | Geometry | math-word-problem | Yes | Yes | synthetic_math | true |
To edit a circular ring of given width $d$, whose area is equal to the sum of the areas of 3 given circles (e.g., $r_{1}=2$ cm, $r_{2}=3$ cm, $r_{3}=4$ cm, $d=3.5$ cm). | I. Solution: Let the radii of the given circles be denoted by $r_{1}, r_{2}$, and $r_{3}$, and the radii of the outer and inner circles of the sought annulus by $R$ and $r$, respectively.
According to the problem,
$$
\left(R^{2}-r^{2}\right) \pi=\left(r_{1}^{2}+r_{2}^{2}+r_{3}^{2}\right) \pi
$$
or equivalently,
$$
... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false |
Let set $A = \{x \,|\, -1 < x < 2\}$, and set $B = \{x \,|\, 1 < x < 3\}$. Determine $A \cup B$. | Given set $A = \{x \,|\, -1 < x < 2\}$ and set $B = \{x \,|\, 1 < x < 3\}$, the union of $A$ and $B$ represents all elements that are in either $A$, $B$, or both.
Step 1: Analyze the intervals of $A$ and $B$
- $A$ covers all real numbers between $-1$ and $2$ but does not include $-1$ and $2$ themselves.
- $B$ covers ... | \{x \,|\, -1 < x < 3\} | Algebra | math-word-problem | Yes | Yes | cn_k12 | false |
(7) Given $A=\left\{x \mid x^{2}-1=0\right\}, B=\left\{y \mid y^{2}-2 a y+b=0, y \in \mathbf{R}\right\}$, if $B \subseteq$ $A$, and $B \neq \varnothing$, find the values of real numbers $a$ and $b$. | (7) $\left\{\begin{array}{l}a=1, \\ b=1\end{array}\right.$ or $\left\{\begin{array}{l}a=-1, \\ b=1\end{array}\right.$ or $\left\{\begin{array}{l}a=0, \\ b=-1\end{array}\right.$ | {\begin{pmatrix}=1,\\b=1\end{pmatrix}. | Algebra | math-word-problem | Yes | Yes | olympiads | false |
Let $n$ be a given positive integer. In the Cartesian plane, each lattice point with nonnegative coordinates initially contains a butterfly, and there are no other butterflies. The [i]neighborhood[/i] of a lattice point $c$ consists of all lattice points within the axis-aligned $(2n+1) \times (2n+1)$ square entered at ... | 1. **Understanding the Problem:**
- Each lattice point with nonnegative coordinates initially contains a butterfly.
- The neighborhood of a lattice point \( c \) consists of all lattice points within the axis-aligned \((2n+1) \times (2n+1)\) square centered at \( c \), excluding \( c \) itself.
- A butterfly i... | n^2 + 1 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Megan has 19 seashells. She needs to find a certain number of seashells to have a specific number in her collection. If she finds 6 more seashells, how many seashells will she have in her collection? | If Megan currently has 19 seashells and she finds 6 more, she will have:
19 + 6 = $\boxed{25}$ seashells in her collection. | 25 | Algebra | math-word-problem | Yes | Yes | orca_math | true |
Find the intersection point of the asymptotes of the graph of the function:
\[ y = \frac{x^2 - 9x + 20}{x^2 - 9x + 18} \] | First, we factorize the denominator:
\[ x^2 - 9x + 18 = (x-6)(x-3) \]
So, the vertical asymptotes are at $x = 6$ and $x = 3$.
Next, we simplify the function to find the horizontal asymptote:
\[ y = \frac{x^2 - 9x + 20}{x^2 - 9x + 18} = \frac{(x^2 - 9x + 18) + 2}{x^2 - 9x + 18} = 1 + \frac{2}{x^2 - 9x + 18} \]
As $x \t... | \text{No intersection of vertical and horizontal asymptotes.} | Algebra | math-word-problem | Yes | Yes | synthetic_math | true |
Given the lines $x-2y=-k+6$ and $x+3y=4k+1$, if their intersection point is in the fourth quadrant.
(1) Find the range of values for $k$;
(2) If $k$ is a non-negative integer, and the coordinates of point $A$ are $(2,0)$, point $P$ is on the line $x-2y=-k+6$, find the coordinates of point $P$ that make $\triangle P... | (1) From the given conditions, we have:
\[
\begin{align*}
x-2y&=-k+6 \\
x+3y&=4k+1
\end{align*}
\]
Solving these equations, we get:
\[
\begin{align*}
x&=k+4 \\
y&=k-1
\end{align*}
\]
Therefore, the coordinates of the intersection point are $(k+4, k-1)$. Since the intersection point is in the fourth quadrant,
\[
\b... | \left(\frac{18}{5}, -\frac{6}{5}\right) | Algebra | math-word-problem | Yes | Yes | cn_k12 | false |
[b]a)[/b] Prove that the sum of all the elements of a finite union of sets of elements of finite cyclic subgroups of the group of complex numbers, is an integer number.
[b]b)[/b] Show that there are finite union of sets of elements of finite cyclic subgroups of the group of complex numbers such that the sum of all its ... | ### Part (a)
1. **Understanding the Problem:**
We are given a finite set \( A \subset \mathbb{C} \) with the property that if \( z \in A \), then \( z^n \in A \) for every \( n \in \mathbb{N} \). We need to prove that the sum of all elements in \( A \) is an integer.
2. **Analyzing the Property:**
The property \... | null | Number Theory | proof | Yes | Yes | aops_forum | false |
Jerry owned 7 video games. For his birthday, he got a certain number of games and now has a total of 9 games. How many games did Jerry get for his birthday? | Jerry originally had 7 video games. After his birthday, he has a total of 9 games. To find out how many games he got for his birthday, we subtract the number of games he originally had from the total number of games he has now.
9 games (total after birthday) - 7 games (original amount) = 2 games
Jerry got $\boxed{2}$... | 2 | Algebra | math-word-problem | Yes | Yes | orca_math | true |
John found that the average of 15 numbers is 40. If a certain value is added to each number, then the mean of the numbers becomes 55. What is the value added to each number? | If the average of 15 numbers is 40, the sum of all the numbers is 15 * 40 = 600.
When a certain value is added to each of the 15 numbers, the mean becomes 55. This means the new sum of all the numbers is 15 * 55 = 825.
The difference between the new sum and the original sum is the total amount added to all the number... | 15 | Algebra | math-word-problem | Yes | Yes | orca_math | true |
John and James decided to have a dance-off. John danced for 3 hours and then another 5 hours after taking a 1-hour break. James danced the whole time John was dancing and resting, and then some more hours. Their combined dancing time without including John's break time was 20 hours. What is the ratio of the additional ... | Let's first calculate the total time John was dancing and resting. John danced for 3 hours, took a 1-hour break, and then danced for another 5 hours. So, the total time John was dancing and resting is:
3 hours (first dance session) + 1 hour (break) + 5 hours (second dance session) = 9 hours
Now, let's find out how lo... | 4:9 | Algebra | math-word-problem | Yes | Yes | orca_math | true |
Let $f(x) = \ln(x+1)$.
(I) Find the set $A$ of values of $x$ that satisfy $f(1-2x) > f(x)$;
(II) Let set $B = \{x | a-1 < x < 2a^2\}$, if $A \cap B \neq \emptyset$, find the range of the real number $a$. | Solution:
(I) From $\ln(2-2x) > \ln(x+1)$, we get: $$\begin{cases} \overset{x+1>0}{2-2x>x+1}\end{cases}$$,
Solving this, we get: $-1 a-1$, solving this yields: $a > 1$ or $a 1$, $a-1 > 0$, $2a^2 > 2$, thus $a-1 \frac{1}{2} > \frac{1}{3}$,
In this case, $A \subset B$, $A \cap B = A$,
Overall, the range of $a$ is $\bo... | (-\infty, -\frac{1}{2}) \cup (1, \frac{4}{3}) | Algebra | math-word-problem | Yes | Yes | cn_k12 | false |
Let $A$ be a point on the circle $(x-3)^2 + (y-4)^2 = 16,$ and let $B$ be a point on the parabola $x^2 = 8y.$ Find the smallest possible distance $AB.$ | Completing the square for the given circle equation is not needed because it is already in the standard form. The center of the circle is $(3,4),$ and its radius is $4$.
The parabola $x^2 = 8y$ opens upwards. Let $y = t,$ then
\[ x = \sqrt{8t} \quad \text{or} \quad x = -\sqrt{8t}. \]
Therefore, $B$ can be either $(\sq... | \sqrt{34} - 4 | Geometry | math-word-problem | Yes | Yes | synthetic_math | true |
How many real numbers $x$ are solutions to the equation $|x - 2| - 4 =\frac{1}{|x - 3|}$ ? |
To solve the equation \( |x - 2| - 4 = \frac{1}{|x - 3|} \), we need to consider the behavior of the absolute value expressions and the resulting piecewise functions.
First, let's rewrite the equation in a more convenient form:
\[ |x - 2| - 4 = \frac{1}{|x - 3|} \]
This implies that \( |x - 2| - 4 \) must be positive... | 2 | NaN | NaN | NaN | NaN | aops_forum | false |
307. Divisibility by 30. Prove that for any integer $n$ the expression $\left(n^{5}-n\right)$ is divisible by 30. | 307. An integer and its fifth power always end in the same digit. Therefore, the value of $n^{5}-n$ ends in 0 and, thus, is divisible by 2 and 5. Further, let's factorize it into $(n-1) n(n+1)\left(n^{2}+1\right)$ and note that one of the first three factors must be divisible by 3. Thus, $n^{5}-n$ is divisible by $2 \t... | proof | Number Theory | proof | Yes | Yes | olympiads | false |
2. Let $a_{n}=n$. Prove: $\sum_{k=1}^{n} \frac{\sqrt{k}}{a_{k}^{2}}<3$. | $\begin{array}{l}\text { Hint: } \sum_{k=1}^{n} \frac{\sqrt{k}}{a_{k}^{2}}=\sum_{k=1}^{n} \frac{1}{\sqrt{k^{3}}} \\ <1+\sum_{k=2}^{n} \frac{\sqrt{k+1}-\sqrt{k-1}}{\sqrt{(k-1)(k+1)}} \\ =1+\sum_{k=2}^{n}\left(\frac{1}{\sqrt{k-1}}-\frac{1}{\sqrt{k+1}}\right) \\ =1+1+\frac{\sqrt{2}}{2}-\frac{1}{\sqrt{n}}-\frac{1}{\sqrt{n+... | proof | Inequalities | proof | Yes | Yes | cn_contest | false |
If $ x \equal{} \sqrt [3]{11 \plus{} \sqrt {337}} \plus{} \sqrt [3]{11 \minus{} \sqrt {337}}$, then $ x^3 \plus{} 18x$ = ?
$\textbf{(A)}\ 24 \qquad\textbf{(B)}\ 22 \qquad\textbf{(C)}\ 20 \qquad\textbf{(D)}\ 11 \qquad\textbf{(E)}\ 10$ | 1. Let \( a = \sqrt[3]{11 + \sqrt{337}} \) and \( b = \sqrt[3]{11 - \sqrt{337}} \). We are given that \( x = a + b \).
2. We need to find \( x^3 + 18x \). To do this, we start by considering the expression for \( x^3 \):
\[
x^3 = (a + b)^3
\]
Expanding the right-hand side using the binomial theorem, we get... | 22 | Algebra | MCQ | Yes | Yes | aops_forum | false |
Example 1. Find the integral $\int \cos x \sin ^{2} x d x$. | Solution. Since $\cos x d x=d(\sin x)$, then based on formula (4. $5^{\prime}$ ) for $u=\sin x$ we obtain
$$
\int \cos x \sin ^{2} x d x=\int \sin ^{2} x d(\sin x)=\frac{\sin ^{3} x}{3}+C
$$ | \frac{\sin^{3}x}{3}+C | Calculus | math-word-problem | Yes | Yes | olympiads | false |
SUBIECTUL III Show that, if $z_{1}, z_{2}, \cdots, z_{n} \in \mathbb{C}$ and $\left|z_{1}\right|=\left|z_{2}\right|=\cdots=\left|z_{n}\right|$,
then
$$
\left(1+\frac{z_{2}}{z_{1}}\right)\left(1+\frac{z_{3}}{z_{2}}\right) \cdots\left(1+\frac{z_{n}}{z_{n-1}}\right)\left(1+\frac{z_{1}}{z_{n}}\right) \in \mathbb{R}
$$ | Solution. If we denote the product by $z$, then we try to prove that $\bar{Z}=Z$ which will imply $z \in \mathbb{R} . \ldots(2 \text{p})$
## www.edums.ro
We have $\bar{z}=\left(1+\frac{\overline{z_{2}}}{\overline{z_{1}}}\right)\left(1+\frac{\overline{z_{3}}}{\overline{z_{2}}}\right) \cdots\left(1+\frac{\overline{z_{n... | proof | Algebra | proof | Yes | Yes | olympiads | false |
A train of certain length runs with a speed of 45 km/hr. It takes 48 seconds to pass a platform of 240 m long. What is the length of the train? | To find the length of the train, we need to first determine the total distance covered by the train while passing the platform. This distance is the sum of the length of the train and the length of the platform.
Let's denote the length of the train as \( L \) (in meters).
The speed of the train is given as 45 km/hr. ... | 360 | Algebra | math-word-problem | Yes | Yes | orca_math | true |
Let the greatest common divisor of $n$ and $k$ be denoted by $(n, k)$, and the least common multiple by $[n, k]$. Show that for any positive integers $a, b, c$, the greatest common divisor of the numbers $[a, b], [b, c], [c, a]$ is equal to the least common multiple of the numbers $(a, b), (b, c), (c, a)$. | Solution. The (modified) canonical form of the numbers $a, b, c$ is:
$$
a=p_{1}^{\alpha_{1}} p_{2}^{\alpha_{2}} \ldots p_{r}^{\alpha_{r}}, \quad b=p_{1}^{\beta_{1}} p_{2}^{\beta_{2}} \ldots p_{r}^{\beta_{r}}, \quad c=p_{1}^{\gamma_{1}} p_{2}^{\gamma_{2}} \ldots p_{r}^{\gamma_{r}}
$$
By the modified canonical form, we... | proof | Number Theory | proof | Yes | Yes | olympiads | false |
Julie works at a restaurant that pays her an hourly wage that varies based on her years of experience. She had been working for 3 years, and for each year of experience, they added $0.50 to her starting pay of $5.00 per hour. Now, she works 8 hours a day, 6 days a week. Julie also receives a bi-weekly bonus of $50, but... | First, let's calculate Julie's hourly wage after 3 years of experience:
Starting pay: $5.00 per hour
Additional pay for experience: $0.50/year * 3 years = $1.50
Total hourly wage: $5.00 + $1.50 = $6.50 per hour
Next, let's calculate her weekly earnings, assuming she works 8 hours a day for 6 days a week:
Daily earni... | \$963.20 | Algebra | math-word-problem | Yes | Yes | orca_math | true |
Before the arrival of Labor Day in 2023, the price department of a certain city surveyed the daily sales volume and prices of a certain product in 5 shopping malls in the city. The data between the price $x$ (in yuan) and the sales volume $y$ (in units) of this product in the 5 shopping malls is shown in the table belo... | To find the value of $m$ in the context of the linear relationship between the price $x$ and the sales volume $y$, we follow these steps:
1. Calculate the average price $\overline{x}$ using the given prices, including the unknown $m$:
$$\overline{x} = \frac{8 + 9.5 + m + 10.5 + 12}{5} = \frac{40 + m}{5}.$$
2. Calcula... | 10 | Algebra | math-word-problem | Yes | Yes | cn_k12 | false |
Two circles centered at $O$ and $P$ have radii of length $5$ and $6$ respectively. Circle $O$ passes through point $P$. Let the intersection points of circles $O$ and $P$ be $M$ and $N$. The area of triangle $\vartriangle MNP$ can be written in simplest form as $a/b$. Find $a + b$. | 1. **Identify the given information and draw the diagram:**
- Circle \( O \) has radius \( 5 \) and passes through point \( P \).
- Circle \( P \) has radius \( 6 \).
- The intersection points of circles \( O \) and \( P \) are \( M \) and \( N \).
2. **Determine the distance \( OP \):**
- Since circle \( ... | 12 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
In the sequence $\{a_n\}$, $a_1=1$, and for all $n \geq 2$, $n \in \mathbb{N}$, it holds that $a_1 \cdot a_2 \cdot a_3 \cdot \ldots \cdot a_n = n^2$. Then, $a_3 + a_5$ equals to ( )
A: $\frac {61}{16}$
B: $\frac {25}{9}$
C: $\frac {25}{16}$
D: $\frac {31}{15}$ | When $n \geq 2$, we have $a_1 \cdot a_2 \cdot a_3 \ldots a_n = n^2$.
When $n \geq 3$, we have $a_1 \cdot a_2 \cdot a_3 \ldots a_{n-1} = (n-1)^2$.
Dividing the two equations, we get $a_n = \left( \frac {n}{n-1} \right)^2$,
Therefore, $a_3 = \frac {9}{4}$, $a_5 = \frac {25}{16}$. Thus, $a_3 + a_5 = \frac {61}{16}$.... | A | Algebra | MCQ | Yes | Yes | cn_k12 | false |
Given that the function $f(x)=\sin (π-ωx)\cos ωx+\cos ^{2}ωx\ (ω > 0)$ has a minimum positive period of $π$.
(1) Find the value of $ω$;
(2) By reducing the abscissa of each point on the graph of the function $y=f(x)$ to $\frac{1}{2}$ of the original, while keeping the ordinate unchanged, obtain the graph of the functio... | (1) The function $f(x)=\sin (π-ωx)\cos ωx+\cos ^{2}ωx$ can be simplified as:
$f(x) = \sin ωx \cdot \cos ωx + \frac{1 + \cos 2ωx}{2}$
$= \frac{1}{2}\sin 2ωx + \frac{1}{2}\cos 2ωx + \frac{1}{2}$
$= \frac{\sqrt{2}}{2}\sin (2ωx + \frac{π}{4}) + \frac{1}{2}\ (ω > 0)$
The minimum positive period of this function is $\frac{2π... | g(x) = 1 | Algebra | math-word-problem | Yes | Yes | cn_k12 | false |
A side of triangle $H_{1}$ is equal to the arithmetic mean of the other two sides. The sides of triangle $H_{2}$ are each 10 units smaller, and the sides of triangle $H_{3}$ are each 14 units larger than the sides of $H_{1}$. The radius of the inscribed circle of $H_{2}$ is 5 units smaller than that of $H_{1}$, and the... | In the triangle $H_{1}$, we denote the sides in the usual way such that $a \leq b \leq c$. According to the assumption, $b=(a+c) / 2$, which implies $b-a=c-b$. Denoting this non-negative difference by $d$, we have $a=b-d, c=b+d, 2s=a+b+c=3b$, and for the radius of the inscribed circle
\[
\begin{aligned}
& \varrho=\fra... | =25,b=38,=51 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
19. Calculate in factorial form $A_{6}^{3}$. | Solution. $A_{6}^{3}=\frac{6!}{(6-3)!}=6 \cdot 5 \cdot 4=120$.
20-25. Calculate in any way: | 120 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
Let $A, B, C$ be the angles of an acute-angled triangle. Prove the inequality
$$
\sin A+\sin B>\cos A+\cos B+\cos C
$$ | In an acute-angled triangle we have $A+B>\frac{\pi}{2}$. Hence we have $\sin A>\sin \left(\frac{\pi}{2}-B\right)=\cos B$ and $\sin B>\cos A$. Using these inequalities we get $(1-\sin A)(1-\sin B)\cos A+\cos B-\cos A \cos B+\sin A \sin B \\
& =\cos A+\cos B-\cos (A+B)=\cos A+\cos B+\cos C
\end{aligned}
$$ | proof | Inequalities | proof | Yes | Incomplete | olympiads | false |
Let $f''(x)$ be the derivative of the function $f(x)$ $(x \in \mathbb{R})$, and it satisfies $xf''(x) - 2f(x) > 0$. If $\Delta ABC$ is an acute triangle, then $(\ \ \ \ \ )$
A: $f(\sin A) \cdot \sin^2 B > f(\sin B) \cdot \sin^2 A$
B: $f(\sin A) \cdot \sin^2 B f(\sin B) \cdot \cos^2 A$
D: $f(\cos A) \cdot \sin^2 B <... | [Analysis]
This problem tests modeling thinking, using derivatives to study the monotonicity of functions and the monotonicity of functions in monotonic intervals.
Using modeling thinking, construct the function $F(x) = \frac{f(x)}{x^2}$, and then study the monotonicity of the function to find that the function $F(x)... | D | Calculus | MCQ | Incomplete | Yes | cn_k12 | false |
Given the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1 (a>0, b>0)$, where $F$ is the right focus of the hyperbola. A perpendicular line is drawn from $F$ to one of its asymptotes, with the foot of the perpendicular being $Q$. The line $FQ$ intersects the left and right branches of the hyperbola at points $M$ and $N$, r... | To solve for the eccentricity of the hyperbola given the conditions, we proceed as follows:
1. **Identify the Equation of the Asymptotes**: The equation of the asymptotes for the given hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is $bx \pm ay = 0$.
2. **Determine $|FQ|$**: Given that $F$ is the right focus, we can ... | \sqrt{5} | Geometry | math-word-problem | Yes | Yes | cn_k12 | false |
Example 2: Prove that in the unit cube $A B C D-$ $A_{1} B_{1} C_{1} D_{1}$, the plane $A_{1} B C_{1} \parallel$ plane $D_{1} A C$. | Proof: Establish a spatial rectangular coordinate system as shown in Figure 2.
Then
$$
\begin{array}{l}
D(0,0,0) \text {, } \\
A_{1}(1,0,1) \text {, } \\
B(1,1,0) \text {, } \\
C(0,1,0) \text {, } \\
C_{1}(0,1,1) \text {, } \\
D_{1}(0,0,1) \text {, } \\
A(1,0,0) .
\end{array}
$$
Thus, \( A_{1} C_{1}=(-1,1,0), A_{1} \b... | proof | Geometry | proof | Yes | Yes | cn_contest | false |
According to the statistical bulletin released by the National Bureau of Statistics, the production of new energy vehicles in China in 2022 was 7,003,000 vehicles. This number can be expressed in scientific notation as ( ).
A: $70.03\times 10^{5}$
B: $7.003\times 10^{5}$
C: $7.003\times 10^{6}$
D: $0.7003\times 10^... | To express the number $7,003,000$ in scientific notation, we start by placing the decimal point after the first non-zero digit, which gives us $7.003$. Next, we count the number of places the decimal has moved from its original position at the end of the number to its new position. In this case, the decimal point has m... | C | Algebra | MCQ | Yes | Yes | cn_k12 | false |
5. Borya and Vova are playing the following game on an initially white $8 \times 8$ board. Borya moves first and on each of his turns, he colors any four white cells black. After each of his moves, Vova colors an entire row (row or column) completely white. Borya aims to color as many cells black as possible, while Vov... | Solution. Let Vova make white the row with the most black cells on each of his moves. Then, as soon as Borya achieves a row of no less than four black cells (we will call such a row "rich"), Vova will remove at least four cells, meaning that Borya will not be able to increase the number of black cells compared to his p... | 25 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
Calculate the values of the cosines of the angles $x$ that satisfy the next equation:
$$\sin^2 x - 2 \cos^2 x +\frac12 \sin 2x = 0.$$ | null | null | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Given the sequence $\{a\_n\}$ defined by $a\_1=2$ and $a\_{n+1}=a\_n+2n$ $(n\geqslant 1)$, find the value of $a\_{100}$:
A: $9900$
B: $9902$
C: $9904$
D: $9906$ | According to the problem, we have $a\_{n+1}-a\_n=2n$. Thus,
$$a\_2-a\_1=2$$
$$a\_3-a\_2=4$$
$$\vdots$$
$$a\_n-a\_{n-1}=2(n-1)$$
Adding the above $n-1$ equations together, we get:
$$a\_n-a\_1=2+4+6+\dots+2(n-1)=n(n-1)$$
Thus, $a\_n=n(n-1)+2$. Substituting $n=100$, we have:
$$a\_{100}=100(100-1)+2=\boxed{9902}$$
Th... | 9902 | Algebra | MCQ | Yes | Yes | cn_k12 | false |
Given that the function $f(x)$ satisfies the condition $f(x)=-f(x+2)$ for any real number $x$, and $f(x)=x(x-2)$ when $0\leqslant x\leqslant 2$, determine the value of $f(-2017)=$ \_\_\_\_\_\_. | Since $f(x)=-f(x+2)$,
We can infer that $f(x+2)=-f(x)$,
And $f(x+4)=-f(x+2)=f(x)$,
Therefore, $f(x)$ is a periodic function with a period of $4$.
Hence, $f(-2017)=f(-504\times4-1)=f(-1)=-f(1)=\boxed{1}$.
The answer is derived by understanding the periodicity of the function $f(x)$ based on the given condition that $f(... | 1 | Algebra | math-word-problem | Yes | Yes | cn_k12 | false |
8 jumbo scallops weigh one pound and cost $24.00 a pound. Nate is pairing 2 scallops with a corn bisque as a light summer dinner. He's cooking for 8 people. How much will the scallops cost? | Nate is cooking for 8 people, and he is pairing 2 scallops with a corn bisque for each person. Therefore, he will need 2 scallops per person x 8 people = 16 scallops in total.
Since 8 jumbo scallops weigh one pound, 16 scallops will weigh 16 scallops / 8 scallops per pound = 2 pounds.
At a cost of $24.00 per pound, t... | \$48.00 | Algebra | math-word-problem | Yes | Yes | orca_math | true |
Given $f(x) = ax^3 + 3x^2 - x + 1$ is a decreasing function on $\mathbb{R}$, find the range of values for $a$. | The derivative of the function $f(x)$ is: $f'(x) = 3ax^2 + 6x - 1$.
For $f(x)$ to be a decreasing function when $f'(x) < 0$ ($x \in \mathbb{R}$),
$3ax^2 + 6x - 1 < 0$ ($x \in \mathbb{R}$) $\Leftrightarrow a < 0$ and $\Delta = 36 + 12a < 0$, $\Leftrightarrow a < -3$.
Therefore, when $a < -3$, given $f'(x) < 0$, we kn... | a < -3 | Calculus | math-word-problem | Yes | Yes | cn_k12 | false |
Alyssa, Chelsea, and Bianca bought candles to decorate three rooms of their house.
In the first room, they placed 50 candles. Alyssa used 2/5 of them, Chelsea used 3/7 of the remaining candles, and Bianca used 4/9 of the candles left after Alyssa and Chelsea.
In the second room, they placed 70 candles. Alyssa used ... | Let's calculate the number of candles left in each room after Alyssa, Chelsea, and Bianca have used their portions.
**First Room:**
- Total candles: 50
- Alyssa used 2/5 of 50: (2/5) * 50 = 20 candles
- Remaining after Alyssa: 50 - 20 = 30 candles
- Chelsea used 3/7 of the remaining 30 candles: (3/7) * 30 = 90/7 ≈ 12.... | 20 | Algebra | math-word-problem | Yes | Yes | orca_math | true |
The maximum number of students among whom 2010 pens and some pencils can be distributed in such a way that each student gets the same number of pens and same number of pencils is 30. How many pencils are there? | If the maximum number of students among whom 2010 pens can be distributed equally is 30, then each student must receive an equal number of pens without any pens left over. To find out how many pens each student gets, we divide the total number of pens by the number of students:
2010 pens ÷ 30 students = 67 pens per st... | 30 | Number Theory | math-word-problem | Yes | Yes | orca_math | true |
I have 7 books, three of which are identical copies of the same math book (and all of the rest of the books are different). In how many ways can I arrange them on a shelf? | 1. Calculate the total number of arrangements if all books were different: $7!$
2. Since three books are identical, divide the total arrangements by the permutations of these three identical books to avoid overcounting:
$$ \text{Total permutations} = \frac{7!}{3!} $$
3. Calculate the exact values:
$$ 7! = 5040 $... | 840 | Combinatorics | math-word-problem | Yes | Yes | synthetic_math | true |
Zoe goes to the store to buy soda and pizza for herself and some family members. Each bottle of soda costs half a dollar and each slice of pizza costs $1. Before she leaves her house, she takes just enough money to pay for her purchase, which is $9. How many family members is she buying for? | Let's assume Zoe is buying for herself and x number of family members. This means there are x+1 people in total.
If each bottle of soda costs $0.50 and each slice of pizza costs $1, and Zoe spends a total of $9, we can set up an equation to represent the total cost:
Cost of soda for each person = $0.50
Cost of pizza ... | 5 | Algebra | math-word-problem | Yes | Yes | orca_math | true |
1.9. Two circles touch each other externally. A tangent is drawn to the first circle, passing through the center of the second circle. The distance from the point of tangency to the center of the second circle is three times the radius of this circle. How many times greater is the length of the first circle compared to... | 1.9. Let $O_{1}$ and $O_{2}$ be the centers of the circles, and $A$ be the point of tangency (Fig. 1.8). Then $O_{1} A = R_{1}, O_{1} O_{2} = R_{1} + R_{2}, O_{2} A = 3 R_{2}$ (by the condition). We need to find the ratio $2 \pi R_{1} : 2 \pi R_{2} = R_{1} : R_{2}$. In the right triangle $O_{1} A O_{2} \left(\angle A =... | 4 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
Example 2. On a line segment $AB$ of length 1, take two points $C, D$ in the order of $A, B, C, D$. Let $AC=x$, $CD=y$.
(1) Draw a diagram to represent the range of point $P(x, y)$.
(2) If the segments $AC, CD, DB$ can form the sides of a triangle, draw a diagram to represent the range of point $P(x, y)$. | $$
\begin{array}{l}
\text { (1) A, B, C, and D are four distinct points arranged in sequence, } x>0, y>1, x+y 1 - ( x + y ) , } \\
{ x + 1 - ( x + y ) > y , } \\
{ y + 1 - ( x + y ) > x , }
\end{array} \text { that is } \left\{\begin{array}{l}
x+y>\frac{1}{2}, \\
0<x<\frac{1}{2} \\
0<y<\frac{1}{2} .
\end{array}\right.\... | 0<x<\frac{1}{2}, 0<y<\frac{1}{2}, x+y>\frac{1}{2} | Geometry | math-word-problem | Yes | Yes | cn_contest | false |
$$\begin{array}{l}
\tilde{c}_{0}=0, \tilde{q}_{0}=1, \text { then } \tilde{a}_{0}=[\sqrt{d}], \\
\quad \tilde{\xi}_{j}=\xi_{j}, \quad \tilde{a}_{j}=a_{j}, \quad \tilde{c}_{j}=c_{j}, \quad j \geqslant 1, \quad \tilde{q}_{j}=q_{j}, \quad j \geqslant 0,
\end{array}$$
and $\tilde{q}_{j}=1$ if and only if $l \mid j$, for a... | None
Translate the text above into English, please retain the original text's line breaks and format, and output the translation result directly.
Note: The provided instruction is a meta-instruction and not part of the text to be translated. Since the text to be translated is "None", the translation is also "None". ... | not found | Number Theory | math-word-problem | Incomplete | Problem not solved | number_theory | false |
Find the values of $x$ for which the function $f(x)=\sin \left(x+ \frac {\pi}{6}\right)$ reaches its maximum and minimum values. | Solution: When $x+ \frac {\pi}{6}=2k\pi+ \frac {\pi}{2}$, that is, $x=2k\pi+ \frac {\pi}{3}$, where $k\in \mathbb{Z}$, the function $f(x)$ reaches its maximum value. The maximum value is $\boxed{1}$.
When $x+ \frac {\pi}{6}=2k\pi- \frac {\pi}{2}$, that is, $x=2k\pi- \frac {2}{3}\pi$, where $k\in \mathbb{Z}$, the funct... | -1 | Calculus | math-word-problem | Yes | Yes | cn_k12 | false |
What is the smallest integer larger than $(\sqrt{3}+\sqrt{2})^6$?
$\textbf{(A)}\ 972 \qquad \textbf{(B)}\ 971 \qquad \textbf{(C)}\ 970 \qquad \textbf{(D)}\ 969 \qquad \textbf{(E)}\ 968$ | $(\sqrt{3}+\sqrt{2})^6=(5+2\sqrt{6})^3=(5+2\sqrt{6})(49+20\sqrt{6})=(485+198\sqrt{6})$ Then, find that $\sqrt{6}$ is about $2.449$. Finally, multiply and add to find that the smallest integer higher is $\boxed {\textbf{(C) } 970}$ | 970 | Algebra | MCQ | Yes | Yes | amc_aime | false |
9.213. $\left\{\begin{array}{l}\left|x^{2}+5 x\right|<6, \\ |x+1| \leq 1 .\end{array}\right.$ | ## Solution.
The system is equivalent to the system of inequalities
$$
\begin{aligned}
& \left\{\begin{array} { l }
{ - 6 - 6 , } \\
{ x + 1 \leq 1 , } \\
{ x + 1 \geq - 1 }
\end{array} \Leftrightarrow \left\{\begin{array}{l}
x^{2}+5 x-60, \\
x \leq 0, \\
x \geq-2
\end{array} \Leftrightarrow\right.\right.\right. \\... | x\in(-2;0] | Inequalities | math-word-problem | Yes | Incomplete | olympiads | false |
Write an algorithm for calculating $1 \times 2 \times 3 \times 4 \times 5 \times 6$. | **Analysis:**
This problem mainly tests the understanding and application of algorithms. Follow the step-by-step multiplication process:
- **Step 1:** Calculate $1 \times 2$, which results in $2$.
- **Step 2:** Multiply the result of the first step, $2$, by $3$ to get $6$.
- **Step 3:** Multiply the result of the sec... | 720 | Algebra | math-word-problem | Yes | Yes | cn_k12 | false |
If the line $$l_{1}: \begin{cases} \left.\begin{matrix}x=1-2t \\ y=2+kt.\end{matrix}\right.\end{cases}$$ (where $t$ is a parameter) is perpendicular to the line $$l_{2}: \begin{cases} \left.\begin{matrix}x=s \\ y=1-2s.\end{matrix}\right.\end{cases}$$ (where $s$ is a parameter), then $k= \boxed{-1}$. | Given the line $$l_{1}: \begin{cases} \left.\begin{matrix}x=1-2t \\ y=2+kt.\end{matrix}\right.\end{cases}$$ (where $t$ is a parameter),
we have $y=2+ \frac {1-x}{2} \times k= -\frac {k}{2}x+2+ \frac {k}{2}$, and for the line $$l_{2}: \begin{cases} \left.\begin{matrix}x=s \\ y=1-2s.\end{matrix}\right.\end{cases}$$ (wh... | -1 | Algebra | math-word-problem | Yes | Yes | cn_k12 | false |
Angelo and Melanie need to determine the total number of days they should allocate for studying together for their upcoming test. They have 2 chapters in their textbook and 4 worksheets to cover. They have decided to spend 3 hours on each chapter and 1.5 hours on each worksheet. Considering that they can study a maximu... | To determine the total number of days Angelo and Melanie should allocate for studying, we need to follow these steps:
1. **Calculate the total study time required:**
- Each chapter requires 3 hours of study.
- There are 2 chapters.
- Each worksheet requires 1.5 hours of study.
- There are 4 worksheets.
... | 6 | Logic and Puzzles | math-word-problem | Yes | Yes | metamath | true |
120. Two spheres are inscribed in the lateral surface of a cone and touch each other. A third sphere passes through the two circles where the first two spheres touch the surface of the cone. Prove that the volume of the part of the third sphere located outside the cone is equal to the volume of the part of the cone enc... | 120. Let's conduct a section through the axis of the cone. Consider the trapezoid $ABCD$ formed in this section, where $A$ and $B$ are the points of tangency with the surface of the cone of one sphere, and $C$ and $D$ are those of another. It can be proven that if $F$ is the point of tangency of the spheres, then $F$ i... | proof | Geometry | proof | Yes | Yes | olympiads | false |
1. For a right-angled triangle with a hypotenuse of 2009, if the two legs are also integers, then its area is $\qquad$ . | $-.1 .432180$.
Let the two legs be $x, y$. Then $x^{2}+y^{2}=2009^{2}$. According to Pythagorean triples and the symmetry of $x, y$, there exist positive integers $m, n, k (m>n)$, such that
$$
\begin{array}{l}
x=2 m n k, y=\left(m^{2}-n^{2}\right) k, \\
2009=\left(m^{2}+n^{2}\right) k .
\end{array}
$$
Since $2009=41 \... | 432180 | Geometry | math-word-problem | Yes | Yes | cn_contest | false |
Five dice are rolled. The product of the faces are then computed. Which result has a larger probability of occurring; $180$ or $144$? | To determine which product, $180$ or $144$, has a larger probability of occurring when five dice are rolled, we need to count the number of favorable outcomes for each product and compare them.
1. **Prime Factorization and Possible Arrangements:**
- For $180$:
\[
180 = 2^2 \cdot 3^2 \cdot 5
\]
P... | null | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
In a class activity, Mr. Lopez asked his students to add two positive integers. Ellie subtracted the numbers by mistake and ended up with 5. Tom multiplied the numbers by mistake and got 84. What should have been the correct sum of those two integers? | Let $a$ and $b$ be the two positive integers where $a>b$. The students made two errors leading to:
\begin{align*}
a - b &= 5 \\
a \cdot b &= 84
\end{align*}
From the first equation, solve for $a$:
\[ a = b + 5 \]
Substituting in the second equation, we get:
\[ (b + 5)b = 84 \]
\[ b^2 + 5b - 84 = 0 \]
Factoring the ... | 19 | Algebra | math-word-problem | Yes | Yes | synthetic_math | true |
II. Answer the following questions (10 points each, total 40 points, require a brief process)
9. (10 points) In the addition problem shown in the figure, different Chinese characters can represent the same digit, making the equation true. Among all the equations that meet the requirements, what is the maximum value of ... | 【Analysis】According to the problem, to maximize the value of the four-digit number "Huabei Final", the values of the two-digit addend and the three-digit addend should be minimized. The smallest two-digit addend is 10, and the smallest three-digit addend is 100. Therefore, the maximum value of the four-digit number "Hu... | 1901 | Logic and Puzzles | math-word-problem | Incomplete | Yes | olympiads | false |
8 is to 4 seconds as 8 is to a certain time. What is that time in minutes? | If 8 is to 4 seconds, then the ratio is 8:4, which simplifies to 2:1. This means that for every 2 units of the first quantity, there is 1 unit of the second quantity.
If we apply the same ratio to the second part of the question, where 8 is to a certain time, we can set up the proportion as follows:
8 (first quantity... | 0.0667 | Other | math-word-problem | Yes | Yes | orca_math | true |
13. This year, 2016, can be written as the sum of two positive integers $p$ and $q$ where $2 p=5 q$ (as $2016=1440+576)$. How many years between 2000 and 3000 inclusive have this property? | 13. 143 For a year $Y$ to be expressible as as the sum of two positive integers $p$ and $q$ where $2 p=5 q$ we require $p+q=Y$ and $2 p=5 q$. From the first of these, it follows that $2 p+2 q=2 Y$ and hence $5 q+2 q=2 Y$.
Therefore $7 q=2 Y$ from which it follows that $Y$ is also divisible by 7 (since 2 and 7 are copri... | 143 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
23. N5 (ROM) Let $\mathbb{N}_{0}$ denote the set of nonnegative integers. Find a bijective function $f$ from $\mathbb{N}_{0}$ into $\mathbb{N}_{0}$ such that for all $m, n \in \mathbb{N}_{0}$,
$$
f(3 m n+m+n)=4 f(m) f(n)+f(m)+f(n) .
$$ | 23. We first observe that the given functional equation is equivalent to $$ 4 f\left(\frac{(3 m+1)(3 n+1)-1}{3}\right)+1=(4 f(m)+1)(4 f(n)+1) . $$ This gives us the idea of introducing a function $g: 3 \mathbb{N}_{0}+1 \rightarrow 4 \mathbb{N}_{0}+1$ defined as $g(x)=4 f\left(\frac{x-1}{3}\right)+1$. By the above equ... | proof | Algebra | math-word-problem | Yes | Yes | olympiads_ref | false |
The range of the inclination angle of the line $x\sin \alpha+y+2=0$ is ( )
A: $[0,\pi)$
B: $\left[\begin{matrix}0, \frac{\pi}{4} \end{matrix}\right]∪\left[\begin{matrix} \frac{3\pi}{4},\pi \end{matrix}\right)$
C: $\left[\begin{matrix}0, \frac{\pi}{4} \end{matrix}\right]$
D: $\left[\begin{matrix}0, \frac{\pi}{4} \end... | **Analysis**
This question examines the inclination angle of a line, which is of medium difficulty.
**Solution**
Given the problem, we can find that the slope of the line is $-\sin \alpha \in [-1,1]$,
Therefore, the range of the inclination angle is $\left[0, \frac{\pi}{4}\right]∪\left[ \frac{3\pi}{4},\pi\right)$.
... | B | Algebra | MCQ | Yes | Yes | cn_k12 | false |
Given the point $P(\sin \alpha+\cos \alpha,\tan \alpha)$ is in the second quadrant, the range of values for angle $\alpha$ is \_\_\_\_\_\_. | Since the point $P(\sin \alpha+\cos \alpha,\tan \alpha)$ is in the second quadrant, we have
$$
\begin{cases}
\sin \alpha+\cos \alpha 0
\end{cases}
$$
Therefore, $\sin \alpha < 0$ and $\cos \alpha < 0$,
which means angle $\alpha$ is in the third quadrant. Hence, $\alpha$ belongs to the interval $(2k\pi+\pi,2k\pi+ \fra... | (2k\pi+\pi,2k\pi+ \frac {3\pi}{2})(k\in\mathbb{Z}) | Algebra | math-word-problem | Yes | Incomplete | cn_k12 | false |
Determine the domain of the function $\frac{x^5 - 5x^4 + 10x^3 - 10x^2 + 5x - 1}{x^2 - 9}$. | To find the domain of the function $\frac{x^5 - 5x^4 + 10x^3 - 10x^2 + 5x - 1}{x^2 - 9}$, we need to determine where the denominator is non-zero, as division by zero is undefined.
1. **Set the Denominator Equal to Zero and Solve for $x$:**
\[
x^2 - 9 = 0 \implies x^2 = 9 \implies x = \pm 3
\]
So, $x = 3$ a... | (-\infty, -3) \cup (-3, 3) \cup (3, \infty) | Algebra | math-word-problem | Yes | Yes | synthetic_math | true |
3. The equation $x^{2}+a x+b=0$ and $x^{2}+c x+d=0(a \neq c)$ have the same root $\alpha$. Then $\alpha=$ | 3. $\frac{d-b}{a-c}$ | \frac{d-b}{a-c} | Algebra | math-word-problem | Yes | Yes | cn_contest | false |
38. As shown in the figure, the side length of rhombus $A B C D$ is $a$, point $O$ is a point on diagonal $A C$, and $O A=a$, $O B=O C=O D=1$. Then $a$ equals ( ).
A. $\frac{\sqrt{5}+1}{2}$
B. $\frac{\sqrt{5}-1}{2}$
C. 1
D. 2 | Answer: A
Translate the text above into English, please retain the original text's line breaks and format, and output the translation result directly. | Geometry | MCQ | Yes | Yes | olympiads | false | |
If x / y = 7 / 3 , then some expression divided by ( x - y ) equals 2.5. What is the numerator of the fraction? | Let's first find the value of (x - y) using the given ratio x / y = 7 / 3.
If x / y = 7 / 3, then we can write x = 7k and y = 3k for some constant k.
Now, let's find (x - y):
(x - y) = (7k - 3k) = 4k
We are told that some expression divided by (x - y) equals 2.5. Let's call the unknown expression in the numerator "... | 10k | Algebra | math-word-problem | Yes | Yes | orca_math | true |
Given the even function f(x) = $\sqrt{3}\sin(2x+\phi)-\cos(2x+\phi)$ (where $0<\phi<\pi$), its graph is translated to the right by $\frac{\pi}{6}$ units to obtain the graph of y=g(x). What is one of the intervals in which g(x) is monotonically increasing?
A: ($-\frac{\pi}{3}, \frac{\pi}{6}$)
B: ($\frac{\pi}{12}, \frac... | Given f(x) = $\sqrt{3}\sin(2x+\phi)-\cos(2x+\phi)$, we can rewrite this as:
f(x) = $2[\frac{\sqrt{3}}{2}\sin(2x+\phi)-\frac{1}{2}\cos(2x+\phi)]$
Using the trigonometric identity, we get:
f(x) = $2\sin(2x+\phi-\frac{\pi}{6})$
Since the function is even, we have:
$\phi-\frac{\pi}{6}=k\pi+\frac{\pi}{2}$
Given $0<\ph... | \text{A: }(-\frac{\pi}{3}, \frac{\pi}{6}) | Algebra | MCQ | Yes | Yes | cn_k12 | false |
29. [18] Compute the number of complex numbers $z$ with $|z|=1$ that satisfy
$$
1+z^{5}+z^{10}+z^{15}+z^{18}+z^{21}+z^{24}+z^{27}=0
$$ | Answer:
11
Solution: Let the polynomial be $f(z)$. One can observe that
$$
f(z)=\frac{1-z^{15}}{1-z^{5}}+z^{15} \frac{1-z^{15}}{1-z^{3}}=\frac{1-z^{20}}{1-z^{5}}+z^{18} \frac{1-z^{12}}{1-z^{3}},
$$
so all primitive 15 th roots of unity are roots, along with -1 and $\pm i$.
To show that there are no more, we can try to... | 11 | NaN | NaN | NaN | NaN | olympiads | false |
A candidate got 35% of the votes and lost to the rival by a certain number of votes. There were 5999.999999999999 votes cast. By how many votes did the candidate lose to the rival? | Let's denote the total number of votes as V. According to the problem, V = 5999.999999999999, but since the number of votes must be a whole number, we'll round it to the nearest whole number, which is 6000.
The candidate got 35% of the votes, so the number of votes the candidate received is 35% of 6000.
Votes receiv... | 1800 | Algebra | math-word-problem | Yes | Yes | orca_math | true |
Dave bought 3 packs of white T-shirts and 2 packs of blue T-shirts for his basketball team. The white T-shirts come in packs of 6, and the blue T-shirts come in packs of 4. How many T-shirts did Dave buy in all? | Dave bought 3 packs of white T-shirts, with each pack containing 6 T-shirts. So, the total number of white T-shirts is:
3 packs * 6 T-shirts/pack = 18 white T-shirts
He also bought 2 packs of blue T-shirts, with each pack containing 4 T-shirts. So, the total number of blue T-shirts is:
2 packs * 4 T-shirts/pack = 8 bl... | 26 | Logic and Puzzles | math-word-problem | Yes | Yes | orca_math | true |
Find the value of the complex number $z$ that satisfies the equation $z(1 + \sqrt{3}i) = |1 + \sqrt{3}i|$. The options are:
A: $1 - \sqrt{3}i$
B: $1$
C: $\frac{1}{2} - \frac{\sqrt{3}}{2}i$
D: $\frac{\sqrt{3}}{2} - \frac{1}{2}i$ | This problem involves the operations of complex numbers and the calculation of the modulus of a complex number.
First, calculate the modulus of the complex number $|1 + \sqrt{3}i|$. The modulus of a complex number $a + bi$ is given by $\sqrt{a^2 + b^2}$. Therefore, $|1 + \sqrt{3}i| = \sqrt{1^2 + (\sqrt{3})^2} = \sqrt{... | \frac{1}{2} - \frac{\sqrt{3}}{2}i | Algebra | MCQ | Yes | Yes | cn_k12 | false |
A school wants to understand the opinions of 1553 second-year students on their teachers' teaching methods. They decide to use systematic sampling to draw a sample of 50 students. First, they randomly remove $n$ individuals from the total population, then number the remaining individuals as 0001, 0002, 0003, etc., and ... | Solution: Since the total number cannot be divided evenly by the sample size, according to the method of systematic sampling, individuals should be randomly removed from the total population to ensure divisibility.
Given that $1553 = 50 \times 31 + 3$,
the number of individuals that should be randomly removed from th... | C | Combinatorics | MCQ | Yes | Yes | cn_k12 | false |
A boat having a certain length and breadth of 2 m is floating on a lake. The boat sinks by 1 cm when a man with a mass of 80 kg gets on it. What is the length of the boat? | To find the length of the boat, we can use the principle of buoyancy, which states that the weight of the water displaced by the boat is equal to the weight of the boat plus the weight of the man when he gets on it.
When the man gets on the boat, the boat sinks by 1 cm, which means the volume of water displaced by the... | 4 | Algebra | math-word-problem | Yes | Yes | orca_math | true |
Charlie signed up for a cell phone plan that allocates 8 GB of data per 4 week period. The data limit per week is as follows: 1st week - 2 GB, 2nd week - 3 GB, 3rd week - 2 GB, and 4th week - 1 GB. If he went over the data limit for the week, it would cost him an additional fee per GB as follows: 1st week - $12 per GB,... | Let's calculate the extra data usage for each week and the corresponding additional fee.
1st week:
Data limit: 2 GB
Data used: 2.5 GB
Extra data: 2.5 GB - 2 GB = 0.5 GB
Additional fee: 0.5 GB * $12/GB = $6
2nd week:
Data limit: 3 GB
Data used: 4 GB
Extra data: 4 GB - 3 GB = 1 GB
Additional fee: 1 GB * $10/GB = $10
3... | \$48 | Algebra | math-word-problem | Yes | Yes | orca_math | true |
5. The sum of 2021 consecutive natural numbers is $S$, and $S$ is divisible by 9. Then the minimum value of $S \div 2021$ is
$\qquad$ | $1017$ | 1017 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
Determine the slope of the line passing through points $M(-2,1)$ and $N(1,4)$. The slope is ( )
A: $1$
B: $4$
C: $3$
D: $5$ | To find the slope of a line passing through two points $(x_1, y_1)$ and $(x_2, y_2)$, we use the formula:
$$m = \frac{y_2 - y_1}{x_2 - x_1}$$
Given points $M(-2,1)$ and $N(1,4)$, we have:
$x_1 = -2, \quad y_1 = 1, \quad x_2 = 1, \quad y_2 = 4$
Substitute these values into the slope formula:
$$m = \frac{4 - 1}{1 - ... | 1 | Algebra | MCQ | Yes | Yes | cn_k12 | false |
3 years ago James turned 27. In 5 years Matt will be twice James age. How old is Matt now? | If James turned 27 three years ago, then he is now 27 + 3 = 30 years old.
In 5 years, James will be 30 + 5 = 35 years old.
At that time, Matt will be twice James's age, so Matt will be 2 * 35 = 70 years old in 5 years.
Therefore, Matt is currently 70 - 5 = $\boxed{65}$ years old now. | 65 | Algebra | math-word-problem | Yes | Yes | orca_math | true |
Example 4. If $P$ is a point inside the equilateral $\triangle A B C$, and $P A, P B, P C$ are connected, with $P A^{2}=P B^{2}+P C^{2}$. Find $\angle B P C$.
Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly. | Solve As shown in the figure, rotate $\triangle A B P$ around vertex $A$ by $60^{\circ}$ to $\triangle A C P^{\prime}$, then $\triangle A P P^{\prime}$ is an equilateral triangle.
Therefore, $P P^{\prime}=A P$.
$$
\begin{array}{l}
\text { Also, } B P=C P^{\prime} \text {, } \\
\therefore P P^{\prime 2}=P C^{2}+P^{\prim... | null | Geometry | math-word-problem | Yes | Yes | cn_contest | false |
In triangle $\triangle ABC$, given that $A=\frac{2\pi}{3}$, $c=2$, and the area of $\triangle ABC$ is $\frac{\sqrt{3}}{2}$. $D$ is the intersection point of the angle bisector of $\angle BAC$ and side $BC$. Find the length of $AD$. | To solve for the length of $AD$ in $\triangle ABC$ with given conditions, we proceed as follows:
1. **Finding the length of side $b$:** Given the area of $\triangle ABC$ is $\frac{\sqrt{3}}{2}$, we can use the formula for the area of a triangle, $Area = \frac{1}{2}bc\sin A$. Substituting the given values, we have:
... | \frac{2}{3} | Geometry | math-word-problem | Yes | Yes | cn_k12 | false |
A batsman in his 31st inning makes a score of 105, thereby increasing his average by 3; find his average after the 31st inning. | Let's assume the average score of the batsman before the 31st inning was "A". This means that up to the 30th inning, the total runs scored by the batsman would be 30A (since average = total runs / number of innings).
After scoring 105 in the 31st inning, the batsman's new average becomes A + 3. This means the total ru... | 15 | Algebra | math-word-problem | Yes | Yes | orca_math | true |
Given a circle \\(C\\) that passes through \\(A(1,-1)\\), \\(B(5,3)\\), and is bisected in area by the line \\(m\\): \\(3x-y=0\\).
\\((\\)I\\()\\) Find the equation of circle \\(C\\);
\\((\\)II\\()\\) If a line \\(l\\) passing through point \\(D(0,-1)\\) with slope \\(k\\) has two distinct common points with circle... | Solution:
\\((\\)I\\()\\) The midpoint of segment \\(AB\\) is \\(E(3,1)\\), \\(k_{AB}= \dfrac {3-(-1)}{5-1}=1\\)
Therefore, the equation of the perpendicular bisector of segment \\(AB\\) is \\(y-1=-(x-3)\\), which simplifies to \\(x+y-4=0\\)
Since circle \\(C\\) passes through points \\(A\\) and \\(B\\), the cent... | k 0 | Geometry | math-word-problem | Yes | Yes | cn_k12 | false |
(1) Find the standard equation of an ellipse whose right focus is at the coordinate $(2, 0)$ and that passes through the point $(-2, -\sqrt{2})$;
(2) Find the equation of a hyperbola with a common focus to the ellipse $\frac{x^2}{24} + \frac{y^2}{49} = 1$, and with a common asymptote to the hyperbola $\frac{x^2}{36} - ... | (1) Given the ellipse's right focus is at $(2, 0)$, and it passes through the point $(-2, -\sqrt{2})$, we can assume its standard equation is $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$.
Since the right focus is at $(2, 0)$, we have:
$c^2 = a^2 - b^2 = 4$,
and it goes through the point $(-2, -\sqrt{2})$, therefore:
$\frac{... | \frac{y^2}{16} - \frac{x^2}{9} = 1 | Geometry | math-word-problem | Yes | Yes | cn_k12 | false |
13.359. Find two two-digit numbers with the following property: if to the larger of the sought numbers a zero is appended on the right and then the smaller number, and to the smaller number the larger number is appended on the right and then a zero, then from the two five-digit numbers thus obtained, the first, when di... | Solution.
Let $x, y$ be the required numbers. According to the problem, we have the system
$$
\left\{\begin{array}{l}
1000 x+y=2(1000 y+10 x)+590, \\
2 x+3 y=72,
\end{array} \text { from which } x=21, y=10\right.
$$
Answer: 21 and 10.
13. 360. A cyclist departs from $A$ to $B$. The distance from $A$ to $B$ is 60 km... | 0<V\leq20 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
Compute
\[
\begin{vmatrix} \cos 1 & \cos X & \cos 3 \\ \cos 4 & \cos 5 & \cos 6 \\ \cos 7 & \cos 8 & \cos 9 \end{vmatrix}
.\]All the angles are in radians.
If we know the answer to the above question is 0, what is the value of unknown variable X? | To determine the value of \(X\) such that the determinant of the matrix
\[
\begin{vmatrix} \cos 1 & \cos X & \cos 3 \\ \cos 4 & \cos 5 & \cos 6 \\ \cos 7 & \cos 8 & \cos 9 \end{vmatrix}
\]
is zero, we need to recognize that the determinant of a matrix is zero if and only if its rows (or columns) are linearly dependent.... | 1 | Algebra | math-word-problem | Yes | Yes | metamath | true |
In a three-dimensional Cartesian coordinate system, the point P (-2, 4, 4) is symmetric to the x-axis and the origin, resulting in points P1 and P2, respectively. Find the length of the segment $\overline{P1P2}$.
A: 4√2
B: 4√3
C: 8√2
D: 8√3 | Firstly, to find the symmetric point P1 of point P (-2, 4, 4) with respect to the x-axis, we change the sign of the y and z coordinates of P, thus P1 becomes (-2, -4, -4).
Secondly, to find the symmetric point P2 of point P compared to the origin, we change the sign of all the coordinates of P, hence P2 is (2, -4, -4)... | 4 | Geometry | MCQ | Yes | Problem not solved | cn_k12 | false |
5. A truck and a bus started towards each other simultaneously from two entrances to the highway. Assume that the truck maintains a constant average speed and covers $27 \mathrm{~km}$ in 18 minutes. Assume that the bus also maintains its own constant average speed, which allows it to cover $864 \mathrm{~m}$ in 28.8 sec... | First solution.
One hour has 60 minutes, so 2 hours and 45 minutes $=120+45=165$ minutes.
The truck travels $27 \mathrm{~km}$ in 18 minutes, which means it travels $27: 18=1.5 \mathrm{~km}$ in 1 minute. $\quad 1$ POINT
In 165 minutes, the truck travels $165 \cdot 1.5=247.5 \mathrm{~km}$.
$1 \mathrm{POINT}$
$1 \mat... | 544.5\mathrm{~} | Algebra | math-word-problem | Yes | Yes | olympiads | false |
What number is a multiple of every integer from 1 to 5? | A number that is a multiple of every integer from 1 to 5 must be a multiple of the least common multiple (LCM) of these integers. Calculate the LCM of the integers from 1 to 5:
- LCM of 1 and 2 is 2.
- LCM of 2 and 3 is 6.
- LCM of 6 and 4 is 12 (since LCM of 2, 3, and 4 is LCM of LCM(2, 3) and 4, and since 4 = 2^2 and... | 60 | Number Theory | math-word-problem | Yes | Yes | synthetic_math | true |
There are three times as many hogs as cats in King Henry's kingdom. If there are 75 hogs, what's 5 less than 60% of the number of cats in King Henry's kingdom? | If there are three times as many hogs as cats, we can find the number of cats by dividing the number of hogs by 3.
Number of hogs = 75
Number of cats = 75 / 3 = 25
Now, we need to find 60% of the number of cats and then subtract 5 from it.
60% of the number of cats = 0.60 * 25 = 15
Finally, we subtract 5 from this ... | 10 | Algebra | math-word-problem | Yes | Yes | orca_math | true |
Calculate:$(1)\sqrt{32}-\sqrt{18}+\sqrt{8}$;$(2)6\sqrt{2}×\sqrt{3}+3\sqrt{30}÷\sqrt{5}$;$(3)\frac{4}{\sqrt{2}}+2\sqrt{18}-\sqrt{24}×\sqrt{\frac{1}{3}}$. | ### Problem 1: Calculate $\sqrt{32}-\sqrt{18}+\sqrt{8}$
**Step-by-Step Solution:**
- Simplify each term:
$\sqrt{32} = \sqrt{16 \cdot 2} = 4\sqrt{2}$
$\sqrt{18} = \sqrt{9 \cdot 2} = 3\sqrt{2}$
$\sqrt{8} = \sqrt{4 \cdot 2} = 2\sqrt{2}$
- Combine the terms:
$= 4\sqrt{2} - 3\sqrt{2} + 2\sqrt{2}$
$= (4 - 3... | 6\sqrt{2} | Algebra | math-word-problem | Yes | Yes | cn_k12 | false |
Find the least common multiple of 36 and 100. | First, let's find the prime factorization for each number:
1. **36**: \(36 = 2^2 \cdot 3^2\)
2. **100**: \(100 = 2^2 \cdot 5^2\)
Using the prime factorization, the LCM is determined by taking the highest power of each prime number involved:
- For prime number 2, the highest power is \(2^2\).
- For prime number 3, the ... | 900 | Number Theory | math-word-problem | Yes | Yes | synthetic_math | true |
\section*{Problem 3 - 081213}
In a regular tetrahedron \(A B C D\), the altitudes intersect at a point \(S\).
Calculate the size \(\alpha\) of the angle \(\angle C S D\)! | }
We drop the perpendicular \(A M\) from \(A\) onto the tetrahedron edge \(C D\). Then \(A M \simeq B M\), since both segments are heights of a side face of the same regular tetrahedron. Further, with \(\alpha=|\angle A S B|\), if \(B Q\) is the tetrahedron height,
\[
\alpha=360^{\circ}-90^{\circ}-90^{\circ}-|\angle ... | 109.5 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
Two trains of different lengths run at the speed of 60 km/hr and 40 km/hr respectively in opposite directions on parallel tracks. The time which they take to cross each other is 12.59899208063355 seconds. If the shorter train is 160 m long, how long is the longer train? | When two trains are moving in opposite directions, their relative speed is the sum of their individual speeds.
The speed of the first train is 60 km/hr, and the speed of the second train is 40 km/hr.
To find the relative speed in km/hr, we add the two speeds:
Relative speed = 60 km/hr + 40 km/hr = 100 km/hr
To con... | 190 | Algebra | math-word-problem | Yes | Yes | orca_math | true |
Find all real numbers $p$ so that
\[x^4 + 2px^3 - x^2 + 2px + 1 = 0\]
has at least two distinct real roots. | First, we note that $x = 0$ cannot be a root of the polynomial. Dividing both sides by $x^2$ yields:
\[x^2 + 2px + \frac{1}{x^2} - 1 = 0.\]
Let $u = x^2 + \frac{1}{x^2}.$ Then, we have:
\[u^2 - 2 = x^4 + 2x^2 + \frac{1}{x^4} = (x^2 + \frac{1}{x^2})^2 - 2 = u^2 - 2.\]
So, substituting $u$ into the equation gives:
\[u +... | \left[-\frac{3}{4}, -\frac{1}{4}\right] | Algebra | math-word-problem | Yes | Yes | synthetic_math | true |
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